large nonanonymous repeated games

Ž .Games and Economic Behavior 37, 26�39 2001doi:10.1006�game.2000.0826, available online at http:��www.idealibrary.com on

Large Nonanonymous Repeated Games

Nabil I. Al-Najjar

Department of Managerial Economics and Decision Sciences, Kellogg School of Management,Northwestern Uni�ersity, E�anston, Illinois 60208

E-mail: [email protected]

and

Rann Smorodinsky

Da�idson Faculty of Industrial Engineering and Management, Technion,Haifa 32000, Israel

E-mail: [email protected]

Received December 30, 1998; published online August 28, 2001

Ž . ŽE. J. Green 1980, J. Econ. Theory 22, 155�182 and H. Sabourian 1990, J..Econ. Theory 51, 92�110 studied repeated games where a player’s payoff depends

on his actions and an anonymous aggregate outcome, and show that long-runplayers behave myopically in any equilibrium of such games. In this paper weextend these results to games where the aggregate outcome is not necessarily ananonymous function of players’ actions, and where players’ strategies may dependnonanonymously on signals of other players’ behavior. Our argument also providesa conceptually simpler proof of Green and Sabourian’s results, showing how theiranalysis is driven by general bounds on the number of pivotal players in a game.� 2001 Academic Press

1. INTRODUCTION

Long-term strategic interactions among a large number of players arecharacterized by two competing forces. Since players in a repeated settingconsider the long-term consequences of their actions, their behavior maysubstantially differ from the one predicted by short-term, myopic consider-ations. On the other hand, a player facing a large number of opponentsmay judge his impact on future play to be too small to offset the loss fromnot taking a short-term optimal action.

260899-8256�01 $35.00Copyright � 2001 by Academic PressAll rights of reproduction in any form reserved.

LARGE NONANONYMOUS REPEATED GAMES 27

Ž .Green 1980 studied the effects of these two competing forces in anattempt to provide adequate foundation for competitive behavior in mar-kets where firms interact strategically in a repeated setting. SabourianŽ .1990 extended Green’s model in several important directions. In particu-lar, Sabourian drops Green’s restriction to trigger strategies and reliessolely on Green’s continuity assumption. Both papers consider an anony-mous repeated game with random outcomes, where a player’s stage gamepayoff may depend on his own action as well as an aggregate outcome thatmay be influenced by other players’ actions. Sabourian proves that if themapping from the action profile to outcomes is anonymous and continuous

Ž .in an appropriate sense, then in any subgame perfect equilibrium of thegame, players play their myopic �-best reply at each node of the game.

In this paper we extend Green’s and Sabourian’s results to the casewhere outcomes are not necessarily anonymous functions of players’actions, and where players may condition on individualized signals of theiropponents not reflected in the collective outcome. In addition, we providea conceptually simpler proof, which clarifies the intuition underlying Greenand Sabourian’s analysis, and a potentially useful tool for carrying it out innew contexts.

Specifically, we consider a setting in which players observe noisy signalsof their opponents’ actions, as well as a collective outcome that is anarbitrary function of the signal realizations. Players’ payoffs depend ontheir own actions and the collective outcome, but their strategies may alsodepend on their opponents’ signals. We establish an analogue of Greenand Sabourian’s result that most players play non-strategically. We do thisusing a notion of influence that measures the impact of a unilateral changeof one agent’s signal on the expected value of the collective outcome. Fora given threshold � � 0, a player is �-pivotal if his influence exceeds � .

ŽWe use a nonpivotalness result established in earlier work Al-Najjar and.Smorodinsky, 2000 to conclude that the number of players who are

�-pivotal relative to the collective outcome in a given period is bounded bysome integer K �. This bound holds uniformly over all strategy profiles, and�

independently of the total number of players in the game.Our version of Green and Sabourian’s result then follows from this

nonpivotalness theorem: for any equilibrium and at any node of therepeated game, no more than K � players are �-pivotal; i.e., they believe�

their actions have important enough impact on the common outcome tooffset the cost of deviating from taking an approximate myopic bestresponse. Thus, in any equilibrium and at any node of the repeated game,the remaining N � K � players play approximate one-shot best responses.�

This, in essense, is Green and Sabourian’s point, except that their conclu-sion that ‘‘all players’’ play myopic �-best reply is slightly weakened to ‘‘allbut a vanishing fraction of players.’’

AL-NAJJAR AND SMORODINSKY28

Aside from its simplicity, this argument also shows that Green andSabourian’s result holds when the payoff-relevant outcome dependnonanonymously on the players’ signals, or when players condition theirbehavior on signals of their opponents’ actions. The key feature needed topreserve their result is that a player’s signals affect other players’ payoffsonly indirectly through the common outcome. The relevance of thisextension may be seen in the competitive industry example that motivatedGreen’s original work. It is quite natural to allow for the possibility thatfirms have considerable, disaggregated information about the behavior ofother firms in their industry. Our model allows firms to condition in anonanonymous manner on such signals, provided they affect payoffs onlyindirectly through common outcomes such as market price, state of de-mand and technology, and so on. While Green�Sabourian analysis rulesout the possibility of firms conditioning their actions on payoff-irrelevantsignals of their competitors, we show that it is an implication of equilib-rium that this added flexibility has essentially no effect on the behavior ofmost players. Thus, Green and Sabourian’s requirements that firms condi-tion only on a common outcome, and that this outcome depends only onthe players’ frequencies of actions, are unnecessary.

The paper proceeds as follows. Section 2 introduces the repeated gamemodel, which is essentially that in Sabourian. Section 3 contains the mainresult for the special case of a finite outcome set, as well as an extension toa model with a large outcome space. Specifically, we define a family ofrepeated games for which the number of players who play strategically isbounded by an integer that holds uniformly over all games in that family.Finally, Section 4 contains concluding remarks which relate our results tothe literature and suggests directions for future work.

2. MODEL

The model we study is an adaptation of a model introduced by GreenŽ . Ž .1980 and later studied by Sabourian 1990 . In the original model theaction spaces and the outcome spaces are continuous. We on the otherhand assume that all that the action spaces are finite. As for the outcomespace, we provide results for the finite case as well as for the continuouscase. We begin by introducing the stage game and move later to therepeated game.

2.1. Stage Game

Ž .A stage game H is defined as the tuple H � KK, A, S, C, X, F, � .

� � 4KK � 1, 2, . . . , K is the set of players;


�K k kA � Ł A , where A is a finite set of actions available to play-k�1

er k;�

K k kS � Ł S , where S is a finite set of signals of player k ’s actions;k�1

�k K k k k� 4 Ž .C � C , where C : A � � S is a mapping from actions tok�1

probability distributions over signals1;�

2X is a finite set of aggregate outcomes ;� Ž .F: S � � X maps signal profiles into outcome distributions.�

k K k k� 4� � � , where � : A � X � � denotes player k ’s payoffk�1function.

Note that a player’s payoff depends on the actions taken by his oppo-nents only via the outcome x.

Ž . k Ž k .A player’s mixed strategy is a probability distribution � � � A . Weshall often abuse notation and write ak to denote the Dirac measure on

k Ž 1 2 k . Ž .a . Let � � � , � , . . . , � � � A denote the strategy profile. As usualwe denote by ��k the strategy profile of all players but player k. Astrategy profile � induces a measure over A � S � X, denoted � . Player�

� Ž k .�k ’s expected payoff from the strategy profile � is E � a , x .��

k Ž . �kDEFINITION 1. A strategy � is a best response BR to � ifk k k kE � a , x � E � a , x � 0 �� A . 1Ž . Ž . Ž . Ž .� �k �kŽ� , � . �

Ž .It is an �-BR if the right-hand side of 1 is replaced with an � � 0.

DEFINITION 2. Let J be a nonnegative integer and let � � 0. AŽ 1 k . Ž . kstrategy profile � � � , . . . , � is called a J, � -equilibrium for H is �

is not an �-BR for at most J players.

Ž .In other words, in a J, � -equilibrium, all players but at most J, playŽ .their �-BR, to their opponents’ strategy profile. A 0, 0 -equilibrium is

Ž .commonly known as a stage game Nash equilibrium.

2.2. Repeated Game Ž 1 K .Given a stage game H, let H � H , . . . , denote the infinitely

repeated game with discount factor k for player k.A pure strategy for player k is an assignment of an action ak at time t,

which may depend on players’ signals in the previous t � 1 stages.3 A

1 Ž .For any finite set Y we denote by � Y the set of all probability distributions on Y.2 Section 3.2 extends the model to large outcome spaces.3 In our model F is deterministic. Therefore we did not formally allow for strategies to

depend on past outcomes, as this is redundant given the dependence on past signals.However, one can easily extend our model to the case where F may be random and allow forstrategies to depend on past outcomes as well.


behavior strategy, �k, for player k is an assignment of a stage game mixedstrategy instead.4 Formally,

k t k 5� : S � � A .Ž .�

t�0

Note that players are assumed not to observe opponents’ strategies or evenstage game actions.

� k4KThe strategy profile � � � induces a probability distribution overk�1Ž . A � S � X , denoted � . We use � to denote the marginal over S .� �

Let � and � denote the corresponding marginals over the t th coor-�, t �, t

dinate. The expected stage game payoff at time t to player k is therefore� Ž k .�E � a , x . The total expected payoff to player k from a strategy�� , t

Ž 1 k . kŽ . Ž k . t � Ž k .�profile � � � , . . . , � is U � � Ý E � a , x .t�0 �� , t

A Nash equilibrium of the repeated game, H, is a strategy profile1 k k k ˆ k �kŽ . Ž . Ž .� � � , . . . , � satisfying U � U � , � for any k � K and any

ˆ kbehavior strategy � of player k.

2.3. A Uniform Family of Games

In this paper we prove results that hold uniformly for families of gamesparameterized by Q � , � 1, M � , � � 0 as

� k � k � �HH Q, , M , � � H : � � Q, � �k , X � M ,�Ž .

C k ak � �� Sk �k and all ak � Ak ,4Ž . Ž .

� Ž k . Ž k . kwhere � S � S denotes the set of all probability distributions on Swhich assign probability at least � to any sk � Sk.

In words, a repeated game H in such a family is one where stage payoffscannot exceed Q � , players’ discount factors are bounded from above by � 1, the cardinality of the outcome set is bounded by M � , andprobabilities of signals are bounded away from zero in all games, where Q, , M, � are uniform over the entire family.

4 In this paper we restrict attention to behavior strategies. This restriction, from theseemingly more general mixed strategies, defined as probability distributions over the set of

Ž .all pure strategies, is without loss of generality see Kuhn, 1953; and Aumann, 1964 .5 With some expositional complications the model and results of this paper can be

extended to the case where a player’s strategy may depend on his own past actions.


Ž .Note that for any H � HH Q, , M, � , any strategy profile � assigns a t 6positive probability to every finite history s � � S .t�1

3. RESULTS

We begin this section with our main result which refers to the modelintroduced in the previous section. We then show how to extend this resultto the case of a large outcome space, and finally we make a briefcomparison with Sabourian’s results.

k tFor a behavior strategy � and a finite history s � � S of length t,t�1k 7 Ž .we denote by � the continuation strategy after the history s. A J, � -s

myopic equilibrium of H is a strategy profile � such that for any finitethistory s � S , which has positive probability according to � , the stage�

Ž . Ž .game strategy profile � � is a J, � -equilibrium of H.s

3.1. Main Result

The following theorem states that in a Nash equilibrium of any game Ž .H � HH Q, , M, � most players play myopically.

THEOREM 1. For any Q, , M, � and any � � 0, there exists an integerŽ . Ž .J � J Q, , M, � , � such that if H � HH Q, , M, � then any Nash equilib-

Ž . rium of H is a J, � -myopic equilibrium of H .

The sense in which ‘‘most players play myopically’’ is formalized in thetheorem by asserting the existence of an upper bound, independent of K ,for the number of players who do not play their �-BR of the stage game.

Theorem 1 follows from the next lemma regarding influence. Let �k bea behavior strategy for k. Denote by �k the behavior strategy which takesŽa.the pure action a � Ak at stage 0, and otherwise follows �k. We denote byŽ �k k .� , � the profile of strategies where all players j � k play theŽa.

strategy � j and player k plays �k .Ža.

LEMMA 1. For any Q, , M, � and any � � 0, there exists an integerˆˆ ˆ Ž . Ž .J � J Q, , M, � , � such that if H � HH Q, , M, � and � is an arbitraryˆ

ˆprofile of beha�ior strategies, then for any stage t there exists a subset K of attˆ ˆ kmost J players such that for any k � K and pair of actions a, b � A ,t

k k kE � d , x � E � d , x � � �d � A .Ž . Ž . ˆ� ��k k �k kŽ� , � . , t Ž� , � . , tŽa. Žb.

6 Consequently, any Nash equilibrium is also a subgame perfect equilibrium. However, hadwe considered the slightly more general model where players’ strategies may depend on ownpast signals, we would have needed to differentiate between the two solution concepts.

7 k k tŽ . Ž . ˜Namely, Ý s � Ý ss , for any finite history, s � � S , of length t and where˜ ˜ ˜s t�0 tŽ . ˜ss � � S denotes a new history of length t � t generated by concatenation.˜ t�0


ˆIn other words, Lemma 1 asserts that for all but J players a change ofactions at the beginning of the game can only have a limited effect on theirexpected payoff at stage t, as long as the action at stage t is leftunchanged.

The proof of Lemma 1, which builds on a result from an earlier workŽ .Al-Najjar and Smorodinsky, 2000 , is postponed to the Appendix.

Ž .Proof of Theorem 1. If � is a Nash equilibrium for H � HH Q, , M, � t Žthen � is also a Nash equilibrium for H , for any s � S remember thats

.any history is assigned positive probability according to � . Therefore, in�

order to prove Theorem 1 it is sufficient to show that for any Nashequilibrium the induced stage game strategy for the first stage of the game

Ž .is a J, � -equilibrium. In other words, it is sufficient to prove thatŽ . � kŽ .4K Ž .� � � � � is a J, � -equilibrium of the stage game H.k�1This follows from two observations. First, as the discount factors and

stage payoffs are uniformly bounded by and Q respectively one can findŽ .a stage T � T Q, , � , independent of K , such that the discounted sum

� �� of stage payoffs after T lies in the interval � , for any sequence of4 4

action tuples taken by the players. Thus, a change in action taken by playerk at the initial stage of the game cannot change his accumulated dis-counted payoff from time T on by more than ��2.

ˆSecond, by Lemma 1 there are only J players for which a change ofaction at stage 0 can increase their payoff at any stage t by more than �̂

� ˆ� . Therefore we conclude that at most J � TJ players can increase2T

their accumulated payoff between stages 1 and T , by more than ��2, by achange of action at stage 0.

We conclude that for all but J players, a change of action at stage 0 willnot change the total discounted sum from stage 1 on by more than � . As

k �k kŽ .� is a best response to � we learn that the stage game strategy � ��k Ž .must be an �-BR to � � for all but J players. This implies that

� kŽ .4K Ž .� � is a J, � -equilibrium for H. �k�1

3.2. Large Outcome Spaces

In this section we extend the results of the model to the case of a large� . 8outcome space, X. Specifically, we shall assume X � 0, 1 . Theorem 1

8 The use of the half open interval is without loss of generality, and is done for notationalconvenience, as becomes clear in the proof of Theorem 2.


does not hold in this case without additional assumptions, as shown in thefollowing example:

k k � 4 kŽ k .EXAMPLE 1. Let A � S � 0, 1 , and let C a assign probability1 � � to sk � ak and � to sk � 1 � ak. Let F: S � X be any one-to-one

1 k1 K kŽ Ž . .function e.g., F s , . . . , s � � s . As F is one-to-one we can allowk 21k k�1 kŽ . Ž .for the payoff function � x � s � 1 � s . In other words, an5

agent’s payoff depends on his predecessor’s signal as well as his own signal.From a myopic perspective players should therefore play ak � 0 at allstages. However, consider the following strategies for all players: at stage 1let ak � 1 and at stage t let ak � 1 if and only if sk�1 was equal to 1, atall periods 1, . . . , t � 1. Note that, independently of the number of players,

Žthese strategies constitute a Nash equilibrium if players are sufficiently.patient ; yet all players do not take myopic best responses at the first stage.

To recover an analogue of Theorem 1 we add an assumption regardingthe continuity of the payoff functions.

� � kDEFINITION 3. Fix a function : 0, 1 � � . We say that � is��k k � � Ž . � Ž k . -continuous if for any a � A , � � 0, x � y � � implies � a , x �

Ž k . �� a , y � � .

In order to extend the results of Theorem 1 we replace the restrictionon the cardinality of X with a uniform continuity of payoffs: for fixed

� �parameters � 1, : 0, 1 � � , � � 0, define the family of games:��

k � k �HH Q, , , � � H : � is -continuous, � � Q, and for all k ,�Ž .

k � , C k ak � �� Sk , and ak � Ak .4Ž . Ž .

The force of uniform continuity is that it is required for all games in HH,uniformly in the number the players N.

We now show that players’ equilibrium strategies in this new family ofgames has a myopic nature:

� .THEOREM 2. Assume X � 0, 1 . For any � , , � and any � � 0, thereŽ . Ž .exists an integer J � J , , � , � such that if H � HH Q, , , � then any Ž . Nash equilibrium of H is a J, � -myopic equilibrium of H .

Similar to the proof of Theorem 1, the proof of Theorem 2 builds on aresult on influence, which is equivalent to Lemma 1.

LEMMA 2. For any , , � and any � � 0, there exists an integerˆˆ ˆ Ž . Ž .J � J , , � , � such that if H � HH Q, , , � and � is an arbitrary profileˆ


ˆ ˆof beha�ior strategies, then for any stage t there exists a subset K of at most Jtˆ kplayers such that for any k � K and pair of actions a, b � A ,t

k k kE � d , x � E � d , x � � �d � A .Ž . Ž . ˆ� ��k k �k kŽ� , � . , t Ž� , � . , tŽa. Žb.

The proof of Lemma 2 is postponed to the Appendix.Given Lemma 2 the proof of Theorem 2 follows exactly in the steps of

the proof of Theorem 1, and it is therefore omitted.

3.3. A Comparison with Sabourian’s Results

Ž .Our main result is different than Sabourian’s 1990 result in a few ways.It is weaker in some ways and stronger in another way. The relative

Ž .weaknesses are: a We assume that agents’ actions are imperfectly ob-served by the aggregating mechanism, whereas Sabourian allows for play-

Ž .ers to recall their own actions. b In order to consider a large outcomespace we need to impose a continuity assumption on the payoff function.Ž .c The action spaces are assumed finite in our model. We conjecture thatthis restriction is not necessary.

On the other hand our result holds without the constraint posed byˆSabourian regarding the structure of the mechanism F. In particular we do

ˆnot require that F be anonymous as Sabourian does. Furthermore weallow for all players to perfectly observe the history of signals of allplayers, whereas Sabourian assumes players can only recall their ownactions.

4. CONCLUDING REMARKS

In this paper we provided an extension of Green and Sabourian’sanalysis to games with nonanonymous outcomes and strategies. Usingrecent nonpivotalness results, we establish powerful bounds on how manyplayers might act strategically. This provides a transparent and simpleargument that sheds light on the forces driving Green and Sabourian’sanalysis.

Two related strands of literature should be mentioned. First, nonpivotal-ness theorems have been used in the literature to address a variety of

Ž .other questions. These include, most notably, Rob’s 1989 and MailathŽ .and Postlewaite’s 1990 asymptotic inefficiency results in economies with

externalities and public goods. Another related work is Fudenberg et al.Ž .1998 who examine agency, reputation, and repeated games with a largeplayer. While similar in spirit, the techniques developed in these papers donot allow for the derivation of a bound on the number of �-pivotal players


Ž � .independent of N as we do here K defined above . This bound allows us�

to develop a clean statement of the result on approximate reversion tomyopic play and provides sharp bounds on the rates of convergence.9

The second related strand of literature is that concerned with refiningequilibria in dynamic games by appealing to Markovian restrictions on

Ž .strategies. See Maskin and Tirole 1994 for discussion and bibliography.The analytical and practical appeal of Markovian restrictions is obvious.However, justifying Markovian behavior hinges on the plausibility of theassumption that players believe their opponents limit their strategy choicesto ones that condition only on payoff-relevant aspects of the environment.In our context, with no state variables, Markovian behavior reduces toplaying an equilibrium of the one-shot game. In this case, our analysis maybe restated by saying that most players optimally ignore payoff-irrelevantaspects of the environment in choosing their actions.10 Our paper makesexplicit what sort of conditions are needed to justify players makingapproximate Markovian assumptions about the behavior of their oppo-

Žnents e.g., noisy information about actions, large number of players,interdependence of payoffs through a common collective outcome, and so

.on . The advantage of this analysis is that it gives a set of criteria forŽ .judging when the often controversial Markovian assumption is justified.

We hope to pursue this in greater generality in the future.

APPENDIX

The object of this appendix is to prove Lemmas 1 and 2. The proofsprovided are based on an earlier result from Al-Najjar and SmorodinskyŽ .2000 . We begin by recalling this result.

� �Let F: S � 0, 1 be an arbitrary assignment from the signal space intothe unit interval. Let �k be an arbitrary probability measure on Sk

k kŽ .satisfying � s � � for any player k and any signal s � S . We denote by

9 Ž .Fudenberg et al. 1998, p. 65 informally suggest the possibility of a result in the spirit ofour main results. Two issues should be noted, however, First, their analysis seems to requirethat the set of outcomes be finite while we are able to deal, in Theorem 2, with the

Žcontinuous outcome case the outcome set in our model corresponds to the set of actions ofthe large players in theirs, which they require to be finite�see, for instance, the statement in

.their abstract, p. 46 . Second, their conjecture seems to claim that the a�erage loss over allplayers from not playing myopically becomes small as the number of players increases. This isstrictly weaker than our results, as this conjecture would be consistent with an unboundednumber of players playing nonmyopically.

10 Note that Green and Sabourian’s analysis cannot address this issue as it assumes thatplayers can condition only on payoff-relevant outcomes.


� the product probability distribution of the �ks over S. Player k ’sinfluence on F is

k k kV � max E F � s � s � min E F � s � s .Ž . Ž .� �k ks�S s�S

k Ž .Note that V bounds the change in E F induced by a unilateral changein the marginal �k. The result of Theorem 2 in Al-Najjar and SmorodinskyŽ .2000 focuses on the number of players for which this bound exceeds agiven threshold 1 � � � 0:

� Ž .PROPOSITION 1. There exists an integer J � such that for any K � ,�

� k4Kany signal space S, and any �ector of probability measures, � ,k�1

� k 4 �� k : V � � � J � .Ž .�

Throughout the rest of the Appendix various parameters of the modelare fixed:

� The parameters Q � , � 1, M � , , � � 0 defining the fami-Ž . Ž .lies of games HH Q, , M, � and HH Q, , , � are fixed.

� Ž . Ž .For any game H � HH Q, , M, � or in HH Q, , , � and for any

player in H we fix an arbitrary action at stage zero, an arbitrary strategy�k, and an arbitrary stage t.

Note that we do not bound the number of players in the game.These parameters, in turn, fix, for each game, the probability distribu-

Ž .tions � over S. As it is assumed that for each H � HH Q, , M, � or�, t Ž . kŽ k . � Ž k .H � HH Q, , , � the distribution C a is in � S , we conclude that

� Ž k .� � � S as well.�, t

Ž .Let � be an arbitrary distribution over S. We shall denote by F � thecorresponding distribution over X.

Proof of Lemma 1

Ž .Following Al-Najjar and Smorodinsky 2000 we define the influence ofkŽ .player k, V H , by

k 11�k k �k kV H � max F � � F � ,Ž . Ž . Ž .Ž� , � . , t � , � . , tŽa. Žb.ka, b�A

� � Ž . 12where � denotes the sup-norm metric on � X .

11 A minor difference between the original notion of influence and the one used here, isthat here we take the maximum over pairs of distributions induced by actions in Ak, whereasin the original definition the maximum was over all pairs of Dirac measures. One should notethat the influence defined here is, generally, smaller than the original notion.

12 Ž . � �The sup-norm metric is defined as follows. For F , F � � X , let F � F �1 2 1 2� Ž . Ž . � � � � Ž . Ž . �max F B � F B . Note that 2 F � F � Ý F x � F x .B X 1 2 1 2 x � X 1 2


Ž .The number of �-pivotal players in H , denoted J H , is�

k J H � � k : V H � � .� 4Ž . Ž .�

Ž .Al-Najjar and Smorodinsky 2000, Theorem 2 provide a bound, inde-Ž .pendent of the number of players K , on J H :�

�PROPOSITION 2. For any � � 0 there exists an integer J such that if� �Ž . Ž .H � HH Q, , M, � then J H � J .� �

�In other words, what Proposition 2 says is that all players but at most J�13of them cannot change the distribution over X by more than � .

We now turn to show that small changes in the distribution over Xcause small changes in the expected payoffs:

k Ž .PROPOSITION 3. For any � � 0 and any action d � A , if F , F � � Xˆ 1 2� � � kŽ . kŽ . �satisfy F � F � ��2Q then E � d, x � E � d, x � � .ˆ ˆ1 2 F F1 2

Proof.

k kE � d , x � E � d , xŽ . Ž .F F1 2

� � d , x F x � F xŽ . Ž . Ž .Ý 1 2x�X

� �� Q F x � F x � 2Q F � F � � .Ž . Ž . ˆÝ 1 2 1 2x�X

�ˆWe can finally prove Lemma 1. Let � � ��2Q and let J � J , as inˆ �

Proposition 2. This ensures that there exists a subset of players, denotedˆK , of size less than or equal to J, such that if k � K then for any pair oft t

k � Ž . Ž .��k k �k kactions a, b � A , F � � F � � � . Proposition 3,Ž� , � ., t Ž� , � ., tŽa. Žb.

in turn, ensures that for k � K t

k k kE � d , x � E � d , x � � �a, b , d � A ,Ž . Ž . ˆF Ž � . F Ž � .�k k �k kŽ� , � . , t Ž� , � . , tŽa. Žb.

as required.

13 Two comments are in place: First, because of the argument made in footnote 11 theproposition applies to the current notion of influence. Second, the statement of the result inour earlier paper applies to the cases Q � 1 and M � 2. However, as indicated by footnotes4 and 8 in that paper, the generalization to arbitrary Q and M is straightforward.


Proof of Lemma 2

The analog of Proposition 2 for the large outcome space is:

PROPOSITION 4. For any , , � , any � � 0, and any measurable subsetˆ Ž . ŽX X, there exists an integer J � J , , � , � such that if H � HH Q, , ,

.� and � is an arbitrary profile of beha�ior strategies, then for any stage t thereˆ ˆexists a subset of players, K , of at most J players such that for any k � K andt t

pair of actions a, b � Ak,

ˆ ˆ�k k �k kF � x � X � F � x � X � � .Ž . Ž .Ž . Ž .Ž� , � , t Ž� , � . , tŽa. Žb.

In other words, most players’ actions at the first stage of the game haveno effect on the probability that the outcome at time t will be in anarbitrary subset of the outcome space. The claim of Proposition 4 is

Ž .actually a rephrasing of Theorem 2 of Al-Najjar and Smorodinsky 2000 .Consequently the proof is omitted.

We turn to the analog of Proposition 3 for the large outcome space. Fixan arbitrary � � 0. Let M be an integer such that for any H � HH and anyˆ

1 k k� � � Ž .player, k, in H , if x, y � X satisfy x � y � then � a , x �Mk k 'Ž . �� a , y � � . Note that M is a function of the parameters of theˆ

family of games and is independent of the number of players in the game.m � 1 mˆŽ . � .Denote by X m � , , m � 1, . . . , M.M M

k Ž .PROPOSITION 5. For any � � 0 and any action d � A , if F , F � � Xˆ 1 2'ˆ ˆ� Ž Ž .. Ž Ž .. �satisfy F X m � F X m � � �M, for all m � 1, . . . , M, thenˆ1 2

k kE � d , x � E � d , x � � .Ž . Ž . ˆF F1 2

Proof.

k kE � d , x � E � d , xŽ . Ž .F F1 2

Mˆ ˆ� F X m � F X mŽ . Ž .Ž . Ž .Ý 1 2

m�1

� max � k d , x � min � k d , xŽ . Ž .ž /ˆ ˆŽ . Ž .x�X m x�X m

M M '�̂' 'ˆ ˆ� F X m � F X m � � � � � .Ž . Ž . ˆ ˆ ˆŽ . Ž .Ý Ý1 2 Mm�1 m�1

�


Ž .We can finally prove Lemma 2. Let M and X m , m � 1, . . . , M, be as'�̂in the claim of Proposition 5. Let � � . By Proposition 4 there existM

Ž . Ž .positive finite integers J m � J m , m � 1, . . . , M such that for each m�

Ž .there exists a subset of players, denoted K m , of size less than or equaltkŽ . Ž .to J m , such that if k � K m then for any pair of actions a, b � A ,t

ˆ ˆ� Ž .Ž Ž .. Ž .Ž Ž .. ��k k �k kF � x � X m � F � x � X m � � . Let K �Ž� , � ., t Ž� , � ., t tŽa. Žb.M MˆŽ . Ž .� K m . Note that K consists of at most J � Ý J m players, andm� 1 t t m�1

ˆobviously J does not depend on the number of players in the game.Proposition 5, in turn, ensures that for k � K t

k k kE � d , x � E � d , x � � �a, b , d � A ,Ž . Ž . ˆF Ž � . F Ž � .�k k �k kŽ� , � . , t Ž� , � . , tŽa. Žb.

as required.

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Ž .Aumann, R. J. 1964 . ‘‘Mixed and Behavior Strategies in Infinite Extensive Games,’’ inŽ .Ad�ances in Game Theory M. Dresher, L. S. Shapley, and A. W. Tucker, Eds. , Annals of

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large nonanonymous repeated games

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