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Behavioral Spillovers, Learning,

and Institutional Path Dependence

Jenna Bednar and Scott E Page

November 19, 2013

DRAFT

Abstract

How does the order that laws and other institutions are introduced affect their per-formance, and under what conditions will sequencing matter? To gain a foothold onthis problem, we construct a formal model of institutional path dependence. We explorehow learning combined with behavioral spillovers in initial action and strategy choicesproduces path dependent behaviors, outcomes, and optimal institutional choices. We

consider a population of individuals who play a sequence of games drawn from a familyof games. When confronted with a new game, individuals choices of initial strategiesdepend on behavior in similar games. In the subsequent plays of the new game, in-dividuals choose the strategy that performs best given the distribution of strategies.Within this framework, we derive sufficient conditions for outcomes in new games tobe influenced by the sequence of previous games. We show that larger spillovers implygreater path dependence and derive conditions for there to exist a sequencing of gamesthat produces the most efficient outcomes for each game. We then show that optimalsequencing, somewhat paradoxically, implies maximizing future path dependence. Wealso demonstrate how our analytic framework can provide insights into the theory ofendogenous institutional change, and show how weaker incentives to punish reduce

path dependence.

Keywords: Learning, mechanism design, path dependence, equilibrium selection, quasi-

parameters

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often referred to in the literature as exceptionalism: similar institutions often produce dra-

matically different outcomes across countries. Social scientists are able to measure cultural

variation (eg, Inglehart), and isolate its effects on behavior (eg Putnam 1993, Licht et al

2007, Engelhardt and Freytag 2013). When knowledge of educational, psychological, socio-

logical, and cultural attributes is available, it can often enable one to anticipate deviations

from rationality (Nisbett 2003). However models of social forces such as culture are rare

and generally assume that culture is an unchanging exogenous force acting on immediate

behavior (Williamson 2000), perhaps by altering goals, beliefs, or incentives.

Recently a few models have challenged the orthodoxy of culture as primordial and un-

explainable. In prior work, we define culture as patterns of behavior, and demonstrate

theoretically and in experiments that behavioral patterns emerge as a product of the set

of institutions that structure interaction in a community of agents. Behavioral spillovers

explain deviations from predicted behavior if one theorizes about a game form in isolation;

when considering the broader institutional context, the behavioral response becomes under-

standable (Bednar and Page 2007, Bednar et al 2012). In this paper we continue this line of

research by investigating how individuals interact within a sequence of related institutional

settings in the presence of behavioral spillovers.

We stake no claim to the originality of the idea that understanding human behavior in

a novel setting can be improved by considering the larger context. Cognitive psychologists

have demonstrated the role of extant knowledge in guiding search (see Warner et al 2009).

Whats new in this paper is demonstrating how those contextual effects can be embedded in

a tractable model of institutions. Our framework creates a larger burden for those who study

institutions as we claim that in order to understand behavior within a particular institution,

one should view a community of individuals interacting not in just in that setting but in

other settings as well. Long (1958) refers to these multiple settings as an ecology of games

while Bednar and Page (2007) call them game ensembles.

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The notion of institutional path dependencethat the performance of an institution

depends on preexisting institutions, and the order that those institutions were introduced

appears in an array of literatures, from models of institutional performance and change, to

applications including developmental economics (eg., Mansfield and Snyder 2005, Levy and

Fukuyama 2010), market performance, and the feasibility of democracy (Alexander 2001).

We derive seven main results. First, we find that behavioral spillovers induced by the

initial strategy choices can result in suboptimal equilibrium strategies. Here were leveraging

Folk Theorem logic which enables multiple strategies, including Pareto inferior strategies, to

be equilibria of repeated games. Second, we demonstrate that any set of games that contains

at least one game that supports multiple equilibria can be ordered in at least two ways that

produce different outcomes in at least one game. In other words, any set of institutions

will be subject to some degree of path dependence unless those institutions all have unique

equilibria. Third, we characterize those institutional contexts that generate greater path

dependence than others. This result relates the size of the behavioral spillover and the set

of previous games to the extent of path dependence. Fourth, we show that initial game

dependence, a type of path dependence, increases in the size of the behavioral spillover.

Fifth, we show that the most efficient paths include more diverse games earlier in the

sequence. This result implies that while contexts that exogenously allow path dependence

are optimally avoided, but once one has to choose institutions, one would like to maintain

as much path dependence as possible. Sixth, by applying our framework to the framework

of endogenous institutional change constructed by Greif and Laitin (2004), we can show

that negatively reinforced institutional drift leads to institutional change at an inefficient

moment, and in some cases at the most inefficient moment. And last, we show that optimal

institutional choice implies strong incentives to choose an equilibrium but weak punishments

for deviating.

The remainder of the paper consists of six parts. We first present our modeling framework.

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former. Viewed with a wider lens, our assumption of a type who base their actions on the

most similar previous game placed can be seen as a form of cased based strategic choice

(Gilboa and Schmeidler 1995).1 We differ slightly in that we use similarity-based reasoning

to derive initial strategies.

Our assumption of how the behavioral spillovers arise more closely aligns with the games

theory framework of Bednar and Page (2007) in which individuals apply common strategies

across multiple games. Recent experimental evidence on subjects playing multiple games

reveals evidence of behavioral spillovers from previous games. Individuals introduced to

a new game apply strategies that worked well in a similar game that theyve just played

(Bednar, et al 2012, Cason et al 2012). The spillovers may be partly due to a desire to lower

cognitive costs.2 for the logic of our model to apply, we need only that some individuals

choose strategies that are in some way based on what has happened in the past. We do

not require the efficiency minded individuals. In fact, our results carry through even more

powerfully, if we assume that the non nearest game priortype chooses strategies randomly.

Finally as for our third assumption, that individuals learn their way to an equilibrium,

that assumption is generally considered to hold for simpler games. Experiments demonstrate

that individuals do tend toward equilibrium and often do so quickly (Camerer 2003). In the

paper, we assume individuals learn which strategies to play using best response learning

(Nash 1951). We differ from Nashs assumption in that we make explicit assumptions about

the initial strategies and link those assumptions to specific equilibria. Whats most germane

question is whether our choice of learning rule matters. Learning rules can matter, but only

dramatically so when there exists a non equilibrium strategy thats a best response for large

portions of the space (Golman and Page 2010).

1Our construction results in behavior similar to those found when individuals apply the analogous gamesconcept and place games in equivalence classes (Jehiel 2005).

2Samuleson (2001) demonstrates in a theoretical model how cognitive costs can effect strategies but doesnot consider the possibility of applying a single strategy in multiple games.

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interested in both identifying path dependence and comparing it across contexts.

To accomplish the later, we require additional definitions. Define a continuation path of

lengthm, Cm, to be a sequence ofm games and let m ={Cm : Cm = (g1, g2, gm)} denote

the set of all continuations of lengthm. Define acontext and pathas a context together with

a continuation path of length m. Denote this by (, Cm).

Given two next outcome equivalent contexts, and, context exhibits greater path

dependence than context if and only if either of the following two equivalent conditions

hold.7

(i) For any game, the set of continuation paths that changes the outcome in gameg in context

strictly contains the set of continuation paths that change the outcome in context.

Cm m: (g) = (, Cm)(g)

{Cm m: (g) = (, Cm)(g)} g G m 1

(ii) For any game, along any continuation path, the number of times the outcome ing changes

in context is greater than or equal to the number of times it changes in context, with a

strict inequality for some paths.

Those games whose outcomes can change along a continuation, we refer to assusceptible

given the context. Those games that are not susceptible, we refer to as immune. Finally,

in contexts with large behavioral spillovers, the first game can have a large effect on future

outcomes. We define the extent of initial game dependencefor a context to be the prob-

ability that the outcome of a game in the susceptible region will be the same as that of the

initial game in the context.

7To prove that these definitions are equivalent is straightforward. Suppose that the first condition doesnot hold. Then the continuation path for which a game changes under and not contradicts the secondcondition. If the second condition is violated, then choose a continuation path under which a game switchesoutcomes more often in context .

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2 Two Examples: Games of Tradition and Trust

To elucidate the primary concepts in our framework, we first derive results for two familiar

classes of games that can be parameterized along a single dimension. The first family of

games considers situations in which individuals can choose a traditional action or an innova-

tive action. This is an example of acoordination game. Deferring to or challenging authority

such as the rule of law would also produce a coordination game as might using a market

or reciprocal exchange (Kranton 1996). The second family of games considers situations

in which individuals can either take a safe action or a trusting action. Both sets of games

involve only pairwise interactions. The analysis could be extended to more general cases.

Upholding Tradition or Being Innovative

In the first family of games, individuals must decide to whether to stick to tradition or to

adopt an innovation. The payoffs to each action are determined by a parameter [0, 16]. If

both players stick to tradition, each gets a payoff of (16 ). If both challenge the tradition

and play an innovative new action, each gets a payoff of . These games have two pure

strategy equilibria for [4, 12]: sticking to tradition (T) and innovating (I). Sticking to

tradition will be efficient if 8 and innovating is efficient if 8. If the two players

choose opposite actions then each receives a payoff of zero.

Traditional (T) Innovate (I)

Traditional (T) 16 , 16 4, 4

Innovate (I) 4, 4 ,

For clarity of presentation, we will assume that the behavioral spillover, , has value 34

,

implying that three-fourths of the population will begin by playing the equilibrium action

from the closest game. Note that we can identify a game by the parameter . Assume that

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the first game in a sequence of games has 1= 7. In the model, we assume that the outcome

in the first game will be efficient, so individuals will choose the traditional strategy. Assume

that for the second game, 2 = 9. By construction, in the first period three fourths of the

population will play the traditional strategy and one fourth choose to be innovative. This

one fourth of the population can be thought of as the people who see the logic of the game

and play the efficient equilibrium strategy. Note that these individuals are notrationalper

se, in that if they knew the percentage of people playing each action, they might not choose

to take the innovative action. Therefore, we might better think of them as people who can

solve for the efficient equilibriumandwho believe that a sufficient percentage of others will

play it to make it optimal.

The payoffs for the two strategies in the population can be calculated as follows:

Traditional (T): 34

(7) + 14

(4) = 254

Innovative (I): 34

(4) + 14

(9) = 214

Thus, even though if everyone were to be innovative they would earn higher payoffs,

initially the payoff to being traditional is higher given the proportion of the population whose

initial response is to apply behavior developed for a similar game. Furthermore, if people

learn to play the strategy with the higher payoff, then the traditional strategy will come

to dominate.8 Thus, in the resulting equilibrium everyone chooses the traditional action.

Suppose though that the first game in the sequence had produce innovative strategies, i.e.

that 1 > 8). If so, the outcome in the second game with 2 = 9 would also have been

innovative.

The outcome in the game 2= 9 depends on the games that precede it, So we call shall

refers to it as susceptible. Note that in order for a game to have a path dependent outcome

8This can be shown formally by calculating the payoff to each strategy as the percentage of individualschoosing the traditional strategy increases.

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TraditionalImmune Susceptible Innovative Immune

= 0 = 16= 6 = 10

Figure 1: Susceptible and Immune Regions in Tradition/Innovative Game as a Function of

it must be susceptible. In this example, the sequence of games (1 = 9, 2 = 7) produces

innovative outcomes in both games, where as we just showed, the sequence (1 = 7, 2 = 9)

produces traditional outcomes in both games. Hence, outcome depend not just on the set

of games, i.e. set dependence (Page 2006), but also on the order in which those games are

played.

Not all games will be susceptible. Ifis sufficiently high (resp. low) then the outcome will

be innovative (resp. traditional) regardless of the previous games, as depicted in figure 1. To

see why, suppose that the first game in a sequence produces an efficient, traditional outcome,

e.g. 1 < 8. It can be shown that if the second game has 2 > 10, then the outcome will

be that both players choose innovative actions regardless of the previous games.9

A similar

calculation shows that for t < 6, the strategy chosen will be traditional regardless of the

previous games played. We will refer to all games with 6 or 10 as immune, as their

outcomes do not depend on the set of previous games.10 The derivation of these thresholds

enables us to divide the parameter space into three regions: one in which the traditional

action is immune, one in which being innovative is immune, and one in which outcomes

depend on the previous games, what we will call the susceptible regiongiven the context.

9The payoff to the traditional action equals 34(16 2) + 14(4) = 13

342. The payoff to the innovative

action equals 34(4) + 14(2) = 3 +

142. The latter exceeds the former if and only if2 10.

10Note that using the folk theorem, one can support an equilibrium for traditional actions as long as 12.

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SafeImmune Susceptible Trust Immune

= 0 = 16= 8 = 12

Figure 2: Susceptible and Immune Regions in Safe/Trusting Game as a Function of

Trust Games

We next consider a family of trust games in which there exist two actions, a safeaction and

trustingaction. This family of games generalizes the familiarstag huntgame. In this family

of games, if 8 then the both players choosing the safe action is the efficient equilibrium

and if 8 both players choosing the trusting equilibrium is efficient.

Safe Trusting

Safe 16 , 16 6, 2

Trusting 2, 6 ,

In this family of games, the initial susceptible regions are as shown in Figure 2.11

Notice that the immune region for the safe action equals the efficient region, implying that

for any game in which playing the safe action is efficient, the safe action will be played. This

is decidedly not true for the trusting action. In fact, the trusting action has a relatively small

immune region. Consider the following set of games {7, 9, 10, 11, 14}. If game = 7 occurs

first, then the only possible sequence that obtains the efficient outcomes in all remaining

games is (7, 14, 11, 10, 9). In contrast, for the set of games {9, 7, 6, 5, 2}, if game = 9

appears first, any sequencing of the remaining games results in efficient outcomes. Trust,

therefore, will be relatively harder to produce. This occurs because playing safe (in the stag

11To solve for the boundary of the immune region for the trusting action, choose so that the trustingstrategy receives a higher payoff even if three fourths of the individuals play the safe action. Formally, set so that 34(16 2) +

14(6)

34(2) +

14(B). Solving gives the threshold at = 12. A similar calculation gives

that the threshold for the safe action as = 8.

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hunt, going for the rabbit) is risk dominant and generally speaking learning advantages risk

dominant institutions (Samuelson 1997). To ensure the trusting outcome in a game with

a value of near eight, there must have been a previous game that produced trust thats

closer to it than a game in which people played safely. Given the parameterization of this

example, trust must be built, but playing safe need not be. Playing safe will always occur

when doing so is efficient.

3 Results: One-Dimensional Coordination and Trust

Games

We now state general results for a family of games that includes coordination games and

trust games. Each game has two actions, which we denote by A and B. These games can

be indexed by a single one-dimensional parameter. And for any value of that parameter,

the payoff maximizing equilibrium will either be for both players to choose Aor for both to

choose B. Our results will rely on the concepts and intuitions developed in the examples.

We make the following formal assumptions.

Assumption 1 There exists a family of symmetric two by two games indexed by a one

dimensional real valued parameter: G() with [L, U] with two pure strategies denoted

by A and B. Both players choosing the same strategy is payoff maximizing for al l , with

both choosingA payoff maximizing atL and both choosingB payoff maximizing atU

Assumption 2The payoff to playingB increases in and the payoff to playingA decreases

in . These marginal effects increase in magnitude when the other individual chooses the

same action.12

12Formally, this can be written as BB ()

() > BA()

() and AA()

() < AB()

() , whereij() equals the payoff

to an individual playingi whose opponent plays j.

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Assumptions 1 and 2 together implies that there exists an = such that for any game

=, always choosing A is the payoff maximizing equilibrium strategy, and for any

> =, always choosing B is the payoff maximizing equilibrium strategy. To simplify the

presentation, we define A() and B() to denote the boundaries of the initial susceptible

region. Thus, strategy A is immune for any game with < A() and strategy B is immune

for any game with > B(). If there exists no immune region for strategy A (resp. B)

then we set A =L (resp. B =U).

We first state a claim that the size of the initial susceptible region increases in the size

of the spillover. This claim will prove useful in characterizing the relative levels of path

dependence. The result implies that the stronger the spillover, the more likely that the

individuals will choose the inefficient equilibrium. The proof of this and all claims are in the

appendix.

Claim 1. The size of the susceptible region weakly increases in , i.e. A() (resp. B())

weakly decreases (increases) in

We next state a lemma that will prove useful in deriving and understanding many of our

subsequent results. The lemma states that at the end of any sequence of games, there will

exist a threshold Tsuch that in the next game, the strategy A will be played if < T and

B will be played if > T. In what follows, we will refer to the equilibrium strategy that

emerges from a game as the outcome.

Lemma 1. In each epoch t > 1, there exists a threshold Tt such that if t < Tt, A will be

the outcome and ift > Tt, B will be the outcome .

In the proof of the lemma that the threshold equals the average of the largest that

produces an outcome ofA and the smallest that produces an outcome ofB , provided that

the average lies in the susceptible region. This fact will become important in the subsequent

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discussion of our results. Notice also that the threshold will depend on both the spillover

parameter and the first game. We now state a corollary that states that given a context

where the outcome in game 1 is A. For any continuation, the threshold will be increasing

in both the size of the behavioral spillover and 1.

Corollary 1. Given = {, (1)}, where 1 < =, for any continuation (2, 3. , , , k), the

threshold at timek, Tk weakly increases in both and1.

Path Dependence and Initial Game Dependence

We now demonstrate how the extent of path dependence depends on parameters of the modeland how levels of path dependence depend on the context in which a continuation occurs.

We first state a sufficient condition for institutional path dependence to exist.

Claim 2. Any set of games that contains at least one susceptible game and two games with

different efficient equilibrium outcomes exhibits path dependence.

The claim has a straightforward corollary.

Corollary 2. For any set of games that contains at least one susceptible game and two

games with distinct efficient outcomes, for either outcome there exists an ordering of the

games such that all susceptible games produce that outcome.

Intuitively, increasing the size of the susceptible region should produce more path depen-

dence. As an example, suppose that lies in the interval [0, 9] and that the boundaries of the

susceptible region are A = 2 and B = 7. Consider the following set of four s: {1, 3, 6, 8}

and the following two sequences: (1, 6, 8, 3) and (1, 8, 6, 3). In the first sequence, 1 = 1, and

the outcome will beA. This moves the threshold T2 to 7, and therefore the outcome for the

game 2 = 6 will also be A. The outcome for the third game will be B, and this will move

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set T4 = 7, the average of 5, the largest with an outcome ofA and 8, the smallest with

an outcome ofB .

Next, consider the sequence (1, 8, 6, 3). The first outcome will be A and the threshold

will become 7. The second outcome will be B, so the new threshold will be 4.5, the average

of 1 and 8. The third outcome will then be B as well and the new threshold will be 3.5, the

average of 1 and 6. Thus, the final outcome will also be B. By switching the second and

third games, the outcomes of two games switch from A to B. In contrast, suppose that we

change the boundaries of the susceptible region to A = 4 and B = 5, none of the outcomes

depend on the path

However, this intuition doesnt hold in all cases. Our next claims links the size of the

susceptible region to the extent of path dependence. First, we show that a larger susceptible

region need not imply greater path dependence

Claim 3. There exist outcome equivalent contexts and such that the susceptible region

for contains the susceptible region or, such that context does not exhibit greater path

dependence

We next show that the intuition that a larger susceptible region implies greater path

dependence in the case where there has existed at least one outcome of each type in both

contexts

Claim 4. Given any two next outcome equivalent contexts andthat include one outcome

of each type, if the susceptible region for contains the susceptible region or, then context

exhibits greater path dependence than context .

The previous claim implies that when deciding between two games with the same efficient

equilibrium, that choosing a game with clearer incentives, i.e a further from the threshold,

will result in greater future path dependence. We state this as a corollary.

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Corollary 3. Given any context, and two games that produce the same outcome in that

context, choosing the game further from the threshold results in context with greater path

dependence than choosing the game closer to the threshold.

Given Claim 1 and Claim 4, it follows that the degree of path dependence is determined

by once there has existed a game that produces each outcome. However, in the limit

as approaches one the susceptible region will approach the entire space. In those cases,

whichever strategy is played first will then be played in all games. So there will be sensitivity

to the initial game, but no dependence at all on the subsequent path. Recall that the extent

of initial game dependence equals the probability that the a game on a continuation will

have the same outcome as the first game in the context. More generally, the extent of initial

game dependence increases strictly increases in the level of behavioral spillovers.

Claim 5. The extent of initial game dependence strictly increases inand approaches one

as the level of behavioral spillovers approaches one.

The extent of path dependence as a function of the size of the susceptible region (and

implicitly) can be seen in numerical simulations. In the two figures that follow, we set the

parameterto lie in the interval [0, 1] and set= = 0.5.In the first figure, we letA = 0.2 and

B = 0.8. We then randomly drew sequences of twenty games from a uniform distribution

over [0, 1]. In figure 2 below, we report the final threshold at the end of twenty epochs from

200 numerical simulations.13 As can be seen in the figure 2, approximately one quarter of

the final thresholds wind up at the boundary of the immune regions but a majority end up

in the susceptible region.

In figure 3, we enlarge the susceptible region. We let A = 0.05, B = 0.95. Now, nearly

half of all runs result in final thresholds at either 0.05 or 0.95. Thus, the larger susceptible

region results in final thresholds that depend more on the initial game, less on the path,

13The simulations were written in Excel and can be duplicated easily

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Figure 3: Distribution of Threshold After Twenty Uniformly Drawn Games in [0, 1] withA = 0.2, B = 0.8, = = 0.5. (200 runs)

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(#

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# #*" #*% #*& #*' #*( #*) #*+ #*, #*- "

Figure 4: Distribution of Threshold After Twenty Uniformly Drawn Games in [0, 1] withA = 0.05, B = 0.95, = = 0.5.(200 runs)

and tend to be more extreme, and therefore less efficient. Thus, quite counter to intuition,

evidence of path dependence could correlate positively with efficiency and not negatively.

This is because if the outcomes depend on the path, it means that both types of equilibria

are still in play. If it does not depend on the path, then the system has locked into one

equilibrium for almost all games.

Efficient Paths

We next consider efficiency. We first show that for some sets of games there may not exist a

sequence that produces efficient outcomes in each game. We say that a sequencing isefficient

if every game produces an efficient outcome.

Claim 6. There exists sets of games such that for any sequencing of the games, the equilib-

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rium selected will not be efficient in at least one game.

We next derive a necessary and sufficient condition for a set of games to have a sequence

that produces the efficient outcome in each game. Given any set of games {1, 2, ...N},

we can reorder the s from smallest to largest. We can then relabel thoses that are less

than = by 1 to R, where j < j+1. Similarly, we can relabel the s that are greater

that = by1 throughMwhere i > i+1. We next define the other equilibrium index. For

a susceptible game j, this equals the number of games labelled which are further from

j than its distance from j1. For an immune game, the index equals M. Imagine the

games being introduced in increasing order 1, then 2 and so on. The other equilibrium

index for the game j is how many of the games that could already exist in the sequence

and still have the game j produce the efficient equilibrium. Formally, we write this as:

I(j) = max i s.t. (i j)> (j j1) ifj > A

= M otherwise

I(i) = max j s.t. (i j)> (i1 i) ifi < B

= R otherwise

Note that the sequences of indexes need not be increasing. Suppose that= = 50,A = 5

and B = 95. Consider the set of games {1, 3, 6, 30, 31, 44, 45, 52, 96}. The indices of the

games will be as follows I(1) =I(3) =I(6) = 2. I(30) = 1, I(31) = 2, I(44) = 1, I(45) = 2,

I(52) = 3, and I(96) = 7. We now state a necessary and sufficient condition for there to

exist an efficient sequencing.

In what follows, we say that a game j (i) exceeds balanced sequencing if j > I(j)

(i > I(i)). Notice that if no game exceeds balanced sequencing, then the games can be

introduced in an alternating sequence and the efficient outcome will occur in each game.

The logic is straightforward: if game 4 has an index of four or more, this means that games

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1 through 4 can occur ahead of it in the sequence and 4 will still produce the efficient

outcome. If game 4 exceeds balanced sequencing, say it has an index of two, then it must

occur in the sequence prior to game 3. This implies that game 3 must have an index of

at least four. We can now state a formal claim giving necessary and sufficient conditions for

the existence of an efficient sequence.

Claim 7. Given a set of games {1, 2,..R, M, M1,..1}, there exists a sequencing of

the games that produces efficient outcomes in every game if and only if the following two

conditions hold

(i) Ifj > I(j), then for anyi s.t. i > I(j), I(i) j.

(ii) If i > I(i), then for anyj s..t. j > I(i), I(j) i.

This claim implies that if an efficient sequencing exists that one way to produce it is to

begin from the most diverse institutions (as determined by ) and to work inward. Thus,

societies that experience more diverse sets of institutions in earlier epochs may perform

better.

That insight holds more generally. Suppose that the conditions of the claim and that

there does not exist an efficient outcome. A natural question to ask is what is true of

sequences that lead to more efficient outcomes. In figure 2, these are the sequences that

produced thresholds near one half, or more precisely, what is true of sequences that are more

likely to produce the correct outcome. We can then state the following claim.

Claim 8. Given any set of games relabeled as 1 < 2.. < R < = < M < M1.. < 1,

any game sequence in which there exists aj > j wherej (resp. j) appears prior to anj

(resp. j) produces inefficient outcomes in at least as many games as an alternative sequence

in which gamej appears beforej (resp i appears beforej).

We next describe an example that clarifies the logical argument. Let = = 10 and let

A = 6 and B = 14. Consider the following sequence of s: (18,8,16,4). To define the

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outcomes, we list a sequence of outcome threshold pairs. In the first epoch, the outcome

will be B and the new threshold will be 4. We write this as (18B, 6), we place an asterisk

to denote that the outcome is efficient. We obtain the following sequence of outcomes

and thresholds (8B, 6), (16B, 6), and (4A, 6) Thus, two of the games produce inefficient

outcomes. To improve the sequence identify twos that are in the incorrect order: 8 and

4. Apply the following algorithm: First, move any remaining games that produceB as an

outcome ahead of the game with = 8. In the example, this is the game with = 16. This

move has no effect on the game with = 8, which was already producing the inefficient

outcome. Nor can it have any effect on the game with = 16, which was already producing

B. Next, switch the game with = 8 and the game with = 4, which is immune. After the

switch, the new sequence of outcomes and thresholds will be as follows: (18B, 6) (16B, 6),

(4A, 10), and (8A, 12).

Linking Path Dependence and Efficiency

Our results suggests a subtle logical linkage between path dependence and efficiency. On

the one hand, increasing results in more path dependency and the potential for much

worse outcomes. Based on this logic, people often equate path dependency with inefficiency.

However, we also show that when choosing games, one would like to choose games that have

clearer incentives, i.e. are further from the threshold. But as we showed in Corollary 2, these

choices imply greater future path dependence. In other words, choosing institutions with

clearer incentives earlier in the sequence limits the extent of the behavioral spillovers resulting

in greater efficiency. However, doing so also makes the future more path dependent becausethose institutions that have moderate incentives could still produce either equilibrium. In

contrast, if a game with weak incentives, a game with a in the middle of the susceptible

regions was chosen first, the result will be less path dependence in the future, but also less

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efficiency.

Therefore, our model suggests that contexts thatexogenouslyexhibit greater path depen-

dency should be avoided, but once placed in a context, one should endogenouslymaintaing

the potential for path dependence in order to keep paths to efficiency open.

4 Endogenous Institutional Change

Greif and Laitin (2004) describe a process of endogenous institutional change. Translating

their quasi-parameter to our model, the game index parameter, , changes over time. As

changes institutions can become reinforced or more fragile depending on whether the

endogenous change is reinforcing or degrading. One can reinterpret our framework to provide

insights into their model by considering each new epoch not as introducing a new institution

but as an endogenously changing the quasi-parameter of an existing institution.

Following their construction we will say that endogenous institutional change is degrading

if the direction of the change in is toward the outcome that was initially inefficient. In our

model, this means that the initial institution had A as an efficient outcome but that over

time, increases, and when the population continues to play A, institutional performance

becomes less efficient. We can then state the following claim.

Claim 9. A degrading quasi-parameter will produce institutional change only when the quasi-

parameter enters the immune domain for the alternative strategy.

pf. By symmetry, assume that the current outcome is A for all institutions. As increases,

the outcome will remain A until the system passes into the immune region for B.

This result implies that when institutions do change, its only because that change has

become inevitable and the institution has far outlived its effectiveness.

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We can then link this result to our first claim and state a corollary. Recall that the first

claim said that as spillovers increase, so does the size of the susceptible region.

Corollary 4. If the payoffs when individuals play distinct strategies do not depend on ,

as behavioral spillovers increase, institutional change occurs more slowly. At the point of

change, the existing institutions are less efficient than if the strategies did depend on

The corollary implies that larger spillovers implies more rigidity and less efficiency. One

can interpret the result as saying that stronger behavioral norms result in less efficient

outcomes.

5 Results for General Classes of Games

We now extend our model to consider much broader classes of games. To demonstrate

how the logic applies within a broader class of games, we first parameterize the class of

all two by two symmetric games and show how the model can be extended to that class.

We then describe a more general characterization of games and strategies. Within the

more general framework, we show that when spillovers are large, new institutions that have

weak punishments for deviation are more likely to produce efficient outcomes. We find that

creating strong incentives to choose the correct strategy by creating high equilibrium payoffs

improves the likelihood of efficient outcomes but it is not as important as not punishing too

much when individuals do not coordinate. Too much punishment favors the equilibria from

previous games.

Two by Two Symmetric Games

To show how the model can be extended to more general classes of games, we first parametrize

all two by two symmetric games as follows:

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B=0 + (1 )M

Therefore, the efficient outcome will arise in the second game if and only if (1 ) >

(1 2)M. Note that this is trivially satisfied if M equals zero or if < 12

(given that

M 0). But suppose that the off diagonal payoffs sum to a large negative number so that

M < 0. Now, it could be that A will no longer be the equilibrium outcome in the second

game. Thats more likely to be the case as increases. Thus, a really low min max payoff

lowers the possibility of getting the efficient strategy for the new game.

This intuition holds more generally. Suppose that we have an arbitrary family of gamesG = {G} with a well defined distance measure, d : G G [0, ). Also assume

that there exists a well defined distance measure between games within G. Consider the

introduction of some new game GT in the Tth epoch. When that game gets introduced,

a proportion of the population will play the strategy used in the previous game thats

closest according to the distance measure. That strategy may or may not be an equilibrium

strategy. If it is an equilibrium strategy, we will assume that it is played. If it is not, we will

assume that those individuals who do not choose the efficient strategy play minmax.

Claim 10. Let Gdenote the previous game in the sequence of games closest to GT given

d. Denote the payoff in the efficient repeated game equilibrium inGT byAT, Adenote the

payoff inGTfrom playing the equilibrium strategy used inG, and letMdenote the minmax

payoff in GT. The efficient equilibrium will be chosen in GT if and only if the following

holds:

AT > A+ (A M)(2 1)

(1 )

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pf. The payoff from playing the efficient strategy in GT equals M+ (1 )AT. The payoff

from playing the equilibrium strategy used in the G equals A + (1 )M. The first

expression is larger than the second if and only if (2 1)M+ (1 )AT > A. This can

be rewritten as (2 1)(A+M A) + (1 )AT > A. Rearranging terms gives the

result.

The claim implies that when the spillover term is large, i.e close to one, then there

are two ways to get the payoff maximizing outcome. You can either make the payoffs from

choosing the correct outcome very high, or you can make M, the minmax payoff, close to

the payoff from playing the equilibrium strategy from the nearest game. This finding echoes

some of the argumentation found in Bednar (2009) which suggests that at times weaker

punishments can improve institutional performance because they encourage experimentation.

Here, we find a similar logic holds. If a new institution creates incentives that allow for large

punishments, in the form of a small minmax payoff, then those people trying to entice the

majority to choose a new more efficient behavior will suffer the brunt of the punishment.

Mild punishments, in contrast, enable exploration. Ideally, the punishment yields a payoff

nearly as high as the payoff from playing the strategy from the nearest previous game in

the sequence. That will trivially be the case if the that previous games strategy isnt an

equilibrium.

Therefore, the model provides the following advice for building institutional incentives.

You must either create an institution within which all existing behaviors are not self re-

enforcing, i.e. equilibria, or you must not allow for much punishment. Punishment works

against innovations. This result further refines the previous result that one should choose

institutions with clear incentives by showing that institutions should be chosen that reward

correct behavior but do not punish strongly out of equilibrium.

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6 Discussion

In the paper, weve first shown that if individuals choose initial strategies based on their past

experiences, then we should expect to see path dependence in their behavior. Relatedly, the

level of inefficiency caused by this path dependence should correlate with the extent of this

behavioral spillover. Neither of these results should be especially surprising. We have then

shown that under rather mild conditions path dependence is inevitable.

We also found that large susceptible regions imply greater path dependence but much

of that path dependence may be initial game dependence. If there is almost no immune

region, then the ultimate threshold will tend to be in one of either two locations, implyingthat the choice of the first institution matters a lot in such cases. Given that, if we see path

dependence, it implies that theres contingency, that multiple behaviors are in play. Thats

generally a good thing. A lack of path dependence along with a lack of behavioral diversity

suggests lock-in and less efficiency.

Related to this finding, given a peel of spillovers, one should actually choose to maintain

path dependence in each epoch. Reducing path dependence implies choosing a game that

locks in future behavior that is often inefficient.

This last insight follows from our results on the optimal procedure for introducing institu-

tions, namely starting from both extremes and working ones way toward institutions where

outcomes are more contingent on the past. Thus, paradoxically, the way to reduce path

dependence is to keep its possibility alive for as long as possible. And the way to do that

is to introduce as many diverse behaviors as early as possible. This will not occur through

drift. Drift moves in a single direction and will only produce behavioral change when the

quasi-parameter moves into what we call the insulated region. Our model suggests that you

cannot expect behavior to right itself until the change was almost inevitable.

Relatedly, when considering a much more general class of games, we are able to show

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that weakening the ability to punish increases the likelihood of efficient outcomes. Weak-

ening punishment encourages exploration. Taken together all of these results suggest two

fundamental insights: First, if individuals learn from their past behavior, the maintaining

a diversity of strategies increases the likelihood of efficient outcomes by creating a more

a system with greater adaptive capacity. Second, the empirical relationship between path

dependence and efficiency is much more complicated than previously thought. In our frame-

work, dependence on the path implies the presence of the behavioral diversity that produces

good outcomes. Thus, paradoxically, robustness, and in this context we mean the ability to

produce good outcomes in a variety of institutional settings, may require the possibility of

contingency.

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References

Bednar, Jenna. 2009. The Robust Federation: Principles of Design. New York: Cambridge

University Press.

Bednar, J., Y. Chen, Y, X. Liu, and S.E. Page (2012) Behavioral spillovers and cognitive

load in multiple games: An experimental study. Game and Economic Behavior74:1

pp 12-31.

Bednar, Jenna. and Scott E. Page (2007) Can Game(s) Theory Explain Culture? The

Emergence of Cultural Behavior Within Multiple Games, Rationality and Society,

19(1): pp 6597.

Camerer, Colin, (2003). Behavioral Game Theory: Experiments on Strategic Interac- tion.

Princeton University Press, Princeton.

Camerer, Colin, and Tek Ho (1999) Experience-Weighted Attraction Learning in Normal

Form Games, Econometrica67, no. 4: 827-874.

Cason, Tim, Anya Savikhin and Roman Sheremeta (2012) Behavioral Spillovers in Coordi-

nation Games, European Economic Review56, pp. 233-245.

Easterly, W. (2001)The Elusive Quest for Growth: Economists Adventures and Misadven-

tures in the TropicsMIT Press. Cambridge MA.

Easterly, W. (2006) The White Mans Burden: Why the Wests Efforts to Aid the Rest

Have Done So Much Ill and So Little GoodPenguin, New York.

Englehardt, Sebastian v. and Andreas Freytag. 2013. Institutions, Culture, and Open

Source. Journal of Economic Behavior and Organization95(Nov):90110.

32


33/42

Gilboa, I. and Schmeidler, D. (1995), 11Case-based decision theory, The Quarterly Journal

of Economics , Vol. 110,, pp. 605-639.

Golman, Russell and Scott E Page (2010) Basins of Attraction and Equilibrium Selection

Under Different Learning Rules.Evolutionary Economics49-72.

Greif, Avner and David Laiton (2004) Theory of Endogenous Institutional Change.

American Political Science Review4: 633-652.

Henrich, Joseph, Robert Boyd, Samuel Bowles, Herbert Gintis, Ernst Fehr, and Colin

Camerer (eds) (2004) Foundations of Human Sociality: Economic Experiments and

Ethnographic Evidence in Fifteen Small-Scale SocietiesOxford University Press.

Hurwicz, Leonid (1994) Economic Design, Adjustment Processes, Mechanisms, and Insti-

tutions, Economic Design, Vol. 1, No. 1, August 1994, pp. 1-14.

Jehiel, P (2005) Analogy-based Expectation Equilibrium, Journal of Economic Theory

123: pp 81-104.

Kranton, R. (1996) Reciprocal Exchange: A Self-Sustaining System,American Economic

Review,86 (4), pp. 830-851.

Licht, Amir N., Chanan Goldschmidt, and Shalom H. Schwartz. 2007. Culture Rules:

The Foundations of the Rule of Law and Other Norms of Governance. Journal of

Comparative Economics35:659688.

Long, N. E. (1958) The local community as an ecology of games. American Journal ofSociologyVol. 64, No. 3, 251-261.

Maskin, E. (2007) Mechanism Design: How to Implement Social Goals: Nobel Memorial

Lecture

33


34/42

Mount, K and S. Reiter (2002) Computation and Complexity in Economic Behavior and

OrganizationCambridge University Press, Cambridge England.

Nisbett, Richard (2003) The Geography of Thought : How Asians and Westerners Think

Differently...and WhyFree Press, New York, NY.

Ostrom, Elinor (2005) Understanding Institutional Diversity The Free Press. New York,

NY.

Myerson, R. (1992) Revelation Principle, in New Palgrave Dictionary of Money and

Finance, edited by J. Eatwell, M. Milgate, and P. Newman, New York: Stockton

Press.

Nash, John F. (1951) Non-cooperative games. Annuals Mathematics54: 286295.

Ostrom, Elinor (2005)Understanding Institutional Diversity, Princeton University Press.

Page, Scott E (2006) Essay: Path DependenceQuarterly Journal of Political ScienceVol

1: 87-115

Page, Scott E. (2012) A Complexity Perspective on Institutional Design. Politics, Phi-

losophy and EconomicsVol 11: 5-25.

Putnam, Robert, Robert Leonardi and Raffaella Nannetti. 1993. Making Democracy Work:

Civic Traditions in Modern Italy

Samuelson, Larry (1997) Evolutionary Games and Equilibrium SelectionMIT Press, Cam-

bridge, MA.

Samuelson, Larry. 2001. Analogies, Adaptations, and Anomalies. Journal of Economic

Theory97:320-367.

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Appendix

Proof of Claim 1: Leti() denote the payoff if both individuals choose strategy i and let

iD denote the payoff to the individual who plays strategy i when the other player choosesthe opposite. A game is immune provided that the expected payoff from Aexceeds the payofffromB . If there exists no immune region, the result follows immediately. Therefore, assumethere exists an immune region for strategy A. Given that A() equals the boundary of theimmune region, it must satisfy the following equation:

(1 )A(A()) +AD(

A) = (1 )BD(A) +B(

A())

Which can be rewritten as

A(A()) BD(

A) =

(1 )

B(

A()) AD(A)

By construction, both individuals choosing strategy A is an equilibrium at A(). Therefore,A(

A()> BD(A). Given that the left hand side is strictly greater than zero, this implies

that B(A()) > AD(

A). Suppose then that we increase to + . This increases thecoefficient on the right hand side of the equation. Note that by assumption 2, the left handside of the equation increases as A decreases and the right hand side decreases. Therefore,it follows that A(+)< A(). A similar argument holds for B() strictly increasing in.

Proof of Lemma 1: It suffices to consider the case where 1 < =. It follows that T2

will equal B as any susceptible game produces outcome A. Until there exists a k such that

k B

, the threshold will remainB

. Therefore, assume2 < B

, thenT3= B

. If2 B

,thenT3=

1

2(1 + 2), provided that

1

2(1 + 2) lies in the interval (

A, B). If 12

(1 + 2) A,

then T3 = A, and if 1

2(1+ 2) B, then T3 = B. To determine the threshold for all

subsequent periods, leta equal largest k for k < t that produces outcome A let b be the

smallest k for k < t that produces outcome B. The threshold will equal the average ofa

and b provided that lies in the susceptible region. Otherwise, it will equal whichever ofA

or B is closest to that average.

Proof of Claim 2: It suffices to show that the outcome in the first two games in a sequencecan differ. Let S denote the susceptible game. Without loss of generality assume that Ais payoff maximizing in the susceptible game. Let o denote a game in which B is payoff

maximizing. By assumption, such a game must exist in the set. Consider first the sequencethat begins with gameo followed byS. The outcomes will beB in both games. In contrast,consider the sequence that begins with game S followed by o. The outcome in the firstgame will be A. The outcome in the second game will be A ifo < B and B otherwise. Ineither case, the outcome in at least on game depends on the order of the games.

Proof of Corollary 1: Assume


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At () denote the largest i for i= 1 to t that produces the outcome Agiven , and Bt ()

denote the smallest i fori = 1 to t that produces the outcomeB given. If there exists noi that produces outcome B , set

Bt () = . The proof relies on induction. By assumption

Bt (): By construction, Tt+1() = Tt() and Tt+1() =

Tt(), so inequality (i) holds. Inequalities (ii) and (iii) hold because jt+1() =

jt () and

jt+1() =jt () for j=A, B.

Case 2: At () < t+1 < Tt(): First, consider the case where Bt () = . In this case,

Tt+1() =Tt() =B

() (Recall thatB

() denotes the boundary for the immune region forB.) Therefore, by construction, Tt+1()

B() Tt+1(). The other two inequalities holdtrivially. Therefore, we restrict attention to the case where Bt () < . By the inductionhypothesis,Tt() Tt(), therefore the outcome in the game t is A for both spillover rates.Therefore,Bt () =

Bt+1() and

Bt () =

Bt+1() so (iii) holds. To see that inequality (ii)

holds, note first that by assumption At+1() =t+1. There are two possibilities to consider.First, ift+1 <

At , then (ii) holds strictly. Otherwise, t+1 =

At () and

At () =

At (),

so (ii) holds weakly.. To show that (i) holds, we first solve for Tt+1():

Tt+1() = t+1+

Bt ()

2

To solve for Tt+1(), let = max{t+1, At ()}

Tt+1() = +Bt ()

2

By the induction assumption,Bt Bt () and by construction,

t+1, which completesthis case.

Case 3: Tt() < t+1 < Bt (): First, consider the case where t+1 < Tt(), then the

outcome is B for andA for and all three inequalities hold via straightforward arguments.Ift+1 Tt(), then the outcome is B for both and . By assumption

Bt+1= t+1 and

Tt+1() = At () +t+1

2

To solve for Tt+1(), let = min {t+1,

Bt ()}

Tt+1() = At () +

2

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By the induction assumption, At At () and by construction,

t+1, which completesthis case and the proof.

Proof of Claim 3: The proof is by counterexample using our first example of traditional and

innovative strategies. Assume that context = {0.8, (0.1, 0.9)} Let context = {0.75, ()},the initial context that we covered in the example. Initially, the threshold in both contexts

equals eight, i.e. T = T = 8. Given this construction, the susceptible region in context will be larger than the susceptible region for context . The continuation (0.1), will have noeffect on the context , but it will move the threshold in context , T to 10. Therefore, cannot be more path dependent.

Proof of Claim 4: By assumption of next outcome equivalence, the thresholds for the twocontexts are the same. We denote these by T and T. Let a equal the largest k in context that produces outcome A. Define a similarly for context . Similarly, let b equal thesmallest k in context that produces outcome B. The interval [L, U] can be partitioned

into six intervals [L, a

) , [a

,a

), [a

, T), [T,b

), [b

, b

), and [b

), U].Without loss of generality, assume that for the next game that arises < T so that theoutcome in the game will be A. We first state a lemma that simplifies the remainder of theproof.

Lemma 2. The introduction of the first new game movesT, the threshold in context, atleast as far as it moveT, the threshold in context.

The proof considers distinct cases. Note that both context has produced a B outcome,then so has . If [L,

a), then neither threshold moves so the result holds. If [a,a),then only Tmoves, so the result holds. Finally, if [a, T), then the thresholds move to+b

2

and +b

2

in contexts and respectively. Given that b b, the result follows.

Given the lemma, it follows that after the introduction of the game , the set of games willnow have different outcomes is larger in context that in context . This follows becauseof the fact that the threshold has moved further in context . Therefore, after one game,context producesmorepathdependencethan context . We make the following observationwhich plays an important role in the remainder of the paper.

Observation 1. Following any continuation, the susceptible region of is at least as largeas the susceptible region of.

We now state another lemma that generalizes the previous lemma.

Lemma 3. If contexts and have both produced both types of outcomes and if has alarger susceptible region, then any new game will move theTat least as far as it movesT

pf. Assume that in the new game < T. The previous lemma covers the case where thethreshold are the same. Suppose first that T T. If [L, ), then Tdoes not change, sothe result holds. If the interval [a, T) is not empty and contains then the thresholds move

to +b

2 and +

b

2 in contexts and respectively. Given that b b, the result follows.

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Next suppose that T T. As before, if If [L,a), then T does not change as

before, so the result again holds. If [a, T) then the thresholds move to +b

2 and +

b

2 in

contexts and respectively. Given that b b, the result follows. Finally, suppose that [T , T). Now the outcomes in the two contexts differ. The outcome in context is A but

the outcome in context is B. The thresholds therefore move to +b

2 and +

a

2 in contexts

and respectively. In context , the threshold moves a distance 12

( a). In context

, the threshold moves a distance 12

( b). Given that T is the midpoint ofa and b, the

result follows from the fact that | b | , and second, that there exists a continuation that produces a differentoutcome give but not given . It suffices to show for the case where the first outcome isA. In any continuation all outcomes are A off and only ifi <

B(), the boundary of theimmune region for B given . The result follows from the fact that that B() > B().Next to show that there exists a continuation that produces an outcome ofB for some gameunder but not under , consider the single game continuation, 2 (

B(), B()). It hasoutcome B in the context defined by and outcome Ain the context defined by .

The proof that in the limit asapproaches one, that the extent of initial game dependenceconverges to one, follows directly from Assumptions 1 and 2.

Proof of Claim 6: To simplify notation, we write B() as B and define A similarly.

Choose 1 in the interval (A, =) and 2 in the interval (=, B). By construction, bothare susceptible. In the sequencing (1, 2), A will be chosen in both games, resulting in aninefficient outcome in game 2. In the sequencing (2, 1), B will be chosen in both games,resulting in an inefficient outcome in game 1.

Proof of Claim 7: We first prove that the condition is sufficient. First, suppose nogames exceed balanced sequencing. It suffices to consider the case where R < M. Then thesequence, (1, 1, 2, 2, ...R, R, ....M) results in efficient outcomes for each game. In whatfollows, we refer to this as the alternating sequence. When game j occurs in the sequence

j 1 of the games have been added to the sequence. By assumption j 1 < I(j),

which implies that the efficient outcome occurs in gamej. Similarly, when i occurs in thesequencei of the games have been added to the sequence. By assumptioni I(i), whichimplies that the efficient outcome occurs in game i.

Next assume that there exists games that exceed balanced sequencing. Let I(j) havethe smallest value among the s that exceed balanced sequencing and let I(i) have thesmallest value among s that do. Note first that I(j) cannot equal I(i). If it did, giventhat i > I(i) =I(j), which by condition (1) implies that I(i) j

, but I(i) =I(j)

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By the previous lemma, given that game j produces outcome B , only games labelled as can produce the inefficient outcome in either sequence. Furthermore, at that point in thesequence where there exist previous games in the sequence labelled as that produce bothoutcomes, subsequent thresholds are determined entirely by the outcomes in games labelled

ass. Formally, the threshold will be the average of the largest that produces an outcomeofA and the smallest that produces an outcome ofB .

Consider the game that occurs immediately after j in the new sequence, call this game. The threshold at the time that this game arises in the original sequence of games equalsthe average of the largestkcorresponding to an earlier game that produced outcome A andthe largest k corresponding to an earlier game that produced an outcome B. Note firstthat by assumption, the outcomes in games i and j are unchanged in the two sequences.Note next that all other games labelled as that occur prior tohave identical prior gamesin the two sequences, so their outcomes must also be unchanged. Therefore, the thresholdin effect at the time that arises is the same for the two sequences. By induction, all

remaining games also face the same thresholds.Last, suppose that the outcome in game j becomes A in the new sequence. This is theefficient outcome. Consider the game, that follows the game j in the new sequence.If its threshold is unchanged, then there exists a game mthat occurs prior to j in bothsequences, that is labelled as an , produces outcome B, and satisfies m j. If suchan m exists, then by the argument in the previous paragraph, outcomes in and allsubsequent games are unchanged. Suppose that no such m exists. It follows that in theoriginal sequence j was the smallest producing an outcome of B. This implies that thethreshold for game tau will be higher in the new sequence than the original sequence. Aninductive argument shows that the same result holds for all subsequent games in the newsequence. Therefore, at least as many games in the new sequence must produce the efficient

outcome, thus completing the proof.

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