Punishment, Compliance, and Anger

in Equilibrium

JOB MARKET PAPER

Robert J. Akerlof∗

November 18, 2010

Abstract

This paper develops an alternative to a repeated-game approach to thinking aboutnorms. It instead conceives of norms as views regarding how people should behave.Norms matter because they determine what people consider to be fair, and when in-dividuals feel unfairly treated, they become angry. Norms are (potentially) followedbecause a failure to do so may provoke anger. The paper considers a two-periodpsychological game in which a first player chooses whether to “comply” and a sec-ond player potentially punishes noncompliance, where the motivation for punishmentis anger. Because anger depends upon the expected level of compliance, there maybe multiple equilibria. This gives an explanation of why some organizations, despiteprofessed disapproval of corruption, have high levels of corruption and punishment ofcorruption is relatively mild, while other organizations have low levels of corruption andsevere punishment of corruption when it occurs. Because norms are contextual, thepaper explains why in market contexts it is often viewed as fair for a seller to set a pricebased upon economic self interest while transacting according to economic self-interestfrequently provokes anger within a firm/organization. The paper gives foundations forHart and Moore’s (2008) model of aggrievement since signing a contract can be viewedas establishing norms. An extension to the basic model shows that “institutionalizingpunishment”–that is, creating norms regarding how noncompliance should be punishedand who should do the punishing–can increase the provision of punishment, leading togreater compliance. (JEL: D02, D03, D74.)

∗Massachusetts Institue of Technology, akerlof@mit.edu. I would like to thank Robert Gibbons, Philippe Aghion,Oliver Hart, Richard Holden, George Akerlof, and seminar participants at Berkeley and MIT for helpful commentsand suggestions.

1 Introduction

This paper develops an alternative to a repeated-game approach to thinking about

norms. It instead conceives of norms as views regarding how people should behave. Norms

matter because they determine what people consider to be fair, and when individuals feel

unfairly treated, they become angry. Norms are (potentially) followed because a failure to

do so may provoke anger.1

Consider one relevant example: a firm in which a manager gives orders to employees.

If an employee fails to follow orders—shirks—her coworkers might be angry with her if they

are inconvenienced and there is a sense among the coworkers (a norm) that the manager’s

orders should be followed. The norm is crucial to generating the anger: if instead the

coworkers felt that the employee was entitled to violate the manager’s orders they would

not feel angry. Anger may lead the coworkers to punish the shirking worker themselves or

alternatively report the shirking to the manager (giving the manager the ability to inflict

punishment). Hence, when a manager is unable to observe whether employees follow orders

directly, the ability to provide incentives to follow orders will depend crucially on whether

a norm exists that orders should be followed.

The paper considers a two-period psychological game. A first player chooses whether

to “comply” at a cost. A second player, who values compliance, potentially punishes

noncompliance out of anger.

Both players have views as to whether there is a duty to comply: we will refer to these

views as “norms.” The norms play two roles: (i) if player 1 feels a sense of duty to comply,

this motivates her to comply; (ii) player 2’s view as to whether player 1 has a duty to

comply plays a role in determining whether player 2 will be angry over noncompliance.

The key assumption of the paper regards what makes people feel angry. We assume that

anger arises when a person feels she has been harmed because someone lacks an appropriate

sense of duty. Hence, in the model, player 2 feels angry if she thinks player 1 failed to

1See Baker, Gibbons, and Murphy (1999, 2002), or Kandori (1992) for examples of repeated-game ap-proaches to norms.

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comply, but a person with an appropriate sense of duty would have complied.

There are sometimes multiple equilibria. There might be one equilibrium in which

the degree of compliance is high and there is considerable anger over and punishment of

noncompliance; and there might be another equilibrium in which the degree of compliance

is low and there is only mild anger over and punishment of noncompliance. This explains

why some organizations, despite professed disapproval of corruption, have high levels of

corruption and punishment of corruption is relatively mild, while other organizations have

low levels of corruption and severe punishment of corruption when it occurs.2

The theory captures the finding of many researchers that fairness attitudes—what makes

people angry—differ greatly across contexts (see especially Elster (1992), Walzer (1983),

Kahneman, Knetsch, and Thaler (1986), and Young (1994)).3 These contextual differences

are accounted for by the presence of different norms. Consider one example of the impor-

tance of context. In markets it is often considered fair or appropriate for buyers and sellers

to act according to their self interest. In contrast, within firms and other organizations,

transacting according to self interest is rarely seen as fair or appropriate.

The paper gives foundations for Hart and Moore’s (2008) model of aggrievement, in

which the contracts that parties sign ex ante determine what makes the parties angry (or

“aggrieved”) ex post. Hart and Moore (2008) can be interpreted as follows to relate it

to this paper: when parties in their model sign a contract, this establishes a norm. It

establishes a norm that the parties should meet the terms of the contract. Hart and Moore

(2008) have shown that anger is important for organizational economics, since it is a key

reason for ex post inefficiencies such as haggling, rent seeking, and shading of performance.4

In an extension to the basic model, we will show that the provision of punishment can po-

tentially be greatly increased (and compliance obtained) by “institutionalizing punishment”—

2There are other papers that, taking very different approaches from the approach of this paper, obtain asa result the existence of high corruption and low corruption equilibria. See, especially, Andvig and Moene(1990), Andvig (1991), Tirole (1996), and Cadot (1987). For a review of the corruption literature, seeBardhan (1997). Also see Shleifer and Vishny (1993).

3For a review of the literature on fairness and justice, see Konow (2003).4See Gibbons (2005), Williamson (1971, 1979, 1985), and Klein et al. (1978) for further discussion of the

importance of ex post inefficiencies for organization theory. See also Hart and Moore (2007).

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that is, by creating norms regarding how noncompliance should be punished and who should

do the punishing. We will show that the key condition for institutionalization to be helpful

is that a punishment of size p can be inflicted at a cost to the punisher of less than p (a

condition which would seem to hold in many circumstances).

This paper is related to several other papers that have developed psychological game

models of fairness: Rabin (1993), Dufwenberg and Kirchsteiger (2004), and Falk and Fis-

chbacher (2006). These papers, like this one, all recognize the importance of beliefs others’

intentions/motivations in the formation of fairness attitudes. For this reason, they all sim-

ilarly employ psychological game models.5 Indeed, there is a great deal of evidence that

beliefs regarding others’ intentions/motivation matter crucially to whether one feels one has

been treated fairly (see, for example, Falk, Fehr, and Fischbacher, 2008).

But there are three important respects in which this paper differs from Rabin (1993),

Dufwenberg and Kirchsteiger (2004), and Falk and Fischbacher (2006). First, this paper

accounts for Ostrom’s (1990) finding that in most contexts there are “graduated sanctions”:

there is more punishment for repeat offenders. The previous models cannot explain this

fact because anger depends upon the type of strategy an actor follows but not upon the

type of the actor.

Second, these other papers do not explore contextual differences in what is considered

fair. Following up on the previous example, they do not explain why it might be acceptable

to act in a self-interested manner in a market but unacceptable within a firm.

Finally, Rabin (1993), Dufwenberg and Kirchsteiger (2004), and Falk and Fischbacher

(2006) model both negative and positive reciprocity, treating them as flip sides of the same

thing. This paper, in contrast, focuses on negative reciprocity: anger and punishment

rather than gratitude and reward. Section 5 explains the reason for treating positive

reciprocity separately and hints at how it might be modeled.

It is important to point out that this paper examines what makes people angry, taking

norms as exogenous. It does not explain why certain norms prevail in certain contexts.

5Psychological games were first discussed by Geanakoplos, Pearce, and Stacchetti (1989). These papersand this one use variants of their equilibrium concept.

3

As we will see, just understanding what will be regarded as fair taking norms as exogenous

is not a trivial problem. That said, it is important to endogenize the norms: understand

how they form and change. This is a topic that I have explored in other work that is

complementary with this paper.6

The paper will proceed as follows. Section 2 develops the setup of the model and

defines a punishment-compliance equilibrium of the game. Section 3 gives the main results

of the paper, describing the punishment-compliance equilibria. Section 4 extends the model

developed in section 2 to show that institutionalizing punishment can improve the provision

of punishment and make it easier to obtain compliance. Section 5 relates the theory

developed in this paper to Hart and Moore (2008) as well as Rabin (1993), Dufwenberg and

Kirchsteiger (2004), and Falk and Fischbacher (2006). Section 6 concludes.

2 The Model

We will consider a two-period game with two players, player 1 and player 2. At time

1, player 1 chooses an action a ∈ {0, 1}.7 We will refer to a = 1 as “compliance” and a = 0

as “noncompliance.” At time 2, after observing a, player 2 chooses how much to punish

player 1, p ∈ [0,∞).We will proceed as follows. We will begin by discussing player 1’s choice of action

a. Then, we will discuss player 2’s choice of p. In discussing player 2’s choice, we will

introduce the concept of mistreatment. Anger is caused by a feeling on player 2’s part that

she has been mistreated by player 1, and anger is what motivates player 2 to punish player

1. Finally, we will define an equilibrium concept for the game, which will be similar to the

equilibrium concept developed by Battigalli and Dufwenberg (2009).

6For an overview of my ideas on norm formation and change, see my “Research Statement” (Akerlof(2010c)). See also Akerlof (2008), Akerlof (2010a), and Akerlof (2010b).

7The results of the model look similar when individual 1 can choose any action a ∈ [0,∞). We willdiscuss this case below. The reason for making 1’s choice discrete for the time being is to simplify analysis.

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2.1 Player 1’s Choice

Player 1 maximizes:

U1 = −p(a)− Ca−Dmax(I1 − a, 0)

For simplicity, we assume that a = 1 if player 1 is otherwise indifferent.

The first term of the utility function is the disutility associated with player 2’s punish-

ment, where p(a) ∈ [0,∞) is individual 1’s belief concerning the punishment action a will

receive. The second term reflects a cost C > 0 of compliance (choosing a = 1 rather than

a = 0). The final term reflects a disutility to player 1 of D > 0 if player 1 feels that she has

failed to do her duty. We assume I1 ∈ {0, 1}. If I1 = 1, player 1 feels that she has a dutyto comply (choose action a = 1) and loses D if she chooses fails to comply. If I1 = 0, player

1 does not feel it is her duty to comply (choose action a = 1) and does not lose utility D

from taking either action. Put another way, player 1 feels entitled to take whichever action

she wants to take. Observe that we can think of I1 as a norm.

Let a∗(C, I1, p(·)) denote the optimal choice of action.8 If player 1 does not feel a sense

of duty to comply (I1 = 0), player 1 will only comply if the cost C of complying is less than

the punishment associated with failure to comply. That is, if I1 = 0, a∗ = 1 if and only if

C ≤ p(0) − p(1). Similarly, if player 1 feels a sense of duty to comply (I1 = 1), player 1

will only comply if the cost C is less than the punishment associated with failure to comply

plus the benefit to player 1 from doing her duty, D. That is, if I1 = 1, a∗ = 1 if and only

if C ≤ D + [p(0)− p(1)].

The following table summarizes:

I1 = 0 I1 = 1

a∗(C, I1, p(·)) = 0 C > p(0)− p(1) C > D + [p(0)− p(1)]

a∗(C, I1, p(·)) = 1 C ≤ p(0)− p(1) C ≤ D + [p(0)− p(1)]

8We choose not to write a∗ as a function of D because, in the analysis that follows, D will have a fixedvalue whereas the values of C, I1, and p(a) may vary.

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What Player 2 Knows About Player 1 Player 2 knows the value of D. However,

player 2 does not know the cost of compliance C or whether player 1 has a sense of duty,

I1. 2’s prior is that C is distributed according to distribution F with support S = (0,∞).2’s prior is that I1 = 1 with probability q and I1 = 0 with probability 1− q. Let I ⊆ {0, 1}be the support of I1. We assume that, if q = 1, I = {1} and if q = 0, I = {0}.9 2 believes

that the values of I1 and C are independent.

2.2 Player 2’s Choice

In this section, we will derive p∗(a, I2, p(·)), the optimal choice of punishment. The

optimal punishment will depend upon whether player 1 complies, a. It will also depend

upon I2, player 2’s view regarding what player 1’s duty is. If I2 = 1, player 2 feels that

player 1 has a duty to comply. If I2 = 0, player 2 feels that player 1 is entitled to take

whichever action she wants to take. Player 2’s view regarding player 1’s duty might differ

from player 1’s own view (I1). I1 and I2 are the two norms present in the model.

Finally, the optimal choice of punishment depends upon p(·), where p(a) ∈ [0,∞) isplayer 2’s belief regarding p(a). When player 2 believes player 1 expects noncompliance

to be punished harshly, she will take a different view of noncompliance from the view she

would have taken if she believed that player 1 expected mild punishment. This final term

will capture the idea, as discussed in the introduction, that there will be less anger (and

hence less punishment) if the level of compliance is expected to be low.

To preview the equilibrium concept, which we will discuss formally in Section 2.3, it will

be roughly as follows: p∗(a, I2, p(·)) = p(a) = p(a). In other words, in equilibrium, there

will be a convergence of belief.

9This assumption implies that, if I1 = 1 (I1 = 0) almost surely—with probability 1—I1 = 1 (I1 = 0) forsure.

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2.2.1 Player 2’s choice

We assume that player 2 cannot commit ex ante to punish player 1. Player 2 chooses

p to maximize:

U2 = va− κ(p)−Φ(M(a, I2, p(·)), p)

The first term of the utility function reflects a benefit v ≥ 0 received when player 1 choosesto comply. The second term is a cost associated with punishing player 1. The third term

is the disutility associated with feeling mistreated by player 1. M , which we will discuss

more presently, denotes player 2’s view of how much player 1 has mistreated her. We will

assume that the disutility associated with mistreatment is lower when player 2 punishes

player 1. Hence, the third term captures player 2’s anger and the corresponding desire to

punish player 1 in the event that player 2 feels mistreated.

We will use the following convenient functional forms for κ and Φ for the remainder of

paper:

κ(p) =p

θ

Φ(M,p) = M log

µkM

p

¶10θ > 0 parameterizes the ability to punish player 1 (higher θ implies a greater ability or

lower cost of punishing). k is a positive constant.11

The amount of punishment player 2 will choose to administer will be:

p∗(a, I2, p(·)) = θM(a, I2, p(·))

Higher θ (a greater ability to punish player 1) leads to more punishment. Greater perceived

mistreatment by 1 leads to more punishment. When player 2 feels fairly treated by player

1 (M = 0), player 2 will not punish player 1.

10M log kMp

is technically undefined whenM = 0. We will assume Φ(0, p) = limM→0+ M log kMp

= 0.11The value of k will have no effect on the amount of punishment p that player 2 will administer. However,

we might want to assume that k ≥ θe−1, which ensures that an increase in M will decrease player 2’s utility.

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Having expressed the optimal punishment, p∗, as a function of mistreatment, we will

now discuss mistreatment. We will derive an expression for M(a, I2, p(·)) in the followingsection, which will in turn give us a new expression for the optimal punishment.

2.2.2 Mistreatment

A key innovation of the paper is the way in which mistreatment is modeled. Perceived

mistreatment, M , is player 2’s expectation of how much better off she would be if player 1

had an appropriate sense of duty.

To be clear about what this means, suppose that player 2 knew the value of C (the

cost of compliance). Then, player 2 would expect player 1 to choose a∗(C, I2, p(·)) ifplayer 1 had an appropriate sense of duty. Player 2 would be better off by an amount

vmax (a∗(C, I2, p(·))− a, 0) if player 1 had an appropriate sense of duty, where a is player

1’s actual choice. Recall that v is the value to player 2 of compliance.

So, if C were known, mistreatment would be given by: M = vmax (a∗(C, I2, p(·))− a, 0).

Observe that, for a sufficiently high value of C (cost of compliance), a∗(C, I2, p(·)) = 0, inwhich case, player 2 would not feel mistreated even if player 1 did not comply (a = 0).

Therefore, a high cost of compliance serves as an excuse for noncompliance. A nice feature

of the model is that it makes precise what it means for someone to have an excuse.

Of course, the cost of compliance C is not necessarily known. More generally:

M(a, I2, p(·)) = vmax (a(a, I2, p(·))− a, 0)

where a is player 2’s expectation of the action player 1 would have taken if player 1 had

had an appropriate sense of duty and a is the actual choice made by player 1.

The following is a formula for a:

a(a, I2, p(·)) = EC,I1(a∗(C, I2, p(·))|a∗(C, I1, p(·)) = a)

= Pr(a∗(C, I2, p(·)) = 1|a∗(C, I1, p(·)) = a)

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a is player 2’s expectation of a∗(C, I2, p(·)). Player 2 conditions this expectation on her

observation of a (player 1 believes, of course, that a∗(C, I1, p(·)) = a).

In the event that a∗(C, I1, p(·)) 6= a for any C ∈ S and I1 ∈ I, a(a, I2, p(·)) is undefined.Otherwise, a(a, I2, p(·)) is defined and, applying Bayes’ rule, we find that:

a(0, 1, p(·)) =[F (D + [p(0)− p(1)])− F (p(0)− p(1))] (1− q)

[1− F (p(0)− p(1))]− q [F (D + [p(0)− p(1)])− F (p(0)− p(1))]

a(1, 1, p(·)) = 1

12

a(0, 0, p(·)) = 0

a(1, 0, p(·)) =F (p(0)− p(1))

F (p(0)− p(1)) + q [F (D + [p(0)− p(1)])− F (p(0)− p(1))]

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2.2.3 The Optimal Punishment

Hence, the amount of punishment that player 2 will administer will be as follows:

p∗(a, I2, p(·)) = θM(a, I2, p(·))

= θvmax (a(a, I2, p(·))− a, 0)

So,

p∗(0, I2, p(·)) = θva(0, I2, p(·))

p∗(1, I2, p(·)) = 0

As we would expect, player 1 will not be punished in the event that she chooses to12 In the event that F (p(0)− p(1)) = 1, this is not the correct expression for a(0, 1, p(·)).13 In the event that F (D + [p(0)− p(1)]) = 1, this is not the correct expression for a(1, 0, p(·)).

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comply (p∗(1, I2, p(·)) = 0). If player 1 fails to comply, she will receive punishment of

θva(0, I2, p(·)). The punishment is increasing in player 2’s ability to punish (θ), the valueof compliance (v), and player 2’s expectation of what a person with an appropriate sense

of duty would have done (a).

What Player 1 Knows About Player 2 We have implicitly assumed, in writing player

1’s belief regarding punishment as p(a) rather than p(a, v, θ, I2), that player 1 knows the

values of v (the value to 2 of compliance), θ (the ability of 2 to punish), and I2 (player 2’s

view regarding player 1’s duty). It simplifies the analysis to make this assumption and not

much insight is lost by doing so. However, it is easy to extend the model to consider cases

where player 1 is uncertain of the values of v, θ, or I2.

2.3 Punishment-Compliance Equilibrium

Having derived a formula for player 1’s optimal choice, a∗(C, I1, p(·)), and a formula forplayer 2’s optimal choice, p∗(a, I2, p(·)), we are now in a position to define an equilibriumof the game.

The equilibrium concept that we will employ is similar to the equilibrium concept de-

veloped by Battigalli and Dufwenberg (2009). Their equilibrium concept is an adaptation

of Kreps and Wilson’s sequential equilibrium concept to psychological games. We will

consider, in contrast, the psychological game equivalent of a perfect Bayesian equilibrium.

The reason we will examine perfect Bayesian equilibria rather than sequential equilibria is

that we can ensure existence of a perfect Bayesian equilibrium under considerably weaker

conditions.

Definition 1 A pair of strategies (a(C, I1), p(a)) is a punishment-compliance equilibrium

if:

(1) a(C, I1) = a∗(C, I1, p(·)).(2) p(a) = p∗(a, I2, p(·)) for all a for which a∗(C, I1, p(·)) = a for some C ∈ S and

I1 ∈ I (that is, all a on the equilibrium path).

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(3) If a∗(C, I1, p(·)) 6= a for any C ∈ S and I1 ∈ I, there exists p(·) such that (i)a = a∗(C, I1, p(·)) for some C ∈ S and I1 ∈ I, and (ii) p(a) = p∗(a, I2, p(·)).

Condition (1) says that p = p: player 1 believes the punishment for action a will be

p(a). Condition (2) says that p(a) = p(a) for a on the equilibrium path: player 2 believes

that player 1 believes the punishment for action a will be p(a). Condition (3) says that,

off the equilibrium path, player 2 forms some belief about what player 1 believes (p) that

is consistent with player 1 having chosen action a and punishes player 1 accordingly.

The following lemma gives conditions under which a punishment-compliance equilibrium

exists.

Lemma 1 A punishment-compliance equilibrium exists if any of the following conditions

is satisfied: (i) F is continuous on (0,D + θv], (ii) I2 = 0, (iii) q = 1, or (iv) C = C with

probability 1. If a punishment-compliance equilibrium exists, it is not necessarily unique.14

As noted in the introduction, there may be multiple punishment-compliance equilibria.

In fact, multiple equilibria can arise under any refinement of the equilibrium concept. So,

the decision to consider perfect Bayesian equilibria (or the psychological game equivalent)

rather than sequential equilibria (a refinement) does not drive the finding that multiple

equilibria exist.

3 Results

In this section, we will examine the punishment-compliance equilibria of the game.

We discussed in the introduction the idea that norms define a context. In what follows,

we will examine what the punishment-compliance equilibria look like in different contexts

(that is, for different values of I1 and I2). Section 3.1 establishes results that hold for any

distribution function F (player 2’s prior regarding C). In Section 3.2, we will characterize

the set of equilibria for two particular choices of F . Section 3.3 considers the case where

14All proofs are provided in an appendix to the paper.

11

player 1 can choose any a ≥ 0 (as opposed to a ∈ {0, 1}) and shows that the set of equilibrialooks very similar to the set of equilibria in the discontinuous case. Section 3.4 considers

the welfare implications of duty and anger.

3.1 General Propositions

Proposition 1, stated below, says that if player 2 feels that player 1 is entitled to do

what she wants (I2 = 0), noncompliance will not be punished. Player 1 will only comply

if her sense of duty compels her to do so (a = 0 unless I1 = 1 and C ≤ D).

Proposition 1 If I2 = 0, a unique punishment-compliance equilibrium exists with:

p(0) = p(1) = 0 and a(C, I1) = 1 if and only if I1 = 1 and C ≤ D.

What drives this result? No matter what action player 2 observes player 1 choosing,

she knows that she is at least as well off as she would be if player 1 had felt entitled not to

comply (I1 = I2 = 0). Hence, it is not possible for player 2 to feel mistreated by player 1

(M = 0). Since player 2 never feels mistreated, player 2 will not punish noncompliance.

Remark: an application to markets Proposition 1 may seem trivial, but it has impor-

tant applications. In particular, it gives us a new way in which to think about markets.

In a market, participants generally feel that sellers are entitled to sell or not sell as they

please at a given price and that buyers, similarly, are entitled to buy or not buy as they

please at a given price. Were we to formally define what a market is, this would surely

be a key part of the definition. While it is certainly true that participants in a market

sometimes feel that buyers have a duty to buy or sellers have a duty to sell, these would

seem to be deviations from the idealized notion of a market.

Proposition 1 suggests that, if buyers in a market indeed feel that sellers are entitled to

sell or not sell as they please, a buyer will not be angry at a seller for refusing to sell at

some price p. Put another way, however a seller prices her goods or services will be seen as

fair by the buyer. Similarly, proposition 1 suggests that if sellers feel buyers are entitled

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to buy or not as they please, whether a buyer is willing to buy or not at some price p, this

will be seen as fair by the seller.

Hence, in an idealized market in which people are entitled to buy and sell as they

please, buying and selling according to one’s self interest will be seen as fair by other

market participants.

This gives some justification to the lack of consideration of fairness in most models of

market behavior. However, it suggests that more careful consideration may need to be

paid to what is seen as fair (a) in non-market contexts, such as within firms, and (b) in

markets that differ substantially from the idealized market. Labor markets typically fit

into category (b) since employees (and potential employees) often feel that employers have

a duty to pay certain wages and have a duty not to fire workers.

Proposition 2 shows that, if player 2 feels player 1 has a duty to comply (I2 = 1) but

is certain that player 1 feels a sense of duty to comply (q = 1), noncompliance will not

be punished. Player 1 will only comply out of a sense of duty (a = 0 unless I1 = 1 and

C ≤ D).

Proposition 2 If I2 = 1 and q = 1 (player 2 is certain I1 = 1), a unique punishment-

compliance equilibrium exists with:

p(0) = p(1) = 0 and a(C, I1) = 1 if and only if I1 = 1 and C ≤ D.

The reason for this result is simple. If player 2 is convinced that player 1 has an

appropriate sense of duty, then player 2 concludes, if she observes noncompliance, that the

reason for noncompliance was not an inadequate sense of duty on player 1’s part but rather

a high cost of compliance C. High C, as we noted earlier, is considered a good excuse for

noncompliance. Since player 2 assumes that there must be a good excuse if player 1 fails

to comply, she will not punish noncompliance.

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3.2 Two Cases

We have considered two instances (I2 = 0 and q = 1) in which noncompliance will

not be punished. We turn now to considering two cases in which noncompliance might be

punished.

In considering these cases, it will be interesting to examine the role played by duty (D)

in obtaining compliance. One way in which duty leads to compliance is that it motivates

player 1 to comply if I1 = 1. We can think of this as duty’s direct effect. There is also an

indirect effect. Punishment of noncompliance only arises when player 2 feels player 1 has

duty D to comply (recall that proposition 1 shows that punishment is absent otherwise).

Punishment—the indirect effect—might also motivate compliance.

Definition 2 We will distinguish between two ways in which duty (D) affects compliance.

(1) The direct effect of D: player 1’s sense of duty, which motivates compliance if I1 = 1.

(2) The indirect effect of D: punishment of noncompliance, which motivates both the

I1 = 1 type and the I1 = 0 type to comply. Punishment of noncompliance only arises when

I2 = 1.

It would be nice to find cases in which the direct effect ofD is not important in generating

compliance while punishment of noncompliance (the indirect effect ofD) is key to generating

compliance. We will show in this section that there are indeed cases where compliance

occurs entirely because of the indirect effect.

The reason that this is a compelling result is that it shows that, even when duty hardly

motivates individuals to comply (the direct effect of duty is small), it may nonetheless gen-

erate considerable punishment of noncompliance (the indirect effect of duty may nonetheless

be large). Hence, duty (one of the two nonstandard ingredients in the model) need not be

very large in order to be important. Its importance may lie more in its ability to generate

punishment of noncompliance than in its ability to directly incentivize player 1 to comply.

The following proposition considers the case where player 2 feels player 1 has a duty

to comply (I2 = 1), player 2 feels that player 1 might potentially lack a sense of duty to

14

comply (q < 1), and the cost of compliance is C with probability 1.

Proposition 3 Suppose I2 = 1, C = C with probability 1, and q < 1 (I1 = 0 with posi-

tive probability). The following is a characterization of the set of punishment-compliance

equilibria.

(1) If C > D, an equilibrium exists with: (i) p(0) = p(1) = 0, and (ii) a(C, I1) = 0 for

I1 = 0, 1.

(2) If C ≤ D + θv, an equilibrium exists with: (i) p(0) = θv and p(1) = 0, and

(ii) a(C, I1) = 0 if and only if I1 = 0 and C > θv.

If C = C for sure (S = {C}), the set of punishment-compliance equilibria will be thesame.

The following figure illustrates proposition 3 for the case where D < θv.

Equilibria:

1: no punishment, no compliance.

2: punishment of noncompliance, compliance by the type only.

3: punishment of noncompliance, compliance by both types.

1123

13

Let us make some observations about proposition 3. First, observe that if C ≤ θv, an

equilibrium exists for anyD > 0 in which both types comply. Hence, even if D is arbitrarily

small, so that the direct effect ofD is essentially zero, the indirect effect ofD (punishment of

noncompliance) will be sufficient to generate compliance. This shows that there are indeed

cases where the direct effect of D is small (not at all important in generating compliance)

but the indirect effect of D is nonetheless large (sufficient to generate compliance).

15

A second and related point is that there are cases in which player 1 will comply even

if player 1 has no sense of duty to comply (I1 = 0). If I1 = 0, player 1’s compliance is

motivated entirely by the desire to avoid punishment.

A third observation is that two equilibria exist if D < C ≤ D+θv: one in which there is

no compliance and no punishment of noncompliance and another in which there is compli-

ance (either by the I1 = 1 type only or by both types) and punishment of noncompliance.

The reason for this result is as follows. If there is harsh punishment of noncompliance

(p(0) = θv), there will be a high rate of compliance (player 1 will always comply if I1 = 1

and C = C). As a result, player 2 will feel angry about noncompliance, which in turn

justifies the harsh punishment. If there is no punishment of noncompliance (p(0) = 0), the

level of compliance will be low (player 1 will not comply if C = C). Hence, player 1 will

not feel angry about noncompliance, which in turn justifies the lack of punishment.

As mentioned in the introduction, this provides a possible explanation of the finding

that in some countries which profess disapproval of corruption (equivalent to I1 = I2 = 1

where a = 0 corresponds to corruption), corruption is nonetheless rampant and punish-

ment of corruption is relatively mild, while in other countries, corruption is uncommon and

punishment of corruption when it occurs is severe.

A fourth and final observation regards the instances where anger and punishment actu-

ally arise. If I1 = 1 (= I2) and C = C, there is no equilibrium in which player 1 is punished

(or player 2 is angry). On the other hand, if I1 = 0 ( 6= I2), C = C, and θv < C ≤ D+ θv

an equilibrium exists in which player 1 is punished (and player 2 is angry). This suggests

that, when player 2 is angry with player 1 and punishes player 1, a likely reason is that

players 1 and 2 have different views regarding player 1’s duty.

There is a second reason why anger and punishment might arise: if player 2 is uncertain

about player 1’s cost of compliance, player 2 might conclude that player 1 is likely not doing

her duty even if she is. Hence, anger might also arise because of a misunderstanding.

The theory of anger developed here can be applied to model conflict—cases where two

parties are potentially angry with one another. Our finding that anger arises either because

16

of (i) different views regarding duties and obligations or (ii) misunderstanding implies that

these are also the key reasons for conflict.

We turn now to an examination of proposition 4. In proposition 3, a player with an

appropriate sense of duty (I1 = 1) complies with probability 1 or fails to comply with

probability 1. Proposition 4 gives us a case where a player 1 with an appropriate sense of

duty only sometimes comply.

Proposition 4 Suppose I2 = 1, C = Clow with probability r (0 < r < 1), C = Chigh > Clow

with probability 1 − r, and q < 1 (I1 = 0 with positive probability). The following is a

characterization of the set of punishment-compliance equilibria.

(1) If Clow > D, an equilibrium exists with: (i) p(0) = p(1) = 0, and (ii) a(Clow, I1) =

a(Chigh, I1) = 0 for I1 = 0, 1.

(2) If Chigh ≤ D + θv, an equilibrium exists with: (i) p(0) = θv and p(1) = 0, and (ii)

a(Clow, 1) = a(Chigh, 1) = 1 and a(C, 0) = 1 if and only if C ≤ θv.

(3) If³r−rq1−rq

´θv < Clow ≤ D+

³r−rq1−rq

´θv < Chigh, an equilibrium exists with: (i) p(0) =³

r−rq1−rq

´θv and p(1) = 0, and (ii) a(Clow, I1) = 1 if and only if I1 = 1 and a(Chigh, I1) = 0

for I1 = 0, 1.15

Cases (1) and (2) of proposition 4 exactly mirror cases (1) and (2) of proposition 3.

Observe that, as before, there may be multiple equilibria.

We also get a third case now, in which neither the I1 = 1 type nor the I1 = 0 type

complies if the cost of compliance is high (C = Chigh) and only the I1 = 1 type complies if

the cost of compliance is low (C = Clow). When noncompliance is observed, one possibility

is that C = Chigh, in which case there is a good excuse for noncompliance. Another

possibility, however, is that C = Clow and I1 = 0: there is no good excuse for noncompliance.

r−rq1−rq is the probability that player 1 does not have a good excuse for noncompliance. As a

result, the punishment for noncompliance is³r−rq1−rq

´θv.

15 If Clow ≤ min D, r−rq1−rq θv and Chigh > D + θv, an equilibrium does not exist. This is consistent

with Lemma 1, since Lemma 1 does not ensure existence in this particular case.

17

Observe that, in case (3), punishment rises as q falls: the more convinced player 2 is

that player 1 lacks a sense of duty, the angrier player 2 will be over noncompliance and the

greater the corresponding punishment will be.

This helps us to understand Ostrom’s (1990) finding that in virtually all organizations,

there are “graduated sanctions.” The first time a rule is violated, punishment is typically

mild. But, repeated violation of a rule leads to more severe punishment. An explanation

of this result is that q falls after someone repeatedly violates a rule. The more the rule is

violated, the more likely it is that the person lacks an appropriate sense of duty and hence

lacks a good excuse for violating the rule. As a result, anger and punishment increase when

there is repeated violation.16

3.3 A Continuous Action Space

A worry regarding the results developed so far is that they might depend greatly upon

the assumption that the action space is discontinuous (a ∈ {0, 1}). In particular, a concernis that the finding that there are multiple equilibria might depend upon the discontinuous

action space.

It turns out that the results look extremely similar when the action space is continuous.

We will develop an analog of proposition 3 for a continuous action space in this section.

We find that the condition for the existence of multiple equilibria is exactly the same as in

proposition 3. In fact, there are many more equilibria when multiple equilibria do exist

than there are in the discontinuous case.

We only need to change our assumptions slightly for the continuous action space case.

We will assume that a ∈ [0,∞) rather than a ∈ {0, 1}. We will assume that I1, I2 ∈ [0,∞)rather than I1, I2 ∈ {0, 1}. We assume that G is player 2’s prior regarding I1.

Proposition 5 Suppose C = C with probability 1 and I1 < I2 with positive probability

(G(I−2 ) > 0). The set of punishment-compliance equilibria is as follows:16To model this more formally, we could build a model in which player 1 repeatedly chooses whether to

comply—with I1 fixed and C drawn repeatedly according to F—and player 2 repeatedly chooses how much topunish player 1.

18

(1) If C > D+θv, an equilibrium exists with: (i) p(a) = 0 for all a, and (ii) a(C, I1) = 0

for all I1.

(2) If C < D, an equilibrium exists with: (i) p(a) = θvmax(I2 − a, 0), and (ii) for

I1 ≤ I2: a(C, I1) = I2 if C ≤ θv and a(C, I1) = I1 if C > θv, for I1 > I2: a(C, I1) = I1.

(3) If D ≤ C ≤ D + θv, for any a ∈ [0, I2], an equilibrium exists with: (i) p(a) =

θvmax(a − a, 0), and (ii) for I1 ≤ a: a(C, I1) = I1 if C ≤ θv and a(C, I1) = a if C > θv,

for I1 > I2: a(C, I1) = a.

As mentioned, the set of equilibria in proposition 5 looks very similar to the set of

equilibria in proposition 3. In both cases, if C > D + θv, a unique equilibrium exists in

which a(C, I1) = 0 for all I1. In both cases, if C < D, a unique equilibrium exists in which

a(C, I2) = I2. In proposition 3, multiple equilibria exist if and only if D ≤ C ≤ D + θv.

There are two equilibria: one in which a(C, I2) = 0 and one in which a(C, I2) = I2. In

proposition 5, we also find that multiple equilibria exist if D ≤ C ≤ D + θv. In this case,

however, there is a dense set of equilibria: one in which a(C, I2) = a for every a ∈ [0, I2].We also find in this case that, even when the direct effect of D is small, the indirect

effect of D (punishment of noncompliance) may be large. In particular, if C ≤ θv, for every

D > 0, an equilibrium exists in which a(C, I1) ≥ I2 for all I1.

3.4 A Brief Discussion of Welfare

It is worth discussing the effect of anger and duty on welfare. A natural comparison

would be to the following case, where anger and duty are absent:

U1 = −p− Ca

U2 = va− κ(p)

Observe that this case is equivalent to the case where I1 = I2 = 0 (both players feel that

player 1 is entitled not to comply).

19

We will compare total welfare (U1 + U2) when there is duty and anger to total welfare

when duty and anger are absent. In comparing welfare in the model to welfare in this case,

we are effectively asking how well off players 1 and 2 are when there are no rules relative

to the case where rules emerge because of duty (or, norms) and anger.

Observe that, when duty and anger are absent, player 1 will choose a = 0 and player 2

will choose p = 0. Hence, total welfare is: U1 + U2 = 0.

It turns out that there are cases where duty and anger improve welfare and there are

also cases where duty and anger harm welfare. For example, according to proposition 3, if

I2 = 1 and C ≤ θv, an equilibrium exists in which both the I1 = 1 type and the I1 = 0 type

comply with probability 1. So, expected total welfare is: v − C. Duty and anger increase

total welfare if v > C. But, if θ > 1, it possible that v < C, in which case duty and anger

reduce total welfare.

This result is intuitive. If v > C and player 2 feels player 1 has a duty to comply, this

is (in some sense) a good rule. It is a rule that, if followed, improves total welfare. Player

2 might feel that player 1 has a duty to comply even if v < C. This is a bad rule (in

the sense that it reduces total welfare). Hence, a key conclusion that we draw is that the

welfare impact of duty and anger depends greatly upon the specific rules/norms that exist.

There are good and bad rules.

There is a second reason why duty and anger may be harmful. Punishment, which is of

course costly to both parties, can occur on the equilibrium path. For example, according to

proposition 3, if I2 = 1, θv < C ≤ θv+D, and q = 0, player 1 will comply with probability

0 and player 2 will administer punishment of θv whenever player 1 fails to comply. In such

a case, all that results from duty and anger is costly punishment and a feeling on player 2’s

part that she has been mistreated.17

17A sufficient condition for total welfare to be lower is that k ≥ θe−1. As mentioned above, the assumptionthat k ≥ θe−1 ensures that an increase in M will decrease player 2’s utility.

20

4 An Extension: the Provision of Punishment

As mentioned in the introduction, there are two nonstandard ingredients in the model:

(1) a sense of duty to comply, and (2) a desire to punish mistreatment.

In the previous section, we showed that (1) could be small. More precisely, we showed

that, even when duty hardly motivates compliance (the direct effect ofD is small), a sense on

player 2’s part that player 1 has a duty to comply may be sufficient to generate considerable

punishment of noncompliance (the indirect effect of D may nonetheless be large).

We turn in this section to arguing that (2) can also be relatively small. More precisely,

even when the desire or ability to punish mistreatment is low, it may be possible to generate

sufficient punishment of noncompliance to achieve compliance. The reason is that, by

“institutionalizing” punishment, it is often possible to greatly increase its provision. By

“institutionalizing” punishment, we mean that norms exist regarding who should punish

noncompliance and how much noncompliance should be punished. For example, player 2

might feel that a third party (a player 3) has a duty to punish noncompliance by player 1.

In Proposition 3, we found that if C > θv, the only equilibrium is one in which player

1 fails to comply with probability 1. The cost of compliance is too high relative to the

desire/ability to punish mistreatment. We will show in this section that, if punishment is

institutionalized, it may be possible to obtain compliance with probability 1 even if C > θv.

We will see that the condition that must be satisfied for institutionalization to be helpful

is that punishment p can be administered at a cost of less than p (κ(p) < p), which, in many

settings, would seem to be a reasonable assumption.

In this section, we will build a model of institutionalization. There will be three players:

a player 1, who complies or fails to comply (as before), a player 3, who potentially punishes

player 1 if 1 fails to comply, and a player 2, who benefits (as before) if player 1 complies

and (in contrast to the original model) potentially punishes player 3 if player 3 fails to

punish player 1’s noncompliance. When player 1 complies, it is to avoid being punished by

player 3. When player 3 punishes player 1, it is to avoid being punished herself by player

21

2. Player 2’s punishment of player 3 is motivated by anger.

The section will proceed as follows. We will elaborate the three-player model just

described. The analysis of the model will be simpler than might be expected because the

player 3-player 2 relationship in this case exactly mirrors the player 1-player 2 relationship

in the original model. We will establish a condition for the existence of an equilibrium in

which player 1 always complies and will show that this condition may be met even if it is

impossible to achieve compliance without institutionalization.

4.1 The Model

To keep the model relatively simple, we will consider a case where player 1’s cost of

compliance is C and this is known to all players.

The timing of events is as follows. Time 1: player 1 chooses a ∈ {0, 1}, which bothplayers 2 and 3 observe. Time 2: if player 1 does not comply (a = 0), player 3 chooses

b ∈ {0, 1}. Player 3’s choice is observed by player 2. b = 1 means that player 3 inflicts

punishment of C on player 1 and b = 0 means that player 3 does not punish player 1. Time

3: if player 1 does not comply (a = 0), player 2 decides how much to punish player 3, p(b),

with p(b) ∈ [0,∞). There is also an event that takes place at time 0, but we will hold offdiscussion of time 0 for the moment.

4.1.1 Player 1’s choice

Player 1 maximizes:

U1 = −E(b(I3)|X) · C(1− a)− Ca

As before, we assume that a = 1 if player 1 is otherwise indifferent.

The second term of the utility function is the cost of compliance. The first term of the

utility function is the disutility associated with player 3’s punishment. E(b(I3)|X) · C is

the expected punishment that will occur when player 1 does not comply (a = 0).

22

E(b(I3)|X) ∈ [0, 1] is player 1’s expectation of b at time 1. I3 ∈ {0, 1} is player 3’s typeand b(I3) ∈ {0, 1} is player 1’s belief regarding the choice a player of type I3 will make.Player 1’s prior is that the probability that I3 = 1 is q < 1. X is the information that

player 1 possesses at time 1 (which we will discuss more presently). In the event that player

1 does not have any information at time 1 (X = ∅), E(b(I3)|X) = qb(1) + (1− q)b(0).

In contrast to the model in the previous section, we assume that player 1 does not

feel a sense of duty to comply. The reason for making this assumption is to show that

institutionalization is helpful even if we eliminate player 1’s sense of duty.

Observe that player 1 will comply if and only if player 1 believes that she will be

punished for noncompliance with probability 1: E(b(I3)|X) = 1. That is, a∗(b(·),X) = 1if E(b(I3)|X) = 1 and a∗(b(·),X) = 0 if E(b(I3)|X) < 1.

4.1.2 Player 3’s choice

If player 1 fails to comply (a = 0), player 3 chooses b at time 2 to maximize:

U3 = −p(b)− κ(C)b−Dmax(I3 − b, 0)

Player 3 chooses b = 1 is she is otherwise indifferent.

The first term is the disutility associated with player 2’s punishment, where p(b) is player

3’s belief concerning the punishment associated with choice b. The second term is the cost

associated with punishing player 1’s noncompliance. Importantly, we assume that the cost

to player 3 of punishing player 1 is the same as the cost to player 2 of punishing player 1 in

the original model. This makes for a fair comparison of this case with the model from the

previous section.

The third term of the utility function is player 3’s sense of duty to punish player 1. If

I3 = 1, player 3 feels duty D to punish player 1. If I3 = 0, player 3 does not feel any duty

to punish player 1.

Let b∗(I3, p(·)) denote the optimal choice of b.

23

A discussion of time 0 Player 2 will only feel mistreated by player 3 if player 2

believes that she has been harmed in some way by player 3—is worse off than she would

be if player 3 had had an appropriate sense of duty. The only way in which player 3 can

harm player 2 is by making a choice which causes player 1 to choose noncompliance (a = 0)

rather than compliance (a = 1) (player 2 loses v if player 1 fails to comply, as in the original

model).

Observe that, if player 3 chooses b = 0, this does not actually cause player 1 to choose

noncompliance: player 1 has already chosen whether to comply. Hence, player 2 will not

construe the choice of b = 0 rather than b = 1 as mistreatment.

What could cause player 1 to choose noncompliance (and hence cause player 2 to feel

mistreated by player 3) is an action by player 3 at time 0 which allows player 1 to discern

whether player 3 will punish noncompliance. Such an action could cause player 1 to choose

noncompliance. And, indeed, individuals are often able to discern, in advance of taking an

action, whether punishment will result from doing so.18

This motivates the following assumption. We assume that, at time 0, player 3 chooses b

and the optimal choice of b is: b = b∗(I3, p(·)). We assume that, with probability μ, player1 observes b. In other words, with probability μ, player 3 takes an action which signals to

player 1 whether noncompliance will be punished. We can think of μ as parameterizing

player 1’s ability to discern how she will be punished.

In the event that player 1 learns b (X = {b}), a∗(X) = b. In the event that player 1

does not learn b (X = ∅), b does not have an effect on player 1’s choice of a. Hence, if

player 3 chooses b = 0 rather than b = 1, this causes harm to player 2 of μv in expectation.

4.1.3 Player 2’s Choice

What player 2 knows As mentioned above, player 2 knows the value of C. Player 2

also knows the value of D. Analogous to the original model, player 2 does not know the

18There are many ways in which player 1 might be able to discern whether player 3 will punish noncom-pliance. For example, consider the case where player 3 is just one of a set of people whose noncomplianceplayer 3 has a duty to punish. Player 1 might be able to discern whether her own noncompliance will bepunished by observing whether player 3 chooses to punish the noncompliance of others.

24

value of I3: like player 1, player 2’s prior is that I3 = 1 with probability q < 1.

As mentioned, player 2 observes a and, if a = 0, player 2 observes b. We do not need

to assume that player 2 observes b (although we could), since b = b and player 2 observes b.

We will assume that player 2 knows the value of μ (the probability that player 1 learns b).

Player 2’s choice If player 1 complies (a = 1), player 2’s utility is:

U2 = va = v

Player 2 receives benefit v ≥ 0 from player 1’s compliance. Since player 1 complied, player

2 does not choose a punishment p for player 3.

If player 1 fails to comply (a = 0), player 2’s utility is:

U2 = va− κ(p)−Φ(M(b, I2, p(·)), p)

= −κ(p)−Φ(M(b, I2, p(·)), p)

The second term is the cost of punishing player 3. Importantly, the cost of punishing player

3 is the same as the cost to player 2 of punishing player 1 in the model from the previous

section. This assumption allows for a fair comparison of this case with the previous model.

The final term is the disutility associated with feeling mistreated by player 3. Perceived

mistreatment is defined in exactly the same way as before: as the expected loss from (in

this case) player 3’s failing to have an appropriate sense of duty to comply. Analogous to

the original model, perceived mistreatment will depend upon b (whether player 3 punishes

player 1’s noncompliance), I2 (which, in this case, is player 2’s view regarding how player

3 ought to behave), and p(·) (player 2’s belief regarding p(·)).The optimal punishment is:

p∗(b, I2, p(·)) = θM(b, I2, p(·))

25

Mistreatment As mentioned above, player 2 is not harmed if player 3 chooses b = 0

rather than b = 1. But, player 2 is harmed if player 3 chooses b = 0 rather than b = 1

(and, of course, b = b). The expected loss to player 2 is μv.

Hence, a formula for player 2’s perceived mistreatment is:

M(b, I2, p(·)) = μvmax(b∗(I2, p(·))− b)

b∗(I2, p(·)) is player 2’s belief regarding the b player 3 would have chosen if player 3 hadhad an appropriate sense of duty. b = b is player 3’s actual choice.

The optimal punishment Hence, the optimal punishment p∗(b, I2, p(·)) will be:

p∗(0, I2, p(·)) = θμvb∗(I2, p(·))

p∗(1, I2, p(·)) = 0

4.1.4 Equilibrium

The equilibrium concept that we will employ is very similar to the equilibrium concept

defined in section 2. We will define an equilibrium as follows.

Definition 3 (a(X), b(I3), b(I3), p(b, I2)) is an equilibrium if and only if:

(1) a(X) = a∗(b(·),X)(2) b(I3) = b(I3) = b∗(I3, p(·))(3) p(b, I2) = p∗(b, I2, p(·)) if b = b∗(I3, p(·)) for some I3 ∈ I.(4) p(b, I2) = p∗(b, I2, p(·)) for some p(·) such that b = b∗(I3, p(·)) for some I3 ∈ I.Conditions (2), (3), and (4) mirror conditions (1), (2), and (3) respectively of the equi-

librium concept defined in section 2. Condition (1) says that b(I3) = b(I3): player 1’s belief

regarding the choice a player 3 of type I3 will make is correct.

26

4.2 Results

As promised, we will now establish a condition for the existence of an equilibrium in

which player 1 always complies. We will show that this condition is potentially weaker

than the corresponding condition when there is no institutionalization of punishment.

Observe that, if player 1 does not learn b (X = ∅), a∗(b(·),X) = 1 if and only if b(I3) = 1for both I3 = 1 and I3 = 0. Therefore, (a(X), b(I3), b(I3), p(b, I2)) is an equilibrium in which

player 1 always complies if and only if player 3 always punishes noncompliance (b(I3) = 1

for both I3 = 1 and I3 = 0).

As mentioned earlier, the relationship between player 3 and player 2 in this case exactly

mirrors the relationship between player 1 and player 2 in the model from the previous

section. Player 3’s choice of b = b is equivalent to player 1’s choice of a in the original

model. The value to player 2 of player 1 choosing a = 1 was v in the original model. In

this case, the value to player 2 of player 3 choosing b = b = 1 is μv.

Proposition 6, which establishes a condition for the existence of an equilibrium in which

player 1 always complies, follows from Proposition 3: b = b is substituted for a, μv is

substituted for v, and κ(C) is substituted for C since the cost to player 3 of choosing

b = b = 1 is κ(C).

Proposition 6 An equilibrium exists in which player 1 always complies if and only if

κ(C) ≤ θμv. Or, substituting for κ:

C ≤ θ2μv

Let us compare this to the model from the previous section. In the original model,

an equilibrium exists in which player 1 always complies if and only if: C ≤ θv. Hence,

institutionalization increases the provision of punishment if θ2μv > θv. Or, simplifying:

μθ > 1

27

If player 1 is very good at discerning how she will be punished (μ = 1), institutionaliza-

tion will improve the provision of punishment if θ > 1. θ > 1 if and only if κ (p) < p (the

condition given in the beginning of the section). As mentioned above, this condition says

that punishment p can be administered at a cost less than p.

Let us consider the intuition behind this result. In order to induce b = 1, player 2 needs

to be willing to administer punishment of κ¡C¢if b = 0. In the model from the previous

section, player 2 needed to administer punishment of C if a = 0 in order to induce a = 1.

Therefore, if κ¡C¢< C, less punishment is required under institutionalization. A concern

is that institutionalization decreases player 2’s willingness to punish. However, we found

that if μ = 1 player 2 will be as motivated to punish player 3 as she was to punish player 1.

Remark: additional levels of institutionalization Additional levels of institution-

alization could, potentially, further increase the provision of punishment. For example,

suppose we were to add a second level of institutionalization: player 4 has a duty to punish

player 3 if player 3 fails to punish noncompliance by player 1. Then, the condition for com-

pliance by player 1 becomes: μ2θ3v > C, where μ2 is player 3’s ability to discern player 4’s

punishment. This might be a weaker condition than the condition for compliance with one

level of institutionalization: μθ2v > C. With n levels of institutionalization, the condition

for compliance becomes: μnθn+1v > C (where μn is player (n−1)’s ability to discern player

n’s punishment). While θn grows exponentially large for θ > 1, we might expect that μn

falls as n increases: it may be very hard to discern how player n would punish player (n−1)for player (n − 1)’s failure to punish player (n − 2) for player (n − 2)’s failure to punishplayer (n − 3), etc. So, it is not clear that, with a sufficiently large number of levels of

institutionalization, compliance can be achieved for an arbitrarily large cost of compliance,

C. Furthermore, the provision of punishment might be greatest for n relatively low.

28

5 Relation to other work on fairness and anger

Having now elaborated a theory of fairness and anger, it is worth discussing briefly its

relation to other theories.

This paper was motivated in part by a desire to give foundations to Hart and Moore’s

(2008) model of aggrievement. In Hart and Moore (2008), what parties consider to be

unfair—what makes them angry/aggrieved—depends upon the contract they sign ex ante.

We can think of the signing of a contract in Hart and Moore (2008) as establishing norms.19

This allows for an interpretation of their model in the terms of the theory of fairness

developed in this paper.

In their model, there is a buyer and a seller who contract ex ante on a range of prices

[p¯, p] at which trade can occur in a later period. They assume that if trade occurs ex

post, the buyer will be angry/aggrieved by an amount v−p¯(v is the buyer’s valuation of

the item being traded) and the seller will be angry/aggrieved by an amount p− c (c is the

seller’s valuation). The parties have the opportunity to shade their performance and do so

in proportion to their anger/aggrievement.

We can view the contract signed ex ante as establishing a sense on the buyer’s part

that the seller is entitled to sell at a price p >p¯, but has a duty to sell at higher prices.

Similarly, we can view the contract as establishing a sense on the seller’s part that the buyer

is entitled not to buy at a price p > p but has a duty to buy at lower prices. This allows

an interpretation in terms of the model developed in this paper. With these norms, it is

possible for a buyer to feel mistreated by an amount v−p¯and a seller to feel mistreated by

an amount p− c, corresponding to the anger/aggrievement in Hart and Moore (2008). We

can view the shading of performance that takes place as punishment of the other party.

It is also profitable to consider how the model developed in this paper relates to a set

19Norms are treated in this paper as exogenous. Clearly, to interpret Hart and Moore (2008) requires amodel in which norms are endogenous. In other work, I have considered how to endogenize norm formationand change. In this work, I conclude that individuals choose norms subject to constraints (see my “ResearchStatement” for a discussion.) Individuals are constrained in that certain views regarding how people shouldbehave seem to be “hard-wired.” Among the norms that seems to be hard-wired is a norm that individualshave a duty to do what they agree/contract to do.

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of papers on fairness and reciprocity: Rabin (1993), Dufwenberg and Kirchsteiger (2004),

and Falk and Fischbacher (2006). These papers are similar to this paper in the sense that

they view fairness as being about intentions, and consequently employ psychological game

models. They differ, however, in three important respects which we will consider in turn.

First, this paper assumes that there are different types of people—they differ in their

sense of duty—and beliefs about a person’s type are important for understanding the anger

or lack of anger that will arise in response to particular actions. For example, Proposition

4 shows that there will be more anger over noncompliance when it is considered more likely

that player 1 lacks a sense of duty to comply. We argued that this gives an explanation

of Ostrom’s (1990) finding of graduated sanctions: there is more punishment and anger

when an offense is repeated. These other papers are unable to capture the phenomenon of

graduated sanctions.

Second, this paper captures contextual differences in fairness attitudes. These other

papers do not explain, for example, why in a market it may be considered fair to buy and

sell according to self interest, while transacting according to self interest is unlikely to be

considered fair in a firm/organization.

Finally, this paper focuses on negative reciprocity—punishment of misbehavior arising

from anger—whereas these other papers model both negative and positive reciprocity, viewing

them as flip sides of the same thing. We can think of positive reciprocity as arising from a

sense of gratitude while negative reciprocity arises from anger.20

One way we might have captured positive reciprocity is as follows. We might have

assumed that M was the harm or benefit resulting from player 1 having a sense of duty I1

different from I2.21 We might have assumed that player 2 feels grateful if M < 0 and is

motivated to reward player 1.

There is a reason that we did not choose to model positive reciprocity and gratitude in

this way—as the flip side of anger. It turns out that gratitude is not exactly the flip side of

20Many sociologists and social psychologists have pointed to gratitude as the key source of positive reci-procity. See especially Simmel (1950).21 In terms of the model, this would mean assuming that M(a, I2, p(·)) = v (a(a, I2, p(·))− a) instead of

our existing assumption that M(a, I2, p(·)) = vmax (a(a, I2, p(·))− a, 0).

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anger.

The following hypothetical situation will help to illustrate this. Imagine players 1 and

2 are neighbors and player 2 feels that player 1 has a duty to keep her front garden tidy, but

not immaculate. If player 1 shared player 2’s sense of duty, she would keep her garden tidy,

but not immaculate. Suppose that player 1 belongs to a gardening club, and as a result,

has a very strong sense of duty to keep an immaculate garden. Her garden is immaculate

in consequence.

Player 2 benefits from the immaculate garden relative to the tidy garden. Hence, this

is a case where player 2 benefits from player 1 having a sense of duty I1 different from I2.

This does not appear to be a case, however, in which player 2 would feel grateful and/or a

desire to reciprocate.

The reason is that player 1 does not keep the garden immaculate for player 2’s sake.

The desire to reciprocate (positively) arises only when: (i) player 1 goes beyond the call of

duty and (ii) does so for player 2’s sake.

There is a sense in which it is appropriate for Rabin (1993), Dufwenberg and Kirchsteiger

(2004), and Falk and Fischbacher (2006) to lump together positive and negative reciprocity,

even if it is inappropriate to do so in this paper. Implicitly, these papers are attempting

to capture a case in which there is a particular sense of duty: a sense of duty to take into

account the interests of the other player (or players.) In such a context, if (i) is satisfied,

(ii) is automatically satisfied. So, gratitude really is the flip side of anger in such a case.

However, it makes sense more generally, when we allow for other notions of what a player’s

duty might be, to assume that gratitude is not precisely the flip side of anger.

6 Conclusion

A key question in organization theory is how rules emerge and how rules are enforced.

While much attention in organization theory is focused on the demand for rules—identifying

the best possible rules, taking the contracting environment as given—many organizational

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phenomena can only be understood by understanding constraints on the supply side.2223

This paper provides one answer to the question of how rules emerge. It stresses the

importance of norms—views regarding duties and obligations that exist within a group—as a

key generator of rules in organizations. An implication is that, in order to understand more

deeply how rules are supplied, it is necessary to understand how norms emerge. Akerlof

(2008, 2010a) attempt to answer this question with precisely this goal in mind.

There are other answers to the question of how rules emerge. Relational contracts, for

example, are another source of rules in organizations.24 An important question is whether

norms account for phenomena that could not otherwise be explained. Akerlof (2010a) iden-

tifies several phenomena that appear to be difficult to explain with an alternative approach.

For example, it is suggested that the firms’ reluctance to hire overqualified workers—a puz-

zle identified by Bewley (1999) which is difficult to explain with standard models—can be

explained by an overqualified worker’s lack of respect for a manager’s “authority” (a low

sense of duty to follow orders).

This paper suggests an interesting approach to modeling conflict. Conflict can be

modeled as two parties who are angry with one another. Akerlof (2010b) on feuding and

cultures of honor examines a setting where conflict can arise. It shows that an initial

provocation can lead to a protracted conflict (feuding) because retaliation by one party

further angers and provokes the other party. It also shows that there is a natural tendency

for arms races to occur and for cultures of honor to form, both of which make conflict worse

and reduce welfare.22Ostrom (1990), Elster (1989), and Bates (1988) stress the importance of understanding how rules and

institutions emerge, for example.23Elster (1989) makes the interesting point that, even if individuals can commit to punish rule violation

(an assumption we did not make), there is a problem in providing incentives in organizations because afree-rider problem may exist: everyone hopes someone else will do the punishing. Anger is one potentialsolution to Elster’s problem, since it creates a desire to punish rule violation for its own sake.24See Baker, Gibbons, and Murphy (1999, 2002).

32

References

Akerlof, Robert. 2008. “A Theory of Social Motivation.” Unpublished Manuscript, MIT.

Akerlof, Robert. 2010. “A Theory of Authority.” Unpublished Manuscript, MIT.

Akerlof, Robert. 2010. “A Model of Feuding and Cultures of Honor.” Unpublished Manu-script, MIT.

Akerlof, Robert. 2010. “Research Statement.” Unpublished Manuscript, MIT.

Andvig, Jens. 1991. “The Economics of Corruption: A Survey.” Studi Economici 43: 57-94.

Andvig, Jens and Karl Moene. 1990. “How Corruption May Corrupt.” Journal of EconomicBehavior and Organization 13: 63-76.

Baker, George, Robert Gibbons and Kevin J. Murphy. “Informal Authority in Organiza-tions.” Journal of Law, Economics, and Organization 15: 56-73.

Baker, George, Robert Gibbons and Kevin J. Murphy. 2002. “Relational Contracts and theTheory of the Firm.” Quarterly Journal of Economics 117: 39-83.

Bardhan, Pranab. 1997. “Corruption and Development: A Review of Issues.” Journal ofEconomic Literature 35: 1320-1346.

Bates, R. H. 1988. “Contractarianism: Some Reflections on the New Institutionalism.” Pol-itics and Society 16: 387-401.

Battigalli, Pierpaolo and Martin Dufwenberg. 2009. “Dynamic Psychological Games.” Jour-nal of Economic Theory 1: 1-35.

Bewley, Truman F. 1999. Why Wages Don’t Fall During a Recession. Cambridge, MA:Harvard University Press.

Cadot, Olivier. 1987. “Corruption as a Gamble.” Journal of Public Economics 33: 223-244.

Dufwenberg, Martin and Kirchsteiger. 2004. “A theory of sequential reciprocity.” Gamesand Economic Behavior 47: 268-298.

Elster, Jon. 1989. The Cement of Society: A Study of Social Order. Cambridge, UK: Cam-bridge University Press.

Elster, Jon. 1992. Local Justice: How Institutions Allocate Scarce Resources and NecessaryBurdens. NY: Russell Sage Foundation.

Falk, Armin and Urs Fischbacher. 2006. “A theory of reciprocity.” Games and EconomicBehavior 54: 293-315.

Falk, Armin, Fehr, Ernst, and Urs Fischbacher. 2008. “Testing theories of fairness–Intentions matter.” Games and Economic Behavior 62: 287-303.

33

Fehr, Ernst and Klaus Schmidt. 1999. “A Theory of Fairness, Competition, and Coopera-tion.” Quarterly Journal of Economics 114: 817-868.

Geanakoplos, John, David Pearce, and Ennio Stacchetti. 1989. “Psychological Games andSequential Rationality.” Games and Economic Behavior 1: 60-79.

Gibbons, Robert. 2005. “Four Formal(izable) Theories of the Firm?” Journal of EconomicBehavior and Organization 58: 200-245.

Hart, Oliver and John Moore. 2007. “Incomplete Contracts and Ownership: Some NewThoughts.” American Economic Review Papers and Proceedings 97: 182-186.

Hart, Oliver and John Moore. 2008. “Contracts as Reference Points.” Quarterly Journal ofEconomics 123: 1-48.

Kahneman, Daniel, Jack L. Knetsch, and Richard Thaler. 1986. “Fairness as a Constrainton Profit Seeking: Entitlements in the Market.” American Economic Review 76: 728-741.

Kandori, Michihiro. 1992. “Social Norms and Community Enforcement.” Review of Eco-nomic Studies 59: 63-80.

Klein, B., R. Crawford, and A. Alchian. 1978. “Vertical Integration, Appropriable Rents,and the Competitive Contracting Process.” Journal of Law and Economics 21: 297-326

Konow, James. 2003. “Which Is the Fairest One of All? A Positive Analysis of JusticeTheories.” Journal of Economic Literature 41: 1188-1239.

Ostrom, Elinor. 1990. Governing the Commons: The Evolution of Institutions for CollectiveAction. Cambridge, UK: Cambridge University Press.

Rabin, Matthew. 1993. “Incorporating Fairness into Game Theory and Economics.” Amer-ican Economic Review 83: 1281-1302.

Shleifer, Andrei and Robert Vishny. 1993. “Corruption.” Quarterly Journal of Economics108: 599-617.

Simmel, Georg. 1950. The Sociology of Georg Simmel. Compiled and translated by KurtWolff, Glencoe, IL: Free Press.

Tirole, Jean. 1996. “A Theory of Collective Reputations.” Review of Economic Studies 63:1-22.

Walzer, Michael. 1983. Spheres of Justice: A Defense of Pluralism and Equality. New York,NY: Basic Books Inc.

Williamson, Oliver E. 1971. “The vertical integration of production: market failure consid-erations.” American Economic Review 61: 112-123.

Williamson, Oliver E. 1979. “Transaction cost economics: the governance of contractualrelations.” Journal of Law and Economics 22: 233-261.

34

Williamson, Oliver E. 1985. The Economic Institutions of Capitalism. New York, NY: TheFree Press.

Young, H. Peyton. 1994. Equity in Theory and Practice. Princeton, NJ: Princeton UniversityPress.

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