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Egalitarianism and Efficiency in Repeated Symmetric Games
Games and Economic Behavior 32, 247–262 (2000) doi:10.1006/game.2000.0810, available online at http://www.idealibrary.com on
Egalitarianism and Efficiency in Repeated Symmetric Games V. Bhaskar1 2 Department of Economics, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom Received September 15, 1997
We analyze the symmetric equilibria of repeated symmetric games where there is a conflict of interests over equilibria—the battle-of-the-sexes or the hawk–dove game are key examples. If one restricts attention to symmetric equilibria, efficient equilibria must be egalitarian. For finitely repeated games, and generic discount factors, there is a unique outcome path which ensures efficiency within the class of symmetric equilibria. This is also true for the infinitely repeated games if the players are sufficiently impatient. Journal of Economic Literature Classification Numbers: C72, C73. © 2000 Academic Press
1. INTRODUCTION This paper analyzes symmetric equilibria in repeated games where the stage game is a symmetric two-player game with a pair of asymmetric pure strategy equilibria. We show that when there is a conflict of interest between the players over the equilibria (as in the battle-of-the-sexes or the hawk–dove games),3 efficient payoffs are obtained when the symmetric repeated game equilibrium is egalitarian. An egalitarian equilibrium is one
1 I thank Sanjeev Goyal, Ted To, and participants at the 17th Arne Ryde Symposium on Focal Points for comments on an earlier draft. I am particularly grateful to an anonymous referee and associate editor for their constructive suggestions. 2 E-mail: [email protected]. 3 The class of games we consider represents several important situations in economics and biology—entry of firms into a natural monopoly industry (Dixit and Shapiro, 1985; Farrell, 1987), product compatibility (Farrell and Saloner, 1988), network externalities (Katz and Shapiro, 1985), and competition between two animals for a scarce resource (Maynard-Smith, 1982).
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v. bhaskar α
β
α
0 0
x 1
β
1 x
y y
FIG. 1. The game G.
which equalizes, as far as possible, the realized payoffs of the two players.4 Moreover, for finitely repeated games, and generic discount factors, there is a unique outcome path which ensures efficiency within the class of symmetric equilibria. This is also true for the infinitely repeated games if the players are sufficiently impatient. For some intuition, consider the battle-of-the sexes, with payoffs as given in Fig. 1. Assume for the moment that x > 1 and y = 0 Hence the stage game has a pair of asymmetric equilibria (α β and β α where player one prefers the former and player two the latter. Let the game be repeated twice and consider symmetric equilibria in the repeated game. In such a symmetric equilibrium, players must choose the same randomization probability in period one, and must also play the mixed equilibrium of the stage game in period two if their realized first period actions coincide. However, symmetry allows the players to play asymmetrically in period two, if their realized actions in period one differ. Consider first the bourgeois convention, where if coordination is achieved in period one upon α β players play the same equilibrium α β in the second period. Under this convention players have a relatively large stake in securing coordination on their own terms, and hence both players play α with high probability. The probability of expost coordination will consequently be relatively low. Compare this with the egalitarian convention, where the players play β α in period two if coordination in period one has taken place on α β This reduces the incentive a player has to try and ensure coordination on her preferred equilibrium. Hence this convention induces the player to play each action with probability close to one-half, so that the probability of expost coordination is higher than under the bourgeois convention—indeed, it will be higher than in mixed equilibrium of the one-shot game. Hence the symmetric equilibrium associated with the egalitarian convention payoff dominates the symmetric equilibrium associated with the bourgeois convention. We show that this idea extends for more general payoffs, and also when the game is repeated many times. We also explore the conditions under which there is a unique egalitarian convention. 4
In a symmetric mixed strategy equilibrium, expected payoffs are always equal for the two players, but realized payoffs can be unequal.
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The crucial restrictions in the analysis are that we consider only symmetric equilibria and also do not allow cheap talk. The restriction to symmetric equilibria in symmetric games reflects the view that coordination on asymmetric equilibria may be impossible and is supported by Harsanyi and Selten’s (1988) principle of symmetry invariance. Farrell (1987) studied symmetric equilibria of the metagame when a game such as G is preceded by pre-play communication. He showed that cheap talk could achieve a degree of asymmetric coordination in the battle-of- sexes, but is completely ineffective in the hawk–dove game.5 We find that repeated interaction is always effective, independent of the extent of conflict in the stage game. 6 The remainder of this paper is organized as follows. Section 2 considers the twice-repeated game and shows that an egalitarian convention is uniquely optimal. Section 3 discusses the case of general finite and infinite repetitions, and the final section concludes. 2. THE TWICE REPEATED GAME We consider a symmetric two-person game, where each player has action set A = α β and payoff function u A2 → R as given in Fig. 1. Assume x > y so that α β and β α are the two pure strategy (strict) Nash equilibria. Observe that G encompasses the class of 2 × 2 symmetric games with a pair of asymmetric strict equilibria—the payoffs 0 and 1 are simply normalizations given that these are von Neumann–Morgenstern utilities. G x−y also has a mixed equilibrium where α is played with probability p = 1+x−y x and the payoff in this mixed equilibrium is u¯ = 1+x−y The game G has equilibrium conflict of interest if x = 1—this case is the main focus of this paper. We shall call the game G the battle-of-the-sexes if y ≤ 1 if y > 1 we call it the hawk–dove game. Despite equilibrium conflict of interest, players still have common interests in achieving coordination on one of the pure strategy Nash equilibria. This is particularly true in the case of the battleof-the-sexes, since in this case either pure equilibrium Pareto-dominates the payoff in the mixed strategy equilibrium (u. ¯ In the hawk–dove game, x > u¯ > 1, so that the mixed equilibrium is not Pareto-dominated by a pure equilibrium. However, if y < x+1 any mixed action profile is inferior to the 2 lottery which yields the two pure equilibria with equal probability, so that there remains some commonality of interest in coordinating behavior. 5 Farrell assumes that the messages come from a commonly understood natural language and selects for equilibria which are consistent with this interpretation. Warneryd (1992) and Banks and Calvert (1992) also consider cheap talk in asymmetric games. 6 Repetition and cheap talk may not be directly comparable, since one may take the view that pre-play communication is always possible whereas repetition may or may not be possible.
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The coordination problem in game G arises from the fact that it is not clear which of the two pure strategy equilibria players should expect to be played. The two equilibria are indistinguishable—any argument one may offer in favor of one can equally well be offered in support of the other. In view of this problem—which has been noted by many including Farrell (1987), Harsanyi and Selten (1988), and Warneryd (1992)—it is reasonable to restrict attention to the symmetric equilibrium. More formally, only the symmetric mixed strategy equilibrium of G satisfies Harsanyi and Selten’s criterion of symmetry invariance, which requires that the selected equilibrium be invariant to re-labelling of the players or strategies. 7 Let the game G be played twice, and let players maximize the sum of stage game payoffs, discounted at rate δ. Since the repeated game is also symmetric, we will restrict attention to symmetric subgame perfect equilibria. Let σ σ be such an equilibrium and consider the prescriptions of this profile at the second stage of this game. Clearly, such an equilibrium must (a) play an equilibrium of the stage game after any pair of first period actions and (b) prescribe identical (mixed) actions for both players if the history is symmetric. Hence in any symmetric equilibrium of the two-period game, both players must play the mixed equilibrium of G after the histories α α and β β Consider now a history where the realized actions of the two players differ, e.g., the history α β where player 1 has played α and player 2 has played β In this case, the history from the point of view of player 1 (which is (α β) differs from the history from the point of view of player 2 (which is (β α Hence the continuation action profile induced in period 2 can well be asymmetric.8 A rule which achieves asymmetric coordination by conditioning upon history is called convention. Hence we have two possible conventions. The bourgeios convention requires that players adopt the same actions in period two in the event that their first period actions are coordinated. The egalitarian convention requires that each player chooses a different action from the one chosen in period one (i.e., the players choose β α in period two if their realized first period actions are α β 9 Our key point is that different conventions generate different incentives for first period actions. To see this, write down the payoff matrix of the row player for first period actions under each of these conventions: 7
The equilibrium α β (where player 1 plays α and player 2 plays β can be transformed into the other equilibrium, β αby exchanging the labels of the players, and hence is not symmetry invariant. 8 The role of history as a correlation device for coordinating future actions was first pointed out by Crawford and Haller (1990). We discuss their work in the concluding section. 9 There is also a symmetric equilibrium which ignores the realized asymmetries and plays the mixed equilibrium in period two, after all histories. Call this convention the mixed convention. In the proof of Proposition 1 we show that this equilibrium is never optimal.
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First period payoffs of the row player under Bourgeois Convention α β
α
β
δu¯ 1+δ
1 + δx δu¯ + y
First period payoffs of the row player under Egalitarian Convention α β
α
β
δu¯ 1 + δx
x+δ δu¯ + y
Observe that the two conventions are not equivalent as long as there is any conflict of interest over equilibria between the players, i.e., as long as x = 1Restricting attention to the symmetric equilibrium of each of these payoff matrices, we see that for any of the above matrices, such an equilibrium always exists and is unique. This symmetric equilibrium is mixed under the bourgeois convention if x − 1 + δy − 1 > 0 otherwise, it is pure and plays α with probability one. Similarly, the symmetric equilibrium is mixed under the egalitarian convention if x − y − δ1−y > 0 otherwise, 1+x−y it is pure and plays β with probability one. Observe that in the battle of the sexes, since y ≤ 1, both conventions imply randomized choices in period 1 This is not necessarily the case in the hawk–dove game, since one or both of these conventions may require deterministic choices in period one. The following proposition shows that in all instances, the egalitarian convention is optimal. Proposition 1. The symmetric equilibrium with the egalitarian convention is always optimal. It is uniquely optimal if x = 1 Proof. Let σ1b resp. σ1e be the probability with which α is played in period one under the bourgeois (resp. egalitarian) convention. Consider first the case where x − 1 + δy − 1 > 0 and x − y − δ1−y > 0 so that 1+x−y both conventions imply randomization in the first period. The equilibrium payoffs are given by V σ b σ b =
δu1 ¯ + δx − δu¯ − y + 1 + δx1 + δ − δu ¯ 1 + δx − δu¯ − y + 1 + δ − δu ¯
(1)
δux ¯ + δ − δu¯ − y + x + δ1 + δx − δu ¯ 1 + δx − δu¯ − y + 1 + δ − δu ¯
(2)
V σ e σ e =
The difference in payoffs is given by V σ e σ e − V σ b σ b =
δx − 12 ≥ 0 1 + δx − δu¯ − y + 1 + δ − δu ¯
(3)
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Hence for this range of parameter values the egalitarian convention is always optimal, and uniquely so if x = 1 Note that if either convention prescribes a pure action, this implies that y > 1 Suppose that both conventions require pure actions in the first period. In this case, the payoff to the bourgeois convention is δu¯ whereas the payoff to the egalitarian convention δu¯ + y which is strictly greater. Suppose that egalitarian convention has pure actions in period one, whereas the bourgeois convention has random actions. In this case V σ e σ e = δu¯ + y while the V σ b σ b is a convex combination of δu¯ + y and 1 + δ with strictly positive weights. Since y > 1 and u¯ > 1 δu¯ + y > 1 + δ and hence σ e is strictly optimal. Finally, if the egalitarian convention has random actions, its payoff is a convex combination of 1 + δx and y + δu ¯ while the bourgeois convention’s payoff is δu ¯ Since 1 + δx > δu¯ and y + δu¯ > δu ¯ the egalitarian convention is also strictly optimal in this case. We now show that the egalitarian convention is strictly better than the symmetric equilibrium, σ m where players play the mixed strategy equilibrium of the G in period 1 and in period 2 at all information sets. To see this, write the payoff matrix to first period actions under this equilibrium, as First period payoffs of the row player under Mixed Convention α β
α
β
δu¯ 1 + δu¯
x + δu¯ δu¯ + y
Suppose that σ1e is pure and plays β with probability one, which implies y > 1 In this case the V σ e σ e = y + δu ¯ whereas V σ m σ m equals the payoff to playing β in the above matrix and is a convex combination of y + δu¯ and 1 + δu¯ and is hence strictly less. Suppose that σ1e is mixed and consider two subcases. Suppose that y ≥ 1 so that y ≥ u¯ ≥ 1 In this case it can be verified that p > σ1e . Consider the payoff to playing β under both these ¯ +1 − py + δu ¯ while V σ e σ e = equilibria. V σ m σ m = p1 + δu e e e e σ1 1 + δx + 1 − σ1 y + δu ¯ Hence V σ σ − V σ m σ m = p − e e σ1 y − 1 + σ1 δx − u ¯ > 0 Suppose that y < 1 so that y < u¯ < 1 Consider the payoff to playing α under both these equilibria. V σ m σ m = pδu¯ +1 − px + δu, ¯ while V σ e σ e = σ1e δu¯ + 1 − σ1e x + δ Since x + δ > x + δu ¯ V σ m σ m ≥ V σ e σ e ⇒ σ1e > p Consider the payoff to playing β under both ¯ +1 − py + δu ¯ while these equilibria. V σ m σ m = p1 + δu V σ e σ e = σ1e 1 + δx + 1 − σ1e y + δu ¯ Since 1 + δx > 1 + δu¯ > y + δu ¯ V σ m σ m ≥ V σ e σ e ⇒ σ1e < p Hence we have a contradiction, and V σ m σ m < V σ e σ e
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Although the egalitarian convention gives rise to the highest payoffs (in the set of symmetric equilibria), it does not always yield the highest probability of expost coordination in period one. To see this, let x − y < 1 which implies that p < 21 Hence α is more risky than β and is played with probability less than one in the mixed equilibrium of the stage game. In this case, the egalitarian convention makes playing α even less attractive, and hence α is played with probability less than p in the first period. On the other hand, the bourgeois convention raises the probability with which α is played and yields a higher probability of expost coordination in period 1. Nevertheless, the payoffs are greater under the egalitarian convention, since the payoffs when not coordinated also matter—for example, if y > 0, the payoffs to the profile β β are greater than to the profile α α and hence reducing the probability of playing α raises payoffs. A referee has pointed out that the egalitarian convention is optimal even if there are some increasing returns or learning by doing in resource use, so that the bourgeois convention is expost optimal. Consider a contest between two players for a fixed resource, which takes the form of the hawk–dove game which is played in each of two periods. Assume that in period one, payoffs are given by the game G in Fig. 1, where the value of the resource is x > 1. These are also the payoffs in period two, in the event of the first period actions being α α or β β However, in the event that the first period action profile is α β the period two payoffs from playing α β again are λx (λ > 1 and 1 to players 1 and 2, respectively, while the payoffs to playing β α are 1 and x 10 Hence there is learning by doing in resource use. In this case, the bourgeois convention is expost efficient since it will result in a larger total surplus in period two, as compared to the egalitarian convention. However, the bourgeois convention results in more aggressive behavior and lower payoffs in period one. The payoffs under the two conventions can be computed using Eq. (6) in the appendix. As an illustrative example, we derive a sufficient condition for the egalitarian convention to be optimal in this case, given that there is no discounting δ = 1 For the egalitarian convention to be optimal, it is sufficient that 1 + x2 ≥ 2x1 + λ i.e., if λ ≤ λ∗ = 1 + x2 /2x This critical value always exceeds one (since x > 1 and is increasing in x For example, if x = 2, λ∗ = 54 and if x = 5 λ∗ = 13 Hence egalitarianism is 5 optimal even if there is a significant amount of learning by doing.11 Our analysis assumes that a player only observes his opponent’s realized action, and not his mixed strategy, and applies more generally, e.g., 10
Period two payoffs in the event of β α being played in period one are given symmetrically. 11 By using similar arguments as in the proof of Proposition 1, one can verify that the egalitarian convention is better than the mixed convention for any value of λ
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to games with incomplete payoff information. For example, in repeated auctions one typically observes the opponent’s bids, but not the map from valuation to bids. Second price common value auctions often have asymmetric equilibria which yield low revenues to the seller as compared to the symmetric equilibrium (see Bikchandani, 1988, and Klemperer, 1998). Our analysis suggests that in a twice-repeated second price common value auction, collusion can be sustained in a symmetric equilibrium, where the loser in round one bids aggressively, while the winner in round one bids less aggressively. This will result in lower seller revenues also in round one, since both players have an incentive to bid less aggressively. 3. MANY REPETITIONS Let GT δ be the T -fold repetition of G, where the discount rate δ ∈ 0 1 if T is finite, and δ ∈ 0 1 if T = ∞ Payoffs in the repeated game are given by the discounted sum of stage game payoffs. A player’s history at period t ht is the sequence a1 b1 at − 1 bt − 1 where aτ denotes a player’s own action in period τ, while bτ denotes his opponent’s action. Let Ht denote the space of all t-period histories for a player. For each ht ∈ Ht let ηht denote the corresponding history for the other player, i.e. ηht is obtained by permuting aτ and bτ for each τ A behavior strategy for a player in GT δ is a sequence of functions st Tt=1 where s1 ∈ A is the player’s first period strategy and for t > 1 st Ht → A Let T be the set of behavior strategies in GT . Write V T σ θ for the payoff of strategy σ against strategy θ in GT δ A strategy profile σ θ is a Nash equilibrium if V T σ θ ≥ V T σ θ∀σ ∈ T and V T θ σ ≥ V T θ σ∀θ ∈ T σ θ is a subgame perfect equilibrium if it is a Nash equilibrium and ∀ht ∈ Ht the profile σht θηht is a Nash equilibrium of GT −t+1 . A subgame perfect equilibrium σ θ is symmetric if σ = θ Let T be the set of subgame perfect equilibria of GT δ, and let W T be the set of payoffs generated by T —each element in this set consists of a pair of payoffs, one for each player. Let TS denote the set of symmetric subgame perfect equilibria, and let WST be the set of payoffs generated by TS It will be convenient to let the elements of WST be a single number which denotes the payoffs of both players. Our focus is on symmetric equilibria, i.e., on the set TS We call a symmetric equilibrium σ σ ∈ TS optimal if V T σ σ ≥ V T θ θ for any θ θ ∈ TS Any such equilibrium gives rise to an outcome path, i.e., a probability distribution over A × AT when it is played. We say that an optimal equilibrium σ σ has a uniquely optimal outcome path if V T σ σ > V T θ θ for any symmetric equilibrium θ θ which has a different outcome path from σ σ
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−1 A convention in a T − 1 period repeated game is a sequence γτ Tτ=1 where γτ ∈ α β β α∀τ Since α β is a Nash equilibrium of the stage game, it is clear that any convention can be supported as the outcome path of a subgame perfect equilibrium—players can simply ignore deviations and continue with the convention. Any payoff vector which is generated by a convention is hence always an element of W T −1 We shall call a convention optimal if it is the continuation path of an optimal equilibrium, conditional upon an asymmetric action profile being played in the first period. Q = q1 q2 q1 = x + Tt=2 δt−1 t x + 1 − t 1 q2 = 1 + TLet t−1 t 1 + 1 − t x t ∈ 0 1 2 ≤ t ≤ T be the set of payoff t=2 δ pairs which arise when players adopt some convention, given that their realized actions are α β at the initial date. Q is a finite set if T is finite, but can be infinite if T = ∞ Let K ∗ L∗ = arg minq1 q2 ∈Q q1 − q2 —this is the payoff pair which arises in an egalitarian convention12 Let V¯ T −1 denote the maximal element in WST −1 i.e., it is the maximal symmetric subgame perfect equilibrium payoff in the T − 1 period repeated game. Consider the following outcome path in the T period game, where the players randomize at the initial date. If their actions coincide, they play an optimal symmetric equilibrium of the T − 1 period game in the next period. If their actions differ, they play the egalitarian convention. Hence the payoffs to first period actions of the row player are given by the derived game (∗ given as
α α β
The Derived Game (∗ β
δ V¯ T −1 δ V¯ T −1 L∗ K ∗
K ∗ L∗ y + δ V¯ T −1 y + δ V¯ T −1
We now show that, under certain conditions, the optimal symmetric equilibrium of the T period game has an outcome path which corresponds to the symmetric completely mixed equilibrium of the derived game (∗ Proposition 2. Suppose that either (i) y ≤ 1 or (ii) y < x+1 δ > 21 and 2 T is sufficiently large. Then in any optimal symmetric equilibrium of GT δ the players adopt egalitarian convention if their initial actions differ. Furthermore, they play an optimal symmetric equilibrium of GT −1 δ if their initial actions coincide. Proof. 12
See the Appendix.
We shall show such an egalitarian convention always exists, i.e., the minimization problem defining it has a solution.
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This proposition establishes that if stage game G is the battle-of-thesexes or if it is a hawk–dove game with sufficient commonality interest and where the future is sufficiently important, the outcome path corresponding to an optimal symmetric equilibrium has a simple characterization. Players randomize symmetrically; if their realized actions differ, they play either α β or β α in each period, where the path is chosen so as to equalize, as far as possible, the payoff streams of the two players. In other words, the optimal convention is egalitarian. If the realized actions in the initial period coincide, the players continue with the optimal symmetric equilibrium of the T − 1 period game.13 We now explicitly solve for an egalitarian convention and investigate the conditions under which this is unique. Our construction is very similar to that used in the folk theorems of repeated games without public randomization, e.g., Fudenberg and Maskin (1991, Lemma 1) and Sorin (1986). Suppose that the realized action profile in the initial period is α β Any convention can be represented by a sequence < zt > where zt ∈ −1 1∀t 2 ≤ t ≤ T where zt = 1 if α β is to be played and zt = −1 if β α is to be played. Let Z T = < zt > zt ∈ −1 1∀t 2 ≤ t ≤ T be the set of all such sequences, and let F Z T → 0 ∞ be defined by T T t Fz = 1 + δ zt (4) t=2 Hence an egalitarian convention minimizes the function F over the set ZT ∞ We now construct an infinite sequence zˆ =< zˆt >∞ t=2 ∈ Z which allows us to construct optimal conventions in finitely and infinitely repeated games. Given any such infinite sequence, define the associated partial sum Eτ by Eτ = 1 +
τ t=2
δt zˆt
(5)
The sequence zˆ is defined as follows: for t ≥ 2 zˆt = 1 if Et − 1 ≤ 0 and zˆt = −1 if Et − 1 > 0 We shall now show that the sequence zˆ defines an optimal convention in the infinitely repeated game, while its T − 1 period truncation defines an optimal convention for the T period repeated game. Intuitively, the convention defined by zˆ keeps track of the discounted sum 13
This proposition does not fully characterize optimal behavior for arbitrary hawk–dove games. However, some cases are straightforward—if 2y ≥ 1 + x it is efficient for the players to play β β. To support this in equilibrium, if player 1 deviates, then the players can revert to playing β α for the remaining periods of the game (this minmaxes player 1). It is routine to verify that the profile can be supported if the players are sufficiently patient and T is sufficiently large.
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of payoffs to the players at each date. Suppose, for example, that x > 1 so that player one’s preferred equilibrium is α β If the realized actions in period one are α β the players play β α in succeeding periods until the total discounted payoffs of player two exceed player one’s total discounted payoffs. At this point players switch to playing α β until player one’s total payoffs exceeds player two’s, and so on. Proposition 3. In the infinitely repeated game, the sequence zˆ defines an egalitarian convention. This convention is the unique egalitarian convention if and only if δ ≤ 1/2 Proof.
See the Appendix.
When T is finite, the finite sequence zˆ T −1 , which is given by the first T − 1 terms of zˆ defines an egalitarian convention Furthermore, it is also easy to show that this is the unique egalitarian convention, for generic values of δ If T is finite, no convention will be able to completely equalize the payoffs K ∗ and L∗ in the derived game (∗ Also, for generic values of the discount factor, no two distinct conventions, which yield distinct streams of future payoffs, will have the same present value. This ensures that there is a unique egalitarian convention.14 Proposition 4. If T is finite, the truncated sequence zˆ T −1 defines an egalitarian convention. This convention is the unique egalitarian convention for almost all values of δ Proof.
See the Appendix.
Our analysis highlights a novel distinction between finitely repeated and infinitely repeated games. In the finitely repeated game, the conflict of interest between the players can be reduced substantially, but is generically never completely eliminated. In consequence, under the conditions set out in Proposition 2, there is a unique optimal convention which minimizes this conflict. In infinitely repeated games, the conflict between players can be completely eliminated if the players are patient, and hence non-uniqueness follows. This is also related to the nature of the Folk theorems for the two types of games. In the infinite case, any individually rational payoff can be exactly achieved if players are sufficiently patient, whereas in the finite case, this payoff can only by approached by taking a sequence of games of increasingly longer length. Hence uniqueness of an optimal convention arises in those contexts where an “exact” Folk theorem does not apply. This in 14
Uniqueness cannot be ensured if players are completely patient, i.e., if δ = 1. If T is an even number, then note that zˆ T −1 defines perfectly egalitarian convention where players alternate between the equilibria. If T ≥ 4, perfect egalitarianism can also be achieved in other ways, e.g., by playing β α in the last T2 periods, and α β before that date.
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turn is related to the fact that mixed strategies are unobservable. The key to an optimal strategy lies in players randomizing initially with probability as close to one-half as possible. If mixed strategies were observable it would be easy to enforce randomization with probability one-half, even in a finitely repeated game. For example, any relatively egalitarian convention could be coupled with punishments—e.g., if one player deviates from this randomization, players could play the other player’s preferred equilibrium thereafter. Since many conventions could ensure the appropriate randomization, non-uniqueness would follow. Hence the unobservability of randomized actions plays an important role in ensuring uniqueness. 4. CONCLUSIONS The role of repeated interaction in solving the coordination problem was first suggested by Crawford and Haller (1990), who considered games of pure coordination, without conflict of interest (in Fig. 1, a pure coordination game is one where x = 1 and y = 015 They argued that the two pure equilibria α β and β α are indistinguishable, but suggested that players could randomize, and in the event of that their realized actions were coordinated, they could “maintain coordination forever once they locate a pair of coordinated actions. They can do this either by repeating those actions or by alternating deterministically between them and the other coordinated pair” (Crawford and Haller, 1990, p. 575, emphasis added). This paper has extended their approach to games which involve a conflict of interest over equilibria, and shown that this conflict of interest can be minimized via egalitarian conventions. The Crawford–Haller approach has been criticized (see Goyal and Janssen, 1996, since there is multiplicity of possible conventions which maintains coordination. With conflict of interests over the equilibria, we find that there often exists a unique optimal convention—this is always the case in the twice repeated game and generically with finitely many repetitions. Our approach to symmetric games relies on the use of history to “break” the symmetry and studies the optimal use of history in this context. Alternatively, in any real context, there may be other cues such as gender, race, or the age of the players, although there may be several such cues and their significance may also not be unambiguous. Consider now a repeated interaction where players also have access to such cues. Our analysis suggests that these cues will be more readily accepted if they are combined with an egalitarian convention, but are less likely to be taken if combined with a bourgeois convention. 15
Kramarz (1996) extends their approach to N person pure coordination games.
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To summarize, our substantive point is that if we restrict attention to symmetric equilibria, efficiency requires egalitarianism. In such contexts, bourgeois conventions, where property rights are entrenched by past precedent, have poor incentive effects. Egalitarian conventions minimize the conflict of interest and provide good incentives for coordination. While the analysis of this paper focuses on repeated games, this point is more general and applies to a variety of contexts such as the allocation of property rights. For example, pollution permits are often allocated on the principle of “grandfathering,” where entitlements are related to past emissions. In a world where property rights are not immutable, such criteria have bad incentive effects, by providing a dynamic incentive to increase pollution APPENDIX Proof of Proposition 2. Consider an arbitrary derived game ( with payoffs as given below, where a = δw b = y + δv for some w v ∈ WST −1 and ∃r s ∈ W T −1 K = x + δr and L = 1 + δs The Derived Game ( α β α β
a a L K
K L b b
Consider the symmetric equilibria of ( If a ≥ L, a is a symmetric equilibrium payoff, and if b ≥ K b is a symmetric equilibrium payoff. If K > b and L > a ( has a pair of asymmetric pure equilibria and a symmetric mixed equilibrium. The payoff in this mixed equilibrium is given by KL − ab (6) K+L−a−b Call ( a battle-of-the-sexes game if minK L ≥ maxa b We have that K − aL − a (7) UK L a b − a = K+L−a−b UK L a b =
K − bL − b (8) K+L−a−b Hence the payoff in the mixed equilibrium of ( is greater than any symmetric payoff of ( (i.e., a or b) provided that the derived game ( is a battle of sexes game. We also have that UK L a b − b =
L − aL − b ∂U = ∂K K + L − a − b2
(9)
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v. bhaskar ∂U K − aK − b = ∂L K + L − a − b2
(10)
∂U K − bL − b = ∂a K + L − a − b2
(11)
K − aL − a ∂U = ∂b K + L − a − b2
(12)
Hence U is increasing in all its arguments if ( is a battle-of-the sexes game. We now show that if minK ∗ L∗ ≥ maxδV¯ T −1 y + δV¯ T −1 so that ∗ ( is a battle-of-the-sexes game, the claim applies, i.e., V¯ T = U ∗ where U ∗ is the mixed equilibrium payoff in the derived game (∗ From (7) and (8) we have that U ∗ is greater than both δ V¯ T −1 and y + δ V¯ T −1 Consider any other derived game ( where a ≤ δV¯ T −1 and b ≤ y + δV¯ T −1 Hence any pure equilibrium payoff of ( is less than U ∗ Furthermore, if ( is a hawk–dove game, then the payoff in the mixed symmetric equilibrium of ( is less than maxδV¯ T −1 y + δV¯ T −1 and hence less than U ∗ If ( is a battle-of-the-sexes game, (9)–(12) and (6) show that this payoff must also be less than U ∗ 1+x Let W˜ T = 1 + x1 − δT +1 /21 − δ if T is finite and W˜ T = 21−δ if T = ∞ We now show that (∗ is a battle of sexes game under the conditions of the proposition. Suppose that y ≤ 1, so that minK ∗ L∗ ≥ minx+ W˜ T − 1 1+ W˜ T − 1 Note that since y < x+1 and 0 < x+1 the per-period 2 2 payoff in any symmetric equilibrium of the T − 1 period repeated game must be less than x+1 Hence W˜ T − 1 > V¯ T −1 Hence minK ∗ L∗ > 2 T −1 T −1 ¯ ¯ y + δV maxδV Consider now the case where y > 1 Clearly δ V¯ T −1 ≤ δW˜ T − 1 and y + δV¯ T −1 ≤ y + δW˜ T − 1 If T = ∞ and δ > 21 Proposition 3 establishes that a perfectly egalitarian convention exists, so that K ∗ = L∗ = W˜ ∞ > y + δW˜ T − 1 Hence (∗ is a battle of the sexes game. If T is finite and δ > 21 , we have that the sum ET (cf. the proof of Proposition 3 ) tends to zero as T → ∞ Hence K ∗ → W˜ T and L∗ → W˜ T as T → ∞ and hence (∗ is a battle of the sexes game if T is sufficiently large. Proof of Proposition 3. If δ ≤ 1/2 Fˆz ≥ 0 and Fz > Fˆz for any other infinite sequence z since this has additional positive terms. Consider now the case δ > 1/2 Let T1 = inft > 1 zt = zˆt+1 Let Tk = inft > Tk−1 zˆt = zˆt+1 We show first that the sequence zˆ changes sign infinitely often if δ > 1/2 so that Tk < ∞ for any k It must clearly change sign once, since
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otherwise the value of F would equal 1 − δ/1 − δ < 0 Hence T1 < ∞. Since ET1 < 0 and ET1 − 1 > 0 we have that 0 > ET1 ≥ δT1 Hence ET1 + δT1 δ/1 − δ > 0 so that T2 < ∞ Extending the same argument, we have that zˆ changes sign infinitely often. signs and ETk = Since ETk T and ETk − 1 have opposite ETk − 1 + δ k it follows that ETk ≤ δTk We now show that Fˆz = 0 which establishes that it is optimal. Given t ETk < 3 if Tk > T Further, for 3 > 0 ∃ T t > T ⇒ δ < 3 Hence Tk < t < Tk+1 , Et < ETk since ETk and E t have the same sign, but Et has added terms which are of the opposite sign of ETk Hence Et < 3∀t > T Hence Fˆz = 0 We now show that there are infinitely many sequences < zt > such that F< zt > = 0 To construct one such sequence, let T > 1 be given and let zt = zˆt ∀t ≤ T and let zT +1 = −zˆT +1 Let wt = zt − zˆt Hence ∞ F< zt > = Fˆz + δT δt−T wt (13) t=T +1
We now chose wt T < t < ∞ from the set −2 2 so that F< zt > = 0 wT +1 is already defined—it equals 2 if zT +1 = −1 and equals −2 if zT +1 = 1 For k > 1 select wT +k = 2zk Hence ∞ F < zt > = Fˆz + 2δT zT 1 + δt zt (14) t=2
= Fˆz +
2δT zT Fˆz
= 0
(15)
Since T was arbitrarily chosen, we have established there are infinitely many sequences < zt > such that F< zt > = 0 Proof of Proposition 4. The proof is by induction. We have proved, in Proposition 1, that the zˆ 1 defines the optimal convention when T = 2 Assume now that zˆ T −1 defines the optimal conventions for GT δ. We show that zˆ T defines the optimal conventions for GT +1 δ Consider an arbitrary sequence z T ∈ Z T and note that Fz T ≥ Ez T −1 + −1k δT where k = 1 if Ez T −1 < 0 and k = 2 if Ez T −1 ≥ 0 We also have that Fzˆ T = Ezˆ T −1 + −1k δT where k is similarly defined. Since Fzˆ T −1 ≤ Fz T −1 it follows that Fzˆ T ≤ Ez T −1 + −1k δT ≤ Fz T zˆ T −1 is also optimal. This implies that eiSupposenow that wT −1 = T T T t−1 t−1 t−1 ther 1 + δ z ˆ = 1 + δ w or 1 + zˆt = −1 + t t t=2 t=2 t=2 δ T t−1 δ w In either case we have that Pδ = 0 where Pδ is a nont t=2 trivial polynomial of degree T − 1 whose coefficients belong to the set −2 0 2 and not all of which are zero. There are finitely many values of δ such that Pδ = 0 Hence we have a unique optimal convention for all but finitely many values of δ
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v. bhaskar REFERENCES
Banks, J., and Calvert, R. (1992). “A Battle-of-the-Sexes Game with Incomplete Information,” Games Econ. Behav. 4, 347–372. Bikchandani, S.(1988). “Reputation in Repeated Second-Price Auctions,” J. Econ. Theory 46, 97–119. Crawford, V., and Haller, H. (1990). “Learning How to Cooperate: Optimal Play in Repeated Coordination Games,” Econometrica 58, 571–595. Dixit, A., and Shapiro, C. (1985). “Entry Dynamics with Mixed Strategies,” in The Economics of Strategic Planning (L.G.Thomas, Ed.), Lexington: Lexington Books. Farrell, J. (1987). “Cheap Talk, Coordination and Entry,” Rand J. Econ. 18, 34–39. Farrell, J., and Saloner, G. (1988). “Coordination through Committees and Markets,” Rand J. Econ. 19, 235–252. Fudenberg, D., and Maskin, E. (1991). “On the Dispensability of Public Randomization in Discounted Repeated Games,” J. Econ. Theory 53, 428–438. Goyal, S., and Janssen, M. (1996). “Can We Rationally Learn to Coordinate?” Theory Decision 40, 29–49. Harsanyi, J., and Selten, R. (1988). A General Theory of Equilibrium Selection in Games. Cambridge, MA: MIT Press. Katz, M., and Shapiro, C. (1985). “Network Externalities, Competition and Compatibility,” Amer. Econ. Rev. 75, 424–440. Klemperer, P. (1998). “Auctions with Almost Common Values: The ‘Wallet Game’ and Its Applications,” Europ. Econ. Rev. 42, 757–769. Kramarz, F. (1996). “Dynamic Focal Points in N-Person Coordination Games,” Theory Decision 40, 277–313. Maynard-Smith, J. (1982) Evolution and the Theory of Games. Cambridge, UK: Cambridge Univ. Press. Sorin, S. (1986). “On Repeated Games with Complete Information,” Math. Oper. Res. 11, 147–160. Warneryd, K. (1992). “Communication, Correlation and Symmetry in Bargaining,” Econ. Lett. 39, 295–300.
Egalitarianism and Efficiency in Repeated Symmetric Games V. Bhaskar1 2 Department of Economics, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, United Kingdom Received September 15, 1997
We analyze the symmetric equilibria of repeated symmetric games where there is a conflict of interests over equilibria—the battle-of-the-sexes or the hawk–dove game are key examples. If one restricts attention to symmetric equilibria, efficient equilibria must be egalitarian. For finitely repeated games, and generic discount factors, there is a unique outcome path which ensures efficiency within the class of symmetric equilibria. This is also true for the infinitely repeated games if the players are sufficiently impatient. Journal of Economic Literature Classification Numbers: C72, C73. © 2000 Academic Press
1. INTRODUCTION This paper analyzes symmetric equilibria in repeated games where the stage game is a symmetric two-player game with a pair of asymmetric pure strategy equilibria. We show that when there is a conflict of interest between the players over the equilibria (as in the battle-of-the-sexes or the hawk–dove games),3 efficient payoffs are obtained when the symmetric repeated game equilibrium is egalitarian. An egalitarian equilibrium is one
1 I thank Sanjeev Goyal, Ted To, and participants at the 17th Arne Ryde Symposium on Focal Points for comments on an earlier draft. I am particularly grateful to an anonymous referee and associate editor for their constructive suggestions. 2 E-mail: [email protected]. 3 The class of games we consider represents several important situations in economics and biology—entry of firms into a natural monopoly industry (Dixit and Shapiro, 1985; Farrell, 1987), product compatibility (Farrell and Saloner, 1988), network externalities (Katz and Shapiro, 1985), and competition between two animals for a scarce resource (Maynard-Smith, 1982).
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v. bhaskar α
β
α
0 0
x 1
β
1 x
y y
FIG. 1. The game G.
which equalizes, as far as possible, the realized payoffs of the two players.4 Moreover, for finitely repeated games, and generic discount factors, there is a unique outcome path which ensures efficiency within the class of symmetric equilibria. This is also true for the infinitely repeated games if the players are sufficiently impatient. For some intuition, consider the battle-of-the sexes, with payoffs as given in Fig. 1. Assume for the moment that x > 1 and y = 0 Hence the stage game has a pair of asymmetric equilibria (α β and β α where player one prefers the former and player two the latter. Let the game be repeated twice and consider symmetric equilibria in the repeated game. In such a symmetric equilibrium, players must choose the same randomization probability in period one, and must also play the mixed equilibrium of the stage game in period two if their realized first period actions coincide. However, symmetry allows the players to play asymmetrically in period two, if their realized actions in period one differ. Consider first the bourgeois convention, where if coordination is achieved in period one upon α β players play the same equilibrium α β in the second period. Under this convention players have a relatively large stake in securing coordination on their own terms, and hence both players play α with high probability. The probability of expost coordination will consequently be relatively low. Compare this with the egalitarian convention, where the players play β α in period two if coordination in period one has taken place on α β This reduces the incentive a player has to try and ensure coordination on her preferred equilibrium. Hence this convention induces the player to play each action with probability close to one-half, so that the probability of expost coordination is higher than under the bourgeois convention—indeed, it will be higher than in mixed equilibrium of the one-shot game. Hence the symmetric equilibrium associated with the egalitarian convention payoff dominates the symmetric equilibrium associated with the bourgeois convention. We show that this idea extends for more general payoffs, and also when the game is repeated many times. We also explore the conditions under which there is a unique egalitarian convention. 4
In a symmetric mixed strategy equilibrium, expected payoffs are always equal for the two players, but realized payoffs can be unequal.
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The crucial restrictions in the analysis are that we consider only symmetric equilibria and also do not allow cheap talk. The restriction to symmetric equilibria in symmetric games reflects the view that coordination on asymmetric equilibria may be impossible and is supported by Harsanyi and Selten’s (1988) principle of symmetry invariance. Farrell (1987) studied symmetric equilibria of the metagame when a game such as G is preceded by pre-play communication. He showed that cheap talk could achieve a degree of asymmetric coordination in the battle-of- sexes, but is completely ineffective in the hawk–dove game.5 We find that repeated interaction is always effective, independent of the extent of conflict in the stage game. 6 The remainder of this paper is organized as follows. Section 2 considers the twice-repeated game and shows that an egalitarian convention is uniquely optimal. Section 3 discusses the case of general finite and infinite repetitions, and the final section concludes. 2. THE TWICE REPEATED GAME We consider a symmetric two-person game, where each player has action set A = α β and payoff function u A2 → R as given in Fig. 1. Assume x > y so that α β and β α are the two pure strategy (strict) Nash equilibria. Observe that G encompasses the class of 2 × 2 symmetric games with a pair of asymmetric strict equilibria—the payoffs 0 and 1 are simply normalizations given that these are von Neumann–Morgenstern utilities. G x−y also has a mixed equilibrium where α is played with probability p = 1+x−y x and the payoff in this mixed equilibrium is u¯ = 1+x−y The game G has equilibrium conflict of interest if x = 1—this case is the main focus of this paper. We shall call the game G the battle-of-the-sexes if y ≤ 1 if y > 1 we call it the hawk–dove game. Despite equilibrium conflict of interest, players still have common interests in achieving coordination on one of the pure strategy Nash equilibria. This is particularly true in the case of the battleof-the-sexes, since in this case either pure equilibrium Pareto-dominates the payoff in the mixed strategy equilibrium (u. ¯ In the hawk–dove game, x > u¯ > 1, so that the mixed equilibrium is not Pareto-dominated by a pure equilibrium. However, if y < x+1 any mixed action profile is inferior to the 2 lottery which yields the two pure equilibria with equal probability, so that there remains some commonality of interest in coordinating behavior. 5 Farrell assumes that the messages come from a commonly understood natural language and selects for equilibria which are consistent with this interpretation. Warneryd (1992) and Banks and Calvert (1992) also consider cheap talk in asymmetric games. 6 Repetition and cheap talk may not be directly comparable, since one may take the view that pre-play communication is always possible whereas repetition may or may not be possible.
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The coordination problem in game G arises from the fact that it is not clear which of the two pure strategy equilibria players should expect to be played. The two equilibria are indistinguishable—any argument one may offer in favor of one can equally well be offered in support of the other. In view of this problem—which has been noted by many including Farrell (1987), Harsanyi and Selten (1988), and Warneryd (1992)—it is reasonable to restrict attention to the symmetric equilibrium. More formally, only the symmetric mixed strategy equilibrium of G satisfies Harsanyi and Selten’s criterion of symmetry invariance, which requires that the selected equilibrium be invariant to re-labelling of the players or strategies. 7 Let the game G be played twice, and let players maximize the sum of stage game payoffs, discounted at rate δ. Since the repeated game is also symmetric, we will restrict attention to symmetric subgame perfect equilibria. Let σ σ be such an equilibrium and consider the prescriptions of this profile at the second stage of this game. Clearly, such an equilibrium must (a) play an equilibrium of the stage game after any pair of first period actions and (b) prescribe identical (mixed) actions for both players if the history is symmetric. Hence in any symmetric equilibrium of the two-period game, both players must play the mixed equilibrium of G after the histories α α and β β Consider now a history where the realized actions of the two players differ, e.g., the history α β where player 1 has played α and player 2 has played β In this case, the history from the point of view of player 1 (which is (α β) differs from the history from the point of view of player 2 (which is (β α Hence the continuation action profile induced in period 2 can well be asymmetric.8 A rule which achieves asymmetric coordination by conditioning upon history is called convention. Hence we have two possible conventions. The bourgeios convention requires that players adopt the same actions in period two in the event that their first period actions are coordinated. The egalitarian convention requires that each player chooses a different action from the one chosen in period one (i.e., the players choose β α in period two if their realized first period actions are α β 9 Our key point is that different conventions generate different incentives for first period actions. To see this, write down the payoff matrix of the row player for first period actions under each of these conventions: 7
The equilibrium α β (where player 1 plays α and player 2 plays β can be transformed into the other equilibrium, β αby exchanging the labels of the players, and hence is not symmetry invariant. 8 The role of history as a correlation device for coordinating future actions was first pointed out by Crawford and Haller (1990). We discuss their work in the concluding section. 9 There is also a symmetric equilibrium which ignores the realized asymmetries and plays the mixed equilibrium in period two, after all histories. Call this convention the mixed convention. In the proof of Proposition 1 we show that this equilibrium is never optimal.
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First period payoffs of the row player under Bourgeois Convention α β
α
β
δu¯ 1+δ
1 + δx δu¯ + y
First period payoffs of the row player under Egalitarian Convention α β
α
β
δu¯ 1 + δx
x+δ δu¯ + y
Observe that the two conventions are not equivalent as long as there is any conflict of interest over equilibria between the players, i.e., as long as x = 1Restricting attention to the symmetric equilibrium of each of these payoff matrices, we see that for any of the above matrices, such an equilibrium always exists and is unique. This symmetric equilibrium is mixed under the bourgeois convention if x − 1 + δy − 1 > 0 otherwise, it is pure and plays α with probability one. Similarly, the symmetric equilibrium is mixed under the egalitarian convention if x − y − δ1−y > 0 otherwise, 1+x−y it is pure and plays β with probability one. Observe that in the battle of the sexes, since y ≤ 1, both conventions imply randomized choices in period 1 This is not necessarily the case in the hawk–dove game, since one or both of these conventions may require deterministic choices in period one. The following proposition shows that in all instances, the egalitarian convention is optimal. Proposition 1. The symmetric equilibrium with the egalitarian convention is always optimal. It is uniquely optimal if x = 1 Proof. Let σ1b resp. σ1e be the probability with which α is played in period one under the bourgeois (resp. egalitarian) convention. Consider first the case where x − 1 + δy − 1 > 0 and x − y − δ1−y > 0 so that 1+x−y both conventions imply randomization in the first period. The equilibrium payoffs are given by V σ b σ b =
δu1 ¯ + δx − δu¯ − y + 1 + δx1 + δ − δu ¯ 1 + δx − δu¯ − y + 1 + δ − δu ¯
(1)
δux ¯ + δ − δu¯ − y + x + δ1 + δx − δu ¯ 1 + δx − δu¯ − y + 1 + δ − δu ¯
(2)
V σ e σ e =
The difference in payoffs is given by V σ e σ e − V σ b σ b =
δx − 12 ≥ 0 1 + δx − δu¯ − y + 1 + δ − δu ¯
(3)
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Hence for this range of parameter values the egalitarian convention is always optimal, and uniquely so if x = 1 Note that if either convention prescribes a pure action, this implies that y > 1 Suppose that both conventions require pure actions in the first period. In this case, the payoff to the bourgeois convention is δu¯ whereas the payoff to the egalitarian convention δu¯ + y which is strictly greater. Suppose that egalitarian convention has pure actions in period one, whereas the bourgeois convention has random actions. In this case V σ e σ e = δu¯ + y while the V σ b σ b is a convex combination of δu¯ + y and 1 + δ with strictly positive weights. Since y > 1 and u¯ > 1 δu¯ + y > 1 + δ and hence σ e is strictly optimal. Finally, if the egalitarian convention has random actions, its payoff is a convex combination of 1 + δx and y + δu ¯ while the bourgeois convention’s payoff is δu ¯ Since 1 + δx > δu¯ and y + δu¯ > δu ¯ the egalitarian convention is also strictly optimal in this case. We now show that the egalitarian convention is strictly better than the symmetric equilibrium, σ m where players play the mixed strategy equilibrium of the G in period 1 and in period 2 at all information sets. To see this, write the payoff matrix to first period actions under this equilibrium, as First period payoffs of the row player under Mixed Convention α β
α
β
δu¯ 1 + δu¯
x + δu¯ δu¯ + y
Suppose that σ1e is pure and plays β with probability one, which implies y > 1 In this case the V σ e σ e = y + δu ¯ whereas V σ m σ m equals the payoff to playing β in the above matrix and is a convex combination of y + δu¯ and 1 + δu¯ and is hence strictly less. Suppose that σ1e is mixed and consider two subcases. Suppose that y ≥ 1 so that y ≥ u¯ ≥ 1 In this case it can be verified that p > σ1e . Consider the payoff to playing β under both these ¯ +1 − py + δu ¯ while V σ e σ e = equilibria. V σ m σ m = p1 + δu e e e e σ1 1 + δx + 1 − σ1 y + δu ¯ Hence V σ σ − V σ m σ m = p − e e σ1 y − 1 + σ1 δx − u ¯ > 0 Suppose that y < 1 so that y < u¯ < 1 Consider the payoff to playing α under both these equilibria. V σ m σ m = pδu¯ +1 − px + δu, ¯ while V σ e σ e = σ1e δu¯ + 1 − σ1e x + δ Since x + δ > x + δu ¯ V σ m σ m ≥ V σ e σ e ⇒ σ1e > p Consider the payoff to playing β under both ¯ +1 − py + δu ¯ while these equilibria. V σ m σ m = p1 + δu V σ e σ e = σ1e 1 + δx + 1 − σ1e y + δu ¯ Since 1 + δx > 1 + δu¯ > y + δu ¯ V σ m σ m ≥ V σ e σ e ⇒ σ1e < p Hence we have a contradiction, and V σ m σ m < V σ e σ e
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Although the egalitarian convention gives rise to the highest payoffs (in the set of symmetric equilibria), it does not always yield the highest probability of expost coordination in period one. To see this, let x − y < 1 which implies that p < 21 Hence α is more risky than β and is played with probability less than one in the mixed equilibrium of the stage game. In this case, the egalitarian convention makes playing α even less attractive, and hence α is played with probability less than p in the first period. On the other hand, the bourgeois convention raises the probability with which α is played and yields a higher probability of expost coordination in period 1. Nevertheless, the payoffs are greater under the egalitarian convention, since the payoffs when not coordinated also matter—for example, if y > 0, the payoffs to the profile β β are greater than to the profile α α and hence reducing the probability of playing α raises payoffs. A referee has pointed out that the egalitarian convention is optimal even if there are some increasing returns or learning by doing in resource use, so that the bourgeois convention is expost optimal. Consider a contest between two players for a fixed resource, which takes the form of the hawk–dove game which is played in each of two periods. Assume that in period one, payoffs are given by the game G in Fig. 1, where the value of the resource is x > 1. These are also the payoffs in period two, in the event of the first period actions being α α or β β However, in the event that the first period action profile is α β the period two payoffs from playing α β again are λx (λ > 1 and 1 to players 1 and 2, respectively, while the payoffs to playing β α are 1 and x 10 Hence there is learning by doing in resource use. In this case, the bourgeois convention is expost efficient since it will result in a larger total surplus in period two, as compared to the egalitarian convention. However, the bourgeois convention results in more aggressive behavior and lower payoffs in period one. The payoffs under the two conventions can be computed using Eq. (6) in the appendix. As an illustrative example, we derive a sufficient condition for the egalitarian convention to be optimal in this case, given that there is no discounting δ = 1 For the egalitarian convention to be optimal, it is sufficient that 1 + x2 ≥ 2x1 + λ i.e., if λ ≤ λ∗ = 1 + x2 /2x This critical value always exceeds one (since x > 1 and is increasing in x For example, if x = 2, λ∗ = 54 and if x = 5 λ∗ = 13 Hence egalitarianism is 5 optimal even if there is a significant amount of learning by doing.11 Our analysis assumes that a player only observes his opponent’s realized action, and not his mixed strategy, and applies more generally, e.g., 10
Period two payoffs in the event of β α being played in period one are given symmetrically. 11 By using similar arguments as in the proof of Proposition 1, one can verify that the egalitarian convention is better than the mixed convention for any value of λ
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to games with incomplete payoff information. For example, in repeated auctions one typically observes the opponent’s bids, but not the map from valuation to bids. Second price common value auctions often have asymmetric equilibria which yield low revenues to the seller as compared to the symmetric equilibrium (see Bikchandani, 1988, and Klemperer, 1998). Our analysis suggests that in a twice-repeated second price common value auction, collusion can be sustained in a symmetric equilibrium, where the loser in round one bids aggressively, while the winner in round one bids less aggressively. This will result in lower seller revenues also in round one, since both players have an incentive to bid less aggressively. 3. MANY REPETITIONS Let GT δ be the T -fold repetition of G, where the discount rate δ ∈ 0 1 if T is finite, and δ ∈ 0 1 if T = ∞ Payoffs in the repeated game are given by the discounted sum of stage game payoffs. A player’s history at period t ht is the sequence a1 b1 at − 1 bt − 1 where aτ denotes a player’s own action in period τ, while bτ denotes his opponent’s action. Let Ht denote the space of all t-period histories for a player. For each ht ∈ Ht let ηht denote the corresponding history for the other player, i.e. ηht is obtained by permuting aτ and bτ for each τ A behavior strategy for a player in GT δ is a sequence of functions st Tt=1 where s1 ∈ A is the player’s first period strategy and for t > 1 st Ht → A Let T be the set of behavior strategies in GT . Write V T σ θ for the payoff of strategy σ against strategy θ in GT δ A strategy profile σ θ is a Nash equilibrium if V T σ θ ≥ V T σ θ∀σ ∈ T and V T θ σ ≥ V T θ σ∀θ ∈ T σ θ is a subgame perfect equilibrium if it is a Nash equilibrium and ∀ht ∈ Ht the profile σht θηht is a Nash equilibrium of GT −t+1 . A subgame perfect equilibrium σ θ is symmetric if σ = θ Let T be the set of subgame perfect equilibria of GT δ, and let W T be the set of payoffs generated by T —each element in this set consists of a pair of payoffs, one for each player. Let TS denote the set of symmetric subgame perfect equilibria, and let WST be the set of payoffs generated by TS It will be convenient to let the elements of WST be a single number which denotes the payoffs of both players. Our focus is on symmetric equilibria, i.e., on the set TS We call a symmetric equilibrium σ σ ∈ TS optimal if V T σ σ ≥ V T θ θ for any θ θ ∈ TS Any such equilibrium gives rise to an outcome path, i.e., a probability distribution over A × AT when it is played. We say that an optimal equilibrium σ σ has a uniquely optimal outcome path if V T σ σ > V T θ θ for any symmetric equilibrium θ θ which has a different outcome path from σ σ
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−1 A convention in a T − 1 period repeated game is a sequence γτ Tτ=1 where γτ ∈ α β β α∀τ Since α β is a Nash equilibrium of the stage game, it is clear that any convention can be supported as the outcome path of a subgame perfect equilibrium—players can simply ignore deviations and continue with the convention. Any payoff vector which is generated by a convention is hence always an element of W T −1 We shall call a convention optimal if it is the continuation path of an optimal equilibrium, conditional upon an asymmetric action profile being played in the first period. Q = q1 q2 q1 = x + Tt=2 δt−1 t x + 1 − t 1 q2 = 1 + TLet t−1 t 1 + 1 − t x t ∈ 0 1 2 ≤ t ≤ T be the set of payoff t=2 δ pairs which arise when players adopt some convention, given that their realized actions are α β at the initial date. Q is a finite set if T is finite, but can be infinite if T = ∞ Let K ∗ L∗ = arg minq1 q2 ∈Q q1 − q2 —this is the payoff pair which arises in an egalitarian convention12 Let V¯ T −1 denote the maximal element in WST −1 i.e., it is the maximal symmetric subgame perfect equilibrium payoff in the T − 1 period repeated game. Consider the following outcome path in the T period game, where the players randomize at the initial date. If their actions coincide, they play an optimal symmetric equilibrium of the T − 1 period game in the next period. If their actions differ, they play the egalitarian convention. Hence the payoffs to first period actions of the row player are given by the derived game (∗ given as
α α β
The Derived Game (∗ β
δ V¯ T −1 δ V¯ T −1 L∗ K ∗
K ∗ L∗ y + δ V¯ T −1 y + δ V¯ T −1
We now show that, under certain conditions, the optimal symmetric equilibrium of the T period game has an outcome path which corresponds to the symmetric completely mixed equilibrium of the derived game (∗ Proposition 2. Suppose that either (i) y ≤ 1 or (ii) y < x+1 δ > 21 and 2 T is sufficiently large. Then in any optimal symmetric equilibrium of GT δ the players adopt egalitarian convention if their initial actions differ. Furthermore, they play an optimal symmetric equilibrium of GT −1 δ if their initial actions coincide. Proof. 12
See the Appendix.
We shall show such an egalitarian convention always exists, i.e., the minimization problem defining it has a solution.
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This proposition establishes that if stage game G is the battle-of-thesexes or if it is a hawk–dove game with sufficient commonality interest and where the future is sufficiently important, the outcome path corresponding to an optimal symmetric equilibrium has a simple characterization. Players randomize symmetrically; if their realized actions differ, they play either α β or β α in each period, where the path is chosen so as to equalize, as far as possible, the payoff streams of the two players. In other words, the optimal convention is egalitarian. If the realized actions in the initial period coincide, the players continue with the optimal symmetric equilibrium of the T − 1 period game.13 We now explicitly solve for an egalitarian convention and investigate the conditions under which this is unique. Our construction is very similar to that used in the folk theorems of repeated games without public randomization, e.g., Fudenberg and Maskin (1991, Lemma 1) and Sorin (1986). Suppose that the realized action profile in the initial period is α β Any convention can be represented by a sequence < zt > where zt ∈ −1 1∀t 2 ≤ t ≤ T where zt = 1 if α β is to be played and zt = −1 if β α is to be played. Let Z T = < zt > zt ∈ −1 1∀t 2 ≤ t ≤ T be the set of all such sequences, and let F Z T → 0 ∞ be defined by T T t Fz = 1 + δ zt (4) t=2 Hence an egalitarian convention minimizes the function F over the set ZT ∞ We now construct an infinite sequence zˆ =< zˆt >∞ t=2 ∈ Z which allows us to construct optimal conventions in finitely and infinitely repeated games. Given any such infinite sequence, define the associated partial sum Eτ by Eτ = 1 +
τ t=2
δt zˆt
(5)
The sequence zˆ is defined as follows: for t ≥ 2 zˆt = 1 if Et − 1 ≤ 0 and zˆt = −1 if Et − 1 > 0 We shall now show that the sequence zˆ defines an optimal convention in the infinitely repeated game, while its T − 1 period truncation defines an optimal convention for the T period repeated game. Intuitively, the convention defined by zˆ keeps track of the discounted sum 13
This proposition does not fully characterize optimal behavior for arbitrary hawk–dove games. However, some cases are straightforward—if 2y ≥ 1 + x it is efficient for the players to play β β. To support this in equilibrium, if player 1 deviates, then the players can revert to playing β α for the remaining periods of the game (this minmaxes player 1). It is routine to verify that the profile can be supported if the players are sufficiently patient and T is sufficiently large.
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of payoffs to the players at each date. Suppose, for example, that x > 1 so that player one’s preferred equilibrium is α β If the realized actions in period one are α β the players play β α in succeeding periods until the total discounted payoffs of player two exceed player one’s total discounted payoffs. At this point players switch to playing α β until player one’s total payoffs exceeds player two’s, and so on. Proposition 3. In the infinitely repeated game, the sequence zˆ defines an egalitarian convention. This convention is the unique egalitarian convention if and only if δ ≤ 1/2 Proof.
See the Appendix.
When T is finite, the finite sequence zˆ T −1 , which is given by the first T − 1 terms of zˆ defines an egalitarian convention Furthermore, it is also easy to show that this is the unique egalitarian convention, for generic values of δ If T is finite, no convention will be able to completely equalize the payoffs K ∗ and L∗ in the derived game (∗ Also, for generic values of the discount factor, no two distinct conventions, which yield distinct streams of future payoffs, will have the same present value. This ensures that there is a unique egalitarian convention.14 Proposition 4. If T is finite, the truncated sequence zˆ T −1 defines an egalitarian convention. This convention is the unique egalitarian convention for almost all values of δ Proof.
See the Appendix.
Our analysis highlights a novel distinction between finitely repeated and infinitely repeated games. In the finitely repeated game, the conflict of interest between the players can be reduced substantially, but is generically never completely eliminated. In consequence, under the conditions set out in Proposition 2, there is a unique optimal convention which minimizes this conflict. In infinitely repeated games, the conflict between players can be completely eliminated if the players are patient, and hence non-uniqueness follows. This is also related to the nature of the Folk theorems for the two types of games. In the infinite case, any individually rational payoff can be exactly achieved if players are sufficiently patient, whereas in the finite case, this payoff can only by approached by taking a sequence of games of increasingly longer length. Hence uniqueness of an optimal convention arises in those contexts where an “exact” Folk theorem does not apply. This in 14
Uniqueness cannot be ensured if players are completely patient, i.e., if δ = 1. If T is an even number, then note that zˆ T −1 defines perfectly egalitarian convention where players alternate between the equilibria. If T ≥ 4, perfect egalitarianism can also be achieved in other ways, e.g., by playing β α in the last T2 periods, and α β before that date.
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turn is related to the fact that mixed strategies are unobservable. The key to an optimal strategy lies in players randomizing initially with probability as close to one-half as possible. If mixed strategies were observable it would be easy to enforce randomization with probability one-half, even in a finitely repeated game. For example, any relatively egalitarian convention could be coupled with punishments—e.g., if one player deviates from this randomization, players could play the other player’s preferred equilibrium thereafter. Since many conventions could ensure the appropriate randomization, non-uniqueness would follow. Hence the unobservability of randomized actions plays an important role in ensuring uniqueness. 4. CONCLUSIONS The role of repeated interaction in solving the coordination problem was first suggested by Crawford and Haller (1990), who considered games of pure coordination, without conflict of interest (in Fig. 1, a pure coordination game is one where x = 1 and y = 015 They argued that the two pure equilibria α β and β α are indistinguishable, but suggested that players could randomize, and in the event of that their realized actions were coordinated, they could “maintain coordination forever once they locate a pair of coordinated actions. They can do this either by repeating those actions or by alternating deterministically between them and the other coordinated pair” (Crawford and Haller, 1990, p. 575, emphasis added). This paper has extended their approach to games which involve a conflict of interest over equilibria, and shown that this conflict of interest can be minimized via egalitarian conventions. The Crawford–Haller approach has been criticized (see Goyal and Janssen, 1996, since there is multiplicity of possible conventions which maintains coordination. With conflict of interests over the equilibria, we find that there often exists a unique optimal convention—this is always the case in the twice repeated game and generically with finitely many repetitions. Our approach to symmetric games relies on the use of history to “break” the symmetry and studies the optimal use of history in this context. Alternatively, in any real context, there may be other cues such as gender, race, or the age of the players, although there may be several such cues and their significance may also not be unambiguous. Consider now a repeated interaction where players also have access to such cues. Our analysis suggests that these cues will be more readily accepted if they are combined with an egalitarian convention, but are less likely to be taken if combined with a bourgeois convention. 15
Kramarz (1996) extends their approach to N person pure coordination games.
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To summarize, our substantive point is that if we restrict attention to symmetric equilibria, efficiency requires egalitarianism. In such contexts, bourgeois conventions, where property rights are entrenched by past precedent, have poor incentive effects. Egalitarian conventions minimize the conflict of interest and provide good incentives for coordination. While the analysis of this paper focuses on repeated games, this point is more general and applies to a variety of contexts such as the allocation of property rights. For example, pollution permits are often allocated on the principle of “grandfathering,” where entitlements are related to past emissions. In a world where property rights are not immutable, such criteria have bad incentive effects, by providing a dynamic incentive to increase pollution APPENDIX Proof of Proposition 2. Consider an arbitrary derived game ( with payoffs as given below, where a = δw b = y + δv for some w v ∈ WST −1 and ∃r s ∈ W T −1 K = x + δr and L = 1 + δs The Derived Game ( α β α β
a a L K
K L b b
Consider the symmetric equilibria of ( If a ≥ L, a is a symmetric equilibrium payoff, and if b ≥ K b is a symmetric equilibrium payoff. If K > b and L > a ( has a pair of asymmetric pure equilibria and a symmetric mixed equilibrium. The payoff in this mixed equilibrium is given by KL − ab (6) K+L−a−b Call ( a battle-of-the-sexes game if minK L ≥ maxa b We have that K − aL − a (7) UK L a b − a = K+L−a−b UK L a b =
K − bL − b (8) K+L−a−b Hence the payoff in the mixed equilibrium of ( is greater than any symmetric payoff of ( (i.e., a or b) provided that the derived game ( is a battle of sexes game. We also have that UK L a b − b =
L − aL − b ∂U = ∂K K + L − a − b2
(9)
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v. bhaskar ∂U K − aK − b = ∂L K + L − a − b2
(10)
∂U K − bL − b = ∂a K + L − a − b2
(11)
K − aL − a ∂U = ∂b K + L − a − b2
(12)
Hence U is increasing in all its arguments if ( is a battle-of-the sexes game. We now show that if minK ∗ L∗ ≥ maxδV¯ T −1 y + δV¯ T −1 so that ∗ ( is a battle-of-the-sexes game, the claim applies, i.e., V¯ T = U ∗ where U ∗ is the mixed equilibrium payoff in the derived game (∗ From (7) and (8) we have that U ∗ is greater than both δ V¯ T −1 and y + δ V¯ T −1 Consider any other derived game ( where a ≤ δV¯ T −1 and b ≤ y + δV¯ T −1 Hence any pure equilibrium payoff of ( is less than U ∗ Furthermore, if ( is a hawk–dove game, then the payoff in the mixed symmetric equilibrium of ( is less than maxδV¯ T −1 y + δV¯ T −1 and hence less than U ∗ If ( is a battle-of-the-sexes game, (9)–(12) and (6) show that this payoff must also be less than U ∗ 1+x Let W˜ T = 1 + x1 − δT +1 /21 − δ if T is finite and W˜ T = 21−δ if T = ∞ We now show that (∗ is a battle of sexes game under the conditions of the proposition. Suppose that y ≤ 1, so that minK ∗ L∗ ≥ minx+ W˜ T − 1 1+ W˜ T − 1 Note that since y < x+1 and 0 < x+1 the per-period 2 2 payoff in any symmetric equilibrium of the T − 1 period repeated game must be less than x+1 Hence W˜ T − 1 > V¯ T −1 Hence minK ∗ L∗ > 2 T −1 T −1 ¯ ¯ y + δV maxδV Consider now the case where y > 1 Clearly δ V¯ T −1 ≤ δW˜ T − 1 and y + δV¯ T −1 ≤ y + δW˜ T − 1 If T = ∞ and δ > 21 Proposition 3 establishes that a perfectly egalitarian convention exists, so that K ∗ = L∗ = W˜ ∞ > y + δW˜ T − 1 Hence (∗ is a battle of the sexes game. If T is finite and δ > 21 , we have that the sum ET (cf. the proof of Proposition 3 ) tends to zero as T → ∞ Hence K ∗ → W˜ T and L∗ → W˜ T as T → ∞ and hence (∗ is a battle of the sexes game if T is sufficiently large. Proof of Proposition 3. If δ ≤ 1/2 Fˆz ≥ 0 and Fz > Fˆz for any other infinite sequence z since this has additional positive terms. Consider now the case δ > 1/2 Let T1 = inft > 1 zt = zˆt+1 Let Tk = inft > Tk−1 zˆt = zˆt+1 We show first that the sequence zˆ changes sign infinitely often if δ > 1/2 so that Tk < ∞ for any k It must clearly change sign once, since
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otherwise the value of F would equal 1 − δ/1 − δ < 0 Hence T1 < ∞. Since ET1 < 0 and ET1 − 1 > 0 we have that 0 > ET1 ≥ δT1 Hence ET1 + δT1 δ/1 − δ > 0 so that T2 < ∞ Extending the same argument, we have that zˆ changes sign infinitely often. signs and ETk = Since ETk T and ETk − 1 have opposite ETk − 1 + δ k it follows that ETk ≤ δTk We now show that Fˆz = 0 which establishes that it is optimal. Given t ETk < 3 if Tk > T Further, for 3 > 0 ∃ T t > T ⇒ δ < 3 Hence Tk < t < Tk+1 , Et < ETk since ETk and E t have the same sign, but Et has added terms which are of the opposite sign of ETk Hence Et < 3∀t > T Hence Fˆz = 0 We now show that there are infinitely many sequences < zt > such that F< zt > = 0 To construct one such sequence, let T > 1 be given and let zt = zˆt ∀t ≤ T and let zT +1 = −zˆT +1 Let wt = zt − zˆt Hence ∞ F< zt > = Fˆz + δT δt−T wt (13) t=T +1
We now chose wt T < t < ∞ from the set −2 2 so that F< zt > = 0 wT +1 is already defined—it equals 2 if zT +1 = −1 and equals −2 if zT +1 = 1 For k > 1 select wT +k = 2zk Hence ∞ F < zt > = Fˆz + 2δT zT 1 + δt zt (14) t=2
= Fˆz +
2δT zT Fˆz
= 0
(15)
Since T was arbitrarily chosen, we have established there are infinitely many sequences < zt > such that F< zt > = 0 Proof of Proposition 4. The proof is by induction. We have proved, in Proposition 1, that the zˆ 1 defines the optimal convention when T = 2 Assume now that zˆ T −1 defines the optimal conventions for GT δ. We show that zˆ T defines the optimal conventions for GT +1 δ Consider an arbitrary sequence z T ∈ Z T and note that Fz T ≥ Ez T −1 + −1k δT where k = 1 if Ez T −1 < 0 and k = 2 if Ez T −1 ≥ 0 We also have that Fzˆ T = Ezˆ T −1 + −1k δT where k is similarly defined. Since Fzˆ T −1 ≤ Fz T −1 it follows that Fzˆ T ≤ Ez T −1 + −1k δT ≤ Fz T zˆ T −1 is also optimal. This implies that eiSupposenow that wT −1 = T T T t−1 t−1 t−1 ther 1 + δ z ˆ = 1 + δ w or 1 + zˆt = −1 + t t t=2 t=2 t=2 δ T t−1 δ w In either case we have that Pδ = 0 where Pδ is a nont t=2 trivial polynomial of degree T − 1 whose coefficients belong to the set −2 0 2 and not all of which are zero. There are finitely many values of δ such that Pδ = 0 Hence we have a unique optimal convention for all but finitely many values of δ
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v. bhaskar REFERENCES
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