Candidates, Credibility, and Re-election Incentives

Richard Van Weelden∗

March 9, 2011

Abstract

I analyze a model of repeated elections in which a representative voter selects among candi-

dates with known policy preferences in each period. In addition to having preferences over policy,

the elected candidate would like to use their position to secure rents at the voter’s expense. I

restrict attention to equilibria which are consistent with retrospective voting in which the candi-

dates’ strategies are stationary. I consider the policies different candidates would implement and

the amount of rent-seeking they would engage in if elected, and, consequently, which candidates

will be elected by the voter. I show that, when utilities are concave over policy choices, the

voter will not want to delegate to candidates who share their policy-preferences. The voter is

better off delegating to candidates with preferences on opposite sides of their own, and using

these candidates to discipline each other. The resulting policy divergence decreases the amount

of rent-seeking by elected candidates, resulting in a better outcome for the voter. I show that

the best equilibrium, from the voter’s perspective, involves two candidates, symmetric about

the median, who are elected at some history of the equilibrium. The policies the candidates

pursue in office do not converge but are more moderate than the candidates’ own preferences.

Keywords: political competition; repeated elections; endogenous candidates; divergent platforms;rent-seeking.

∗Max Weber Fellow, European University Institute and Assistant Professor of Economics, University of Chicago.Phone: +39-346-8354-093. Email: richard.vanweelden@gmail.com. This article is based on Chapter 1 of my Disser-tation from Yale University. This contains all the details and proofs for both the main article and the extensions:a shorter version is available on the author’s website. I am grateful to Dirk Bergemann, Justin Fox, Dino Ger-ardi, Johannes Horner and Larry Samuelson for the extremely helpful advice they gave me. I would also like tothank Evrim Aydin, Marco Battiglini, Alessandro Bonatti, Rahul Deb, Eduardo Faingold, Mikhail Golosov, AnneGuan, Fuhito Kojima, Alessando Lizzeri, Lucas Maestri, Eric Maskin, Massimo Morelli, Stephen Morris, Ben Polak,Herakles Polemarchakis, Arturas Rozenas, Kareen Rozen, Konstantin Sonin, Francesco Squintani, Filippo Taddei,Emanuele Tarantino, Juuso Valimaki, Lin Zhang and various seminar audiences for helpful comments. Financial sup-port from the Cowles Foundation, the Carl Arvid Anderson Fellowship, the Max Weber Programme, and ProfessorLarry Samuelson was greatly appreciated when pursuing this work.

1 Introduction

In elections voters choose between two (or more) competing platforms and candidates. A definingfeature of American elections is that there are usually two candidates – one Republican and oneDemocrat – with a realistic chance of winning election. In addition, in contrast to the median votertheorem, the platforms that the two candidates run on do not converge. In fact, the candidatesnot only run on different platforms but represent different interests and ideologies. In this paper Iprovide a novel explanation for these features of elections: namely that polarization of candidatepreferences creates greater incentives for candidates to act in the voters’ interests when in officein order to secure re-election. Consequently, voters will seek out candidates with preferences awayfrom the median as such candidates will engage in less rent-seeking behavior when in office; whenutilities are concave with respect to policies, the resulting decrease in rent seeking will more thancompensate voters for the policy being pulled away from the center.

Since the seminal results of Downs (1957)1 a large literature has emerged to explain the lackof convergence in candidate platforms.2 These models typically make two key assumptions: theparties are exogenously specified and the parties can make binding commitments to implement anyplatform. As such, the more fundamental question of why parties represent the interests they do,3 isnot addressed. On the other end of the spectrum, citizen candidate models (Osborne and Slivinski1996, Besley and Coate 1997), which endogenize the candidates’ decision of whether or not to runfor office, assume that candidates cannot make binding commitments, and so will always implementtheir ideal policy if elected. The only decision a potential candidate makes in these models, then,is whether or not to run for office. There is then a large gap, in the existing theories of politicalcompetition, between the polar opposite assumptions of perfect commitment with exogenous partiesand endogenous parties with no commitment. In this paper, I develop a model in which both thecandidates and the platforms are endogenously determined.

Neither the assumption that candidates can make binding commitments to any platform or thatcandidates will always implement their most-preferred platform if elected is particularly natural:elected officials have considerable discretion over which policies to pursue when in office, but theirbehavior is surely influenced by the desire to be re-elected. As Persson and Tabellini (2000) describeit, “It is thus somewhat schitzophrenic to study either extreme: where promises have no meaningor where they are all that matter. To bridge the two models is an important challenge.” If theset of platforms that candidates would be willing to implement if elected – and, hence, couldcredibly promise to the voters – is to emerge endogenously, the candidate must have an incentiveto implement platforms other than their ideal point when in office in order to influence futureelections; that is, a platform is credible if the candidate would be willing to implement it to securere-election. This requires a model of repeated elections. In the existing literature on repeatedelections, however, all candidates are identical (e.g. Ferejohn 1986) or the voters know nothing

1See also Hotelling (1929) and Black (1958).

2See Palfrey (1984), Calvert (1985), Roemer (1994) among others.

3As opposed, for example, to having a median voter party. In the original Downsian framework such considerationsdid not apply since parties were motivated purely by the desire to hold office, an assumption later authors relaxed.See Wittman (1983) for an overview.

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about the candidate’s preferences before electing them (e.g. Duggan 2000), and, if the incumbentis not re-elected, the replacement is drawn at random from the population of candidates. As such,while this literature provides interesting insights about how re-election pressures affect the policychoices of incumbent candidates, it cannot address the question of which candidates will be electedin the first place.

In this paper I study the impact of re-election incentives on political competition. I consideran environment in which the preferences of the candidates are known to the voter prior to theelection, and the elected candidate will come up for re-election after implementing the platformfor that period. I can then look at the set of platforms each type of candidate would be willingto implement to secure re-election. This not only provides a theory of the credibility of campaignpromises, but, once we know which platforms a candidate would be willing to implement, wecan consider which candidates would be willing to implement more desirable platforms than othercandidates. This, in turn, determines who the voter should elect. The key insight is that candidateswho have preferences different from the voter are willing to implement more desirable platforms,since the voter can punish such a candidate for a deviation without punishing themselves. As thevoter is indifferent between policies equally spaced from her ideal point – but a candidate withnon-median preferences is not indifferent between those policies – the voter can discipline a non-median candidate with the credible threat of electing a candidate who will implement a policy onthe opposite side of the median, thereby motivating the candidate to engage in less rent-seekingbehavior when in office.

I show that the best equilibrium involves two, and only two, candidates, and that these candi-dates implement different platforms if elected. This provides, to my knowledge, the first result inthe literature with endogenous candidates, diverging platforms, and an ability for the candidatesto implement a platform other than their ideal point.4

I consider a setting in which a (representative) voter chooses from a continuum of potential can-didates with known ideal points in every period. Further, I assume that there is some opportunityfor shirking, such as corruption or lack of effort, on the part of the elected candidate. I model therents that the elected candidate could secure at the voter’s expense as a monetary transfer from theelectorate to the candidate. Since the candidate will come up for re-election, it is possible for thevoter to induce the candidate to engage in less than full rent-seeking by making their re-electionconditional on their behavior in office. If the voter always elects a candidate with the median policypreferences, then all that a deviating candidate can be punished with is the lack of future rents, so,in equilibrium, the amount of rent-seeking must be strictly positive. If utilities are concave withrespect to the implemented policy, however, candidates with a non-median ideal point would bewilling to implement a more desirable platform. Such a candidate could promise to implement apolicy that is slightly in the direction of their ideal point while engaging in significantly less rent-seeking than a median candidate would engage in. If they deviate, the voter would elect a candidate

4Fedderson et al. (1990) allows for endogenous candidates, motivated only by holding office, who can makebinding commitments, and shows that the candidates locate at the median in the pure strategy equilibrium. Roemer(2001) specifies the objective of the party to be that of the average (or median) member, assuming that all citizensare part of one of the two parties. In addition there are several papers (e.g. Palfrey (1984), Callander (2005)) withtwo exogenously specified candidates but the possibility of candidate entry.

2

on the other side of the median and allow them to implement a policy slightly in the direction oftheir ideal point. Since utility functions are concave over policy, the difference between the twopolicies, even if they are very close to the voter’s ideal point, have a significant disciplining effect oncandidates with preferences different than the voter; the disutility to the voter from a non-medianpolicy is then more than off-set by the decrease in rent-seeking behavior. Consequently, there arenon-median candidates who have an incentive to implement more desirable platforms than a mediancandidate would. Further, I show that there are exactly two types of candidates, symmetric aboutthe median, who are willing to implement more desirable platforms than any other candidates.

This framework then provides many interesting insights. First, it provides a theory of thecredibility of campaign promises between the polar opposite assumptions of full commitment inthe Hotelling/Downs framework and the no commitment assumption in citizen candidate models.5

Second, I provide a model of endogenous candidates with a richer set of strategies than whetheror not to run for office. I derive endogenous candidates with non-median policies and, more fun-damentally, non-median preferences. While the policies the candidates implement in office do notconverge, they are more moderate than the candidates would implement in the absence of electoralconcerns. The baseline model also produces an extremely strong incumbency advantage, as the in-cumbent is always re-elected. I show, however, that when the platform implemented is imperfectlyobserved by the voter, the incumbent will be defeated with positive probability.

In the extensions I show that the results continue to hold when some of the assumptions ofthe baseline model are relaxed. First, I show that the results go through with short-lived, oralternatively term-limited, candidates, provided the candidates care about the policy implementedafter leaving office. Similarly, I replace the representative voter with a continuum of voters withheterogeneous preferences, and derive analogous results to the representative voter case. I thenextend the baseline model to allow for imperfect monitoring, so the voter cannot observe perfectlythe platform the candidate implemented while in office, and show that the incumbent is then re-elected with probability greater than 1

2 but less than 1. Finally, I endogenize candidate entry,extending the stage game to allow candidates to decide whether to run. In equilibrium, candidateswho have no possibility of winning the election will not run for office.

While this model and many of its implications are novel, it, of course, incorporates many existingideas in the literature. That repeated elections can induce politicians to implement policies differentfrom their ideal point in order to be re-elected is well known. Barro (1973) and Ferejohn (1986)consider how voters can use re-election incentives to discipline homogenous candidates in a modelof moral hazard. Alesina (1988) considers the moderating effects of re-election incentives in amodel of repeated elections between ideologically motivated, and exogenously specified, candidateswho have preferences on the left and right respectively. Duggan (2000) shows that there exists aunique equilibrium that is consistent with symmetry, candidate-stationarity, retrospective votingand prospective voting in an infinitely repeated elections model where the candidates’ preferencesare private information. Aragones et al. (2007) considers a model of repeated elections withexogenously given candidates, and shows that there exists an equilibrium, using trigger strategies,

5Banks (1990) and Callander and Wilke (2007) also provide an intermediate degree of credibility by assumingthat lying is costly for the candidates.

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in which the candidates announce polices different from their ideal point and keep their promises.Bernhardt et al. (2009) shows that politicians will be willing to implement more moderate policiesto secure re-election if they know that their replacement will have preferences from the other sideof political spectrum, and Bernhardt et al. (2011) extends this model of repeated elections to allowfor heterogeneity in candidates’ competence or valence. Unlike the other papers in this literature,I allow voters to choose among candidates with known, and heterogenous, policy preferences, andso endogenize the candidates. Further, while in the above papers, all else equal, more moderatecandidates are electorally advantaged, I find that voters may actually prefer candidates who holdnon-median policy positions to those who are more centrist.

My results also relate to the literature on the possible benefits of biased agents. Felli andMerlo (2006, 2007) consider a model of endogenous lobbying, where, due to the concavity of utilityfunctions, candidates bargain with lobby groups furthest from their ideal point as those lobbygroups have the greatest incentive to contribute in order to influence the candidates’ policies. Cheand Kartik (2009) show that a decision maker can benefit from consulting an expert with differentbeliefs than herself as such an expert will have greater incentive to acquire information to convincethe decision maker to change her beliefs. Dewatripont and Tirole (1999) consider the advantages ofhaving information acquired by “advocates” rather than an unbiased expert. Finally, Caillaud andTirole (1999) model the internal organization of political parties, and show that an “ideological”party may be preferred to a centrist one, since, though all candidates ultimately locate at thecenter, a candidate for an ideological party is induced to exert more effort in designing a high-quality platform, due the credible threat of the rank and file breaking with the candidate if theydo not.

The paper proceeds as follows: In Section 2 I describe the model; Section 3 provides examplesof subgame perfect equilibria which highlight the benefits of ideological competition, and motivatethe results to come; Section 4 presents the results for the baseline model; Section 5 extends thebaseline model, and Section 6 concludes. The proofs of the results for the baseline model are inthe Appendix A, and the results in the extensions are proven in Appendix B.

2 Model

I assume there is a single “representative” voter, and normalize this voter’s type to be 0. I allowthis voter to choose from a continuum of candidates of known types, κ ∈ [−1, 1], in every period. Ifurther assume, to highlight the fact that the results depend on the heterogeneity of the candidates’preferences as opposed to the additional competition from additional candidates, that there are aninfinite number of candidates of each type.6 Formally, each candidate can be indexed as

k = (κ, i) ∈ [−1, 1]× N,

where κ represents the type and i indexes which type-κ candidate k is.

6In particular this guarantees that the voter always has the option of selecting a different candidate with thesame policy preferences as themselves in every period.

4

In every period the voter chooses one candidate to elect.7 The elected candidate chooses whichpolicy to implement from a one-dimensional policy space, x ∈ [−1, 1]. In addition, there is anopportunity for the elected candidate to secure rents, m ∈ [0,M ], for themselves at the voter’sexpense. These rents could reflect outright corruption, such as taking bribes, or simply a lackof effort – either taking time off work or not addressing difficult but important issues – by thecandidate when in office. For example, the candidate may receive some benefit from holding officebut this benefit is eroded if the candidate works excessively long hours, denies favors to family andfriends, or takes actions which antagonize powerful lobby groups or vocal minorities. I assume thatvoters and candidates have utility functions which are concave over the implemented policy. Tosimplify the analysis I assume utilities are linear in the amount of rents secured, though I discussthe case where utilities are concave over rents in Section 4.3. I will refer to a policy-rents pair,p = (x,m), as a platform.

The utility to the voter, if platform p = (x,m) is implemented in the stage game, is

u(p) = −|x|λ −m,

where λ > 1 reflects the degree of concavity. That utility functions are strictly concave with respectto policy will be the key assumption driving the results.8 The elected candidate also has preferencesover the policy implemented but benefits from securing rents for themselves. So the utility to thecandidate in office if platform p = (x,m) is implemented is

gκ(p) = −|κ− x|λ + γm,

if their type is κ. I assume that γ ≥ 1 so the elected candidate cares about the amount of rentssecured at least as much as the voter and candidates who are not in office. Although not necessaryfor the main results, this reduces the number of cases to consider. Moreover, because the rentsaccrue to the elected candidate while the costs of rent-seeking are distributed across the electorate,this is a natural range of parameters to focus on.

If the candidate is not in office their utility is identical to the voter’s utility, except they haveideal point κ. The candidate’s utility is then

gκ(p) = −|κ− x|λ −m.

In particular, g0(p) = u(p). In specifying the preferences of the candidates out of office to be thesame as the voters, I am following in the citizen-candidate tradition (Osborne and Slivinski 1996,Besley and Coate 1997). As candidates in my model take no action except when elected by the

7I do not allow the candidates to communicate with the voter before the election. Since candidates’ types arecommon knowledge, adding (non-binding) communication would add additional complexity to the model withoutproviding any new insights. I discuss the possibility of communication in Section 4.3

8The assumption that utility functions are concave is one of the most widely made assumptions in economics.While utility functions being concave over income is uncontroversial, and strict concavity of utility functions playsa key role in many models of elections (e.g. Ledyard 1984, Snyder and Ting 2002, Callander and Wilke 2007), it isless clear that utility functions are, in fact, concave over policy (see Osborne 1995 for a discussion). One possibleinterpretation of concavity is that, if there are two issues, if the solution to to the first issue isn’t to an individual’sliking, the individual will get more upset if the other issue goes against them than if they were happy with theresolution to the first issue. Such an assumption seems likely to hold for most voters in most situations.

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voter, what is important for the results is that an elected candidate has preferences over the policyimplemented when in office, and also cares about the policy implemented after leaving office. Whileit seems natural that candidates should care about the policy implemented, the candidate wouldalso have reason to care about the policy implemented after leaving office even if we take a morecynical view of candidate motives: if the candidate cares about policy only to the extent that itcreates a “legacy” for themselves, this legacy is surely eroded if the successor immediately reversesthe implemented policy.9

Notice also that, in this specification of preferences, candidates do not value being in office for itsown sake but as a means to an end: being elected allows the candidate to implement a policy moreto their liking and to secure rents for themselves. The results go through if candidates derive someintrinsic benefit, B, from holding office, provided this benefit is not so large that candidates wouldbe willing to forgo all rent-seeking in order to secure re-election, in which case the the problem ofmotivating candidates to act in the voter’s interest when in office would be trivial.10

The timing of the game is as follows:1. In each period the voter elects a candidate, kt.2. The elected candidate implements platform pt = (xt,mt) for that period.3. The voter observes the pt implemented.4. The game is repeated with the voter deciding whether to re-elect the candidate. If the

incumbent candidate is not re-elected the voter decides which candidate to elect instead.I assume that the game is infinitely repeated with discount factor δ < 1. Since each period in

this model corresponds to an election cycle, we would expect δ to be significantly below 1, and wewill be concerned with more than just the limiting properties as δ → 1.

In each period the voter chooses a candidate, kt, and the elected candidate chooses platformpt = (xt,mt). A history then consists of the previously elected candidates and the platforms theyimplemented when in office. That is, a history is

ht = ((k0, p0), . . . , (kt−1, pt−1)).

Define htk = (ht, k) to be the history in which candidate k is elected after history ht, Ht and Htk to

be the set of all t period histories ht and htk respectively, and H = ∪t≥0Ht and Hk = ∪t≥0H

tk. A

strategy for the voter is to elect one candidate at every history,

sv : H → [−1, 1]× N,

sv(ht) = kt.

In addition, I define sv|ht to be the voter’s strategy in the subgame beginning at history ht. Astrategy for each candidate consists of the platform they would implement, if elected by the voter,

9Some evidence that public officeholders care who succeeds them comes from the timing of judicial retirements:the literature (e.g. Hagle 1993, Spriggs and Walbeck 1995, Hansford et al. 2010, though there are dissenting findings(e.g. Yoon 2006)) shows that judges often time their retirement so that the same party which appointed them canappoint their successor, making it more likely their replacement will share their judicial philosophy.

10Specifically, the results go through provided B < 1−δδγM . This guarantees that there does not exist a subgame

perfect equilibrium in which, at every history, all type-0 candidates implement the voter’s most preferred platformp = (0, 0).

6

at every history,

sk : Hk → [−1, 1]× [0,M ],

sk(htk) = pt.

For any profile of strategies, s, we can then calculate the payoffs in the repeated game for eachplayer. The payoff for the voter is

U(s) = (1− δ)∞∑t=0

δtu(pt),

and for each candidate k the payoff is

Uk(s) = (1− δ)∞∑t=0

δtuk(kt, pt),

where

uk(kt, pt) ={gκ(pt) if k = kt,gκ(pt) if k 6= kt.

For simplicity I restrict attention to subgame perfect equilibria in pure strategies. In addition,since we are concerned not only with the equilibrium platforms, but also with the candidates elected,I introduce the notion of credible candidates. These are the candidates who the voter would electat some history – either on or off the equilibrium path – in a given equilibrium, s∗.

Definition 1 Credible Candidates

A type, κ, of candidate is credible in subgame perfect equilibrium s∗ if a candidate of type κ iselected at some history under strategies s∗.

Before proceeding with the analysis I pause to discuss some of the main elements of the model.The most striking feature is that I have specified a single representative voter, but a continuumof candidates. The assumption of a single voter not only simplifies the analysis, but also biasesagainst finding divergent platforms, since there is no notion of different constituencies that thecandidates could be appealing to.11 In this model policy divergence can appear even when thereis no disagreement among the voters (or moderate voters have mass 1).12 In the extensions,I consider a model with a continuum of voters who have heterogeneous preferences. Becausethe specified preferences are single-peaked, I show that the existence of a representative voter isequivalent to requiring that the voters choose a Condorcet winner in every period. The assumptionof a continuum of candidates, while highly stylized, reflects the idea that there are many potentialcandidates out there who desire to hold office, so, if voters wanted to elect a candidate with acertain set of preferences, such a candidate could be found. In the extensions I make this explicit

11See, for example, Glaeser et al. (2005) and Virag (2008).

12It is often argued that parties represent relatively extreme policies while the voters themselves are more moderate.See Fiorina (2006): “[P]olarized political ideologues are a distinct minority of the American population. For the mostpart Americans continue to be ambivalent, moderate, and pragmatic, in contrast to the cocksure extremists andideologues who dominate our political life.” (Preface to 2nd Edition)

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by treating this as the set of potential candidates who make the decision of whether or not to runfor office.

Finally, as the game is infinitely repeated, candidates in this model are not term-limited andso can be re-elected forever. While there are many offices, such as the Presidency, which are termlimited, there are many others which are not. There is no limit on the number of terms whichcan be served in the U.S. House or Senate, or, in some states, as Governor.13 In the extensionsI consider the case where the candidates are short-lived or, alternatively, term limited, and showthat the results go through if the candidates seek to see candidates of the same type elected infuture periods.

3 Examples

In this section I present examples to highlight the benefit of delegating to candidates with het-erogenous preferences. As this is a repeated game it will, of course, admit many subgame perfectequilibria. Many of these equilibria, however, do not incorporate the benefits of competition betweenthe candidates. In this section I show that the candidate stationary equilibrium with convergingcandidates and platforms provides lower utility to the voter than equilibria with divergence. I thencalculate the amount of divergence that results in the highest equilibrium payoff for the voter – Irefer to this as the Divergent Platforms Equilibrium.

Notice that, if the game were not repeated, all candidates would implement their ideal policywith maximal rent-seeking, M , if elected; the voter would then elect a candidate with the samepolicy preferences as themself. This would, of course, form a subgame perfect equilibrium if playedat every history. This equilibrium does not, however, take advantage of the competition betweenthe candidates. If the voter held the candidates to a more rigorous standard for re-election thecandidates would be willing to implement more desirable platforms. In what follows I will restrictattention to equilibria which satisfy retrospective voting – so the incumbent is re-elected if and onlyif the platform they implemented gives the voter sufficient utility – and platform stationarity – sothe platform the candidate implements is independent of the history at which they are elected. Iwill make these notions precise in the next section.

I begin by considering equilibria where the voter always elects candidates who share their policy-preferences, κ = 0, who then implement the median policy, x = 0. For the voter to be willing tocondition their re-election decision on the candidate’s behavior in office they must be indifferentbetween re-electing the incumbent and kicking them out of office. Hence the level of rent-seekingwill not be affected by whether or not the incumbent is re-elected. We can then calculate theminimum amount of rent-seeking that can be supported from the equation

g0(0,m0) = (1− δ)γM + δg0(0,m0).

A type-0 candidate is indifferent between being elected and securing rents (1 − δ)γM/(γ + δ) inevery period and securing maximal rents today with some other candidate engaging in that level

13There are no term-limits for the governors of Connecticut, Idaho, Illinois, Iowa, Massachusetts, Minnesota, NewHampshire, New York, North Dakota, Texas, Vermont, Washington, or Wisconsin.

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of rent-seeking in all future periods. So, if the voter always delegates to candidates who share theirpolicy preferences, the following is the best such equilibrium for the voter.

Median Candidates EquilibriumThe following strategies constitute a subgame perfect equilibrium:The voter elects a candidate of type-0 at every history. If the voter elected a candidate of type-0in the previous period the voter will re-elect that candidate if and only if the platform they imple-mented gave the voter utility at least u(0, (1−δ)γM

γ+δ ) in the previous period, and elects a differenttype-0 candidate otherwise.Candidates of type κ = 0 implement their ideal policy while securing rents (1− δ)γM/(γ + δ), andall other candidates implement their ideal policy with full rent-seeking.

While this takes advantage of the competition between candidates it does not exploit the ad-vantages from the heterogeneity in the candidates’ preferences. Consider a candidate of type κ > 0,where κ is small. I now consider the most desirable platform the candidate would be willing to im-plement to secure re-election if any deviation is punished with reversion to the Median CandidatesEquilibrium. Note that the most desirable platform, from the voter’s perspective, would trade offpolicy and rents secured by the candidate in the same ratio for the candidate and the voter. Asutilities are linear with respect to rents the ratio of marginal utilities over rent-seeking is a constant,γ. So we must have the ratio of the marginal utilities for the candidate and the voter with respectto policy equal to γ. This implies that the candidate must implement a platform (x,m) whichsatisfies,

−∂gκ(x,m)∂x

∂u(x,m)∂x

= (κ− xx

)λ−1 = γ.

Consequently, candidate κ will implement policy

xκ =κ

1 + γ1

λ−1

=γ

11−λκ

1 + γ1

1−λ.

Now, if the continuation equilibrium if the candidate is not re-elected is the Median CandidatesEquilibrium, the candidate would be willing to limit rent-seeking to level m in order to securere-election provided

gκ(xκ,m) ≥ (1− δ)γM + δgκ(0,(1− δ)γMγ + δ

).

We can determine the minimum level of rent-seeking, m′κ, supportable by substituting into theabove equation. Note that the higher κ the greater the gain to the candidate from deviating inthis period, as the implemented policy is different than their ideal point, but also the harsherthe punishment for deviating – as the utility difference between policies xκ and 0 is greater for acandidate with preferences κ than for the voter. When the players are patient the future is moreimportant and the voter’s utility will be increasing in κ.

Simplifying the algebra, the utility to the voter is

w(κ) = −m′κ − xλκ = −(1− δ)γMγ + δ

+1γ

[δ − (1 + γ1

1−λ )1−λ]κλ.

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The voter’s welfare is then increasing in κ provided δ > δ ≡ (1 + γ1

1−λ )1−λ.

The welfare of the voter would be increasing even with less patience, if, instead, the continuationequilibrium from not securing re-election is a candidate of type −κ being elected and implementingpolicy −κ/(1+γ

1λ−1 ). We can calculate the lowest level of rent-seeking that can be supported from

the equationgκ(xκ,mκ) = (1− δ)γM + δgκ(−xκ,mκ).

Solving for the level of rent-seeking, mκ, the voter’s utility from candidates of type κ is then,

w(κ) = −(1− δ)γMγ + δ

+1

γ + δ(1 + γ

11−λ )−λ[δ((1 + 2γ

11−λ )λ − γ

λ1−λ )− (1 + γ

11−λ )]κλ.

The utility to the voter is then increasing in κ whenever δ ≥ δ ≡ (1+2γ1

1−λ )1−λ.14 As γ is positive,and λ is greater than 1, when

δ = (1 + 2γ1

1−λ )1−λ,

δ = (1 + γ1

1−λ )1−λ,

it is clear that 0 < δ < δ < 1. In Appendix A I calculate the values of δ and δ for differentparameters γ and λ. For reasonable parameters there is a benefit to moving away from the centereven when δ is much lower than 1.

Delegating to candidates of type κ > 0, has both advantages and disadvantages to the voter.First, the candidate will implement policy xκ > 0 in every period, resulting in a utility loss to thevoter. Additionally, the elected candidate has greater potential gains from deviating along the policydimension as the equilibrium policy will not correspond to the candidate’s ideal policy, increasingthe amount of rents required to prevent the candidate from deviating. However, the candidate alsohas more to lose by deviating, since if they lose election the policy in the future moves from xκ

to −xκ. This increased disciplining effect makes the incumbent willing to decrease the amount ofrent-seeking to secure re-election. Because the implemented policy, xκ, is proportional to κ theadvantages and disadvantages are both proportional to κλ. So, when δ ≥ δ, the welfare of the voteris increasing in κ, and will continue to increase in κ, until either we reach maximum polarization,or the rents secured by the elected candidate are driven to 0.15

We can define

κ = min{κ : gκ(xκ, 0) ≥ (1− δ)γM + δgκ(−xκ, 0)} > 0,

to be the minimum degree of heterogeneity in candidate preferences necessary for the rents to bedriven to 0. When κ < 1, we have that for all κ ∈ [κ, 1] the level of rent-seeking will be driven to0. We must then solve for the most moderate policy that can be supported. I then define x(κ) –the policy which makes a type κ candidate indifferent between implementing x(κ) in every period

14The minimum δ required for this equation to be increasing in κ is lower than δ. By assuming that δ ≥ δ we

ensure that, if the continuation policy is −κ/(1 + γ1

1−λ ) if not re-elected, no candidate of type on the interval (0, κ)would be willing to implement a more desirable platform than a candidate of type κ. This guarantees that there willonly be two credible types in the best equilibrium when δ ≥ δ.

15This is due to the fact that rents enter the utility function linearly. I discuss this more in section 4.4.

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without securing any rents, and implementing their ideal point today with full rent-seeking andhaving −x(κ) implemented in all future periods – from the equation

gκ(x(κ), 0) = (1− δ)γM + δgκ(−x(κ), 0).

I show in Appendix A that x(κ) has a unique minimizer on [κ, 1], which I characterize. Define thedegree of divergence in candidate preferences as

κ∗ ={

arg minκ∈[κ,1] x(κ) if κ < 1,1 otherwise.

(1)

Similarly, define the divergence in implemented policies,

x∗ =

{x(κ∗) if κ < 1,

1

1+γ1

λ−1otherwise. (2)

Now define the amount of rent-seeking supported in equilibrium,

m∗ =

0 if κ < 1,

1γ+δ{(1− δ)γM −

δ(1+2γ1

1−λ )λ−1

(1+γ1

1−λ )λ} otherwise.

(3)

Finally, letu∗ = u(x∗,m∗) = −(x∗)λ −m∗, (4)

be the utility to the voter when the policy implemented is x∗ and the elected candidate securesrents m∗. This determines the optimal divergence in platforms and candidate preferences.

Divergent Platforms EquilibriumSuppose δ ≥ δ. The following strategies constitute a subgame perfect equilibrium:At time-0 the voter elects a candidate of type κ∗. The incumbent candidate is re-elected if and onlyif the platform the candidate implements gives utility at least u∗ and the candidate is of type κ∗ or−κ∗. If the incumbent is of type κ∗(−κ∗) and the utility is less than u∗ the voter elects candidateof type −κ∗(κ∗) in the next period. At any history where the incumbent is type κ /∈ {κ∗,−κ∗} thevoter elects candidate of type κ∗ in the next period.

Candidates of type κ∗ choose platform (x∗,m∗), candidates of type−κ∗ choose platform (−x∗,m∗),and all other candidates choose their ideal points while securing full rents in any period the voterelects them.

The Divergent Platforms Equilibrium then highlights the advantage of having candidates withheterogeneous preferences. When δ ≥ δ the voter receives strictly higher payoff in this equilibriumthan in the Median Candidates Equilibrium. Note that in this equilibrium there are two crediblecandidates who implement non-converging platforms if elected, with this polarization benefitingthe voter. In the next section I show that, within the class of platform-stationary subgame perfectequilibria with retrospective voting, this equilibrium is the best one for the voter.

Notice also that, in addition to providing higher utility, the Divergent Platforms Equilibrium,unlike the Median Candidates Equilibrium, is robust to other candidates promising to implement

11

more desirable platforms using the specified equilibrium as a punishment for deviating. This corre-sponds to a renegotiation proofness requirement. This provides an alternative justification for theDivergent Platforms Equilibrium as I will show that it is the only subgame perfect equilibrium inthis class – up to symmetry and the behavior of candidates who are not credible – to be robust toa mutually profitable renegotiation.

4 Results

4.1 Efficiency

As is standard in the literature (e.g. Duggan 2000, Bernhardt et al. 2009, Banks and Duggan2008) I restrict attention to equilibria in which the candidates’ strategies are stationary. My firstrestriction is that the platform the candidate implements is independent of the history at whichthey are elected.

Definition 2 Platform-Stationarity

A subgame perfect equilibrium satisfies platform-stationarity if, for all candidates k, the platformthey implement if elected does not depend on the history at which they are elected. That is, s∗k(h

tk)

is a constant function.

Aside from simplifying the analysis, platform-stationary equilibria have many attractive fea-tures. First, they reflect an environment in which the candidates (i.e. parties) have platforms thatare stable over time – something observed in advanced democracies. It also rules out equilibria inwhich the voter induces good behavior by the candidate with the threat of extremely bad outcomesthat harm both the voter and the candidate in future periods. I discuss non-stationary equilibriain the Appendix A.16

In addition, I restrict attention to equilibria where the voter votes retrospectively. Intuitively,retrospective voting means that the incumbent is re-elected if and only if they provide the voterwith some threshold utility level. Retrospective voting is an important concept in political science(see Fiorina 1981),17 as it provides a simple strategy that closely approximates how voters arebelieved to behave. Achens and Bartels (2004b) summarize the theory of retrospective voting asfollows: “The best current defense of democracy is the theory of retrospective voting. Citizens maynot know much about the issues, the argument goes, but they can tell good from bad outcomes,and that allows them to remove incompetent or corrupt incumbents. Moreover, knowing that thevoters use that rule, every government will have every incentive to do what they want, thus fulfillingthe promise of democratic theory.” Because of this, retrospective voting is heavily studied in the

16With reasonable patience it is possible to support any feasible payoff for the voter in a subgame perfect equi-librium, including equilibria where the voter’s most preferred platform is implemented in every period along theequilibrium path. Even without considering stationary equilibria there is still an advantage to delegating to candi-dates with non-median candidates, however. There is a range of δ under which it is only possible to support thevoter’s most preferred platform in all periods if a non-median candidate is elected along the equilibrium path. Seesection 8,2 in the Appendix for the details.

17Recall also the famous Reagan question from the 1980 Presidential election: “Are you better off than you werefour years ago?”

12

theoretical literature (e.g. Barro 1973, Ferejohn 1986) and it is standard in the repeated electionsliterature to restrict attention to this class of equilibria (e.g. Duggan 2000, Bernhardt et al. 2009,Banks and Duggan 2008).

One subtlety is that the simplest definition – the incumbent is re-elected at any history providedthe platform they implemented in the previous period gives the voter utility u – could specify non-equilibrium behavior off the equilibrium path. Suppose an “extreme” type, who would alwaysimplement an undesirable platform if elected, were to be elected and implement a platform whichgives the voter a high utility; it would not be equilibrium behavior for the voter to re-elect thiscandidate at such a history.18,19 So I begin by defining retrospective voting for the candidate whois supposed to be elected at that history.

Definition 3 Retrospective Voting (R)

A subgame perfect equilibrium, s∗, satisfies retrospective voting (R), if there exists a feasible utilitylevel u such that, for all histories ht, if kt = s∗v(h

t), then the incumbent candidate is re-elected (thatis s∗v(h

t, (kt, pt)) = kt) if and only ifu(pt) ≥ u.

Further, for any platforms pt, p′t with I(u(pt) ≥ u) = I(u(p′t) ≥ u)),

s∗v|(ht,(kt,pt)) = s∗v|(ht,(kt,p′t)).

That is, the candidate is re-elected if and only if they provide the voter with some utilitystandard, with the voter selecting a different candidate otherwise. The voter evaluates the perfor-mance of the candidate, and conditions their behavior, only on whether this threshold utility levelis achieved – who the voter will elect in future periods is not affected by how much the candidateexceeds or falls short of the utility threshold. Retrospective voting (R) rules out equilibria in whichthe voter re-elects the candidate after some platform then rejects them after a platform which givesthe voter higher utility. The main result of this section is that Divergent Platforms Equilibrium isthe best equilibrium to satisfy platform stationarity and retrospective voting (R).20

Recall the definitions of κ∗, x∗, m∗, and u∗ from equations (1)− (4) in the construction of theDivergent Platforms Equilibrium. I consider the strategy profile, s∗D, with divergence, to be theplatform-stationary subgame perfect equilibrium in which:

18To see the difficulties, consider the case where δ = 0 and so any candidate must implement their ideal pointwhile securing full rents at every history in any subgame perfect equilibrium, so the voter must elect a candidate oftype κ = 0 at every history. Now, suppose we are at a history in which a candidate of type κ 6= 0 had implemented apolicy which gives the voter sufficiently high utility in the previous period. The voter’s strategy cannot call for themto re-elect the candidate at such a history.

19In papers such as Duggan (2000), Bernhardt et al. (2009), Banks and Duggan (2008) this does not arise becausethe candidate type is private information.

20Platform stationarity plays a much larger role in the results than retrospective voting. Retrospective votingrules out equilibria where the voter rejects an incumbent for implementing a more desirable platform than one thatwould have secured re-election (see Appendix A). If the amount of rent-seeking is driven to 0, which will happen whenκ < 1, there is only one relevant dimension, and there is no other equilibrium which gives the voter higher payoffthan the Divergent Platforms Equilibrium. I focus on equilibria which satisfy retrospective voting for robustness tocases with positive rent-seeking, because it will play a key role in the Uniqueness section, and because it describesnatural voting behavior.

13

(a) The voter elects a candidate of type κ ∈ {−κ∗, κ∗} at every history.

(b) For all histories ht, if kt = s∗v(ht) they are re-elected (s∗v((h

t, (kt, pt))) = kt) if and only ifu(pt) ≥ u∗; if u(pt) < u∗ then if kt is type κ∗ (−κ∗) the voter elects a candidate of type −κ∗

(respectively κ∗) at history (ht, (kt, pt)).

(c) The strategy for at least one candidate of type κ∗ is to implement platform (x∗,m∗) at anyhistory in which they are elected; the strategy for at least one candidate of type −κ∗ is toimplement platform (−x∗,m∗) at any history in which they are elected.

(d) Candidates of type κ /∈ {−κ∗, κ∗} implement a platform that gives the voter payoff strictlyless than u∗ if elected.

Notice that this strategy profile is unique up to the candidate elected in the first period, andthe behavior – for both the voter and candidate – after an out of equilibrium election decision. Allthat is required is that behavior be consistent with equilibrium at these histories. I now state themain result of this section.

Theorem 1 Efficiency

If δ ≥ δ, then:

(1) there does not exist a subgame perfect equilibrium satisfying retrospective voting (R) andplatform-stationarity which gives the voter payoff higher than u∗.

(2) any subgame perfect equilibrium satisfying retrospective voting (R) and platform-stationarityin which the voter achieves payoff u∗ must be of the form s∗D.

So we have that the best equilibrium in this class has only candidates of type κ∗ and −κ∗ electedat any history. All other candidates must implement a platform that gives the voter strictly lowerutility if elected: such candidates would strictly prefer to implement their ideal point while securingfull rents in this period followed by any continuation equilibrium that gives the voter utility u∗ toimplementing any platform which gives the voter payoff u∗ and securing re-election for themselves.As such, the voter will have a strict incentive not to elect any candidate of type κ /∈ {−κ∗, κ∗} atall histories. So we have the following corollary.

Corollary 1 The best subgame perfect equilibrium satisfying retrospective voting (R) and platform-stationarity involves exactly two credible types of candidates.

So the “best” equilibrium involves two type of candidates implementing non-converging plat-forms. Notice that the punishment for each candidate deviating is that the other candidate iselected: the candidates both punish each other. Voters break the tie between candidates they areindifferent between in order to discipline politicians. This provides a reason for voters to changewhich party they elect from period to period even in the absence of shifts in preferences of eitherthe party or the voters.21

21In the baseline model, of course, the candidate always secures re-election for themselves and so the punishmentof switching parties is never implemented.

14

Notice one more attractive feature of the Divergent Platforms Equilibrium. Because the candi-dates’ strategies are stationary, the voter must be choosing a static best response at every history.As such, the discount factor of the voter is irrelevant: even if the voter were completely myopicthe results would go through unchanged. All that matters for the analysis is that the voter prefersrelatively extreme candidates, recognizing that they engage in less rent-seeking behavior when inoffice. That more centrist candidates also engage in more rent-seeking is an implication of thismodel which could be tested empirically. Candidates with moderate policy preferences are lesssensitive to differences in the policy implemented and so are unwilling to forgo the opportunity tosecure rents for themselves in order to be able to influence the policy implemented in the future.This lack of “trust” in candidates with moderate policies has received much attention in the recentliterature; it has been proposed that non-median policies can signal character/honesty (Kartik andMcAfee 2007, Callander and Wilke 2007) or competence (Carrillo and Castanheira 2008).

This model is not a signaling story, however, and the effect identified here is quite different.While the previous literature has argued that voters may be suspicious of candidates who claim tobe moderate, as announcing a moderate policy could also be a sign of dishonesty, I have shown that acentrist voter may rationally prefer a non-centrist candidate to one they know to be moderate, evenif the non-centrist candidate is neither more competent nor more honest than the more moderateone. Rather, simply because of the incentives provided by the desire to secure re-election, candidateswith preferences less similar to the voter may actually choose policies more to the voter’s liking.

4.2 Uniqueness

In this section I provide a second justification for considering the Divergent Platforms Equilibriumbased on the possibility of joint deviations between a candidate and the voter. I will then proceedto comparative statics in the next section.

While the previous section identified the best equilibrium satisfying the stationarity and ret-rospective voting requirements there are many other equilibria which satisfy those requirements.This paper introduces re-election incentives so as to consider which platforms candidates couldpromise that they would have the necessary incentives to implement once elected. In static modelsof political competition we have an equilibrium when no candidate can offer a more desirable plat-form to the voter and win election. In this model, candidates must have an incentive to implementthe specified platform once elected. However, candidates only have an incentive to implement adesirable platform if the voter rewards them with re-election. If the voter’s strategy is such thatthey would never elect a certain candidate, that candidate would have no incentive to engage inless than full rent-seeking. The voter would then be justified in not electing them. As such, it ispossible to support many equilibria with a differential treatment of candidates. For example, wecould specify equilibria where only candidates of type κ′ are ever elected at any history, where κ′

is arbitrary. Other types could never secure re-election and so would engage in full rent-seekingif elected (see Appendix A for details). Many of these candidates would be willing to meet thestandard for re-election, however, if rewarded with re-election. Hence the candidate elected in thisequilibrium is determined entirely by the differences in the standard for re-election the voter setsfor different candidates.

15

The first task is then to identify equilibria which are not driven by this differential treatment ofcandidates. I do this by strengthening the definition of retrospective voting to ensure all candidatesare held to the same standard. In addition, we would like to identify equilibria which are robustto other candidates offering more desirable platforms to the voter in exchange for moving to acontinuation equilibrium in which they are elected. Such equilibria are a natural analog of theequilibria in static models. This corresponds to a renegotiation proofness requirement (Ray 1994),and in this section I show that the Divergent Platforms Equilibrium is – up to symmetry and thebehavior of candidates who will never be elected – the only one to satisfy it.

I begin by restricting the voter’s strategy after an out of equilibrium candidate is elected, andrequire the voter to hold such candidates to the same standard for re-election. I strengthen thedefinition of retrospective voting to incorporate this requirement. First, define

U−k (ht|s∗) = min{Uk(s∗|(ht,(s∗v(ht),p))) : p ∈ [−1, 1]× [0,M ]}.

That is, U−k (ht|s∗) is the lowest continuation payoff to candidate k if the expected candidate iselected at that history. This allows the voter to punish the off-path candidate with any equilibriumreward or punishment for the candidate elected on the equilibrium path. From the definition of(R) the set of continuation equilibria will have two elements. I now strengthen the definition ofretrospective voting.

Definition 4 Retrospective Voting (R∗)

A subgame perfect equilibrium, s∗, satisfying (R) satisfies (R∗) if, for all histories ht and candidateskt = (κ, i) 6= s∗v(h

t), if there exists a platform p′t such that

u(p′t) ≥ u,

andgκ(p′t) ≥ (1− δ)γM + δU−kt(h

t|s∗),

then, if kt is elected at ht, they are re-elected (s∗v(ht, (kt, pt)) = kt) if and only if

u(pt) ≥ u.

So this means that if the voter holds the elected candidate to a certain standard for re-election,the voter must also be willing to re-elect a different candidate if, off the equilibrium path, they areelected and would be willing to meet that standard to secure re-election when the punishment fornot securing re-election comes from the specified equilibrium. This then rules out equilibria drivenby a differential treatment of candidates.

This says nothing, however, of what the standard for re-election is. Many equilibria can thenbe supported depending on this standard. If the standard is easily met many types of candidateswould be willing to secure re-election for themselves and which candidate is elected is arbitrary. Forexample, there exist equilibria which satisfy (R∗) where the threshold for re-election is the static-Nash payoff, −M . In this equilibrium, provided the players are reasonably patient, any candidatewho is elected would be willing to secure re-election by implementing a platform which gives thevoter payoff exactly −M . The voter is then indifferent over candidates, and the candidate who is

16

elected is arbitrary (see Appendix A for details). Such an equilibrium is unnatural in the followingsense: a median candidate who was not elected could offer to engage in less than full rent-seekingtoday with the understanding that, if they do so, they would be rewarded with election in the nextperiod. The candidate would then prefer to implement this platform to deviating and being kickedout of office in the next election. As such, this proposed deviation is credible and would benefit boththe voter and candidate making it. So we should expect the candidate to offer that platform, for thevoter to believe them, and that candidate to be elected. We want to focus on equilibria in which it isnever possible for a candidate to credibly offer a different platform that benefits both the voter andthe candidate proposing it. This is a renegotiation-proofness requirement. Although renegotiationproofness requirements are only defined for two-player games, I present analogous definitions byrestricting attention to “active” players – which I take to be the voter and the candidate who iselected, as these are the only players who take action in this period of the game. If the voter andthe elected candidate agree to choose a different platform the candidates who are not elected wouldnot have the ability to prevent such a renegotiation.22 I adapt the notion of internal renegotiationproofness from Ray (1994).

Definition 5 Internal Renegotiation Proofness (I)

A subgame perfect equilibrium, s∗, satisfies (I) if there does not exist a history ht, actions (kt, pt)and (k′t, p

′t), and a platform p′′t+1 such that for candidate kt+1 = s∗v(h

t, (kt, pt)):

(1)

(1− δ)u(p′′t+1) + δU(s∗|(ht,(kt,pt))) > U(s∗|(ht,(k′t,p′t))),

(1− δ)gκ(p′′t+1) + δUkt+1(s∗|(ht,(kt,pt))) > Ukt+1(s∗|(ht,(k′t,p′t))).

(2) It is a best response for candidate kt+1 to implement p′′t+1 if the continuation equilibriumfrom implementing p′′t+1 is s∗|(ht,(kt,pt)) and the continuation equilibrium from implementingany other platform is s∗|(ht,(k′t,p′t)).

Part (1) of the definition asks whether a proposed renegotiation – candidate kt+1 is elected todayand implements platform p′′t+1, followed by continuation equilibrium s∗|(ht,(kt,pt)) instead of playingcontinuation equilibrium s∗|(ht,(k′t,p′t)) – strictly benefits both the voter and the candidate kt+1 whowould be elected in this renegotiation. Part (2) guarantees that candidate kt+1 would, in fact, havethe necessary incentive to implement platform p′′t+1 when elected if the reward for doing so is thatcontinuation equilibrium s∗|(ht,(kt,pt)) is played and the punishment for deviating is s∗|(ht,(k′t,p′t)).Internal renegotiation proofness (I) then requires that it is not possible for a candidate to crediblyoffer the voter a more desirable platform in exchange for moving to a more desirable continuationhistory for that candidate. Notice that this will rule out equilibria in which the candidate strictlyprefers securing re-election to deviating to their ideal point. Such a candidate could deviate andthen credibly offer the voter a more desirable platform to ignore the deviation.

22The definitions of renegotiation-proofness rely on Pareto dominance, so both players have a veto over therenegotiation. In our setting we look for dominance for the voter and the elected candidate as other candidatescannot veto the decision not to elect them or the platform implemented by the elected candidate.

17

I now look for equilibria which satisfy platform-stationarity, (R∗) and (I). I define a profile ofstrategies, s∗U , which satisfy s∗D and:

(e) At all histories ht, if the elected candidate kt is of type κ∗ (or −κ∗) they are re-elected(s∗v(h

t, (kt, pt)) = kt) if and only if u(pt) ≥ u∗; the voter elects a candidate of type −κ∗

(respectively κ∗) at history (ht, (kt, pt)) otherwise. In addition, all candidates of type κ∗

implement platform (x∗,m∗) and all candidates of type −κ∗ implement (−x∗,m∗) at anyhistory at which they are elected.

The only additional element from the previous section is that, in order to satisfy (R∗), allcandidates of type κ∗ or −κ∗ – even if they were not supposed to be elected under the voter’sstrategy – will be held to the standard u∗ for re-election if they are ever elected. The strategies arethen unique up to candidate elected in the initial period, and the behavior of candidates of typeκ /∈ {−κ∗, κ∗}, and the voter’s choice at the (off-path) histories in which the incumbent is of typeκ /∈ {−κ∗, κ∗}. The next result is that a subgame perfect equilibrium satisfies platform-stationarity,(R∗) and (I) if and only if it is of the form s∗U .

Theorem 2 Uniqueness

(1) For δ ≥ δ subgame perfect equilibria of the form s∗U satisfies (R∗), (I) and platform-stationarity.

(2) For δ > δ any subgame perfect equilibrium satisfying (R∗), (I) and platform-stationaritymust be of the form s∗U .

So we get that there are exactly two platforms, symmetric about the voters’ ideal point, thatwill be implemented by any candidate elected at any history of any equilibrium satisfying (R∗) and(I). These platforms correspond to the best equilibrium from the previous section. We have animmediate corollary.

Corollary 2 When δ > δ candidates of type κ∗ and −κ∗ are the only credible candidates in anysubgame perfect equilibrium satisfying (R∗),(I) and platform-stationarity.

When δ ∈ [δ, δ] the Divergent Platforms Equilibrium is not the unique equilibrium to satisfythose conditions: the Median Candidates Equilibrium also satisfies platform-stationarity, (R∗) and(I). In addition, when δ < δ the Median Candidates Equilibrium is the only one to satisfy theseconditions.

4.3 Comparative Statics

I have now provided two justifications for considering the Divergent Platforms Equilibrium. Now weexamine the properties of this equilibrium. First note that, in order to get non-median candidates,we must have sufficient patience. The greater the concavity, that is the larger λ, the lower thebound on δ as small changes in policy have a greater relative effect. The more candidates benefitfrom securing rents the more patient candidates must be, as greater differences in policy are neededto induce the elected candidate to forgo rents.

18

Theorem 3 Discounting

(1) Divergence is more easily achieved the more concave utilities are with respect to policy,

∂δ

∂λ< 0;

∂δ

∂λ< 0,

but more difficult to achieve with rent-motivated candidates,

∂δ

∂γ> 0;

∂δ

∂γ> 0.

(2) Concavity drives divergence:limλ→1

δ = limλ→1

δ = 1.

(3) Divergent platforms can always be supported if utilities are sufficiently concave:

limλ→∞

δ = limλ→∞

δ = 0.

This result then means that it is the concavity of utility functions that drives the divergentplatforms.23 Interestingly, in static models, for example in the literature on probabilistic voting(e.g. Ledyard 1984, Kamada and Kojima 2010), concavity often creates additional pressures forconvergence. My results show, however, that in a dynamic environment, concavity is the keyassumption which generates divergent platforms. Further, regardless of the level of discounting,we get a benefit to divergence provided utilities are sufficiently concave. I now consider how theparameters of the model affect the candidates elected and the platform implemented.

23When γ < 1, divergent platforms can be supported even without λ > 1. Notice also that while the assumedpatience δ ≥ δ is more than is necessary for an equilibrium with divergence to be better than the Median CandidateEquilibrium it is straightforward to show that, for any δ, the voter’s welfare is decreasing in κ, when λ is sufficientlyclose to 1.

19

Theorem 4 Properties of Divergent Platforms Equilibrium

If δ > δ, then:

(1) if κ > 1 (i.e. m∗ > 0), the amount of rent-seeking is increasing in the opportunity tosecure rents and decreasing in patience,

∂m∗

∂M> 0;

∂m∗

∂δ< 0.

(2) the amount of divergence is increasing in the opportunity for rent-seeking,

∂κ∗

∂M> 0;

∂x∗

∂M> 0,

and candidates’ taste for rents,

∂κ∗

∂γ> 0;

∂x∗

∂γ> 0,

but decreasing in patience,

∂κ∗

∂δ< 0;

∂x∗

∂δ< 0,

when κ∗ < 1.

So we see that when there is more opportunity for the elected candidate to secure rents forthemselves (M larger) there will be more rents secured by the candidates – provided the equilibriumrent-seeking is positive.24 Perhaps more interesting is that greater opportunity for rent-seeking leadsto greater polarization of both candidates preferences and policies.25 Similarly, if the candidateshave greater taste for rents, relative to policy, the result will be greater divergence – though, byTheorem 3, more patience will be required to support any divergence at all. Unsurprisingly, morepatient candidates will engage in less rent-seeking and implement more moderate policies. Thesepolicies are, in fact, implemented by more moderate candidates as well.

While κ∗ is decreasing in δ, it is bounded away from 0. As δ → 1 the payoff to the voter willconverge to their ideal point, but the preferences of the elected candidates will not converge to thepreferences of the voter. That the voter’s utility goes to 0 as δ goes to 1 is not surprising as thiswas true even without heterogeneity of candidate preferences. With divergent platforms, however,the utility of the voter approaches the utility from the voter’s ideal point at a much faster rate: theratio of the utility loss to the voter from the Divergent Platform Equilibrium to the utility loss inthe Median Candidates Equilibrium goes to 0.

24The stark 0 level of m on a range is a function of the fact that preferences are specified as quasi-linear.

25In the years since World War II the size of United States government has increased substantially and the partieshave become more polarized (e.g. McCarty, Poole, and Rosenthal 2006). If we think that an increase in the sizeand scope of government would create additional opportunities for elected officials to secure rents for themselves (Mincreased) then we could interpret the increased polarization as a response to that.

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Theorem 5 Patient Players

(1) Platforms converge to the voter’s ideal point as candidates become patient:

limδ→1

(x∗,m∗) = (0, 0).

(2) The candidates elected in equilibrium do not converge:

limδ→1

κ∗ = min{((λ− 1)γM)1λ , 1} > 0.

(3) There is faster convergence to the voter’s ideal point with divergence than median candi-dates:

limδ→1

u(0, 0)− u∗

u(0, 0)− u(0, (1−δ)γMγ+δ )

= 0.

4.4 Discussion

We have identified an equilibrium which incorporates many central features of American elections.There are exactly two types of candidates, who implement non-converging platforms, who are everelected. While the platforms do not converge, the candidates implement more moderate platformsthan they would in the absence of re-election incentives.26 I provided two justifications for thisequilibrium. First, among the stationary equilibria with retrospective voting this is the equilibriumwhich gives the voter the highest utility. That is, a two party outcome leads to more efficientoutcomes. In this equilibrium the voter is choosing the standard for re-election which results in thehighest payoff – at least among pure strategy equilibria. This equilibrium then is similar to the oneconsidered by Ferejohn (1986), who considers the optimal standard for re-election when candidatesare homogenous. In my paper, the voter’s decision is more complicated, as the voter not onlyneeds to determine the standard for re-election but also which candidates are willing to meet it. InFerejohn’s model the threatened punishment of not re-electing the incumbent is always credible:since the candidates are identical the voter is indifferent between the candidates. Here, there isheterogeneity in the candidates, but the voter must still be indifferent between the candidates whoare elected at different histories. This is an equilibrium result, rather than an assumption, in mymodel however – if the voter strictly preferred to re-elect the incumbent the threat of switchingcandidates would not be credible and the candidate could engage in full rent-seeking with impunity.

Second, I showed that this equilibrium is the only one to satisfy a strong notion of retrospectivevoting (R∗), which is immune to the possibility of renegotiation (I). Any other equilibrium, then,must either be based on a differential treatment of candidates or would not be robust to othercandidates offering the voter more desirable platforms in order to be elected – something considereda central feature of elections. The types κ∗ and −κ∗ are, by construction, the candidates who arewilling to implement the most desirable platforms given that the reward is re-election and the

26While the non-convergence of candidate platforms is well-established, comparing the policy a candidate imple-ments to their most preferred policy is more difficult. Still, the literature on legislator behavior (e.g. Levitt (1996))finds that both legislator ideology and their constituent’s preferences have a significant effect on the legislator’s votingrecord.

21

punishment is that a candidate on the other side is elected. No other candidate would be willingto implement as desirable a platform if this was the associated punishment from being thrownout of office. If any other type of candidate were elected at any history, or if the standard forre-election were more easily met, other types would have a strict incentive to meet the standardto secure re-election. Since these candidates would be willing to secure re-election for themselves,condition (R∗) says that the voter must be willing to re-elect them if they ever elect them in the firstplace. Hence, there are continuation histories where such a candidate is elected. Since the incentiveconstraints are slack, however, such a candidate would be able to credibly promise to implement aplatform which gives the voter a strictly higher payoff in order to move to the continuation historyat which they are elected, violating condition (I). As such, only candidates of types κ∗ and −κ∗

can be elected at any history of any equilibrium which satisfies conditions (R∗) and (I).In this model I have abstracted away from campaign communications: the voter simply infers

what the candidate will do if they are elected. As the candidates have no private information,any equilibrium with communication is also an equilibrium without communication. It would bepossible, of course, to extend the game to allow candidates to announce the platform they intendto implement before the election takes place; in equilibrium the announcement would only be “be-lieved” if it corresponds to the candidate’s equilibrium action. One possible role of communicationin this environment is in facilitating a renegotiation by the candidate and the voter. In real-worldelections, of course, candidates spend an enormous amount of time (and money) communicatingtheir platforms to voters. In the midst of this communication, then, it should be possible for candi-dates to communicate to voters that there is a mutually profitable renegotiation. In this sense, thecommunication we observe in elections strengthens the rationale for considering renegotiation-proofequilibria.

While the baseline model provides many interesting implications there are other implicationswhich seem less reasonable. Under a range of parameters the level of rent-seeking in the bestequilibrium will be driven to 0. This is driven by the assumption that preferences over the amountof rent-seeking enter the utility functions of the players linearly – if utilities are concave overrents, the amount of rent-seeking will not be driven to 0 even if the heterogeneity in preferences isunbounded. There is, however, still a benefit to policy divergence. Consider utility functions,

gk(p) = −|κ− x|λ + γmν1 ,

gk(p) = −|κ− x|λ −mν2 ,

andu(p) = g0(p) = −|x|λ −mν2 .

When ν1 = ν2 = 1 the payoffs are linear in m. Utility functions are concave over rents if

ν1 < 1 < ν2.

First, note that we would still get divergent platforms if the players are reasonably patient. Definem0 to be the minimum level of rents that can be supported without heterogeneity,

g0(0,m0) = (1− δ)γM + g0(0,m0).

22

We can then define γ0 to be the ratio of marginal utilities in rent-seeking between the candidateand the voter when the level of rent-seeking is m0,

γ0 = γν1

ν2mν1−ν2

0

By the previous argument, the utility of the voter would then be increasing in κ (at 0) when

δ ≥ (1 + 2γ1

1−λ0 )1−λ.

Note first that, from the above equation, for any δ, γ, M , ν1, and ν2, divergence is beneficial providedλ is sufficiently large. Next, notice that m0 is increasing in M . This means that divergence is moreeasily supported when there are greater the opportunities for rent-seeking behavior. However, asm0 goes to 0 as δ goes to 1, it is not always the case that divergence will be beneficial when theplayers are extremely patient. Still, it is straightforward to show that when λ − 1 > ν2 − ν1, sothat utility functions are sufficiently concave over policy relative to rents, divergence is beneficialwhenever the players are sufficiently patient.

Notice, however, that the level of rent-seeking would never be driven to 0. In any equilibriumsatisfying retrospective voting the voter conditions their re-election decision only on the utility theyreceived in the previous period. Therefore, the candidate must implement a platform that tradesoff policy and rents in the same ratio as the voter. So we must have

(κ− xκxκ

)λ−1 = γν1

ν2mν1−ν2κ .

As ν1 < ν2 this guarantees that mκ > 0, and so the amount of rent-seeking will be positive.In addition, although it is true that there is an incumbency advantage in elections,27 the in-

cumbency advantage described in the baseline model is unreasonably strong with the incumbentcandidate re-elected forever. This is due to the perfect information assumption. Since we arerestricting attention to pure strategies and perfect information it is possible for all players to an-ticipate what will happen in all subsequent periods. In particular, the candidate knows exactlywhat they must do to secure re-election. Since the voter would never want to elect a candidatewho would fall short of the standard for re-election, the candidate elected in the first period willbe re-elected forever along the equilibrium path. If we relax the perfect monitoring assumption –so that the voter does not perfectly observe what platform the candidate implemented – this is nolonger true. Even if the candidate does what the voter expects there is still some possibility of apoor outcome resulting. Still, in order to maintain incentives for the candidate to implement desir-able platforms, the voter must condition their re-election decision on the outcome in the previousperiod. Consequently, there will be turnover in equilibrium. I consider this in the extensions.

5 Extensions

In this section I consider four extensions of the baseline model. In the first extension I considershort-lived candidates who can only hold office for one term. This allows me to address the behavior

27See, for example, Ansolabehere and Snyder (2002).

23

of candidates who are term-limited and so cannot be rewarded with re-election. I then relax theassumption of a representative voter, and consider a model with a continuum of voters who haveheterogenous preferences. I show that, under natural conditions about who the voters’ would elect,their choices can be represented by those of a single voter. I then return to the baseline model, butallow for imperfect monitoring. As discussed above, this introduces a source of randomness intothe model, generating turnover in equilibrium. After introducing imperfect monitoring, I extendthe stage game to allow candidates to make the decision of whether or not to run for office. I showthat there will then be two candidates who run for office in every period – with both candidateselected with positive probability.

Since the conditions – platform stationarity, retrospective voting, renegotiation proofness – aredefined analogously in the new environments I leave the formal definitions to Appendix B. In eachsection I discuss how the model adapts and then state the main result of the section. The proofsof these results, which closely mimic the proofs for the baseline results, are in Appendix B.

5.1 Short-Lived Players

I now consider what would happen if the candidates only run in one election and cannot be re-elected. This could be either because they live for only one period or they are term limited.28 Whileparties are long-lived entities I assume for this section that parties cannot induce the candidateto implement a specified platform.29 This doesn’t necessarily preclude equilibria in which thecandidates select platforms other than their ideal point, however, if the platform the candidateimplements affects which candidate is elected in the next period, and the candidate cares about theplatform implemented in the future. This requires that voters reward and punish candidates withtheir choice of candidate after the incumbent leaves office. There is at least some evidence that thishappens. Brownstein (2007), describes the difficulty for the Republicans in the 2008 election: “Bushwont be on the ballot in 2008, of course, but throughout American history outgoing presidents havecast a long shadow over the campaign to succeed them. And when the departing president hasbeen as unpopular as Bush is now, his party has usually lost the White House in the next election.”

The candidates can then be indexed by (κ, t) where κ denotes the candidate’s type and t refersto the period in which they are alive.30 The strategies and payoffs can be defined precisely as in thebaseline model except that at each history the voter is restricted to elect a candidate who is alivein that period (sv(ht) ∈ [−1, 1]× {t}). I again restrict attention to platform-stationary equilibria,which I define to be equilibria in which all candidates of the same type implement the same platformregardless of the period they are alive. I then adapt the notion of retrospective voting to refer tothe voter electing a candidate of the same type in the next period if the incumbent meets thespecified utility standard, u = u∗s. Similarly, the notion of renegotiation proofness can be adapted

28In the most common examples of term limits, most notably the U.S. Presidency, the number of terms is two.There are some examples, such as the Virginia Governorship and Mexican Presidency, where the candidate can onlybe elected to one (consecutive) term.

29Alesina and Spear (1988) consider a model in which parties can make transfers to candidates after they leaveoffice in order to induce them to take action in the party’s interest when in office. Such transfers are not possible inthis model, and, I show, not necessary for divergent platforms to be beneficial.

30With short-lived candidates there is no reason to have multiple candidates of the same type.

24

to short-lived candidates as well. This gives conditions (Rs), (R∗s), and (Is) as the short-livedanalogues of conditions (R), (R∗), and (I) respectively. The details are in Section 9.1.

When candidates are short-lived, since they cannot receive rents in future periods, the amountof rent-seeking in equilibrium will be higher with short-lived than long-lived candidates. Noticealso that, though the rents from being in office can only be received in one period, policy changescan persist indefinitely. In addition, note that, in order to induce the same persistent change inpolicy, a short-lived candidate would be willing to decrease the amount of rent-seeking in one periodby more than a long-lived candidate would be willing to reduce rent-seeking in this and all futureperiods. As such, by inducing polarization, the amount of rent-seeking in equilibrium will decreasemore quickly with short-lived candidates than long lived ones, and so less patience will be necessaryfor divergence to be beneficial. So we can define δs with δs < δ. As in the baseline model, defines∗D to be the platform-stationary subgame perfect equilibrium strategies in which:

(a) At time-0 the voter elects candidate {(−κ∗s, 0), (κ∗s, 0)}, where κ∗s > 0.

(b) If, in period t, the incumbent is (κ∗, t) then the voter elects candidate (κ∗s, t + 1) in thenext period if the platform the candidate implements gives the voter utility at least u∗s, andcandidate (−κ∗s, t+ 1) otherwise. If the incumbent is (−κ∗s, t) then the voter elects candidate(−κ∗s, t+1) in the next period if the platform the candidate implements gives the voter utilityat least u∗s, and candidate (κ∗s, t+ 1) otherwise. If the incumbent is type κ /∈ {−κ∗s, κ∗s} thenthe voter elects a candidate from the set {(−κ∗s, t+ 1), (κ∗s, t+ 1)} in the next period.

(c) Candidates of type κ∗s choose platform (x∗s,m∗s), candidates of type −κ∗s choose platform

(−x∗s,m∗s) in any period they are elected.

(d) Candidates of type κ /∈ {−κ∗s, κ∗s} choose a platform which gives the voter payoff strictly lowerthan u∗s in any period they are elected.

The above strategy profile is almost identical to the strategies with long-lived candidates. If δ ≥δ there will be sufficient patience to support divergence with either short or long-lived candidatesand the degree of divergence will be the same in either case. However, if the amount rent-seekingis not driven to 0, short-lived candidates will engage in more rent-seeking.

Theorem 6 Divergent Platforms with Short-Lived Candidates

(1) For δ ≥ δs subgame perfect equilibria of the form s∗D satisfies (R∗s), (Is) and platform-stationarity.

(2) For δ ≥ δs there is no other subgame perfect equilibrium which satisfies (Rs) and platform-stationarity and gives the voter higher utility than s∗D.

(3) For δ > δs any subgame perfect equilibrium satisfying (R∗s), (Is) and platform-stationaritymust be of the form s∗D.

25

When candidates are only in office for one term they cannot be rewarded with future rents frombeing in office, so, in the absence of policy divergence, there is no reason for the candidate not tosecure full rents. However, with divergent platforms, they can be rewarded for good behavior witha more desirable policy in future periods. If we interpret candidates with the same preferencesas a party, then candidates will implement moderate policies and engage in less rent-seeking tohelp elect their party’s candidate in the future.31 This would hold even without the possibility ofcoercion from the party. The value of policy divergence is much greater with short lived candidatesthan with infinitely lived ones. With no policy divergence the utility to the voter is −M regardlessof δ. With divergence, however,

limδ→1

u(x∗s,m∗s) = 0.

Another nice feature of the short-lived candidate interpretation is that the equilibrium platformsare not affected if the candidates derive an intrinsic benefit, B, from holding office – even if B getsarbitrarily large. So even if candidates would be willing to promise anything to be elected, theinability to make binding commitments means that the candidates will still implement non-medianpolicies.32

5.2 Continuum of Voters

So far I have assumed a single “representative” voter who selects a candidate in every period. Inthis section I adapt the model to allow for a continuum of voters with ideal points drawn from theinterval [−1, 1]. A voter of type v ∈ [−1, 1] derives utility in the stage game

uv(p) = −|x− v|λ −m.

I assume that the preferences of voters are drawn from distribution G on the interval [−1, 1] withG(0) = 1

2 and G′ > 0 everywhere. Hence the preferences of the median voter in this model corre-spond to the preferences of the representative voter in the baseline model. While I analyze voters asdistinct from candidates we could allow for all players to be voters and candidates simultaneouslyas in citizen candidate models.

A model with a continuum of voters raises additional issues about how the voters coordinateon specific candidates. With a continuum of voters the voters cannot cast their vote in order toinfluence the outcome of elections as no voter is ever pivotal. If the voters are not attempting toinfluence the outcome of elections it is not clear how they coordinate on one candidate or the other.Rather than specify the strategies of each individual voter I instead assume that the voters canelect a candidate at every history. That is, I assume there exists a choice function, φ, which mapshistories into the candidate elected by the voters at that history. For any choice function, φ, and

31That polarization of parties can provide incentives to short-lived candidates is also considered by Testa (2005).She considers a model in which long-lived parties must delegate to short-lived candidates in every period. Candidatesalways implement their ideal policy but decide whether to implement it at low cost or at a high cost, securing rentsfor themselves in the process. She shows that the candidates will only implement the policy at low-cost if there issufficient polarization between the candidates selected by each party.

32In the case where candidates live forever we still get non-median candidates provided B < (1−δ)δ

γM .

26

profile of candidate strategies, s, we can then define the payoff to voter v as

Uv(φ, s) = (1− δ)∞∑t=0

δtuv(pt),

with the candidate payoffs defined exactly as in the baseline model. The key to this section isthat, because preferences are single peaked, there is no possibility of Condorcet cycles. In the stagegame, voters choosing a candidate from the Condorcet win set is then equivalent to a representativevoter playing a best response. I define a collective best response as a choice function for which nodeviation would make more than half the voters strictly better off.

Definition 6 Collective Best Response

A choice function, φ, is a collective best response to the candidates’ strategies if, given the candi-dates’ strategies, s, there does not exist a choice function φ

′such that

G({v : Uv(φ′, s) > Uv(φ, s)}) >

12.

I then define a subgame perfect equilibrium in the game with a continuum of voters exactly as inthe baseline model except replacing the requirement that a “representative voter” is playing a bestresponse with the requirement that the voters as a whole are playing a collective best response tothe candidates’ strategies. As in the baseline model I maintain the assumption that the candidates’strategies are platform-stationary.33 As in the baseline model, the voters can then make theirchoice at each history without concern for how this will affect play in future periods. Given thatpreferences are single-peaked, a majority of voters will prefer one stage-game outcome to another ifand only if the median voter prefers that outcome. Requiring that the voters are playing a collectivebest response then means that the voters always choose a (weak) Condorcet winner at every history.As in the baseline model, however, I allow which candidate is elected from this set to depend onthe platform implemented in the previous period.

I begin by adapting the notion of retrospective voting, which I now define in terms of sets ofplatforms rather than utility levels, as different voters will receive different utilities from a givenplatform. I then assume that there exists a set of platforms P , with ∅ 6= P j [−1, 1]× [0,M ], suchthat the incumbent is re-elected if and only if they implement a platform in P . I require that thisset of platforms, P , be monotone. That is, for all platforms p, p′ ∈ [−1, 1] × [0,M ], if p ∈ P andG({v : uv(p′) ≥ uv(p)}) ≥ 1

2 then p′ ∈ P .The requirement that the set of platforms that will secure re-election is monotone rules out

equilibria where the candidate would be re-elected if they implement a certain platform but rejectedfor implementing a platform which gives higher utility to more than half the voters. So, in essence,retrospective voting (Rc) means that the incumbent will be re-elected if and only if a majority ofthe voters were sufficiently happy with the candidate’s performance in the previous period. Again,by single-peakedness, a majority of voters will “approve” of the candidate’s behavior if and only ifthe median voter does.

33As preferences are single-peaked when the candidates’ strategies are stationary a collective best response mustexist.

27

As in the baseline model, the notion of retrospective voting can be strengthened to deal withoff-path candidates, (R∗c). The renegotiation proofness notion can also be adapted to allow for acontinuum of voters. While different voters may disagree on whether a proposed renegotiation isprofitable, condition (Ic) is satisfied if there is no renegotiation which benefits the elected candidateand more than half the voters. Once again, the details are left to Section 9.1.

I consider strategies where behavior of the candidates and the choices of the voters are identicalto the baseline model. As before I refer to this platform-stationay subgame perfect equilibriumstrategy profile as s∗D:

(a) The voters elect a candidate of type κ ∈ {−κ∗, κ∗} at every history.

(b) For all histories ht, if the elected candidate kt is type κ∗ (−κ∗) they are re-elected (s∗v(ht, (kt, pt)) =

kt) if and only if pt ∈ P ∗ ≡ {p : u0(p) ≥ u∗}; if pt /∈ P ∗ the voter selects a candidate of type−κ∗ (respectively κ∗) at history (ht, (kt, pt)).

(c) Candidates of type κ∗ implement platform (x∗,m∗) at any history at which they are elected.Candidates of type −κ∗ implement platform (−x∗,m∗) at any history at which they areelected.

(d) All candidates of type κ /∈ {−κ∗, κ∗} implement a platform pt /∈ P ∗ at any history at whichthey are elected.

I now present the result for the model with a continuum of voters.

Theorem 7 Divergent Platforms with a Continuum of Voters

(1) For δ ≥ δ subgame perfect equilibria of the form s∗D satisfies (R∗c), (Ic) and platform-stationarity.

(2) For δ ≥ δ there is no other subgame perfect equilibrium which satisfies (Rc) and platform-stationarity and gives higher utility to at least half the voters than s∗D.

(3) For δ > δ any subgame perfect equilibrium satisfying (R∗c), (Ic) and platform-stationaritymust be of the form s∗D.

So we see that the representative voter can be replaced with a continuum of voters who alwayschoose from the set of Condorcet winners. The voters decide which candidate from this set to electbased on the performance of the candidate in office in the previous period. I do not address howthe voters coordinate on this outcome. Notice also that the discussion of welfare is more nuancedwith more voters. While a majority of the voters benefit from the polarization of platforms, witha utilitarian welfare function, the welfare effects can be ambiguous. Depending on the distributionof voter preferences, the disutility to voters on the opposite side of the spectrum from the electedcandidate can be larger than the utility gains to all voters from the decrease in rent-seeking. VanWeelden (2011b) provides a fuller analysis of the welfare implications of policy divergence withheterogenous voters. In particular, with sufficient patience, divergence is welfare enhancing for anysymmetric distribution of voter ideal points.

28

5.3 Imperfect Monitoring

In the previous analysis I assumed that voters knew the candidates’ types before electing themand observed precisely the platform that the elected candidate implemented. The limitation ofthis is that, with pure strategies and perfect information, there is no source of randomness in themodel. The incumbent candidate is then re-elected forever. I now consider what would happenif, instead, the utility the voter received from the platform implemented were not deterministic.So while more moderate policies and less rent-seeking by elected candidates lead to higher utility,on average, there are still shocks (such as shocks to the economy) that affect the voter’s utility.I assume that the voter does not observe the platform the candidate implemented but only theutility they derived from it. That is, the voter observes

ut = u(pt) + εt,

where εt is an i.i.d. draw from some distribution F(y).Histories and strategies can then be defined as in the baseline model, except that the voter

only observes the past utilities and not the policies, and the players seek to maximize expectedpayoffs. As in the baseline model I restrict attention to equilibria where the candidates’ strategiesare independent of history. Since the voter only observes the public history I consider pure strategypublic perfect equilibria in which the candidates’ strategies satisfy platform-stationarity.

If the candidates are to have an incentive to implement any platform other than their idealpoint the voter must still condition their decision on the outcome in the previous period. Thenotion of retrospective voting is easily adapted – the voter already conditioned their behavioronly on the utility level in the baseline model. Retrospective voting remains natural behavioreven if voters are imperfectly informed.34 As Cost (2008) describes it, “The average voter doesn’tunderstand the intricacies of economic policy. Heck, when you think about it, nobody reallyunderstands the economy. So, voters often rely on simple yet sensible metrics to make politicaldecisions about the economy. One of them has been more or less operative since the election of1840: if the economy tanks during a Republican administration, vote Democrat. If it tanks duringa Democratic administration, vote Republican. Applying this rule to 2008, we get the following.McCain, because he is of the incumbent party, gets the political harm. Obama, because he is ofthe out party, gets the political benefit. That’s all there is to it.”

As we did with perfect monitoring we can now consider the best equilibrium satisfying ret-rospective voting (Ru) and platform-stationarity. In order to make the analysis tractable, andto parameterize the noise to allow for comparative statics, I assume the noise has the followingfunctional form,

Pr(εt ≤ y) = F (y) =

0, if y < −β,12(β+y

β )2, if y ∈ [−β, 0],1− 1

2(β−yβ )2, if y ∈ [0, β],1, if y > β.

34As the utility to the voter is non-deterministic but the voter conditions their re-election decision on their ownutility this means that candidates are rewarded for luck. Political scientists argue that this, in fact, happens. See,for example, Achen and Bartels (2004a, 2004b) and Wolfers (2006).

29

Therefore, the associated density function is

f(y) = F ′(y) =

{β−|y|β2 , if y ∈ [−β, β],

0, otherwise.

Notice that εt is then a symmetric, mean-0 random variable and that the probability of observinga specific utility level, ut, given implemented platform, pt, is decreasing in |u(pt) − ut|. Theparameter β determines the amount of noise in policy-making, with higher β corresponding to asetting where the voter’s utility is more random. That εt does not have full support will greatlysimplify the analysis since, provided β is not too large, it will be possible to support equilibria inwhich a candidate who engages is full rent-seeking will be defeated with certainty. In addition, noticethat by using a noise without full support I am biasing against generating turnover in equilibrium– if the noise had full support then, regardless of platform implemented and the standard forre-election, the incumbent would be defeated with positive probability.35

I begin by considering the best equilibrium involving only type-0 candidates. If there is nottoo much noise (β not too large) it will be possible to support equilibria with less than full rent-seeking. Notice that in any platform-stationary public perfect equilibrium each candidate who iselected at any history must implement a platform which gives the voter the same utility. As theutility standard for re-election is history independent this implies that the probability of re-election,q, is the same in every period.

Theorem 8 Median Candidates Equilibrium with Imperfect Monitoring

For any δ ∈ (0, 1), there exists β∗(δ) > 0 such that for all β ∈ (0, β∗(δ)) there exist public perfectequilibria satisfying platform-stationarity and retrospective voting (Ru) where only candidates oftype κ = 0 are elected and these candidates engage in rent-seeking m < M . In any equilibrium withless than full rent-seeking the incumbent will be re-elected with probability q ∈ (1

2 , 1). Further, thebest equilibrium involves level of rents-seeking m0

β and probability of re-election q0β with,

∂m0β

∂β> 0;

∂q0β

∂β< 0,

and

limβ→0

m0β =

(1− δ)γMγ + δ

; limβ→0

q0β = 1.

First notice that, since the rents enter the utility function linearly, there cannot be an equilib-rium with less than full rent-seeking in which the incumbent is re-elected with probability less thanor equal to 1

2 . When the elected candidate decides how much rent-seeking to engage in they face atrade-off between the benefits of additional rents today against the associated decrease in re-electionprobability this induces. If the standard for re-election, uβ, is such that u(0,m0

β) ≤ uβ, the gains

35When the noise has finite support, if there are signals which would never be observed without a deviation, thepunishments supporting the equilibrium could be off the equilibrium path. Here the assumption of a finite supportnoise is made only to simplify the algebra and the results do not rely on such a construction. In particular, the onlypunishment the voter has of the candidate – of not re-electing them – will be implemented with positive probabilityin equilibrium.

30

to the candidate from deviating are linear in the amount of the deviation, but the marginal costof deviating is decreasing in the amount of the deviation, since the probability the voter receivesa specific utility level ut in each period is decreasing in |u(0,m) − ut|. Hence, if the candidatecannot profit from a marginal deviation – a necessary condition for the candidate to be optimizing– they would strictly prefer to deviate by a large amount. Hence, in any equilibrium we musthave u(0,m0

β) > uβ, which implies that the incumbent will be re-elected with probability greaterthan 1

2 . As the noise increases the outcome becomes more random and the re-election decisionis then less responsive to the amount of rent-seeking. Consequently the incumbent will engage inmore rent-seeking. As the noise disappears (β goes to 0), the equilibrium converges to the MedianCandidates Equilibrium, with the incumbent never defeated along the equilibrium path.

I now consider equilibria involving candidates of type κ 6= 0. That is, I consider the bestequilibrium that can be supported using only candidates of type κ ≥ 0 and −κ. Note that whenκ = 0 we have converging candidates and platforms as in the equilibrium in the previous theorem.I then look for the best κ for the voter. Unlike the baseline model I am assuming that there are atmost two types of candidates who are ever elected.

Since the voter only observes the utility level, the implemented platform must trade rents andpolicy in the same ratio for the candidate and the voter. When κ and β are not too large, becauserents enter the payoff linearly, the implemented policies will be the same as in the perfect monitoringcase, ακ and −ακ. We can then determine the lowest level of rent-seeking that can be supportedwith candidates κ and −κ. When δ > δ the voter’s welfare will be increasing in κ provided thenoise is not too large. As in the perfect monitoring case, the welfare of the voter increases until therents are driven to 0 or we reach maximum polarization. We can then define κ∗β to be unique degreeof polarization that maximizes the voter’s utility, with (x∗β,m

∗β) the associated platform candidate

κ∗β would implement. See Section 9.1 for details.I now define the following strategy profile, which I refer to as s∗D as in the previous sections, to

be a platform-stationary subgame perfect equilibrium in which:

(a) The voter only ever elects candidates of type κ ∈ {−κ∗β, κ∗β}.

(b) Candidates of type κ∗β implement platform (x∗β,m∗β) and candidates of type −κ∗β implement

platform (−x∗β,m∗β). All other candidates implement platforms which give the voter utilityno higher than u∗β.

(c) If the incumbent is type κ∗β (respectively −κ∗β) the voter re-elects the candidates as long as thevoter’s payoff in that period is at least uβ and elects a candidate of type −κ∗β (κ∗β) otherwise.

Letting q∗β = F (u(x∗β,m∗β) − uβ), the associated probability of re-election, we have the main

result of this section.

Theorem 9 Divergent Platforms with Imperfect Monitoring

When δ > δ there exists β(δ) > 0 such that, for all β ∈ (0, β(δ)), the best two-type equilibriumsatisfying platform-stationarity and (Ru) is of the form s∗D. The incumbent will be re-elected withprobability q∗β ∈ (1

2 , 1) in all periods. Further:

31

(1) greater noise results in lower welfare and more turnover,

∂u∗β∂β

< 0;∂q∗β∂β

< 0.

(2) as the noise disappears the equilibrium converges to the perfect monitoring case,

limβ→0

(x∗β,m∗β) = (x∗,m∗); lim

β→0κ∗β = κ∗; lim

β→0q∗β = 1.

So we see that when there isn’t too much noise the voter still benefits from divergence in plat-forms and candidate preferences. Among the class of “two-type” equilibria the optimal divergencecan be uniquely determined. With positive noise there will be turnover in equilibrium with theincumbent re-elected with probability between 1

2 and 1. So there is an incumbency advantage butone less extreme than with perfect monitoring. Higher levels of noise leads to worse platforms beingimplemented and a higher probability of the incumbent being defeated. As the noise disappearsthe best equilibrium converges to the Divergent Platforms Equilibrium from the perfect monitoringcase.

5.4 Candidates’ Decision to Run for Office

So far, the candidates have not made the decision of whether or not to run for office: the votercould choose any candidate. In this section I allow candidates to decide whether or not to run.The candidates have lexiographic preferences: they first seek to maximize payoffs as in the previousanalysis, but prefer to be elected to not running, and prefer not running to being defeated. I onlyconsider the candidates’ decision of whether to run for the case in which there is positive noise,β > 0, so that the outcome of future elections will be uncertain. I assume that candidates mustmake the decision of whether or not to run for office before the realization of the noise is revealed– given the length of the election process in the United States this is a reasonable assumption.

In period-0, the voter elects any candidate in [−1, 1] × N, and in all subsequent periods thevoter chooses among the candidates who decided in the previous period to run. Let Kt be the setof candidates who decided to run for office in period t. This is a repeated extensive form game withthe following order of play in each stage game.1. The voter elects candidate kt ∈ Kt, where K0 ≡ [−1, 1]× N.2. The elected candidate, kt, implements platform pt; all candidates simultaneously decide whetherto run for office (R) or not (N) in the next period, where Kt+1 is the set of candidates who chooseto run.3. ut = u(pt) + εt is publicly observed.4. This game is then repeated with the voter deciding which candidate in Kt+1 to elect.To rule out the case where no candidate runs for office, assume that if Kt = ∅ then the voter cannotelect anyone and the payoff to all players is −∞. Histories and strategies are defined as before,except that the voter can only elect candidates who have run for office. See Section 9.1 for details.

I assume that candidates have payoffs identical to the previous section, but give them lexio-graphic preferences regarding their decision to run. In particular, I assume that two strategies

32

which give the same expected utility are evaluated based on the discounted number of periodsbeing elected less the discounted number of unsuccessful runs for office,

E[(1− δ)∞∑t=0

δt(I(k = kt)− I(k ∈ Kt \ kt))].

We can then define a public perfect equilibrium as before. I maintain the assumption that can-didates’ strategies are platform stationary. That is, although I allow the candidates to conditiontheir decision of whether to run for office on the history, the platform they implement if electedmust still be history independent.

As before I restrict attention to equilibria where the voter votes retrospectively. Of course, thevoter can only elect a candidate if they have decided to run for office. I then specify retrospectivevoting to be that the voter re-elects the incumbent if they meet the utility threshold and run forre-election in the next period. In addition I incorporate into the notion of retrospective voting (Re)the requirement that voters’ choices are not influenced by unsuccessful runs for office by candidates.This rules out situations where candidates run for office knowing they wont win in order to affectwho will be elected, either in this period or in the future.

I now look for two-candidate equilibria as in the previous section – equilibria in which the voterwould always elect a candidate of type κ ≥ 0 or −κ if such a candidate is available, and in whichat least one candidate of type κ and −κ runs for office in every period. I further assume that allcandidates of the same type would implement the same platform if elected. That is, for any twocandidates k1 = (κ, i), and k2 = (κ, j) and any history ht,

sk1(htk1) = sk2(htk2).

I consider the best equilibrium to satisfy these requirements. I can define the following platform-stationary equilibrium strategy profile, s∗D:

(a) The voter only ever elects candidates of type κ ∈ {−κ∗β, κ∗β}, provided such a candidate isavailable.

(b) Candidates of type κ∗β implement platform (x∗β,m∗β) and candidates of type −κ∗β implement

platform (−x∗β,m∗β) if elected.

(c) If the incumbent is type κ ∈ {−κ∗β, κ∗β} they will run for re-election. If the incumbent is typeκ∗β (respectively −κ∗β) the voter re-elects the candidates as long as their payoff in that periodis at least uβ and elects a candidate of type −κ∗β (κ∗β), if available, otherwise.

(d) At every history, one candidate of type κ∗β and one candidate of type −κ∗β run for officeand are both elected with positive probability; if m∗β > 0, if the incumbent is defeated inany period they will not run for office again along the equilibrium path. Candidates of typeκ /∈ {−κ∗β, κ∗β} do not run for office at any history.

I now state the main result from this section.

33

Theorem 10 Divergent Platforms with Candidate Entry

When δ > δ, for β ∈ (0, β(δ)), the best two-type equilibrium satisfying platform-stationarity and(Re) is of the form s∗D.

So we see that when there is positive noise and the candidates make the decision of whether ornot to run for office, there will be one candidate of both types who runs for office in each period. Theincumbent will be re-elected with probability greater than 1

2 , but the challenger will win electionwith positive probability as well.

6 Conclusion

In this paper I presented a model of repeated elections in order to consider which platforms can-didates would be willing to implement in order to secure re-election. I then showed that electingnon-median candidates results in a better outcome for the voter. I derived an equilibrium with two,endogenously determined, candidates, symmetric about the median, who implement non-convergingplatforms and have non-converging policy-preferences. These predictions are consistent with whatis observed in US elections: two candidates with different ideal points running on non-convergingplatforms that are, in fact, more moderate than the candidates’ ideal points. It is also, to myknowledge, the first model to derive all of these implications. Even though candidates never devi-ate, if there is imperfect monitoring, candidates will not always be re-elected, giving us a reasonablemodel of incumbency advantage and turnover in elections.

Models of political competition generally make very stark assumptions: candidates can makebinding commitments to any platform or will always implement their ideal point if elected; candi-dates are exogenously specified and compete by choosing platforms or candidates are endogenousbut can only implement their ideal point. This paper bridges the gap between these literatures.By studying a model of repeated elections we can consider which platforms candidates would bewilling to implement in order to secure re-election, and, in turn, which candidates would be willingto implement the most desirable ones. Therefore, by bridging the gap between full commitmentand no commitment we can also bridge the gap between the models of endogenous candidates andstrategic platform selection.

Finally, while I have only considered a model of voters and candidates the key insights could beapplied in other settings. For example, when delegating to members of a committee, the principalcould delegate to the more extreme members of the committee – in contrast to the “ally principle”– in order to induce more effort. This avenue is left for future research.

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8 Appendix A: Results for the Baseline Model

8.1 Definition of κ∗ and x∗

For any κ ≥ κ we can define x(κ) to be the unique solution to

(1− δ)γM − δ(κ+ x(κ))λ = −(κ− x(κ))λ.

Differentiating with respect to κ gives

δ(κ+ x(κ))λ−1(1 + x′(κ)) = (κ− x(κ))λ−1(1− x′(κ)),

and re-arranging

[δ(κ+ x(κ))λ−1 + (κ− x(κ))λ−1]x′(κ) = (κ− x(κ))λ−1 − δ(κ+ x(κ))λ−1.

So we have that x′(κ) = 0 if and only if

δ(κ+ x(κ))λ−1 = (κ− x(κ))λ−1,

or equivalently when

x(κ∗∗) =1− δ

1λ−1

1 + δ1

λ−1

κ∗∗.

This then implies that

κ∗∗ + x(κ∗∗) =2

1 + δ1

λ−1

κ∗∗,

κ∗∗ − x(κ∗∗) =2δ

1λ−1

1 + δ1

λ−1

κ∗∗,

38

and so

(1− δ)γM =2λ(δ − δ

λλ−1 )

(1 + δ1

λ−1 )λ(κ∗∗)λ,

which has a unique solution

κ∗∗ =1 + δ

1λ−1

2(

(1− δ)γMδ(1− δ

1λ−1 )

)1λ .

Hence, x′(κ) = 0 at one and only one point on [κ,∞).Next, note that taking derivative with respect to κ again we get

[δ(κ+ x(κ))λ−1 + (κ− x(κ))λ−1]x′′(κ) = (λ− 1)[(κ− x(κ))λ−2 − δ(κ+ x(κ))λ−2] > 0,

at κ = κ∗∗. So x′′(κ) > 0 and we have a local min. Combining these two observations, we can

conclude that x(κ) is decreasing for κ < κ∗∗ and increasing for κ > κ∗∗. Hence we know that thereexist a unique minimizer,

κ∗ = arg minκ∈[κ,1]

x(κ).

8.2 Non-stationary Equilibria and Equilibrium Selection

In this section I provide examples which demonstrate the benefits of restricting attention to theclass of equilibria we have considered. I showed that the Divergent Platforms Equilibrium is thebest equilibrium to satisfy platform-stationarity and retrospective voting (R). I first consider theimportance of the platform-stationary restriction. With reasonable patience it is possible to supportany feasible payoff for the voter in equilibrium, which I now demonstrate. I will then demonstratethat the degree of patience necessary to support the voter’s most preferred platform is lower withnon-median candidates elected along the equilibrium path. As is standard in the literature in re-peated games the first step is to determine the harshest possible punishment that can be supported.In order to reduce the number of cases to consider I will assume that γM > 1 – that is, the avail-able rents are large enough that a median candidate would prefer to implement any platform whilesecuring full rents to implementing their most preferred policy with no rent-seeking – although thisis not necessary for the results.

Example: The Worst Equilibrium Suppose δ ≥ 11+(γ+1)M . Then the following strategies con-

stitute a SPE:At time-0 the voter elects a median candidate. At all histories the incumbent candidate is re-electedif and only if they implement at policy x ∈ {−1, 1}, and elect a candidate of type-0 otherwise. Allcandidates of type κ ≤ 0 implement platform (−1,M) at all histories in which they are elected,and all candidates of type κ > 0 implement platform (1,M) at all histories.

Notice that the lowest possible payoff to the voter in any stage game comes from policyx ∈ {−1, 1} and full rent-seeking m = M . As this platform is implemented in every period thisis the worst possible equilibrium for the voter. Note also that it is the worst possible equilibriumfor any candidate of type κ ≥ 0. I now look at the degree of patience necessary to make to every

39

possible payoff supportable in equilibrium.

Example: Supportable Payoffs with Median CandidatesSuppose δ ≥ γM

1+(γ+1)M . Then for all u ∈ {u(p) : p ∈ [−1, 1] × [0,M ]} the following strategiesconstitute a SPE:The voter elects candidate (0, 1) in the first period, and re-elects them as long as the candidateimplements a platform with u(pt) = u. Candidate (0, 1) implements p′ = arg max{g0(p) : u(p) = u}if they have always been elected in past and implemented platform p′, and (−1,M) at any otherhistory in which they are elected. Any deviation is punished with reversion to the Worst Equilib-rium, with a candidate other than (0, 1) elected in the first period. All candidates other than (0, 1)implement their strategy from the Worst Equilibrium.

Notice that the bound δ ≥ γM1+(γ+1)M is calculated from the amount of patience that makes a

type-0 candidate indifferent between implementing platform (0, 0) in every period and engaging infull rent-seeking in this period given that their least preferred platform will be implemented in all fu-ture periods. The candidate would then have a strict incentive to implement a platform which givesthe voter any other payoff, when γM > 1. Notice that the voter’s most preferred platform can besupported along the equilibrium path, with median candidates elected, if and only if δ ≥ γM

1+(γ+1)M .I now show that less patience is required to induce a non-median candidate to implement thevoter’s most preferred platform along the equilibrium path. The reason for this is quite simple. Byselecting a candidate of type κ > 0 there are greater gains to deviate in terms of policy (from 0 toκ), but the punishment is harsher if defeated (moving from x = 0 to x = −1 has a larger impacton a candidate of type κ than a candidate of type 0). When κ is small, the disciplining effect isfirst order while the increased benefits to deviating is second order, so less patience will be required.

Example: First Best Supportable only with Non-median Candidates on Path Let κ ≥ 0and define δ∗(κ) to be the minimum δ such that

gκ(0, 0) ≥ (1− δ)γM + δgκ(−1,M).

That is, δ∗(κ) is the minimum discounting necessary to support the voter’s most preferred platformin every period in an equilibrium with the Worst Possible Equilibrium used as punishment. As thisis the harshest possible punishment, this determines whether the voter’s most preferred platformcan be supported in any equilibrium with a candidate of type κ elected in the first period. Notethat

δ∗(κ) =γM + κλ

(γ + 1)M + (1 + κ)λ

Notice that δ∗(0) = γM1+(γ+1)M and

∂δ∗(0)∂κ

= −λ γM

(1 + (γ + 1)M)2< 0.

So there exist δ < γM1+(γ+1)M and κ′ > 0 such that the following strategies result in the voter’s most

preferred platform implemented in every period:

40

The voter elects candidate (κ′, 1) in the first period, and re-elects them as long as the candidateimplements pt = (0, 0). Candidate (κ′, 1) implements (0, 0) if they have always been elected in pastand implemented platform (0, 0), and (1,M) at any other history in which they are elected. Anydeviation is punished with reversion to the Worst Equilibrium. All candidates other than (κ′, 1)implement their strategy from the Worst Equilibrium.

So we see that, even without putting any restrictions on the equilibria considered, by delegat-ing to candidates with non-median preferences we can achieve better outcomes than with mediancandidates. Restricting attention to platform stationary equilibria, however, rules out equilibriawhere extremely good payoffs (for the voter) are supported by reversion to equilibria which areextremely bad for both the voter and the candidate who is thrown out of office. As any payoffcan be supported, often with many possible candidates who could be elected on the equilibriumpath, this makes prediction difficult. Second, the best equilibria are supported by using the worstequilibria as punishments and such an equilibrium seems unnatural in an electoral setting. So inthis paper I restrict attention to equilibria satisfying platform stationarity. I now demonstrate theimportance of restricting attention to equilibria which satisfy retrospective voting.

Example: Importance of (R) Suppose κ > 1 and δ ≥ δ. Recall that then x∗ = 1

1+γ1

λ−1and

m∗ > 0. For any x ≥ 0 define m(x) to be the solution to

γm− (1− x)λ = (1− δ)γM − δ(m+ (1 + x)λ).

Let w(x) = u(x,m(x)) = −m(x)− xλ. Note that g(x∗) = u∗ and

w′(x∗) = − 1

γ + δ(−λ(1− x∗)λ−1 − δλ(1 + x∗)λ−1)− (γ + δ)λ(x∗)λ−1)

=λ

γ + δ(δ((1 + x∗)λ−1 − (x∗)λ−1) + (1− x∗)λ−1 − γ(x∗)λ−1)

=λδ

γ + δ((1 + x∗)λ−1 − (x∗)λ−1) > 0.

Thus there exists x′> x∗ such that the following strategies constitute a subgame perfect equilib-

rium satisfying platform-stationarity which gives the voter utility greater than u∗:At time-0 the voter selects a candidate of type 1 The incumbent candidate is re-elected if and only ifthe platform the candidate selects is (x

′,m(x

′)) or (−x′ ,m(x

′)) and the candidate is of type 1 or −1.

If the incumbent is of type 1(−1) and the platform implemented is not in {(−x′ ,m(x′)), (x

′,m(x

′))}

the voter elects candidate of type −1(1) in the next period. At any history where the voter electeda candidate of type κ /∈ {1,−1} the voter elects candidate of type 1 in the next period.Candidates of type 1 implement platform (x

′,m(x

′)) and candidates of of type −1 implement plat-

form (−x′ ,m(x′)) at any history they are elected. All other types of candidates implement their

ideal point with full rents at any history.

Note that the strategies require that the voter rejects the candidate if they implement a platformdifferent than the one specified, even if it gives the voter higher utility. In fact, there are platforms

41

which give the both the voter and the candidate higher utility but which are not selected inequilibrium because of the voter’s strategy. We want to restrict attention to equilibria in which,if the voter would re-elect the candidate if they implement a given policy, they would also re-electthem if they chose a more desirable policy hence we restrict attention to equilibria which satisfy(R).

However, condition (R) will not rule out equilibria that are supported by a differential treat-ment of candidates. For this we require condition (R∗).

Example: Importance of (R∗) Let κ be any type such that |κ|λ < δ(1+γ)γ+δ M . The following

strategies constitute a subgame perfect equilibria which satisfies platform-stationarity and (R):The voter selects a candidate k0 = (κ, 1) in the first period. If the incumbent candidate is type κthe voter and re-elects them if and only if they implement a platform which gives the voter utilityat least −M . At any history in which the incumbent is not re-elected the voter selects the lowestindexed candidate of type κ who has not previously been elected. Candidates of type κ implementplatform (M − ( |κ|

1+γ1

λ−1)λ, κ

1+γ1

λ−1). Any type κ 6= κ implements their ideal policy with full rent-

seeking if elected.

Notice that in this equilibrium the candidate elected is arbitrary and supported by a differentialtreatment of candidates: no candidate who is not type κ could ever secure re-election for themselvesand so would have no incentive to implement desirable platforms, and so the voter would never electthem. By assuming condition (R∗) we rule out equilibria supported by the differential treatmentof candidates and ensure that all candidates are held to the same utility standard for re-election.Retrospective voting, however, says nothing about what that standard is. We then demonstratethe importance of the renegotiation proofness assumption.

Example: Importance of (I) Suppose δ is sufficiently high that, for all types κ ∈ [−1, 1],

γ(M − (|κ|

1 + γ1

λ−1

)λ)− (|κ|

1 + γ1

1−λ)λ ≥ (1− δ)γM − δ(M + |κ|λ).

Then the following strategies constitute a subgame perfect equilibrium satisfying platform-stationarityand (R∗):Let κ′ ∈ [−1, 1] be arbitrary. The voter elects a candidate k0 = (κ

′, 1) in the first period. At every

history the incumbent is re-elected if and only if they implement a platform which gives the voterutility at least −M . If the incumbent implements a platform which gives the voter payoff less than−M they elect elect candidate (0, 1). For all κ ∈ [−1, 1], candidates of type κ implement platform(M − ( |κ|

1+γ1

λ−1)λ, κ

1+γ1

λ−1) if elected.

The difficulty with this example, of course, is that candidates would be willing to implementmore desirable platforms than they are called on to implement to secure re-election. Since thevoter doesn’t care which candidate holds office – they all give the voter utility −M – they wouldbe willing to move to the history where some other candidate is elected in exchange for a better

42

platform being implemented in this period – something that candidates who are not in office couldcredibly promise.

8.3 Proofs of Results for Baseline Model

Before proceeding to the main results I show a simple Lemma: any candidate not of type κ∗ wouldstrictly prefer to implement their ideal policy while securing full rents in this period with (−x∗,m∗)being implemented in every future period to implementing any platform which gives the voter util-ity u∗ in every period.

Lemma 1 If δ ≥ δ then for all κ 6= κ∗ and any (x,m) such that u(x,m) ≥ u∗,

gκ(x,m) < (1− δ)γM + δgκ(−x∗,m∗).

Proof. There are three cases to consider: candidates of type κ < 0, types κ ∈ [0,min{κ, 1}), andtypes κ ∈ [κ, 1]\{κ∗}.First consider κ ∈ [κ, 1]\{κ∗}. If this set is non-empty this implies that κ∗ ≥ κ and so m∗ = 0. Fur-ther, candidates of type κ are indifferent between implementing platform (x(κ), 0) in every periodand implementing their ideal point with full rent-seeking today and having (−x(κ), 0) implementedin all future periods. In addition, as x(κ) is strictly convex,

x∗ < x(κ) ≤ κ

1 + γ1

1−λ,

from the definition of κ. So for candidates of type κ, for any platform (x,m) with u(x,m) ≥u(x(κ), 0) we have

gκ(x,m) ≤ gκ(x(k), 0) = (1− δ)γM + gκ(−x(κ), 0) < (1− δ)γM + δgκ(−x∗,m∗).

As u(x(κ), 0) < u∗ we are done for κ ∈ [κ, 1]\{κ∗}.Now consider κ ∈ [0,min{κ, 1}). We now consider the most desirable platform, from the voter’s

perspective, that candidate κ would be willing to implement. Note that the payoff from the mostdesirable platform can be no higher than the payoff from policy xκ = κ

1+γ1

λ−1and associated level

of rents given by

gκ(xκ,m′κ) = γm

′κ − (κ− xκ)λ = (1− δ)γM − δ(m∗ + (κ+ x∗)λ) = (1− δ)γM − gκ(−x∗,m∗).

We can then define w(κ) = −m′κ− xλκ as the welfare to the voter from candidate κ. We must showthat w(κ) < u∗. Note that

w(κ) = −(1− δ)M +1γ

[δ((κ+ x∗)λ +m∗)− (1 + γ1

1−λ )1−λκλ].

First consider the case where κ ≥ 1. Then κ∗ = 1 and x∗ = 1

1+γ1

λ−1. Therefore

w′(κ) =

λ

γ[δ(κ+ x∗)λ−1 − (1 + γ

11−λ )1−λκλ−1],

43

and we can conclude that w(κ) is increasing on the interval [0, 1) as

δ ≥ δ = (1 + γ1

1−λ )1−λ(κ

κ+ κ

1+γ1

λ−1

)λ−1 > (1 + γ1

1−λ )1−λ(κ

κ+ x∗)λ−1.

As w(1) = u∗ when κ ≥ 1 we can conclude that w(κ) < u∗ on the interval κ ∈ [0,min{κ, 1}).Finally consider the case where κ < 1. Then we have x∗ ≤ x(κ) = κ

1+γ1

λ−1and m∗ = 0. Then

w(κ)− u∗ = −(1− δ)M + (x∗)λ +1γ

[δ(κ+ x∗)λ − (1 + γ1

1−λ )1−λκλ].

Note that we have seen that when x∗ = κ

1+γ1

λ−1we have w(κ)− u∗ < 0 from the previous analysis.

As w(κ)− u∗ is clearly increasing in x∗ we can conclude that w(κ)− u∗ < 0 for all κ ∈ [0, κ) whenκ < 1. So we have that w(κ) < u∗ for κ ∈ [0,min{κ, 1}). So we can conclude that for all (x,m)such that u(x,m) ≥ u∗ and all κ ∈ [0,min{D, κ}),

gκ(x,m) < (1− δ)γM + gκ(−x∗,m∗).

Finally note that for all κ < 0 the payoff from platform (−x∗,m∗) in future periods is strictlyhigher than for candidate −κ > 0. As candidate −κ does not strictly prefer any platform whichgives the voter u∗ to their ideal point today followed by (−x∗,m∗) in future periods candidates oftype κ must strictly prefer to implement their ideal point with full corruption today with (−x∗,m∗)in all future periods to any platform which gives the voter utility u∗.

Clearly, by symmetry, this also implies that no type other than −κ∗ is willing to implement aplatform which gives the voter utility u∗ if the punishment is that (x∗,m∗) would be implementedin all subsequent periods.

Proof of Theorem 1. Part (1). Suppose there exists a SPE satisfying (R) and platform-stationarity which gives the voter utility u

′> u∗ > −M . Let s∗ be a SPE such that U(s∗) = u

′.

By platform-stationarity, the voter has the same options available to her at all histories, and somust receive the same payoff, u

′in every period regardless of the history.

First note that the incumbent can never be defeated in such an equilibrium along the equilibriumpath. If the candidate would be re-elected if they implement their ideal point with full rents thenthey would clearly choose that platform and be re-elected forever. If the incumbent is to be defeatedin equilibrium the candidate cannot meet the standard for re-election when they implement theirideal platform with full rent-seeking. By condition (R), then, in any period where the candidatedoes not secure re-election they must implement their ideal point while securing full rents. As thepayoff to the voter in any period where the candidate engages in full rent-seeking is no higher than−M at no history can the voter elect a candidate who will not secure re-election. Further, becausethe payoff to the voter is higher than −M , the candidate cannot be re-elected after implementingtheir ideal policy with full rent-seeking.

I now turn to showing that, given that every continuation equilibrium must give the same payoffto the voter, no candidate will have an incentive to implement a platform which gives the voterpayoff u′ > u∗. WLOG assume the candidate elected in the first period, k0, is of type κ0 ≥ 0. Now

44

consider the continuation equilibrium where that candidate is not re-elected, s∗|(k0,(κ0,M)). Denoteby k1 the candidate elected in the first period of that equilibrium and let κ1 be their type. By theabove arguments this continuation equilibrium must give the voter utility u

′and result in the same

platform (x1,m1) being implemented in all future periods along the equilibrium path. Recall, byLemma 1, that no candidate would be willing to implement a platform that gives the voter utilityu′

if the outcome path from the continuation equilibrium if not elected has (−x∗,m∗) implementedin all future periods so we must have

gκ0(x1,m1) < gκ0(−x∗,m∗).

Combined with the fact that

u(x1,m1) = u′> u∗ = u(x∗,m∗),

we can conclude that x1 < −x∗ and m1 < m∗. This clearly implies that m∗ > 0 and so κ∗ = 1 andx∗ = 1

1+γ1

λ−1. However, since candidate k1 is evaluated strictly on the utility they provide to the

voter, they must select a platform which trades off policy and rent-seeking in the same ratio as thevoter, m1 < M , and we must have κ1 ∈ [−1, 1],

|x1| ≤ |κ1|1 + γ

1λ−1

≤ 1

1 + γ1

λ−1

.

So clearly we cannot have x1 < −x∗, and we have a contradiction. Therefore, there cannot exist aSPE satisfying (R) and platform-stationarity which gives the voter utility greater than u∗.Part (2). Now consider the SPEs which achieve u∗. We can repeat the above argument except wenow have gκ0(x1,m1) ≤ gκ0(−x∗,m∗) and u(x1,m1) = u(−x∗,m∗) which implies that x1 ≤ −x∗ andm1 ≤ m∗. By the above argument we can then conclude that x1 ≥ −x∗ and so (x1,m1) = (−x∗,m∗).Hence, by Lemma 1, we have that κ0 = κ∗ is the only candidate with κ0 ≥ 0 who could be elected.By symmetry, −κ∗ is the only candidate with κ < 0 who can be elected in the first period of anySPE which gives the voter utility u∗. So we can conclude that no type κ /∈ {−κ∗, κ∗} can be electedat any history.

As all players’ utility functions are strictly concave with respect to policy, if a candidate of typeκ ∈ {−κ∗, κ∗} is implementing platforms other than (x∗,m∗) (or (−x∗,m∗)) this platform mustgive the voter utility less than u∗. Since the continuation payoff to the voter must be u∗ at allhistories such candidates can never be elected at any history. Finally, at least one type κ∗ candidateis implementing (x∗,m∗) and at least one type −κ∗ candidate is implementing (−x∗,m∗) in orderfor the threatened punishment if the elected candidate deviates to be credible. �

Before proceeding to the proof of Theorem 2 I prove one more lemma.

Lemma 2 Suppose s∗ is a SPE satisfying platform-stationarity, (R∗) and (I), in which the samecandidate k0 is elected in every period and implements platform (x0,m0) in every period along theequilibrium path. Then for all platforms (x,m) such that u(x,m) ≥ u(x0,m0),

gκ(x,m) ≤ (1− δ)γM + δgκ(x0,m0),

for all κ ∈ [−1, 1].

45

Proof. Let u be the standard for re-election. As the incumbent is re-elected along the equilibriumpath we must have that u(x0,m0) ≥ u. First note that, in fact, we must have that u(x0,m0) = u.If u(x0,m0) > u then, in order for k0 to be optimizing, we must have that (x0,m0) = (κ0,M).Then if any other candidate of type κ0 were elected, by condition (R∗), they would implementplatform (κ0,M) and secure re-election. Note finally that such a candidate could then offer platform(κ0,M − ε) to move to the history at which they are elected, violating condition (I).

Now suppose there exists a κ1 ∈ [−1, 1] and a platform (x1,m1) such that

u((x1,m1)) ≥ u(x0,m0) = u

gκ1((x1,m1)) > (1− δ)γM + δgκ1(x0,m0)

and let k1 be a candidate of type κ1. Now since U−k1(h0) ≤ gκ1((x0,m0)), condition (R∗) guar-antees that candidate k1 would secure re-election by implementing a platform which gives thevoter at least u if elected. By continuity, provided ε is sufficiently small, candidate k1 would bewilling to implement platform ((1 − ε)x1, (1 − ε)m1) to move to history (h0, (k1, (x1,m1))). Asu((1− ε)x1, (1− ε)m1) > u this renegotiation would benefit both the voter and candidate k1, vio-lating condition (I).

Proof of Theorem 2. Part (1). It is immediate that the specified strategies constitute a SPE.Now consider condition (R∗). By Lemma 1 no candidate of type κ /∈ {−κ∗, κ∗} is willing to imple-ment any platform that will give the voter u∗ given any continuation equilibrium of this SPE socondition (R∗) imposes no requirements on those candidates’ strategies. Note that all candidatesof type κ ∈ {−κ∗, κ∗} are willing to meet the standard for re-election and the specified strategiescall for them to be re-elected if they do, consistent with the requirements of (R∗). So we have thatthe equilibrium satisfies (R∗).Finally, consider condition (I). Note that candidates κ∗ and −κ∗ are both, by construction, indif-ferent between securing re-election for themselves and deviating to their ideal point. Hence if theywere to promise any platform which gives the voter higher utility than u∗ in order to move to thehistory at which they are elected they would strictly prefer to deviate. As such, it is not possiblefor a candidate to offer a more desirable platform to move to the continuation equilibria at whichthey are elected, and so this SPE satisfies condition (I). �Part (2).Claim 1: In any SPE satisfying platform-stationarity, (R∗) and (I) the incumbent is re-electedforever along the equilibrium path.It is sufficient to show that there does not exist a SPE satisfying platform-stationarity, (R∗) and(I) in which the incumbent is defeated after the first period. Let k0 be the candidate elected in theinitial period and let κ0 be the type of the candidate k0. Note that if k0 is defeated in equilibriumit must be that they would not be re-elected if they implemented their ideal point with full rent-seeking: otherwise the candidate could implement that platform and be re-elected forever andachieve a strictly higher payoff than in any equilibrium where they are not re-elected. By condition(R) the continuation equilibrium does not depend on the specific platform they implement, socandidate k0 must implement platform p0 = (κ0,M) in the initial period. By stationarity they

46

must implement that platform at any other history as well. Now note that by (R) there mustbe a feasible utility level which would secure the candidate re-election. Hence if k0 implementedplatform p0 = (0, 0) they would be re-elected.

Now, define h0∗ = h0 and recursively define

ht∗ = (ht−1∗ , (k0, (0, 0))),

ht∗∗ = (ht−1∗ , (k0, (κ0,M))).

That is, ht∗ is the history where the candidate elected in period-0 has been elected and implementedplatform (0, 0) in all previous periods, and hence will be re-elected at ht∗. Similarly, ht∗∗ is the historyat where the candidate k0 implemented their ideal policy with full rent-seeking at history (ht−1

∗ , k0),and so will not subsequently be re-elected at ht∗∗. As candidate k0 is not elected in the first periodof continuation equilibrium (s∗|ht∗∗), and the stage game payoff to the candidate is no higher thanγM when in office and 0 if not in office, the payoff to k0 from continuation equilibrium (s∗|ht∗∗)must satisfy

Uk0(s∗|ht∗∗) ≤ (1− δ)0 + δγM = δγM,

for all t. Notice also that, from the candidate’s strategy, we must have

Uk0(s∗|ht∗) = (1− δ)γM + δUk0(s∗|ht+1∗∗

).

Now, recall that by platform-stationarity the voter must receive the same continuation payoff atevery history. In particular, then, for all t we must have

U(s∗|ht∗) = U(s∗|ht∗∗) = u(κ0,M).

Now supposeUk0(s∗|ht∗∗) < Uk0(s∗|ht∗).

Then, for ε > 0 sufficiently small we have

(1− δ)u(κ0,M − ε) + δU(s∗|ht∗) > U(s∗|ht∗∗),

(1− δ)γ(M − ε) + δUk0(s∗|ht∗) > Uk0(s∗|ht∗∗),

and(1− δ)γ(M − ε) + δUk0(s∗|ht∗) > (1− δ)γM + δUk0(s∗|ht∗∗).

This means that a mutually-profitable renegotiation for the candidate and the voter would exist.So we must have that

Uk0(s∗|ht∗∗) ≥ Uk0(s∗|ht∗) = (1− δ)γM + δUk0(s∗|ht+1∗∗

).

This implies that

Uk0(s∗|ht+1∗∗

)− Uk0(s∗|ht∗∗) ≤1− δδ

(Uk0(s∗|ht∗∗)− γM) ≤ −(1− δ)2

δγM,

for all t. However thenlimt→∞

Uk0(s∗|ht∗∗) = −∞.

47

As payoffs are bounded this is a contradiction. Therefore, we can conclude that there does notexist a SPE satisfying platform-stationarity, (R∗) and (I) in which the incumbent is ever defeatedalong the equilibrium path.Claim 2: There does not exist a SPE satisfying platform-stationarity, (R∗) and (I) which gives thevoter payoff greater than u∗.This follows immediately from Theorem 1.Claim 3: There does not exist a SPE satisfying platform-stationarity, (R∗) and (I) which gives thevoter payoff less than u∗.Let s∗ be a SPE satisfying (R∗), (I), and platform-stationarity where U(s∗) = u

′< u∗. By Claim

1 we can restrict attention to SPEs in which the candidate elected in the first period is re-electedforever. Let k0 be the candidate elected at h0, let κ0 be candidate k0’s type, and let (x0,m0) be theplatform k0 implements when elected. Then we must have u(x0,m0) = u

′. WLOG assume x0 ≥ 0.

Suppose u′ ≤ − (1−δ)γM

γ+δ . Then for all κ < 0,

gκ(x0,m0) ≤ gκ(0,(1− δ)γMγ + δ

).

Now, because δ > δ, this means that for κ < 0, and κ small

gκ(κ

1 + γ1

λ−1

,(1− δ)γMγ + δ

− |κ|λ

(1 + γ1

λ−1 )λ) > (1− δ)γM + δgκ(x0,m0),

u(κ

1 + γ1

λ−1

,(1− δ)γMγ + δ

− |κ|λ

(1 + γ1

λ−1 )λ) = u(0,

(1− δ)γMγ + δ

) ≥ u(x0,m0).

So, by Lemma 2, we have a contradiction. So we cannot have a SPE satisfying (R∗), (I) andplatform-stationarity which gives payoff u

′ ≤ − (1−δ)γMγ+δ to the voter.

Now suppose u′ ∈ (− (1−δ)γM

γ+δ , u∗). Notice that candidate k0 could not be re-elected if theyimplement their ideal point while securing full rents. Let k1 = s∗v(k0, (κ0,M)), let κ1 be the type ofcandidate k1, and let (x1,m1) be the platform candidate k1 would implement if elected. By Lemma2 we must have

g−κ∗(−x∗,m∗) ≤ (1− δ)γM + δg−κ∗(x0,m0),

sog−κ∗(x0,m0) ≥ g−κ∗(x∗,m∗).

As u(x0,m0) < u(x∗,m∗) we must then have x0 < x∗ and m0 > m∗. In addition, becauseu(x0,m0) > −M we can conclude that m0 ∈ (0,M) so we are at an interior solution with

x0 =κ0

1 + γ1

λ−1

.

Note first that we must have κ0 ∈ (0,min{κ, 1}). As x0 < x∗ it must be that κ0 < κ andκ0 < κ∗, so κ0 < min{κ, 1}. Now suppose κ0 = 0. Then the candidate must implement a platformwith x0 = 0, so −m0 = u

′. Also, since κ0 = 0, gκ0(x1,m1) = u(x1,m1) = u

′. However, as

u′> − (1−δ)γM

γ+δ ,gκ0(0,m0) < (1− δ)γM + δgκ0(x1,m1),

48

the candidate would have an incentive to deviate. So we can conclude that κ0 ∈ (0,min{κ, 1}).Combining Lemma 2 and candidate optimizing we must have

gκ0(x0,m0) = (1− δ)γM + δgκ0(x1,m1).

We can now find the level of rent-seeking in this equilibrium,

m0 = (1− δ)M +1γ

(κλ0

(1 + γ1

1−λ )λ− δ((κ0 + x1)λ +m1)).

Note that if x1 > 0 we would have

g−κ0(−x0,m0) > (1− δ)γM + δg−κ0(x1,m1),

so we can conclude that x1 ≤ 0. Now consider candidates of type κ ∈ (κ0,min{κ, 1}). Letxκ = κ

1+γ1

λ−1and define m

′κ to be the solution to u(xκ,m

′κ) = u

′. Then define f(κ) to be

f(κ) = gκ(xκ,m′κ)− (1− δ)γM − δgκ(x1,m1).

Note that f(κ0) = 0 and

f ′(κ0) = λ(δ(κ0 + x1)λ−1 − (1 + γ1

1−λ )1−λκλ−10 ) ≥ λκλ−1

0 (δ − (1 + γ1

1−λ )1−λ) > 0,

as δ > δ. So there exists κ ∈ (κ0, D) for which f(κ) > 0 and so

gκ(xκ,m′κ) > (1− δ)γM + δgκ(x1,m1).

By Lemma 2 this is a contradiction. So we can conclude that there does not exist a SPE satisfying(R∗), (I), and platform-stationarity where the payoff to the voter is less than u∗.

Combining claims 1− 3 we can see that in any SPE satisfying platform-stationarity, (R∗) and(I) the voter must receive payoff u∗. Consequently any equilibrium must be of the form identifiedin Theorem 1. In addition, (R∗) guarantees that any candidate of type κ ∈ {−κ∗, κ∗} would bere-elected if they implement a platform which gives the voter utility at least u∗. Hence, thosecandidates must implement a platform which gives the voter utility u∗ at that history, and byplatform-stationarity, at all histories. So we can conclude that all κ∗ candidates implement plat-form (x∗,m∗) and all candidates of type −κ∗ implement platform (−x∗,m∗) and we are done. �

Proof of Theorem 3. We first calculate the comparative statics for part (1). We begin by notingthat log δ = (1− λ) log(1 + γ

11−λ ) and log δ = (1− λ) log(1 + 2γ

11−λ ) so

∂ log δ∂γ

=γ−λ1−λ

1 + γ1

1−λ> 0,

∂ log δ∂γ

=2γ

−λ1−λ

1 + 2γ1

1−λ> 0.

49

Now we consider the derivative with respect of λ,

∂ log δ∂λ

= − log(1 + γ1

1−λ ) +1

1 + γ1

1−λγ

11−λ

log γ(1− λ)

< 0,

∂ log δ∂λ

= − log(1 + 2γ1

1−λ ) +1

1 + 2γ1

1−λ2γ

11−λ

log γ(1− λ)

< 0,

as λ > 1 and γ ≥ 1.We now consider the limits. We begin by showing that δ and δ both go to 1 as λ → 1. Note

that when λ < 2, γ1

1−λ ≤ γ−1 and so

(1 + 2γ−1)1−λ ≤ δ < δ < 1.

As (1 + 2γ−1)1−λ clearly goes to 1 as λ→ 1 we can conclude that

limλ→1

δ = limλ→1

δ = 1.

Finally we show that δ and δ both go to 0 as λ→∞. Note that when λ > 2, γ1

1−λ ≥ γ−1 and so

0 < δ < δ ≤ (1 + γ−1)1−λ.

As (1 + γ−1)1−λ clearly goes to 0 as λ→∞ we can conclude that

limλ→∞

δ = limλ→∞

δ = 0.

�

Proof of Theorem 4. Part (1) is immediate so I focus on part (2). We begin by calculating somecomparative statics on κ∗∗ and x∗∗. The equations

x∗∗ =1− δ

1λ−1

1 + δ1

λ−1

κ∗∗,

κ∗∗ =1 + δ

1λ−1

2(

(1− δ)γMδ(1− δ

1λ−1 )

)1λ ,

immediately imply that

∂κ∗∗

∂M> 0;

∂x(κ∗∗)∂M

> 0,

and that

∂κ∗∗

∂γ> 0;

∂x(κ∗∗)∂γ

> 0.

I now show that κ∗∗ is strictly decreasing in δ. Since x∗∗ = 1−δ1

λ−1

1+δ1

λ−1κ∗∗ this immediately implies

that x∗∗ is also strictly decreasing. It is sufficient to show that

(1− δ)(1 + δ1

λ−1 )λ

δ(1− δ1

λ−1 ),

50

is decreasing in δ. Define ρ = δ1

λ−1 . Taking log and differentiating w.r.t ρ gives

−(λ− 1)1− ρλ−1

ρλ−2 +λ

1 + ρ− λ− 1

ρ+

11− ρ

=−(λ− 1)1− ρλ−1

ρλ−2 +1− λ+ ρ+ λρ

ρ(1− ρ2).

So we have that κ∗∗ is decreasing if

(λ− 1)1− ρλ−1

ρλ−2 >1− λ+ ρ+ λρ

ρ(1− ρ2).

Rearranging we get the condition that

f(ρ) ≡ (λ− 1)(1− ρλ+1)− ρ(λ+ 1)(1− ρλ−1) > 0.

Note that f(1) = 0, and for all ρ ∈ (0, 1),

f ′(ρ) = (λ+ 1)[−(λ− 1)ρλ − 1 + λρλ−1)] < 0,

as −(λ− 1)ρλ − 1 + λρλ−1 in increasing in ρ and equal to 0 when ρ = 1. So f(ρ) > 0 for all ρ < 1and we can conclude that κ∗∗ is decreasing in δ. Hence we can conclude that

∂κ∗∗

∂δ< 0,

∂x∗∗

∂δ< 0.

So we have that k∗∗ and x∗∗ are increasing in M and decreasing in δ. Finally, since κ∗ < 1 wehave that κ∗ = κ∗∗ and x∗ = x∗∗ so we are done. �

Proof of Theorem 5. Now we can consider the limiting case where δ → 1. Since limδ→1 κ = 0,we know that when δ is close to 1 we have κ∗ = min{κ∗∗, 1}, m∗ = 0, with x∗ = x∗∗ if κ∗∗ < 1 andx∗ = x(1) otherwise. Recall that

(κ∗∗)λ = γM(1− δ)(1 + δ

1λ−1 )λ

2λδ(1− δ1

λ−1 ),

and since the top and bottom of the fraction both converge to 0 as δ goes to 1 we can use L’Hopital’srule to get

limδ→1

(κ∗∗)λ = limδ→1

γM

2λ−(1 + δ

1λ−1 )λ + λ(1− δ)(1 + δ

1λ−1 )λ−1 1

λ−1δ−λλ−1

1− δ1

λ−1 λλ−1

=γM

2λ(2λ)(λ−1) = γM(λ−1).

Consequently, we have

limδ→1

κ∗ = limδ→1

min{κ∗∗, 1} = min{(γM(λ− 1))1λ , 1}.

Next, we consider the limiting policy, x∗. First note that x∗ ≤ x(1) and

−(1− x(1))λ = (1− δ)γM − δ(1 + x(1))λ,

51

so x∗ obviously goes to 0 as δ goes to 1. More interestingly, dividing by m0 = (1 − δ)γM/(γ + δ)we have

−(m−1/λ0 − x(1)m−1/λ

0 )λ = γ + δ − δ(m−1/λ0 + x(1)m−1/λ

0 )λ.

Note that, as m−1/λ0 goes to ∞ as δ goes to 1, this implies that x(1)m−1/λ

0 goes to 0. Hence, wecan conclude that

limδ→1

u(0, 0)− u∗

u(0, 0)− u(0,m0)= lim

δ→1

(x∗)λ

m0= 0.

�

8.4 Definition of Renegotiation Proofness

In this section I present the definitions of renegotiation proofness presented in Farrell and Maskin(1989) and Ray (1994) as they are originally defined. Let G be a two-player game and let s∗ be apure strategy subgame perfect equilibrium.

Definition 7 Weakly Renegotiation Proof

(Farrell and Maskin, 1989) A subgame perfect equilibrium s∗ is weakly renegotiation proof if nocontinuation equilibria of s∗ strictly pareto dominates another.

Note that playing any static Nash equilibrium regardless of history is always weakly renego-tiation proof so a weakly renegotiation proof SPE always exists in mixed strategies. However Irestrict attention to pure strategies in this paper36 and hence in this section as well. Internal rene-gotiation proofness is a strengthening of weak renegotiation proofness which is defined in terms ofsets of payoffs. We can define a set of weakly renegotiation proof payoffs W and look at the set ofpayoffs that can be supported by this set of continuation payoffs. That is, the set of payoffs whichcan be achieved in a SPE using only continuation payoffs from the set W . For any action profilea = (a1, a2) we can define

di(a) = maxa′i

(ui(a′i, a−i)− ui(ai, a−i))

Then we have the following definition.

Definition 8 Supportable Payoffs

The set W of continuation payoffs supports payoff p if there exist actions a and payoffs w,w1, w2 ∈W such that for i = 1, 2:

pi = (1− δ)ui(a) + δwi

and(1− δ)di(a) ≤ δ(wi − wii)

Now we can define θ(W ) to be the set of payoffs that can be supported using only continuationpayoffs in W ,

θ(W ) ={p ∈ R2 : W supports p

}.

36Weak renegotiation proofness is defined for mixed strategies. Internal renegotiation proofness, in contrast, isdefined for pure strategies or observable mixed strategies. See footnote 8 in Ray (1994).

52

We can then define a set of payoffs to be weakly renegotiation proof if every payoff in W canbe supported by W and no element strictly pareto dominates another.

Definition 9 Weakly Renegotiation Proof Sets

A set of payoffs W is weakly renegotiation proof if(a) W j θ(W ),(b) There do not exist w1, w2 ∈W such that w1 >> w2.

We say that the set of payoffs is Internally Renegotiation Proof if it is not possible to supporta pareto superior outcome to any element of W using only elements in W .

Definition 10 Internal Renegotiation Proof Sets

(Ray, 1994) A set W j θ(W ) of payoffs is internally renegotiation proof if there do not existelements w ∈W and p ∈ θ(W ) such that p >> w.

Note that if a set is internally renegotiation proof it must also be weakly renegotiation proof.Ray (1994) shows that there may not exist an internal renegotiation proof set of payoffs for everygame.

8.5 Examples of δ and δ

The following table shows δ and δ for different values of γ and λ. As we can see, provided λ isreasonably large the lower bound on δ is not very restrictive.

γ\λ 2 2.5 3 3.5 4 51 0.50, 0.33 0.35, 0.19 0.25, 0.11 0.18, 0.06 0.13, 0.04 0.06, 0.015 0.83, 0.71 0.64, 0.46 0.48, 0.28 0.35, 0.17 0.25, 0.10 0.13, 0.0310 0.92, 0.83 0.75, 0.58 0.58, 0.38 0.43, 0.23 0.32, 0.14 0.17, 0.0550 0.98, 0.96 0.90, 0.81 0.77, 0.61 0.62, 0.42 0.49, 0.27 0.28, 0.11100 0.99, 0.98 0.93, 0.88 0.83, 0.69 0.69, 0.50 0.56, 0.34 0.33, 0.14500 0.99+ 0.98, 0.95 0.92, 0.84 0.82, 0.68 0.70, 0.51 0.46, 0.24

9 Appendix B: Results for the Extensions

9.1 Definitions for the Extensions

9.1.1 Short-Lived Candidates

With short-lived candidates I define the notion of platform-stationarity so that all candidates of thesame type – regardless of the period they are alive – will implement the same platform if elected.

Definition 11 Platform-Stationarity

A subgame perfect equilibrium satisfies platform stationarity if, for all types κ ∈ [−1, 1] and allhistories ht,

s∗(κ,t)(ht(κ,t)) = s∗(κ,0)(h

0(κ,0)).

Since the candidate only lives for one period it is not possible for them to be rewarded withre-election. Retrospective voting then means that a candidate of the same type is elected in thenext period if and only if the incumbent meets the threshold utility level.

53

Definition 12 Retrospective Voting (Rs)A subgame perfect equilibrium, s∗, satisfies retrospective voting (Rs), if there exists a feasible utilitylevel u such that for all histories ht, if (κ, t) = s∗v(h

t), then the same type of candidate is elected inthe next period (that is s∗v(h

t, ((κ, t), pt)) = (κ, t+ 1)) if and only if

u(pt) ≥ u.

Further, for any platforms pt, p′t with I(u(pt) ≥ u) = I(u(p′t) ≥ u)),

s∗v|(ht,(kt,pt)) = s∗v|(ht,(kt,p′t)).

We can then define U−(κ,t)(ht|s∗) to be the worst continuation payoff for candidate (κ, t) as we

did with long-lived candidates, and define an analog of condition (R∗). We must be careful however– the assumption of platform-stationarity means that candidates cannot influence the behavior ofcandidates elected in future periods, even of candidates of the same type. When players are patienta candidate may prefer less rent-seeking today if it means other candidates would engage in lessrent-seeking in the future. When evaluating candidates after an off-path election decision we thenconsider whether that candidate would be willing to implement a platform that meets the specifiedutility standard given that the continuation play can only depend on whether that standard is met.

Definition 13 Retrospective Voting (R∗s)A subgame perfect equilibrium, s∗, satisfying (Rs) satisfies (R∗s) if, for all histories ht and candidates(κ, t) 6= s∗v(h

t), if there exists a platform p′t such

p′t ∈ arg maxp{gκ(p) : u(p) ≥ u},

and(1− δ)gκ(p′t) + δgκ(p′t) ≥ (1− δ)γM + δU−(κ,t)(h

t|s∗),

then the same type of candidate is elected in the next period (s∗v(ht, ((κ, t), pt) = (κ, t + 1)) if and

only ifu(pt) ≥ u.

Similarly, the notion of (I) adapts with minimal changes.37

37We must be explicit about the time period, however, because the candidates will not be alive in future periods.To simplify the definition I define condition (Is) only for platform-stationary equilibria which satisfy (R∗s).

54

Definition 14 Internal Renegotiation Proofness (Is) A platform-stationary subgame perfectequilibrium, s∗, which satisfies (R∗s) satisfies (Is) if there does not exist a history ht and actions((κ, t), pt) and ((κ′, t), p′t) and a platform p′′t+1 such that for candidate (κ′′, t+ 1) = s∗v(h

t, ((κ, t), pt)):

(1)

(1− δ)u(p′′t+1) + δU(s∗|(ht,((κ,t),pt))) > U(s∗|(ht,((κ′,t),p′t))),

(1− δ)gκ(p′′t+1) + δU(κ′′,t+1)(s∗|(ht,((κ,t),pt),((κ′′,t+1),(0,0)))) > U(κ′′,t+1)(s

∗|(ht,((κ′,t),p′t))).

(2) It is a best response for candidate (κ′′, t+ 1) to select p′′t+1 if the continuation equilibrium

from selecting p′′t+1 is s∗|(ht,((κ,t),pt),((κ′′,t+1),(0,0))) and the continuation equilibrium from select-ing any other platform is s∗|(ht,((κ′,t),p′t),((κ′′′,t+1),(0,0))), where (κ′′′, t+ 1) = s∗v(h

t, ((κ′, t), p′t)).

Now I define the level of patience to support divergence

δs =1 + γ

11−λ

(1 + γ1

1−λ )λ + γλ

1−λ (1 + γ)< δ,

δs =1 + γ

11−λ

(1 + γ1

1−λ )(1 + 2γ1

1−λ )λ−1 + γλ

1−λ (1 + γ)< δ.

For any κ ≥ 0 we can solve for the lowest level of rent-seeking that can be supported in atwo-candidate equilibrium. With δ ≥ δs the welfare of the voter will be increasing until we reachmaximum polarization or rents are driven to 0. Recalling the definitions of κ and x(κ), for anyδ ≥ δs define

κ∗s ={

arg minκ∈[κ,1] x(κ) if κ < 1,1 otherwise,

x∗s =

{x(κ∗) if κ < 1,x1 = 1

1+γ1

λ−1otherwise.

Notice that we will have κ∗s = κ∗ and x∗s when δ ≥ δ. Now I define the amount of rent-seeking,

m∗s =

0, if κ < 1,

1γ(1−δ){(1− δ)γM −

δ(1+2γ1

1−λ )λ−1

(1+γ1

1−λ )λ}, otherwise.

Finally I define the level of utility this gives the voter,

u∗s = u(x∗s,m∗s).

The result for this section is proved in Section 9.2.

9.1.2 Continuum of Voters

With a continuum of voters, retrospective voting requires that the set of platforms that wouldsecure the incumbent re-election be time-invariant and monotone.

55

Definition 15 Retrospective Voting (Rc)A subgame perfect equilibrium (φ∗, s∗) satisfies (Rc) if there exists a monotone set of platforms,P , such that for all histories ht, if kt = s∗v(h

t), then the incumbent candidate is re-elected (that iss∗v(h

t, (kt, pt)) = kt) if and only if pt ∈ P .Further, for any platforms pt, p

′t with I(pt ∈ P ) = I(p′t ∈ P ),

s∗v|(ht,(kt,pt)) = s∗v|(ht,(kt,p′t)).

Strengthening condition (Rc) to (R∗c) is then exactly analogous to the baseline case.

Definition 16 Retrospective Voting (R∗c)A subgame perfect equilibrium (φ∗, s∗) which satisfies (Rc) satisfies (R∗c) if, for all histories ht andcandidates kt = (κ, i) 6= s∗v(h

t), if there exists a platform p′t ∈ P such that and

gκ(p′t) ≥ (1− δ)γM + δU−kt(h

t|s∗)

then, if kt is elected at ht, they are re-elected (s∗v(ht, (kt, pt)) = kt) if and only pt ∈ P .

Finally I adapt the renegotiation proofness notion. While different voters may disagree onwhether a proposed renegotiation is profitable, I require that there is no renegotiation which benefitsthe elected candidate and more than half the voters.

Definition 17 Internal Renegotiation Proofness (Ic)A subgame perfect equilibrium (φ∗, s∗) satisfies (Ic) if there does not exist a history ht, actions(kt, pt) and (k

′t, p′t), and a platform p

′′t+1 such that for candidate kt+1 = s∗v(h

t, (kt, pt)):

(1)

G({v : (1− δ)uv(p′′t+1) + δUv((φ∗, s∗)|(ht,(kt,pt))) > Uv((φ∗, s∗)|(ht,(k′t,p′t)))}) >

12,

(1− δ)gκ(p′′t+1) + δUkt+1((φ∗, s∗)|(ht,(kt,pt))) > Ukt+1((φ∗, s∗)|

(ht,(k′t,p′t))

).

(2) It is a best response for candidate kt+1 to implement p′′t+1 if the continuation equilibrium

from implementing p′′t+1 is (φ∗, s∗)|(ht,(kt,pt)) and the continuation equilibrium from imple-

menting any other platform is (φ∗, s∗)|(ht,(k

′t,p′t)).

So the conditions from the baseline model are easily adapted to the case with a continuum ofvoters. The result for this section is proved in Section 9.3.

9.1.3 Imperfect Monitoring

I first define the public histories as the sequence of candidates elected and voter utilities, ht =((k0, u0), . . . , (kt−1, ut−1)). To simplify the definition of the histories I assume that all candi-dates observe the specific platforms the elected candidates implemented, pt = (p0, . . . , pt−1). Lethtk = ((ht, pt), k), be the candidate-history where candidate k is elected after public history ht

and platforms pt have been implemented. Strategies and payoffs are defined as in the baselinemodel, except that payoffs must be evaluated in terms of expectations. Similarly, the notion ofretrospective voting adapts easily to an imperfect monitoring environment.

56

Definition 18 Retrospective Voting (Ru)A public perfect equilibrium, s∗, satisfies retrospective voting (Ru), if there exists a threshold utilitylevel u such that for all histories ht, if kt = s∗v(h

t), the incumbent candidate is re-elected (that iss∗v(h

t, (kt, ut)) = kt) if and only ifut ≥ u.

Further, for any ut, u′t with I(ut ≥ u) = I(u′t ≥ u),

s∗v|(ht,(kt,ut)) = s∗v|(ht,(kt,u′t)).

First note that for δ > δ and β not too large the welfare to the voter, from the best equilibriuminvolving only candidates of type κ and −κ will be increasing until we reach maximum polarizationor the rents are driven to 0.

Lemma 3 For any δ > δ there exists β(δ) > 0 such that, for any β ∈ (0, β(δ)), there exists κβsuch that for all κ ∈ [0, κβ] the best platform-stationary public perfect equilibrium satisfying (Ru)with only candidates of types κ and −κ ever elected involves policies xκ and −xκ respectively, withlevel of rent-seeking mκ

β, with

∂mκβ

∂β> 0;

∂qκβ∂β

< 0,

and mκβ = 0 when κ = κβ. Further, for all κ ∈ (0, κβ),

∂u(xκ,mκβ)

∂κ> 0.

So we have that when κβ < 1 the welfare of the voter is increasing in κ until we reach maximumpolarization with κ = 1. When κ ∈ (κβ, 1] the amount of rents secured will be driven to 0, and wemust solve for the κ that results in the minimum policy polarization.

Lemma 4 Suppose κβ(δ) < 1. Then, for δ > δ, there exists β(δ) ∈ (0, β(δ)] such that for allβ ∈ (0, β(δ)) and any κ ∈ [κβ, 1] the best platform-stationary public perfect equilibrium satisfying(Ru) in which candidates of types κ and −κ are elected involves rent-seeking m = 0 and policiesxβ(κ) and −xβ(κ) implemented by the respective candidates if elected, where

∂xβ(κ)∂β

> 0.

Further, there exists a unique κ which minimizes xβ(κ) on the interval [κβ, 1].

We can now define the parameters which define the best equilibrium in this class. For eachβ ∈ (0, β(δ)) define,

κ∗β ={

arg minκ∈[κβ ,1] xβ(κ), if κβ < 1,1, otherwise.

x∗β =

{minκ∈[κβ ,1] xβ(κ), if κβ < 1,

1

1+γ1

1−λ, otherwise.

57

m∗β ={

0, if κβ < 1,m1β, otherwise.

Now define the utility of this equilibrium as

u∗β = u(x∗β,m∗β).

Let q∗β be the probability of re-election in the best κ∗β equilibrium and define

uβ = u∗β − F−1(q∗β).

The results for this section are proved in Section 9.4.

9.1.4 Candidates’ Decision to Run for Office

I start by defining histories and strategies for the voters and candidates. As before the voterobserves the candidates elected and the utilities generated; in addition, the voter observes the setof candidates who have run for office. Hence,

ht = ((k0, u0,K1), . . . , (kt−1, ut−1,Kt)).

As in the previous section, let htk be the history at which candidate k is elected after public historyht and sequence of platforms pt. At any history the voter then chooses among the candidates whohave run for office,

sv(ht) = kt ∈ Kt.

Now, in each period, the elected candidate must decide which platform to implement, pt ∈ [−1, 1]×[0,M ]. In addition, all candidates, whether they were elected or not, must decide whether or notto run for office in the next period, dt+1 ∈ {R,N}. The candidates now act both at histories inwhich they are elected as well as histories where they aren’t,

sk(htk′ ) ={

(pt, dkt+1), if k = k′,

dkt+1, if k 6= k′.

The definition of platform stationarity, when candidates can decide whether to run, restrictsonly the platform the candidate would implement in office. It places no restrictions on the decisionof whether or not to run.

Definition 19 Platform-Stationarity

A public perfect equilibrium satisfies platform-stationarity if, for all candidates k, the platform theyimplement if elected does not depend on the history at which they are elected. That is, the platformimplemented at history htk depends only on the elected candidate, k.

The voter can only elect a given candidate if they ran for office. The definition of retrospectivevoting must then be updated accordingly. The last line of the definition says that, if the voter doesnot elect a certain candidate, the voter’s choices are not affected by that candidate’s decision ofwhether or not to run for office.

58

Definition 20 Retrospective Voting (Re)A public perfect equilibrium, s∗, satisfies Retrospective Voting (Re), if there exists a threshold utilitylevel u such that for all histories ht, if kt = s∗v(h

t), if kt ∈ Kt+1 then if

ut ≥ u,

thens∗v(h

t, (kt, ut,Kt+1)) = kt.

Further, for all ut, u′t and Kt+1 with I(ut ≥ u) = I(u′t ≥ u),

s∗v|(ht,(kt,ut,Kt+1)) = s∗v|(ht,(kt,u′t,Kt+1))= s∗v|(ht,(kt,ut,Kt+1\{k

′t+1})

,

for all k′t+1 ∈ Kt+1 \ {s∗v(ht, (kt, ut,Kt+1))}.

9.2 Proofs for Short-Lived Candidates

The proof of the result for short-lived candidates is nearly identical to the one with long-livedcandidates. As with the baseline model we begin with a couple of Lemmas.

Lemma 5 If δ ≥ δs then for all κ 6= κ∗ and any p′

= (x,m) such that p′ ∈ arg maxp{gκ(p) : u(p) ≥

u∗s},(1− δ)gκ(x,m) + δgκ(x,m) < (1− δ)γM + δgκ(−x∗,m∗s).

Proof. There are three cases to consider: candidates of type κ < 0, types κ ∈ [0, 1), and typesκ ∈ [κ, 1]\{κ∗}.First consider κ ∈ [κ, 1]\{κ∗}. If this set is non-empty this implies that κ∗ ≥ κ and so m∗ = 0. Fur-ther, candidates of type κ are indifferent between implementing platform (x(κ), 0) in every periodand implementing their ideal point with full rent-seeking today and having (−x(κ), 0) implementedin all future periods. In addition, as x(κ) has a unique minimizer on [κ, 1], by the definition of κwe have that

x∗ < x(κ) ≤ κ

1 + γ1

1−λ.

So, for candidates of type κ, for any platform (x,m) with u(x,m) ≥ u(x(κ), 0) we have

(1− δ)gκ(x,m) + δgκ(x,m) ≤ gκ(x(k), 0) = (1− δ)γM + gκ(−x(κ), 0) < (1− δ)γM + δgκ(−x∗,m∗s).

As u(x(κ), 0) < u∗s we have that

p′

= arg maxp{gκ(p) : u(p) ≥ u∗s} = (x∗, 0),

andgκ(x∗, 0) < gκ(x(k), 0),

so we are done for κ ∈ [κ, 1]\{κ∗}.Now consider κ ∈ [0,min{κ, 1}). As u∗s > −M the candidate benefits from increased rent-

seeking in this periods so we have thatu(p

′) = u∗s,

59

for any p′ ∈ arg maxp{gκ(p) : u(p) ≥ u∗s}. Note, in addition, that we must have policy xκ = κ

1+γ1

λ−1

or x ≤ κ

1+γ1

λ−1with level rent-seeking equal to 0. We can solve for the rents that give the voter

utility u∗s if the implemented policy is xκ = κ

1+γ1

λ−1from the equation,

u(xκ,m′κ) = u∗s,

to get that

m′κ = −u∗s −

γλ

1−λκλ

(1 + γ1

1−λ )λ.

It is then sufficient to show that

(1− δ)gκ(xκ,m′κ) + δgκ(xκ,m

′κ) < (1− δ)γM + δgκ(−x∗,m∗s).

First consider the case where κ > 1. Then κ∗ = 1 and x∗ = 1

1+γ1

λ−1. Define

w′(κ) = (1− δ)gκ(xκ,m

′κ) + δgκ(x

′κ,m

′κ)− (1− δ)γM − δgκ(−x∗,m∗s).

= (γ − δ(γ + 1))(−u∗ − γλ

1−λκλ

(1 + γ1

1−λ )λ)− κλ

(1 + γ1

1−λ )λ− (1− δ)γM + δ(κ+ x∗)λ +m∗s

Differentiating with respect to κ then gives

w′(κ) = λ[δ((κ+ x∗)λ−1 + (γ + 1)γ

λ1−λ (1 + γ

11−λ )−λκλ−1 − (1 + γ

11−λ )1−λκλ−1]

> λκλ−1(1 + γ1

1−λ )−λ[δ((1 + 2γ1

1−λ )λ−1(1 + γ1

1−λ ) + (γ + 1)γλ

1−λ )− (1 + γ1

1−λ )]

> 0

as δ ≥ δs. So we can conclude that w(κ) is increasing on the interval [0, 1). As w(1) = 0, whenκ > 1 we can conclude that

(1− δ)gκ(xκ,m′κ) + δgκ(xκ,m

′κ) < (1− δ)γM + δgκ(−x∗,m∗s),

on the interval κ ∈ [0,min{κ, 1}).Finally consider the case where κ < 1. Then we have x∗ ≤ κ

1+γ1

λ−1and m∗s = 0. Then

w(κ) = (γ − δ(γ + 1))((x∗)λ − γλ

1−λκλ

(1 + γ1

1−λ )λ)− κλ

(1 + γ1

1−λ )λ− (1− δ)γM + δ(κ+ x∗)λ.

Note that we have seen that when x∗ = κ

1+γ1

λ−1we have w(κ) < 0 from the previous analysis. As

w(κ) is clearly increasing in x∗ we can conclude that w(κ) < 0 for all κ ∈ [0, κ) when κ < 1. Sowe have that w(κ) < 0 for κ ∈ [0,min{κ, 1}). So we can conclude that for all (x,m) such thatu(x,m) ≥ u∗s and all κ ∈ [0,min{κ, 1}),

(1− δ)gκ(x,m) + δgκ(x,m) < (1− δ)γM + gκ(−x∗,m∗s).

Finally note that for all κ < 0 the payoff from platform (−x∗,m∗s) in future periods is strictlyhigher than for candidate −κ > 0. As candidate −κ does not strictly prefer any platform which

60

gives the voter u∗s to their ideal point today followed by (−x∗,m∗s) in future periods candidates oftype κ must strictly prefer to implement their ideal point with full corruption today with (−x∗,m∗s)in all future periods to any platform which gives the voter utility u∗s.

Clearly, by symmetry, this also implies that no type other than −κ∗ is willing to implement aplatform which gives the voter utility u∗s if the punishment is that (x∗,m∗s) would be implementedin all subsequent periods.

Lemma 6 Suppose s∗ is a SPE satisfying platform-stationarity, (R∗s) and (Is), in which the sametype of candidate κ0 is elected and implements platform (x0,m0) in every period along the equilib-rium path. Then, for all κ ∈ [−1, 1] and p

′= (x,m)with p

′ ∈ arg maxp{gκ(p) : u(p) ≥ u(x0,m0)},

gκ(x,m) + (1− δ)gκ(x,m) ≤ (1− δ)γM + δgκ(x0,m0).

Proof Let u be the standard for re-election. As the same type is always elected along the equi-librium path we must have that u(x0,m0) ≥ u. First note that, in fact, we must have thatu(x0,m0) = u. If u(x0,m0) > u then, in order for candidate k0 = (κ0, 0) to be optimizing, wemust have that (x0,m0) = (κ0,M). Suppose κ0 6= 0. Then, if a candidate of type 0 were elected,by condition (R∗), they would implement platform (0,M) and secure election for the type 0 in thenext period. As the voter would prefer this outcome we have a contradiction, so we must havecandidate of type κ0 = 0 elected. Note finally that the standard for re-election must be −M sinceotherwise candidate types κ 6= 0 could secure re-election for their type while implementing platform(κ,M), provided κ is sufficiently small. As this gives payoff to the voter strictly lower than thepayoff from electing candidate κ = 0 this could not be a best response for the voter. Hence wemust have u(x0,m0) = u.

Now suppose there exists a κ1 ∈ [−1, 1] and a platform p′ = (x1,m1) such that

p′ ∈ arg min{gκ(p) : u(p) ≥ u(x0,m0) = u},

(1− δ)gκ1(x1,m1) + δgκ1(x1,m1) > (1− δ)γM + δgκ1(x0,m0),

and let k1 = (κ1, 0). Now since U−k1(h0|s∗) ≤ gκ1((x0,m0)), condition (R∗) guarantees that can-didate k1 would secure election for the same type by implementing a platform which gives thevoter at least u if elected. By continuity, provided ε is sufficiently small, candidate k1 would bewilling to implement platform ((1 − ε)x1, (1 − ε)m1) to move to history (h0, (k1, (x1,m1))). Asu((1− ε)x1, (1− ε)m1) > u this renegotiation would benefit both the voter and candidate k1, vio-lating condition (I). �

Proof of Theorem 6Proof of Part (1):

It is immediate, by construction, that the above strategies constitute a subgame perfect equilib-rium. I now show that this equilibrium satisfies condition (R∗) by showing that for any candidateof type κ /∈ {−κ∗, κ∗} there does not exist a platform p

′t ∈ [−1, 1]× [0,M ] which satisfies

p′t ∈ arg max

p{gκ(p) : u(p) ≥ u∗s},

61

(1− δ)gκ(p′t) + δgκ(p

′t) ≥ (1− δ)γM + δU−(κ,t)(h

t|s∗),

at any history. Without loss of generality we can assume that κ ≥ 0 and so

U−(κ,t)(ht|s∗) ≥ gκ(−x∗s,m∗s),

and the result follows from the previous Lemma. Now we must show that this equilibrium satisfies(I). Notice, however, that since at each history the elected candidate is indifferent between deviatingto their ideal point with full rent-seeking and implementing a platform which gives the voter utilityu∗s, the candidate would strictly prefer deviating to any platform which gives the voter utility higherthan u∗s given any continuation history. So the above equilibrium satisfies condition (Is). �Proof of Part (2):

Suppose there exists a SPE satisfying (R) and platform-stationarity which gives the voter utilityu′> u∗s > −M . Let s∗ be a SPE such that U(s∗) = u

′. By platform-stationarity the voter much

receive the same payoff, u′

in every period regardless of the history.First note that the same type of candidate must be elected in every period along the equilibrium

path. If the same type of candidate would be elected if the incumbent implemented their idealpolicy with full rents then they must be choosing that platform: if the incumbent would prefernot to have the same type elected tomorrow they would have to choose a platform to maximize{gκ0(x,m) : u(x,m) < u ≤ u(κ0,M)}. This maximum could not exist. So if a different type ofcandidate is elected along the equilibrium path the candidate cannot meet the standard to havethe same type elected if they implement their ideal policy with full rent-seeking. By condition (R),then, in any period where the candidate does not secure re-election they must implement theirideal point while securing full rents. As the payoff to the voter in any period where the candidateengages in full re nt-seeking is no higher than −M at no history can the voter elect a candidatewho will not secure re-election. Similarly, if the incumbent implemented their ideal policy with fullrent-seeking the voter they would not secure re-election for the same type.

WLOG assume the candidate elected in the first period, k0 = (κ0, 0) where κ0 ≥ 0. Now considerthe continuation equilibrium where a candidate of the same type is not elected in the next period,s∗|(k0,(κ0,M)). Let k1 = (κ1, 1) be the candidate elected in the first period of that equilibrium. Bythe above arguments this continuation equilibrium must give the voter utility u

′and result in the

same platform (x1,m1) being implemented in all future periods along the equilibrium path. Sincethe incumbent type is re-elected, the continuation play depends only on whether the thresholdutility level is met, and any platform which gives the voter utility level u

′must involve less than

full corruption we must have

gκ0(x0,m0) = maxp{gκ0(p) : u(p) ≥ u′} < max

p{gκ0(p) : u(p) ≥ u∗s}.

Let (x0∗,m

0∗) = arg maxp{gκ0(p) : u(p) ≥ u∗s}, and notice

x0 ≤ x0∗ < κ0.

From the incentive constraints, we must have that

(1− δ)gκ0(x0,m0) + δgκ0(x0,m0) ≥ (1− δ)γM + gκ0(x1,m1).

62

Recall, by Lemma 5, that

(1− δ)gκ0(x0∗,m

0∗) + δgκ0(x0

∗,m0∗) ≤ (1− δ)γM + δgκ0(−x∗,m∗).

Combining these equations tells us we must have

gκ0(x0∗,m

0∗)− gκ0(−x∗,m∗) < gκ0(x0,m0)− gκ0(x1,m1).

As x0∗ ∈ [x0, κ0) and

u(x1,m1) = u(x0,m0) = u′> u∗s = u(−x∗,m∗) = u(x0

∗,m0∗),

this implies that x1 < −x∗ and m1 < m∗. As m∗ > 0, it must be that κ∗ = 1 and x∗ = 1

1+γ1

λ−1. So

the continuation equilibria must have the elected candidate implementing a platform with

x1 <−1

1 + γ1

λ−1

.

However, since the candidate is evaluated strictly on the utility they provide to the voter they mustselect a platform which trades off policy and rent-seeking in the same ratio as the voter. Since wehave m1 < M we must have

|x1| ≤ |κ1|1 + γ

1λ−1

≤ 1

1 + γ1

λ−1

,

which is a contradiction. Therefore, there cannot exist a SPE satisfying (Rs) and platform-stationarity which gives the voter utility greater than u∗s.�

Proof of Part (3):Claim 1: In any SPE satisfying platform-stationarity, (R∗s) and (Is) the same type of candidate isre-elected forever along the equilibrium path.It is sufficient to show that there does not exist a SPE satisfying platform-stationarity, (R∗s) and(Is) in which different types of candidates are elected in the first two periods. Let (κ0, 0) denotecandidate elected in the initial period. Note that if (κ0, 1) is not elected in the next period it mustbe that would not be elected if the incumbent implements their ideal policy with full rent-seeking: ifthe incumbent prefers to implement a different platform not to have the same type elected tomorrowthey would have to choose a platform to maximize {gκ0(x,m) : u(x,m) < u ≤ u(κ0,M)}, but sucha maximum would not exist. By condition (R) the continuation equilibrium does not depend onthe specific platform they implement, so candidate (κ0, 0) must implement platform p0 = (κ0,M)in the initial period. By stationarity all type κ0 candidates must implement that platform at anyother history as well. Now note that by (R) there must be a feasible utility level which would securethe same type re-election. Hence if (κ0, 0) implemented platform p0 = (0, 0) then type (κ0, 1) wouldbe re-elected.

Now, define h0∗ = h0 and recursively define

ht∗ = (ht−1∗ , ((κ0, t− 1), (0, 0))),

andht∗∗ = (ht−1

∗ , ((κ0, t− 1), (κ0,M))).

63

That is, ht∗ is the history where candidates of the type elected in period-0 have been electedand implemented platform (0, 0) in all previous periods, and hence (κ0, t) will be re-elected at ht∗.Similarly, ht∗∗ is the history at which, at history (ht−1

∗ , (κ0, t−1)), candidate (κ0, t−1) implementedtheir ideal policy with full rent-seeking, and so (κ0, t) will not be elected at ht∗∗. As the candidateelected in the first period of the continuation equilibrium (s∗|ht∗∗) is not of type κ0, and the payoffto the voter must be the same in every period, we can conclude that κt = −κ0 6= 0. Hence thepayoff to candidate (κ0, t− 1) from continuation equilibrium (s∗|ht∗∗) must satisfy

U(κ0,t−1)(s∗|ht∗∗) ≤ −M − (1− δ)|2κ0|λ,

for all t. Notice that, by the candidate’s strategy we must have

U(κ0,t−1)(s∗|ht∗) = −(1− δ)M + δU(κ0,t−1)(s∗|ht+1

∗∗).

Now, recall that by platform-stationarity the voter must receive the same continuation payoffat every history. In particular, then, for all t we must have

U(s∗|ht∗) = U(s∗|ht∗∗) = u(κ0,M).

Now supposeU(κ0,t−1)(s

∗|ht∗∗) < U(κ0,t−1)(s∗|ht∗).

Then, for ε > 0 sufficiently small we have

(1− δ)u(κ0,M − ε) + δU(s∗|ht∗) > U(s∗|ht∗∗),

(1− δ)γ(M − ε) + δU(κ0,t−1)(s∗|ht∗) > U(κ0,t−1)(s

∗|ht∗∗),

and(1− δ)γ(M − ε) + δU(κ0,t−1)(s

∗|ht∗) > (1− δ)γM + δU(κ0,t−1)(s∗|ht∗∗).

This means that a mutually-profitable renegotiation for the candidate and the voter would exist.So we must have that

U(κ0,t−1)(s∗|ht∗∗) ≥ Uk0(s∗|ht∗) = (1− δ)γM + δU(κ0,t−1)(s

∗|ht+1∗∗

).

Not also that for all histories ht and t′< t,

U(κ,t′ )(s∗|ht) = U(κ,0)(s

∗|ht),

so we can conclude that

U(κ0,0)(s∗|ht+1∗∗

)− U(κ0,0)(s∗|ht∗∗) ≤

1− δδ

(U(κ0,0)(s∗|ht∗∗) +M) ≤ −(1− δ)2

δ(2κ0)λ,

for all t. Finally, because κ0 cannot equal 0, this means that

limt→∞

U(κ0,0)(s∗|ht∗∗) = −∞.

As payoffs are bounded this is a contradiction. Therefore, we can conclude that there does notexist a SPE satisfying platform-stationarity, (R∗) and (I) in which the incumbent is ever defeatedalong the equilibrium path.

64

Claim 2: There does not exist a SPE satisfying platform-stationarity, (R∗s) and (Is) which gives thevoter payoff greater than u∗s.This follows immediately from Part (2).Claim 3: There does not exist a SPE satisfying platform-stationarity, (R∗s) and (Is) which gives thevoter payoff less than u∗.Let s∗ be a SPE satisfying (R∗s), (Is), and platform-stationarity where U(s∗) = u

′< u∗. By Claim

1 we can restrict attention to SPEs in which the same type of candidate is elected in every period.Let κ0 be the type of candidate elected at h0 and let (x0,m0) be the platform type κ0 candidatesimplement when elected. Then we must have u(x0,m0) = u

′. WLOG assume x0 ≥ 0.

First note that we cannot have u′ ≤ −M . If u

′< −M then type-0 candidates would implement

their ideal point with full rent-seeking if elected, giving the voter utility −M . Any other candidatewould either implement their ideal point with full corruption, or implement a platform which givesthe voter payoff u. As either of these alternatives gives the voter payoff strictly lower than −M , thevoter would have a strict incentive to elect a type-0 candidate. However, by condition (R∗s) thereexists candidates of type κ, where |κ| is sufficiently small, who would be elected at some history.Hence there cannot be an equilibrium with u < −M . Now consider u

′= −M . Then for all κ < 0,

gκ(x0,m0) ≤ gκ(0,M)

Now, because δ > δs, we know that for κ < 0, and κ small

gκ(κ

1 + γ1

λ−1

,M − |κ|λ

(1 + γ1

λ−1 )λ) > (1− δ)γM + δgκ(0,M) ≥ (1− δ)γM + δgκ(x0,m0)

(κ

1 + γ1

λ−1

,M − |κ|λ

(1 + γ1

λ−1 )λ) ∈ arg max

p{gκ(p) : u(p) ≥ u(x0,m0)}

so by Lemma 6 we have a contradiction. So we cannot have a SPE satisfying (R∗s), (Is) andplatform-stationarity which gives payoff u

′ ≤ −M to the voter.Now suppose u

′ ∈ (−M,u∗s). Notice that the voter would not elect a candidate of type κ0

in the next period if the incumbent implements their ideal point while securing full rents. Let(κ1, 1) = s∗v(k0, (κ0,M)), let (x1,m1) be the platform candidates of type κ1 implement if elected.

We begin by showing that we must have κ0 ∈ (0,min{κ, 1}). By Lemma 6 we must have

(1− δ)g−κ∗(−x∗,m∗) + δg−κ∗(−x∗,m∗) ≤ (1− δ)γM + δg−κ∗(x0,m0),

sog−κ∗(x0,m0) ≥ g−κ∗(x∗,m∗s).

As u(x0,m0) < u(x∗,m∗s) we must have x0 < x∗ and m0 > m∗s. We can then conclude that κ0 < κ

and κ0 < κ∗, so κ0 < min{κ, 1}. Now suppose κ0 = 0. Then, since g0(p) = u(p) for all platforms,we must have gκ0(x0,m0) = gκ0(x1,m1). As u(x0,m0) > −M ,

gκ0(x0,m0) < (1− δ)γM

and so(1− δ)gκ0(x0,m0) + δgκ0(x0,m0) < (1− δ)γM + δgκ0(x1,m1),

65

and so the candidate would have an incentive to deviate. So we can conclude that κ0 ∈ (0,min{κ, 1}).Combining Lemma 6 and candidate optimizing we must have

(1− δ)gκ0(x0,m0) + δgκ0(x0,m0) = (1− δ)γM + δgκ0(x1,m1).

Since m0 > m∗ ≥ 0, x0 = κ0

1+γ1

λ−1this becomes

m0 = M +1γ

(κλ0

(1 + γ1

1−λ )λ+m0 − δ((κ0 + x1)λ +m1)).

Note that if x1 > 0 we would have

(1− δ)g−κ0(−x0,m0) + δg−κ0(−x0,m0) > (1− δ)γM + δg−κ0(x1,m1),

so we can conclude that x1 ≤ 0. Now consider candidates of type κ ∈ (κ0,min{κ, 1}). Letxκ = κ

1+γ1

λ−1and define mκ to be the solution to

u(xκ,m′κ) = u

′.

Notice that(xκ,m

′κ) ∈ arg max

p{gκ(p) : u(p) ≥ u′ = u(x1,m1)}.

Then define f(κ) to be

f(κ) = (1− δ)gκ(xκ,m′κ) + δgκ(xκ,m

′κ)− (1− δ)γM − δgκ(x1,m1)

Note that f(κ0) = 0 and

f ′(κ0) = λ(δ(κ0 + x1)λ−1 − (1 + γ1

1−λ )−λκλ−10 (1 + γ

λ1−λ (γ(1− δ)− δ)))

≥ λκλ−10 (δ(1 + (1 + γ

11−λ )−λγ

λ1−λ (1 + γ))− (1 + γ

11−λ )1−λ) > 0

as δ > δs. So there exists κ ∈ (κ0, 1) for which f(κ) > 0 and so

(1− δ)gκ(xκ,m′κ) + δgκ(xκ,m

′κ) > (1− δ)γM + δgκ(x1,m1)

By Lemma 6 this is a contradiction. So we can conclude that there does not exist a SPE satisfying(R∗s), (Is), and platform-stationarity where the payoff to the voter is less than u∗s.

Combining claims 1− 3 we can see that in any SPE satisfying platform-stationarity, (R∗s) and(Is) the voter must receive payoff u∗s. Consequently any equilibrium must be of the form identifiedabove.�

66

9.3 Continuum of Voters

We prove this theorem by relating it to the case with a representative voter. We first prove a simpleLemma.

Lemma 7 For all platforms p and p′,

G({v : uv(p) > uv(p′)}) > 1

2,

if and only ifu0(p) > u0(p

′).

Proof Let p = (x,m) and p′

= (x′,m′) be two platforms and assume, without loss of generality,

that x ≥ x′ . Note that for any v, w ∈ [−1, 1] with w < v, if

uv(p) ≤ uv(p′)

thenuw(p) < uw(p

′).

So there exists a unique cutoff v(p, p′) ∈ [−1, 1] ∪ {−∞,∞} such that

{v ∈ [−1, 1] : uv(p) ≥ uv(p′)} = {v ∈ [−1, 1] : v ≥ v(p, p′)}.

Note then thatG({v : uv(p) > uv(p

′)}) > 1

2,

if and only ifv(p, p

′) > 0,

which holds if and only ifu0(p) > u0(p

′).

�

Proof of Theorem 7Claim 1: If candidates’ strategies, s, satisfy platform-stationarity, φ is a collective best response ifand only if there does not exist a φ

′such that

U0((φ′, s)) > U0((φ, s)).

By platform-stationarity the voters can make their decision at every history without regard to howthis affects the future payoffs, so it is sufficient to show that for all platforms p and p

′,

G({v : uv(p) > uv(p′)}) > 1

2,

if and only ifu0(p) > u0(p

′),

and hence follows from Lemma 7.

67

Claim 2: If a subgame perfect equilibria (φ∗, s∗) satisfies (Rc) then there exists a feasible utilitylevel u such that

P = {p ∈ [−1, 1]× [0,M ] : u0(p) ≥ u}.

Immediate from Lemma 7.Claim 3: There does not exist a platform-stationary SPE satisfying (Rc) in which

U0(φ∗, s∗) > u∗.

Suppose there exists a SPE satisfying (Rc) which gives the voter payoff higher than u∗. Byplatform-stationarity and Claim 1 the payoff to the voter of type-0 must be u∗ in each period.In addition, by Claim 2, the incumbent candidate is evaluated entirely on whether they providethe type-0 voter with some threshold utility level u. Hence the election decisions made by thecontinuum of voters must be consistent with a single voter of type-0 who is playing a best responseto the candidates’ strategies and whose strategy satisfies condition (R). If there exist a profile ofcandidates’ strategies which are consistent with equilibrium, this then the same strategies in thebaseline model would form a SPE satisfying platform-stationarity, (R) and giving the type-0 voterpayoff greater than u∗. This would violate Theorem 1 from the main text. So we can conclude thatthere does not exist a platform-stationary SPE satisfying (Rc) with U0(φ∗, s∗) > u∗.Claim 4: In any subgame perfect equilibrium (φ∗, s∗) which satisfies (R∗c) and (Ic) the incumbentmust be re-elected forever along the equilibrium path.It is sufficient to show that there does not exist a SPE satisfying platform-stationarity, (R∗c) and(Ic) in which the incumbent is defeated after the first period. Let k0 denote candidate elected in theinitial period and let κ0 be the type of the candidate k0. Note that if k0 is defeated in equilibriumit must be that they are not re-elected if they implement their ideal point with full rent-seeking:otherwise the candidate would implement that platform and be re-elected forever. By condition(Rc) the continuation equilibrium does not depend on the specific platform they implement, socandidate k0 must implement platform p0 = (κ0,M) in the initial period. By platform-stationaritythey must implement that platform at any other history as well. By the previous Lemma, since allcandidates are implementing the same platforms at every history, the median voter must receivepayoff u0(κ0,M) in every period. Now note that we could not have an equilibrium with κ0 6= 0.Consider the behavior of a candidate of type κ = 0 if elected. There are two possible casesto consider: either there exists a platform the candidate could implement to secure re-election orthere isn’t. In the first case there exist histories of the equilibrium at which the candidate is elected.As the candidate is evaluated strictly on whether they meet utility u for the median voter, at thathistory they must implement a platform which gives the median voter utility at least −M at thathistory, and, by stationarity, at all histories. If the candidate cannot be re-elected regardless of theplatform they implement, since the candidate is indifferent over the possible continuation histories– a candidate with ideal policy 0 has identical preferences to the median voter if out of office –the candidate will simply implement their ideal point with full corruption. As this is preferredby a strict majority of the voters, electing a candidate of type κ0 6= 0 cannot be a collective bestresponse. Hence, we need only consider equilibria with κ0 = 0. Note also that by (Rc) there mustbe a feasible utility level which would secure the candidate re-election. Hence if k0 implementedplatform p0 = (0, 0) they would be re-elected.

68

Now, define h0∗ = h0 and recursively define

ht∗ = (ht−1∗ , (k0, (0, 0))),

andht∗∗ = (ht−1

∗ , (k0, (0,M))).

That is, ht∗ is the history where the candidate elected in period-0 has been elected and implementedplatform (0, 0) in all previous periods, and hence will be re-elected at ht∗. Similarly, ht∗∗ is the historyat where at history (ht−1

∗ , k0) candidate k0 implemented their ideal policy with full rent-seeking,and so will not subsequently be re-elected at ht∗∗. As candidate k0 is not elected in the first periodof continuation equilibrium (s∗|ht∗∗), and the stage game payoff to the candidate is no higher thanγM when in office and 0 if not in office, the payoff to k0 from continuation equilibrium (s∗|ht∗∗)must satisfy

Uk0(s∗|ht∗∗) ≤ (1− δ)0 + δγM = δγM,

for all t. Notice that, by the candidate’s strategy we must have

Uk0(s∗|ht∗) = (1− δ)γM + δUk0(s∗|ht+1∗∗

).

Now, recall that by platform-stationarity the median voter must receive the same continuationpayoff at every history. In particular, then, for all t we must have

U0(s∗|ht∗) = U0(s∗|ht∗∗) = u0(0,M).

Now supposeUk0(s∗|ht∗∗) < Uk0(s∗|ht∗).

Then, as all continuation equilibria involve the platform (0,M) implemented at all histories, forε > 0 sufficiently small, we have for all v ∈ [−1, 1] and the candidate k0:

(1− δ)uv(0,M − ε) + δUv(s∗|ht∗) > Uv(s∗|ht∗∗),

(1− δ)γ(M − ε) + δUk0(s∗|ht∗) > Uk0(s∗|ht∗∗),

and(1− δ)γ(M − ε) + δUk0(s∗|ht∗) > (1− δ)γM + δUk0(s∗|ht∗∗).

This means that a mutually-profitable renegotiation for the candidate and all voters would exist.So we must have that

Uk0(s∗|ht∗∗) ≥ Uk0(s∗|ht∗) = (1− δ)γM + δUk0(s∗|ht+1∗∗

).

This implies that

Uk0(s∗|ht+1∗∗

)− Uk0(s∗|ht∗∗) ≤1− δδ

(Uk0(s∗|ht∗∗)− γM) ≤ −(1− δ)2

δγM,

for all t. However thenlimt→∞

Uk0(s∗|ht∗∗) = −∞.

69

As payoffs are bounded this is a contradiction. Therefore, we can conclude that there does notexist a SPE satisfying platform-stationarity, (R∗c) and (Ic) in which the incumbent is ever defeatedalong the equilibrium path.Claim 5: In any subgame perfect equilibrium (φ∗, s∗) which satisfies (R∗c) and (Ic) the type-0 votermust receive payoff u∗ in every period.First note, by Claim 3, we cannot have a SPE which gives the type-0 voter payoff greater than u∗.So it remains to be proven that there cannot exist a SPE which gives the type-0 voter payoff lessthan u∗.

Now suppose there exists a SPE which gives the type-0 voter payoff U(s∗) = u′< u∗ which

satisfies platform-stationarity, (R∗c) and (Ic). By claim 4 we can restrict attention to SPEs in whichthe candidate elected in the first period is re-elected forever.

First note that in any SPE satisfying platform-stationarity, (R∗c) and (Ic) in which the sameplatform (x0,m0) is implemented in every period, for all platforms (x,m) such that u0(x,m) ≥u0(x0,m0),

gκ(x,m) ≤ (1− δ)γM + δgκ(x0,m0),

for all κ ∈ [−1, 1]. To see this, let u be the standard for re-election – that is the level of utility thecandidate must provide to the type-0 voter to secure re-election. As the incumbent is re-electedalong the equilibrium path we must have that u(x0,m0) ≥ u. First note that, in fact, we musthave that u(x0,m0) = u. If u(x0,m0) > u then, in order for k0 to be optimizing, we must havethat (x0,m0) = (κ0,M). Then if any other candidate of type κ0 were elected, by condition (R∗),they would implement platform (κ0,M) and secure re-election. Note finally that such a candidatecould then offer platform (κ0,M − ε) – a platform strictly preferred by all voters – to move to thehistory at which they are elected, violating condition (Ic).

Now suppose there exists a type κ1 ∈ [−1, 1] and a platform p = (x1,m1) such that

u(p) ≥ u(x0,m0) = u

gκ1(x1,m1) > (1− δ)γM + δgκ1(x0,m0)

and let k1 be a candidate of type κ1. WLOG assume that x1 ≤ x0. Now, since U−k1(h0|(φ∗, s∗)) ≤gκ1((x0,m0)), condition (R∗) guarantees that candidate k1 would secure re-election by implementinga platform which gives the type-0 voter at least u if elected. By continuity, provided ε is sufficientlysmall, candidate k1 would be willing to implement platform ((1−ε)x1, (1−ε)m1) to move to history(h0, (k1, (x1,m1))). As u0((1− ε)x1, (1− ε)m1) > u = u0(x0,m0), there exists a v > 0 such that forall v < v,

(1− δ)uv((1− ε)x1, (1− ε)m1) + δuv(x1,m1) > uv(x0,m0).

Hence, this renegotiation would benefit candidate k1 and at more than half the voters, violatingcondition (Ic). Hence, we can conclude that for any platform (x,m) with u0(x,m) ≥ u0(x0,m0),

gκ(x,m) ≤ (1− δ)γM + δgκ(x0,m0).

Now we show that no SPE satisfying platform-stationarity, (R∗c), and (Ic) gives the type-0 voterpayoff less than u∗. Let k0 be the candidate elected at h0, let κ0 be candidate k0’s type, and let

70

(x0,m0) be the platform k0 implements when elected. Then we must have u0(x0,m0) = u′. WLOG

assume x0 ≥ 0.Suppose u

′ ≤ − (1−δ)γMγ+δ . Then for all κ < 0,

gκ(x0,m0) ≤ gκ(0,(1− δ)γMγ + δ

)

Now, because δ > δ, we know that for κ < 0, and κ small

gκ(κ

1 + γ1

λ−1

,(1− δ)γMγ + δ

− |κ|λ

(1 + γ1

λ−1 )λ) > (1− δ)γM + δgκ(0,

(1− δ)γMγ + δ

) ≥ (1− δ)γM + δgκ(x0,m0)

u0(κ

1 + γ1

λ−1

,(1− δ)γMγ + δ

− |κ|λ

(1 + γ1

λ−1 )λ) = u0(0,

(1− δ)γMγ + δ

) ≥ u0(x0,m0)

and so we have a contradiction. So we cannot have a SPE satisfying (R∗c), (Ic) and platform-stationarity which gives payoff u

′ ≤ − (1−δ)γMγ+δ to the type-0 voter.

Now suppose u′ ∈ (− (1−δ)γM

γ+δ , u∗). Notice that candidate k0 could not be re-elected if theyimplement their ideal point while securing full rents. Let k1 = s∗v(k0, (κ0,M)), let κ1 be the typeof candidate k1, and let (x1,m1) be the platform the candidate would implement if elected.

Next we note that we must have κ0 ∈ (0,min{κ, 1}). We know that we must have

g−κ∗(−x∗,m∗) ≤ (1− δ)γM + δg−κ∗(x0,m0),

sog−κ∗(x0,m0) ≥ g−κ∗(x∗,m∗).

As u0(x0,m0) < u0(x∗,m∗) we must have x0 < x∗ and m0 > m∗. We can then conclude thatκ0 < κ and κ0 < κ∗, so κ0 < min{κ, 1}. Now suppose κ0 = 0. Then the candidate must implementa platform with x0 = 0, so −m0 = u

′. Also, since κ0 = 0, gκ0(x1,m1) = u0(x1,m1) = u

′. However,

as u′> − (1−δ)γM

γ+δ ,gκ0(0,m0) < (1− δ)γM + δgκ0(x1,m1),

and so the candidate would have an incentive to deviate. So we can conclude that κ0 ∈ (0,min{κ, 1}).As it must be optimal for the incumbent not to deviate we know that

gκ0(x0,m0) ≥ (1− δ)γM + δgκ0(x1,m1).

However, since u(x0,m0) = u(x1,m1), it cannot be the case that gκ0(x0,m0) ≥ (1 − δ)γM +δgκ0(x1,m1). Therefore we know that

gκ0(x0,m0) = (1− δ)γM + δgκ0(x1,m1).

Since m0 > m∗ ≥ 0 the ratio of marginal utilities in terms of policy must be γ so we know thatx0 = κ0

1+γ1

λ−1. We can then determine the amount of rent seeking candidate k0 would engage in,

m0 = (1− δ)M +1γ

(κλ0

(1 + γ1

1−λ )λ− δ((κ0 + x1)λ +m1)).

71

Note that if x1 > 0 we would have

g−κ0(−x0,m0) > (1− δ)γM + δg−κ0(x1,m1),

so we can conclude that x1 ≤ 0. Now consider candidates of type κ ∈ (κ0,min{κ, 1}). Letxκ = κ

1+γ1

λ−1and define m

′κ to be the solution to

u0(xκ,m′κ) = u

′

Then define f(κ) to be

f(κ) = gκ(xκ,m′κ)− (1− δ)γM − δgκ(x1,m1).

Note that f(κ0) = 0 and

f ′(κ0) = λ(δ(κ0 + x1)λ−1 − (1 + γ1

1−λ )1−λκλ−10 ) ≥ λκλ−1

0 (δ − (1 + γ1

1−λ )1−λ) > 0,

as δ > δ. So there exists κ ∈ (κ0, 1) for which f(κ) > 0 and so

gκ(xκ,m′κ) > (1− δ)γM + δgκ(x1,m1),

which is a contradiction. So we can conclude that there does not exist a SPE satisfying (R∗c), (Ic),and platform-stationarity where the payoff to the type-0 voter is less than u∗.

From Claims 1 and 2 any platform-stationary subgame perfect equilibrium which satisfies (Rc)corresponds to platform-stationary equilibrium with a representative voter which satisfies condition(R). Claim 3, combined with the previous Lemma, then proves that there cannot be an equilibriumwhich gives more than half the voters a higher utility level than the one described. This proves part(2) of the Theorem. In addition, Claims 4 and 5 show that in any equilibrium with a continuum ofvoters that satisfies platform-stationarity, (R∗c) and (Ic) the median voter must receive payoff u∗. Asthe incentives faced by the candidates are the same in both models, the set of SPEs (φ∗, s∗) whichsatisfy platform stationarity, (R∗), and (I) corresponds to those with with a representative voterwhich satisfy platform-stationarity, (R∗) and (I). This proves parts (1) and (3) of the Theorem.�

9.4 Imperfect Monitoring

Proof of Theorem 8 We look for equilibria which give the voter payoff higher than the staticNash payoff −M . As the candidates’ strategies are stationary, all candidates who are elected atany history must be implementing platforms which give the voter the same utility level, u(0,mβ).As we are looking for equilibria in which only candidates of type-0 are elected, and since the voteronly observes the utility level, such an equilibrium must have policy x = 0 with level of rent-seekingmβ < M . As the worst possible continuation equilibrium satisfying these conditions for a candidateof type-0 has them never elected at any future history reached with positive probability, we knowthat the continuation payoff to the incumbent if defeated is −mβ.

72

Since the voter’s strategy satisfies Retrospective Voting (Ru), the incumbent is re-elected if andonly if the payoff to the voter is at least uβ. Then the incumbent is re-elected with probability

qβ = Pr(εt ≥ uβ − u(0,mβ)) = F (u(0,mβ)− uβ).

There are then two relevant states for the elected candidate: where they are re-elected and wherethey are defeated. Now define

V (mβ, uβ) = (1− δ)γmβ + δ(qβV (mβ, uβ)− (1− qβ)mβ),

to be the expected payoff to a candidate at the history at which they are elected. Then we have

V (mβ, uβ) +mβ =(1− δ)(γ + 1)mβ

1− δqβ.

Notice that this is the difference in expected utility between the history at which the incumbentis re-elected and the history at which they are defeated. We can then define V (m,mβ, uβ) to bethe expected utility to the elected candidate from engaging in level of rent-seeking m in this periodand engaging in level of rent-seeking mβ in all future histories if elected. Then

V (m,mβ, uβ) = (1− δ)γm+ δ[F (u(0,m)− uβ)V (mβ, uβ)− (1− F (u(0,m)− uβ))mβ].

In order to have an equilibrium we must have that

mβ ∈ arg maxm

V (m,mβ, uβ).

This means that we must have

V (m,mβ, uβ)− V (mβ, uβ) = (1− δ)γ(m−mβ)− δ[F (u(0,m)− uβ)− qβ)(V (mβ, uβ) +mβ] ≤ 0,

for all m ∈ [0,M ].We begin by showing that any equilibrium with mβ < M must have

u(0,mβ) ∈ (uβ, uβ + β).

Taking first order conditions, for mβ to be maximizing V (m,mβ, uβ) we must have that

∂V (mβ,mβ, uβ)∂m

= (1− δ)γ − δf(u(0,mβ)− uβ)(V (mβ, uβ) +mβ) = 0,

so the candidate cannot have a profitable marginal deviation. Note that this implies that f(u(0,mβ)−uβ) > 0 and so mβ < uβ + β. Now consider a deviation to M , or full rent-seeking. From the aboveequality we have

V (M,mβ, uβ)− V (mβ, uβ) = (1− δ)γ(M −mβ) + δ(F (u(0,M)− uβ)− qβ)(V (mβ, uβ) +mβ)

= (M −mβ)δf(u(0,mβ)− uβ)(V (mβ, uβ) +mβ)

+δ(F (u(0,M)− uβ)− qβ)(V (mβ, uβ) +mβ)

= δ(V (mβ, uβ) +mβ)(f(u(0,mβ)− uβ)(M −mβ) + (F (u(0,M)− uβ)− qβ))

= δ(V (mβ, uβ) +mβ)∫ M

mβ

(f(u(0,mβ)− uβ)− f(u(0,m)− uβ))dm

73

Note that if u(0,mβ) ≤ uβ, then f(u(0,m)− uβ)) is decreasing in m on [mβ,M ] so we must havethat V (M,mβ, uβ)− V (mβ, uβ) > 0. So we can conclude that u(0,mβ) ∈ (uβ, uβ + β). Therefore,in any equilibrium with mβ < M the probability of re-election must be qβ ∈ (1

2 , 1).Next note that, if V (M,mβ, uβ)− V (mβ, uβ) ≤ 0,

V (m′,mβ, uβ)− V (mβ, uβ) ≤ 0,

for all m′ ∈ [0,M ]. To see this, note that

V (m′,mβ, uβ)− V (mβ, uβ) = δ(V (mβ, uβ) +mβ)

∫ m′

mβ

(f(u(0,mβ)− uβ)− f(u(0,m)− uβ))dm.

There are two cases to consider, u(0,m′) ≥ uβ or u(0,m

′) < uβ. Consider first the case that

u(0,m′) ≥ uβ. As f(z) is decreasing in |z| we know that f(u(0,m)− uβ) is decreasing in m. Hence∫ m

′

mβ

(f(u(0,mβ)− uβ)− f(u(0,m)− uβ))dm ≤ 0,

and we can conclude that V (m′,mβ, uβ)−V (mβ, uβ) ≤ 0. Now consider the case where u(0,m

′) <

uβ. We will argue, by contradiction, that we cannot have V (m′,mβ, uβ) > V (mβ, uβ). If we had

V (m′,mβ, uβ) > V (mβ, uβ) then

V (M,mβ, uβ)− V (m′,mβ, uβ) = δ(V (mβ, uβ) +mβ)

∫ M

m′(f(u(0,mβ)− uβ)− f(u(0,m)− uβ))dm.

Also

V (m′,mβ, uβ)− V (mβ, uβ) = δ(V (mβ, uβ) +mβ)

(∫ −uβmβ

(f(u(0,mβ)− uβ)− f(u(0,m)− uβ))dm+∫ m

′

uβ

(f(u(0,mβ)− uβ)− f(u(0,m)− uβ))dm).

As we have established that

δ(V (mβ, uβ) +mβ)∫ −uβmβ

(f(u(0,mβ)− uβ)− f(u(0,m)− uβ))dm < 0,

this implies that

(∫ m

′

−uβ(f(u(0,mβ)− uβ)− f(u(0,m)− uβ))dm > 0.

But as f(u(0,m)− uβ) is decreasing in m for (m ∈ −uβ,Mβ), this implies that∫ M

m′(f(u(0,mβ)− uβ)− f(u(0,m)− uβ))dm ≥ 0.

Hence V (M,mβ, uβ) ≥ V (m′,mβ, uβ) > V (mβ,M), which is a contradiction. So we can conclude

that when V (M,mβ, uβ)− V (mβ, uβ) ≤ 0 we have V (m′,mβ, uβ)− V (mβ, uβ) ≤ 0.

So to find the best equilibrium we must find the standard, uβ, to minimize mβ subject to theconstraints that

∂V (mβ,mβ, uβ)∂m

= (1− δ)γ − δf(u(0,mβ)− uβ)(V (mβ, uβ) +mβ) = 0,

74

V (M,mβ, uβ)− V (mβ, uβ) = (1− δ)γ(M −mβ)− δ(F (u(0,M)− uβ)− qβ)(V (mβ, uβ) +mβ) ≤ 0.

We now turn to showing that the optimal solution to the constrained maximizing problem hasless than full rent-seeking when β is not too large. To do this we begin by showing that the solutionis the unique level of rent seeking such that the first order condition holds and the inequality

V (M,mβ, uβ)− V (mβ, uβ) ≤ 0

holds with equality when β < β∗(δ) for some β∗(δ) > 0. Define

z =β − (u(0,mβ)− uβ)

β∈ (0, 1).

Then we have that

qβ = 1− z2

2,

f(u(0,mβ)− uβ) =z

β.

The first order condition then becomes

(1− δ) = δ(z

β

γ + 1γ

mβ −z2

2).

Substituting this into the inequality we get

z

β(M −mβ) ≤ 1− z2

2− F (u(0,M))− uβ).

Now suppose this inequality doesn’t bind. Then we must find uβ to minimize mβ subject to theconstraint that (1− δ) = δ( zβ

γ+1γ mβ − z2

2 ) where z ∈ [0, 1]. Noting that

∂z

∂uβ=

1β,

when ∂mβ∂uβ

= 0 we see that we must have

z =γ + 1γ

mβ

β,

which implies that

z2 =2(1− δ)

δ.

As we must have that z ∈ (0, 1), it is only possible that the second constraint would not bind ifδ > 2

3 . When δ > 23 we must have

z

β(M −mβ) < 1− z2

2− F (u(0,M))− uβ),

where z = γ+1γ

mββ and z2 = 2(1−δ)

δ . Combining those two equations we get

zmβ

β=

2(1− δ)δ

γ

γ + 1.

75

Notice then that we must have

M

β< (

δ

2(1− δ))1/2(1− 1− δ

δ− F (u(0,M))− uβ) +

2(1− δ)δ

γ

γ + 1).

Notice that, since F (u(0,M))− uβ) ≥ 0, and that there exists β1 > 0 such that for all β ∈ (0, β1)

M

β> (

δ

2(1− δ))1/2(1 +

(1− δ)(γ − 1)δ(γ + 1)

),

and the above condition can never be satisfied when β ∈ (0, β1). So it cannot be the case that

z

β(M −mβ) < 1− z2

2− F (u(0,M))− uβ).

So we can conclude that

z

β(M −mβ) = 1− z2

2− F (u(0,M))− uβ).

and we have two equations in two unknowns.Next note that, if we can find a solution with F (u(0,M)) − uβ) = 0, then this would be best

solution. So we look for an solution to the two equations,

1− δ = δ(z

β

γ + 1γ

mβ −z2

2),

z

β(M −mβ) = 1− z2

2,

All that remains is to show that a solution to these equations exist. Note that, by the secondequation, if z ∈ (0, 1), it must be the case that m0

β < M . Equating the two expressions for zβmβ

givesz

βM +

z2

2(1 + γ)=

γ + δ

δ(1 + γ).

Note that the right hand side is constant with respect to z and the left hand side is strictly increasingin z. When z = 0 the left hand side is 0 and

0 <γ + δ

δ(1 + γ).

When z = 1 the right hand side is

M

β+

12(1 + γ)

>γ + δ

δ(1 + γ),

when β < β2 for some β2 > 0. Define β∗(δ) = min{β1, β2}. By the intermediate value theoremthere is then a unique solution, z ∈ (0, 1) to this equation which determines the probability ofre-election q0

β = 1 − z2

2 in the best equilibrium, for all β ∈ (0, β∗(δ)). We can then solve for thelevel of rent-seeking in the best equilibrium, m0

β, from

z

βm0β =

γ

γ + 11− δδ

+z2

2.

76

Now note that the level of rent-seeking is increasing and the probability of re-election is de-creasing in β. First consider the probability of re-election. We have seen that z is the solutionto

z

βM +

z2

2(1 + γ)=

γ + δ

δ(1 + γ).

As the left hand side is strictly increasing in z and strictly decreasing in β we can conclude that

∂z

∂β> 0.

Then, because q0β = 1− z2

2 ,∂q0

β

∂β< 0.

Now I show that the amount of rent-seeking in increasing in β. Note that because ∂z∂β > 0, the

equationz

βM +

z2

2(1 + γ)=

γ + δ

δ(1 + γ),

implies that zβ is decreasing in β. Recall that

z

βm0β =

γ

γ + 11− δδ

+z2

2,

so as zβ is decreasing in β and z is decreasing in β we can conclude that

∂m0β

β> 0,

and the amount of rents secured is increasing in the amount of noise.Finally, consider the limit as β → 0. I first show that the probability of re-election goes to 1.

First note that we must have

limβ→0

δ(z

β

γ + 1γ

m0β −

z2

2) = 1− δ,

which implies thatlimβ→0

z = 0,

as m0β >

(1−δ)γMγ+δ . So we can conclude that

limβ→0

q0β = 1.

Now I show that the amount of rent secured, as β goes to 0 converges to the perfect monitoringcase. Taking the limit of the two equalities as β goes to 0 gives

limβ→0

z

β(M −m0

β) = 1,

andlimβ→0

z

β

γ + 1γ

m0β =

1− δδ

.

77

This implies that

limβ→0

z

βm0β =

(1− δ)γδ(γ + 1)

,

and solimβ→0

z

βM = 1− (1− δ)γ

δ(γ + 1)=

γ + δ

δ(γ + 1).

This implies that

limβ→0

z

β=

γ + δ

δ(γ + 1)1M,

so we can conclude thatlimβ→0

m0β =

(1− δ)γMγ + δ

,

which are the amount of rents secured in the perfect monitoring case.�

Proof of Lemma 3Assume δ > δ. We look for equilibria where the utility to the voter is higher than −M and withcandidates of type −κ or κ elected at any history, where κ 6= 0. I begin by noting that, as thevoter only observes the utility level, in any equilibrium where the elected candidate engages in rentseeking mβ ∈ (0,M) the implemented policy must be

xκ =κ

1 + γ1

λ−1

,

as in the perfect monitoring case. As everything thing is symmetric and the voter must be indifferentbetween re-electing the incumbent and defeating them we can restrict attention to equilibria wherethe implemented policy is xκ (or −xκ). Since the elected candidate always has the option ofdeviating to their ideal point with full rent-seeking, it cannot be an equilibrium for the candidateto engage in less rent-seeking than in the perfect monitoring case, and so the level of rent-seeking ism ∈ (0,M) must be positive for all κ ∈ [0, κ). We seek to find mκ

β, the lowest level of rent-seekingwhich can be supported. This would then characterize the best equilibrium with candidates of typeκ and −κ. As the equilibrium must satisfy retrospective voting there exists a utility standard uβ

such that the incumbent is re-elected if and only if they meet that standard. The probability ofre-election will then be same in every period, qκβ , where

qκβ = Pr(εt ≥ uβ − u(0,mκβ)) = F (u(0,mκ

β)− uβ).

Define

z =β − (u(0,mκ

β)− uβ)β

∈ (0, 1),

so we have

qκβ = 1− z2

2,

f(u(0,mκβ)− uβ) =

z

β,

as in the case where κ = 0.

78

Notice that, in order to give the incumbent candidate the greatest possible incentive to securere-election we can restrict attention to equilibria where, if the incumbent is defeated, they will neverbe elected at any future history on the equilibrium path. There are then three relevant states to anincumbent candidate of type κ: a history at which they are elected, a history at which they havebeen defeated and a type −κ candidate is elected, and a history at which a candidate of type κ iselected in which they have previously been defeated. We can denote the value to each candidateof type κ from each of the three states as V (mκ

β, uβ), V (mκβ, uβ) and V (mκ

β, uβ). In order to makethe punishments as harsh as possible the type of candidate must alternate ever time the incumbentleaves office. The value of being in each state can then be determined by the following system ofequations:

V (mκβ, uβ) = (1− δ)gκ(xκ,mκ

β) + δ(qκβ V (mβ, uβ) + (1− qκβ)V (mκβ, uβ)),

V (mκβ, uβ) = (1− δ)gκ(−xκ,mκ

β) + δ(qκβV (mβ, uβ) + (1− qκβ)V (mκβ, uβ)),

V (mκβ, uβ) = (1− δ)gκ(xκ,mκ

β) + δ(qκβV (mβ, uβ) + (1− qκβ)V (mκβ, uβ)).

Noting that

V (mκβ, uβ)− V (mκ

β, uβ) =(1− δ)(γ + 1)mκ

β

1− δqκβ,

we can then solve that

V (mκβ, uβ)− V (mκ

β, uβ) =(1− δ)(γ + 1)mκ

β

1− δqκβ+

(1− δ)((1 + 2γ1

1−λ )λ − 1)1− δ(2qκβ − 1)

κλ

(1 + γ1

1−λ )λ.

In order to have an equilibrium we must have that for all platforms (x,m) we must have

(1− δ)(gκ(x,m)− gκ(xκ,mκβ)) ≤ δ(qκβ − F (u(x,m)− uβ))(V (mκ

β, uβ)− V (mκβ, uβ)).

Notice that, when mκβ ∈ (0,M), if the candidate were to deviate they would do so first in

terms of rents, then, after reaching full rent-seeking, in terms of policy. Hence, we can repeat theargument from the κ = 0 case and conclude that we must have qκβ ∈ (1

2 , 1), and that we need onlycheck marginal deviations and deviations involving full rent-seeking when β is sufficiently small.Similarly, when β is sufficiently small we can conclude that uβ − β > −M and so the probabilityof securing re-election if deviating to full rent-seeking is then 0. Since the probability of securingre-election is then 0 for any platform with full rents, we need only check that the candidate doesnot have an incentive to deviate to their ideal policy with full rent-seeking. Similarly, we can repeatthe argument from the case without heterogeneity to see that this constraint must bind in the bestequilibrium. We can then conclude that there exists a β1 > 0 such that the constraint that thecandidate does not have an incentive to deviate to their ideal point with full rent-seeking is bindingin the best equilibrium for all β ∈ (0, β1).

We can now characterize the lowest level of rent-seeking, and associated probability of re-election, from the following two equations:

(1− δ)γ = δz

β(V (mκ

β, uβ)− V (mκβ, uβ))

79

(1− δ)(γ(M −mκβ) +

1

(1 + γ1

1−λ )λκλ) = δqβ(V (mκ

β, uβ)− V (mκβ, uβ)).

The second equation can be simplified to

γ + δqβ1− δqβ

mβ = γM +κλ

(1 + γ1

1−λ )λ[1− δqβ

((1 + 2γ1

1−λ )λ − 1)1− δ(2qβ − 1)

],

and so

mκβ =

(1− δqκβ)γMγ + δqκβ

+1

γ + δqκβ

κλ

(1 + γ1

1−λ )λ(1− δqκβ [(1 + 2γ

11−λ )λ + δ(1− qκβ)

((1 + 2γ1

1−λ )λ − 1)1− δ(2qκβ − 1)

]).

Now using the fact that qβ = 1− z2

2 , we can write the above equation as

mκβ =

1γ + δ − δz2/2

((1−δ+δz2/2)γM)+[−(1−z2/2)(δ(1+2γ1

1−λ )λ−1)+z2

2(1+

((1 + 2γ1

1−λ )λ − 1)1− δ + δz2

]κλ

(1 + γ1

1−λ )λ),

and dividing the second equation by the first we get,

(M −mκβ +

1

γ(1 + γ1

1−λ )λκλ)

z

β= 1− z2

2.

So we are left with two equations in two unknowns. The next task is to show that there existsa unique solution to this system of equations for β sufficiently small. To see this note that, byremoving mβ from the equation we get

1− z2

2=z

β

1γ + δ − δz2/2

[(γ + 1)δ(1− z2/2)M

+κλ

(1 + γ1

1−λ )λ[(1− z2/2)(δ(1 + 2γ

11−λ )λ − 1)− z2

2δ(1 +

((1 + 2γ1

1−λ )λ − 1)1− δ + δz2

) + 1 + γ−1δ(1− z2

2)]].

Dividing by 1− z2

2 gives

h(z) ≡ z

β

1γ + δ − δz2/2

[(γ+1)δM+κλ

(1 + γ1

1−λ )λ[δ((1+2γ

11−λ )λ+γ−1)− z2

2− z2

((1 + 2γ1

1−λ )λ − 1)1− δ + δz2

] = 1.

Notice h′(z) > 0, that h(0) = 0 < 1 and that there exists β2 > 0 such that

h(1) =1β

1γ + δ/2

[(γ + 1)δM +κλ

(1 + γ1

1−λ )λ(1 + γ−1)δ)] > 1,

for all β ∈ (0, β2). So we can conclude there these is a unique z with h(z) = 1 for each κ. Notealso that, as β goes to 0 we must have that

limβ→0

z = 0.

Since we have a unique z this then determines the unique level of rent-seeking mβ from the aboveequation. This also implies, since z goes to 0, that the rents go to

limβ→0

mκβ =

(1− δ)γMγ + δ

+κλ

(1 + γ1

1−λ )λ(1− δ(1 + 2γ

11−λ )λ),

80

the level in the perfect monitoring case. Therefore, provided the noise is sufficiently small, theamount of rent-seeking will be less than M and the conditions above for an equilibrium of this formwill all be satisfied. Further, for any κ, as the noise goes to 0 the best equilibrium goes to the bestequilibrium without noise.

I conclude by showing that the welfare of the voter is increasing in κ on (0, κβ) with mβ = 0when κβ = κβ. First note that, by the second equation we must have that the z goes to 0 as βgoes to 0. As δ > δ we have that there exists β3 > 0 such that for all β ∈ (0, β3),

w(κ) = −mκβ − xλκ,

would be increasing in κ if z were held constant, z = z(0). Define β(δ) = min{β1, β2, β3}. Now,since, for all κ,

mβ =1

γ + δ − δz2/2((1−δ+δz2/2)γM)−[(1−z2/2)(δ(1+2γ

11−λ )λ−1)−z

2

2(1+

((1 + 2γ1

1−λ )λ − 1)1− δ + δz2

]κλ

(1 + γ1

1−λ )λ),

which is decreasing in z, if we can show that z(κ) is decreasing in κ we can conclude that thevoter’s welfare is increasing in κ until the level of rent-seeking is driven to 0 or we reach maximumpolarization. To see that z(κ) is decreasing, note that as we must have, for all κ that

h(z, κ) = 1,

we can conclude that∂z

∂κ= −

∂h(z,κ)∂κ

∂h(z,κ)∂z

< 0,

as h is increasing in both κ and z. So we can conclude that the voter’s welfare is increasing in κ

until the level of rent-seeking is driven to 0 or the we reach maximum polarization. Define κβ from

κβ = min{κ : mκβ ≤ 0},

to be the minimum amount of heterogeneity necessary to drive the amount of rents secured to 0.We can then conclude that the welfare of the voter is increasing in κ on the interval [0,min{κβ, 1}].

I now consider the comparative statics. First note that the probability of re-election is decreasingβ and the level of rent-seeking is increasing in β. From the equation

h(z, β) = 1,

we can see that∂z

∂β= −

∂h(z,β)∂β

∂h(z,β)∂z

> 0.

Since we have established that mκβ is increasing in z and qκβ is decreasing, we can then conclude

that∂mκ

β

∂β> 0,

∂qκβ∂β

< 0.

81

Finally, notice that, as mκβ is increasing in κ we must have that

∂κβ∂β

> 0.

�

Proof of Lemma 4When κβ < 1, for κ ∈ (κβ, 1] the elected candidate would have a strict incentive not to deviateif the equilibrium policies where xκ and −xκ. Hence, more moderate policies can be supportedwith no rent-seeking. So, for each κ on this interval we can then solve for the minimum amount ofdivergence that can be supported, xβ(κ). We then find the κ ∈ [κβ, 1] to minimize this divergence.I begin by showing that, for δ > δ and β ∈ (0, β(δ)) and any κ ∈ [κβ, 1] the best platform-stationarypublic perfect equilibrium satisfying (Ru) and candidates of types κ and −κ involves policies xβ(κ)and −xβ(κ) and probability of re-election qκβ = 1 − z2

2 which are given as the unique solution tothe equations

(γM + (κ− xβ)λ) =δ(1− z2

2 )((κ+ xβ)λ − (κ− xβ)λ)1− δ + δz

,

1− z2

2=z

β(

xβκ− xβ

)(λ−1)(γM + (κ− xβ)λ).

First note that there are then two relevant states for a candidate of type κ: the state at whicha candidate of type κ is elected, and the state where a candidate of type −κ is elected. As theincumbent doesn’t secure any rents they are indifferent between whether they are elected or ifanother candidate of the same type is elected. We can then write the payoff to the candidate oftype κ in both states as V (xβ, uβ) and V (xβ, uβ) and, following the same steps as the previousLemma, we can calculate

V (xβ, uβ)− V (xβ, uβ) =(1− δ)((κ+ xβ)λ − (κ− xβ)λ)

1− δ(2qκβ − 1).

In order to have an equilibrium then we must have that, for all platforms (x,m),

(1− δ)(gκ(x,m)− gκ(xβ, 0)) ≤ δ(qκβ − F (u(x,m)− uβ))(V (xβ, uβ)− V (xβ, uβ)).

First consider a marginal deviation. When κ > κβ,

∂gκ(xβ ,0))∂x

−∂u(xβ ,0))∂x

>

∂gκ(xβ ,0))∂m

−∂u(xβ ,0))∂m

,

and so the candidate would deviate first terms of policy. So, in order for there not to be a profitablemarginal deviation, we must have

(1− δ)λ(κ− xβ)λ−1 = δz

βλ(xβ)λ(V (xβ, uβ)− V (xβ, uβ)).

Note that when κ = κβ the candidate would deviate first in terms of rents, but, because the ratioof marginal utilities over policy and rents are the same this condition still holds. Notice also that,as β goes to 0, we must have that z goes to 0. Hence, when β is small the probability of re-election

82

must be greater than 12 . Now, when β is small, the marginal cost will increase then decrease sharply

enough that the only deviation we need consider is to those with full rent-seeking, and hence, to thecandidate’s ideal point. Following an identical argument to the one in the previous cases there thenexists β4 > 0 such that this constraint must bind for β ∈ (0, β4). This gives the second condition,that the candidate is indifferent between implementing the equilibrium policy and deviating to theirideal policy with full rent seeking.

(1− δ)(γM + (κ− xβ)λ = (1− z2

2)δ(V (xβ, uβ)− V (xβ, uβ)).

Dividing the two equations gives

1− z2

2=z

β(

xβκ− xβ

)(λ−1)(γM + (κ− xβ)λ).

Now, substituting in for V (xβ, uβ)− V (xβ, uβ) we get

(γM + (κ− xβ)λ) =δ(1− z2

2 )((κ+ xβ)λ − (κ− xβ)λ)1− δ + δz

.

First note that there is a unique solution to these two equations. Substituting in for (γM+(κ−xβ)λ)and dividing by 1− z2

2 , yields

1 =z

β(

xβκ− xβ

)(λ−1) δ((κ+ xβ)λ − (κ− xβ)λ)1− δ + δz

,

which then implies

(1− δ) = δz

β[−β + (

xβκ− xβ

)(λ−1)((κ+ xβ)λ − (κ− xβ)λ)].

We can now remove z from the first equation by dividing both sides by z/β to get

(δ

1− δ)2[−β + (

xβκ− xβ

)(λ−1)((κ+ xβ)λ − (κ− xβ)λ)]2 − β2

2

= (xβ

κ− xβ)(λ−1)(γM + (κ− xβ)λ)(

δ

1− δ)[β + (

xβκ− xβ

)(λ−1)((κ+ xβ)λ − (κ− xβ)λ)],

which can be simplified to

δ(κ+ xβ(κ))λ − (κ− xβ(κ))λ − (1− δ)γM

= β2(κ− xβxβ

)2(λ−1) (1− δ)2

δ

1(κ+ xβ)λ − (κ− xβ)λ

(1/2− (δ

1− δ)2)

+β(κ− xβxβ

)λ−1[2δ((κ+ xβ)λ − (κ− xβ)λ) + (1− δ)(γM + (κ− xβ)λ)]

which has a unique solution, xβ(κ) ∈ [0, xκβ], for each κ. Substituting in for xβ in the aboveequations then uniquely determines z, and by extension, qκβ for each κ ∈ [κβ, 1]. Notice also thatthe right hand side is increasing in β when β is small, so for all κ, xβ(κ) is increasing in β. Inparticular, this means that xβ(κ) is bounded below by x∗.

83

Now I show that there exists a unique κ which minimizes xβ(κ) on the interval [κβ, 1]. First,define

y(κ) ≡ β(κ− xβxβ

)2(λ−1) (1− δ)2

δ

1(κ+ xβ)λ − (κ− xβ)λ

(1/2− (δ

1− δ)2)

+(κ− xβxβ

)λ−1[2δ((κ+ xβ)λ − (κ− xβ)λ) + (1− δ)(γM + (κ− xβ)λ)].

Note then thatδ(κ+ xβ(κ))λ − (κ− xβ(κ))λ − (1− δ)γM = βy(κ),

and that y(κ), y′(κ) and y′′(κ) are all bounded on the interval [κβ, 1]. Begin by differentiating theabove equation with respect to κ,

λ[δ(κ+ xβ(κ))λ−1 − (κ− xβ(κ))λ−1 + x′β(κ)(δ(κ+ xβ(κ))λ−1 + (κ− xβ(κ))λ−1)] = βy′(κ)

So note that when x′β(κ) = 0 we have that

δ(κ+ xβ(κ)κ− xβ(κ)

)λ−1 = 1 + βy′(κ)

λ(κ− xβ(κ))λ−1.

Differentiating again gives that when x′β(κ) = 0,

x′′β(κ) =1

λ− 1−δ(κ+ xβ(κ))λ−2 + (κ− xβ(κ))λ−2 + βy′′(κ)/λ

(κ+ xβ(κ))λ−1 + (κ− xβ(κ))λ−1.

Notice that, when β is sufficiently small, since xβ(κ) is bounded below,

δ(κ+ xβ(κ)κ− xβ(κ)

)λ−2 < 1 + βy′′(κ)

(κ− xβ(κ))λ−2,

whenever x′β(κ) = 0, and sox′′β(κ) > 0.

So we can define β5 = inf{β : x′′β(κ) ≤ 0}, and

β(δ) = min{β(δ), β4, β5}.

Then we can conclude that for all β ∈ (0, β(δ)), every local extremum on [κβ,∞) is a local minimum.We can then conclude that there exists a unique minimizer on the interval [κβ, 1],

κ∗β = arg minκ∈[κβ ,1]

xκβ.

We can then definex∗β = xβ(κ∗β),

to be the minimum polarization in the best equilibrium. So the candidates and platforms in thebest two-candidate equilibrium can be uniquely determined.

Finally, consider the statics with respect to β. It is immediate, from the fact that xβ(κ) isincreasing in β that the min on the interval is increasing in β. Hence we have that

∂x∗β∂β

< 0.

84

Finally note that qκβ is decreasing in β when β ∈ (0, β(δ). Too see this, recall that

β(1− z2

2) = z(

x∗βκ∗β − x∗β

)(λ−1)(γM + (κ∗β − x∗β)λ),

and so

1− z2

2− zβ ∂z

∂β=∂z

∂β(

x∗βκ∗β − x∗β

)(λ−1)(γM + (κ∗β − x∗β)λ) + z∂

∂β[(

x∗βκ∗β − x∗β

)(λ−1)(γM + (κ∗β − x∗β)λ)].

Since z → 0 as β goes to 0, when β is small we have that

∂z

∂β> 0,

and so∂qκβ∂β

< 0.

�

Proof of Theorem 9Let δ > δ and β ∈ (0, β(δ)). There are then three cases to consider: κβ > 1, κβ < 1 andκβ = 1. Consider first the case where κβ > 1. Then we have established in Lemma 3 that the bestequilibrium involves candidates of type −1 and 1 elected, these candidates implement policies with

x1 and −x1 and level of rent-seeking m1β > 0, and that

∂m1β

∂β > 0 and∂q1β∂β < 0, so

∂u∗β∂β

< 0;∂q∗β∂β

< 0,

and we are done for this case.Now consider the case where κβ < 1. Then we have established in Lemma 4 that there exists

a unique κ∗β such that the best two-type equilibrium involves candidates of type κ∗β and −κ∗β who

implement platforms (x∗β, 0) and (−x∗β, 0), where∂x∗β∂β > 0 and

∂q∗β∂β < 0. So we can conclude that

∂u∗β∂β

< 0;∂q∗β∂β

< 0,

and we are done for this case as well.Finally consider the case where κβ = 1. Since ∂κβ

∂β > 0, we β increases this follows from thefirst case, and as β decreases this follows from the second case, so we are done. �

9.5 Candidates’ Decision to Run for Office

Proof of Theorem 10 We are restricting attention to equilibria in which all candidates of thesame type implement the same platform if elected, where at least one candidate of each type κ and−κ run for office at every history for some κ, and where the voter only elects candidates of typeκ or −κ at any history provided either type is available. Hence the set of platforms which couldbe implemented along the equilibrium path in the possible continuation equilibria is stationary, sothe voter must be myopically optimizing. Note also that the candidates would have the necessary

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incentive to implement any platform they would in the previous section, as they prefer being re-elected to not running, and, as the probability of re-election is greater than 1/2, the incumbent will,in expectation, win more elections than they lose in any equilibrium with less than full rent-seeking.Finding the best such equilibrium then corresponds to finding the best two-type equilibrium fromthe imperfect monitoring section – which we have established consists of candidates of type κ∗β and−κ∗β being elected with positive probability at each history. By the last line of the definition of(Re) we know that, for all candidates, running for office without being elected will not affect theoutcome in future periods. Hence, at most one candidate of type κ∗β and −κ∗β run for office ineach period, and no other type will. Also, as both types of candidates are elected with positiveprobability in each period – and that if one candidate doesn’t run for office the other type would beelected for sure – both candidates have a strict incentive to run for office. It is then an equilibriumfor one, and only one, candidate of each type κβ and −κ∗β to run for office in each period. As theincumbent knows that if they run for office they, and not another candidate of the same type, willbe elected this will consist of the incumbent and one challenger of a different type running for officeat each history. Finally, recall that in the best equilibrium, if m∗β > 0, if the incumbent is defeatedthey will never be elected at any subsequent history. Hence, they will not run for office at anyfuture history. �

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