Elsevier

Games and Economic Behavior

Volume 119, January 2020, Pages 1-14
Games and Economic Behavior

Myopic perception in repeated games

https://doi.org/10.1016/j.geb.2019.10.003Get rights and content

Abstract

In obtaining the celebrated folk theorem, not only everyone must value his future sufficiently high, but also everyone must be perceived so by the others. This common perception of players' time preferences must be maintained even after someone deviates. This paper explores the implications of myopic perception in repeated games with perfect monitoring. Under myopic perception, a deviator will not be perceived as a long-run player in the continuation game, which in turn affects the equilibria in the repeated game before deviation. We develop an algorithm to compute a set that characterizes almost all equilibrium payoffs when the discount factor is sufficiently high. When the stage game has a unique Nash equilibrium and it strictly dominates all other stage-game payoffs, then under myopic perception, the corresponding repeated game has a unique subgame perfect equilibrium.

Introduction

It has been recognized for long that numerous new equilibria emerge when a stage game is played repeatedly. The well-known folk theorem of Fudenberg and Maskin (1986) asserts that under certain conditions, any feasible and strictly individually rational payoff vector of a stage game can be sustained by a subgame perfect equilibrium in the corresponding infinitely repeated game with perfect monitoring when all the players value their future payoffs high enough. It is often the case that the set of equilibrium outcomes grows as the players value their future payoffs more and more. These new equilibria include not only those that are mutually better for all the players than the equilibrium outcomes in the stage game but also many that are worse for everyone. This general result provides little guidance on which equilibrium to adopt when applying repeated game models. It has been a long-standing issue in selecting an equilibrium when applying this model to analyze repeated interactions, and how to systematically discredit the “bad” equilibria while persevering the “good” equilibria in repeated games.1 In this paper, we put forward an idea of myopic perception and examine its impact on the set of equilibria in repeated games.

How do the new equilibria emerge when a (stage) game is played repeatedly? A fundamental force to support these new equilibria in a repeated game is that everyone perceives that every player values his future payoffs high enough, as otherwise, no one would have any incentive to play anything but his best responses in the stage game even if everyone knows that the overall situation is a repeated game. Even after someone has deviated, which should never happen under the perception that every player values the future, equilibrium strategy works only if all the players maintain their initial common perception. This needs to be the case no matter how many times the same player has deviated from what he was supposed to play and the other players have to trust the deviating player to cooperate in his own punishments, which sounds paradoxical.

In this paper, we take a fresh look at players' perceptions of each other's time preference during the course of a prescribed subgame perfect equilibrium. More specifically, in order to support an outcome that is not a Nash equilibrium of the stage game, players must trust each other, in the sense that everyone values his future high enough, in other words, everyone believes that everyone is a long-run player. However, once someone deviates from what was supposed to be played according to the strategy profile, the other players will no longer hold the same perception on the deviating player for being a long-run player. In the continuation game, the deviator will be treated as a short-run player who will always play his myopic best response to whatever the other players play in every future period. On the other hand, if everyone else treats you as a myopic player, you cannot do any better than just behaving as if you were a short-run player. This line of reasoning will limit what can be supported in the continuation games, which in turn affects what can be supported as an equilibrium before the players lose their trust on a deviating player.

One possible way to justify this idea of myopic perception is to consider a repeated game framework with incomplete information about players' time preferences.2 More specifically, consider the case where every player may become a myopic short-run player permanently with some small probability ε>0. Myopic perception can be supported by a simple belief system such that any player who deviates has become a short-run player, regardless of the nature of this player's deviation.3 Consider an equilibrium under myopic perception under either pure strategies or observable mixed strategies. When all (finite) equilibrium conditions hold with strict inequalities, which is the case in most of our constructed subgame perfect equilibria, then for ε>0 small enough, the equilibrium strategies, particularly for the long-run players, will continue to be sequentially rational under the aforementioned simple belief system. Therefore, an equilibrium embedded with myopic perception could be viewed as the limiting case when the probability of any player becoming myopic goes to zero.

Our goal in this paper is to investigate the implication of myopic perception on the equilibrium outcomes in repeated games. To illustrate how powerful this simple and yet intuitive argument of myopic perception can be, consider the following example:

Example 1

In the infinitely repeated game with the following stage game:LRT2,20,1B1,02,2 under myopic perception, playing (T,L) in every period is the unique subgame perfect equilibrium for all discount factor δ(0,1).

First observe that playing (T,L) is the unique Nash equilibrium in the stage game and that the equilibrium payoff vector (2,2) strictly dominates all other feasible payoffs. According to the folk theorem of Fudenberg and Maskin (1986), every feasible payoff vector that strictly dominates the minmax point (0,0) can be supported by a subgame perfect equilibrium in the corresponding repeated game when the discount factor δ is sufficiently close to 1. For example, to support a symmetric payoff vector that is strictly dominated by (2,2), we can adopt a simple strategy profile (see Abreu, 1988) where the two players play their mutual minmax outcome (B,R) for some number of periods followed by the stage-game Nash equilibrium (T,L) forever. This simple strategy profile calls to restart this sequence of plays after any player deviates in any period.

Now consider any possible feasible and strictly individually rational payoff vector other than (2,2), i.e., (T,L) is not played in every period during the repeated game. In some period, one of the two players, say player 1 (the row player), deviates, then player 2 (the column player) will perceive that player 1 becomes a short-run player in the continuation game where player 1 will always play his myopic best response T. This will fundamentally change the continuation game, which is equivalent to a repeated game with the following “reduced” stage game after taking player 1's myopic best response into account:LRT2,20,1 In such a continuation game, there is obviously a unique subgame perfect equilibrium, where (T,L) will be played in every period after player 1 deviates. Accordingly, player 1 must expect to receive no less than 2 in the original subgame perfect equilibrium. By the same argument, player 2 should not receive less than 2 either in any equilibrium of the original repeated game. In the original repeated game, however, there is only one payoff vector, namely (2,2) from playing (T,L) in every period, that fulfills both players' incentive constraints that a player will not receive anything less than 2 in the continuation game after he deviates. Consequently, playing the stage-game Nash equilibrium (T,L) in every period is the only subgame perfect equilibrium in this repeated game. Although this example is quite specific in the sense that every player has a dominant strategy and there is a unique subgame perfect equilibrium outcome in any continuation game after someone deviates, this striking result demonstrated here is quite robust as our Proposition 7 later establishes. 

In this paper, we systematically explore how significant this simple argument of myopic perception is in the infinitely repeated games with discounting under perfect monitoring. As Example 1 illustrates, myopic perception places certain behavioral assumption on the players after someone has deviated from what all players implicitly agreed to follow initially under the perception that everyone is a long-run player with a significantly high discount factor. We first show that any subgame perfect equilibrium will ensure that no one receives less than a newly defined minmax value, so every equilibrium outcome must be at least (weakly) individually rational under this new interpretation. In the same flavor as the traditional folk theorems, we establish a folk theorem result for our repeated game model: when the discount factor is sufficiently close to one, every feasible and strictly individually rational payoff vector can be supported by a subgame perfect equilibrium. Given our first result, our folk theorem characterizes almost all subgame perfect equilibrium outcomes in the repeated games under myopic perception. Payoffs that are not covered by these two results are those that are weakly but not strictly individually rational, and the measure of those payoffs is zero. For stage games with a unique Nash equilibrium payoff vector that strictly dominates all other feasible payoffs, such as Example 1, we show that the corresponding repeated games have a unique subgame perfect equilibrium for any discount factor. We will also present a number of intriguing examples to demonstrate our results.

Although our repeated game model cannot be directly handled as a traditional stochastic game, game dynamics in our model is still traceable according to the set of perceived short-run players. We will follow Fudenberg et al. (1990) to identify what payoffs are deemed feasible for any set of short-run players. Individual rationality, however, has a new interpretation. Given the current situation, we derive each player's least possible equilibrium payoff and use this value to define this player's “individual rationality”. To do this, we first pin down every long-run player's least possible equilibrium payoff because it is determined by the continuation game after he is perceived to be an additional short-run player. Feasibility then limits how much every short-run player can receive. Based on the original stage game and the current set of short-run players, we develop an algorithm to compute every player's least possible equilibrium payoff, still called minmax values for convenience, for all possible circumstances.

The rest of this paper is organized as follows. Section 2 layouts preliminaries of the repeated game model. The formal definition of myopic perception and some simple properties of the equilibrium under myopic perception are provided in Section 3. In Section 4, we look into issues regarding feasibility and, particularly, individual rationality of the equilibrium outcomes, and how myopic perception can affect the set of subgame perfect equilibrium outcomes. In Section 5, we establish a folk theorem for subgame perfect equilibrium outcomes based on our new notions of feasibility and individual rationality and demonstrate the necessity of every condition in this theorem. We conclude in Section 6 with a few remarks.

We adopt a rather simple and yet intuitive form of this idea in the sense that as soon as a player deviates from a prescribed strategy profile, all the other players start to treat the deviating player as a short-run player in the future. If no one else deviates any further, then the continuation game resembles the repeated game model with long-run and short-run players of Fudenberg et al. (1990). Different from their model where the identities of long-run and short-run players are exogenously given, our repeated game model may further evolve as other players deviate to increase their stage game payoffs. Similar to the equilibrium strategies in their model, the perceived myopic players in our model cannot do any better than behaving as myopic players because after a player is perceived as a myopic player, deviating away from his myopic best response may only lower his payoff in the current period without any impact on his future payoffs. In other words, even for a long-run player, once he was perceived as a myopic player, he would have no incentive to behave non-myopically. For a different environment, Ali and Miller (2016) study ostracism, where non-deviating players continue to cooperate among themselves but refuse to trust deviators in a continuum prisoners' dilemma game played on a network. How deviators are treated in this paper resembles ostracism but in standard repeated game setup.

To analyze subgame perfect equilibrium in this model, we need to consider continuation games with all possible sets of short-run players in order to examine players' equilibrium incentive. Our treatment of players' perceptions for each other's time preference is also inspired by Reny, 1992a, Reny, 1992b, who argues that full rationality cannot be common knowledge throughout many dynamic games. Because the continuation game evolves to a different repeated game with a different set of perceived myopic players as soon as some long-run player deviates, one may consider our model as a stochastic game.4 However, most existing work on stochastic games cannot handle our model because how the game dynamics evolves depends on not only players' actions, but also the strategy profile.

One major departure of the model we study in this paper from the standard repeated game model is on players' time preferences. However, our model is different from most recent development in this area. For example, Chade et al. (2008) study repeated games under perfect monitoring with hyperbolic discounting, while Obara and Park (2017), and Li (2019) consider an even more general class of time preferences in repeated games. Hyperbolic discounting has also been analyzed in other dynamic games, such as in bargaining games by Lu (2016). Since the pioneering work of Lehrer and Pauzner (1999), many researchers investigate differential time preferences in repeated games where players may have different time discount rates when the time interval shrinks, such as Gueron et al. (2011), Chen and Takahashi (2012). Lipman and Wang, 2000, Lipman and Wang, 2009 study both finitely and infinitely repeated games with switching cost and analyze the set of equilibrium outcomes as the time interval of every period shrinks. With switching cost, continuation game also depends on the strategy profile under consideration and players' actions in any given period. Awaya (2017) analyzes repeated games when all players become myopic and demonstrates that the set of equilibrium payoffs is not continuous in the limit as the discount factor goes to zero. Myopic behavior or endogenous discounting in repeated games have also been studied in the literature. For example, Bolt and Tieman (2006) and Kochov and Song (2016) study the case where a player's discount factor for the following period is determined by his action in the current period. Neilson and Winter (1996) assume that one player can change the frequency of interaction hence affect everyone's discount factor systematically. In a repeated prisoners' dilemma game where the players have incomplete information regarding each other's discount factors, Maor and Solan (2015) provide the necessary and sufficient conditions that allow full cooperation in equilibrium that is composed of grim trigger strategies. Gradwohl and Smorodinsky (2019) characterize the set of equilibrium payoffs in two-player repeated games with two-sided incomplete information about time preference, where each player can be either extremely patient in terms of lim-inf criteria or extremely impatient in terms of lexicographic criteria. All these models are different from ours since their mechanism of an endogenous discount factor does not depend on the strategy profiles.

Section snippets

Preliminaries

Given the extensive literature on repeated games with perfect monitoring, we will keep our description of the model brief but informative. Let G=N,{Ai}iN,{ui}iN be an n-player normal-form stage game, where for all iN={1,,n}, Ai is the set of player i's (pure) actions and ui:A=×jNAjR is player i's payoff function. Assume that Ai is compact, ui is continuous, and G has, at least, one Nash equilibrium.5

Myopic perception

In obtaining the standard folk theorem, not only each player iN has to evaluate his future high enough with a sufficiently high discount factor, everyone else must also perceive that player i has a sufficiently high discount factor. In other words, player i is a long-run player and is also perceived to be a long-run player by the other players. Even after player i deviates from what is supposed to be played, every player must continue to perceive that the deviating player i is a long-run

Feasibility and “individual rationality”

Under myopic perception, the set of perceived short-run players evolves as players deviate. We need to reformulate feasible payoffs as well as the least possible SPE payoff to every player in this class of dynamic games, which defines a new notation of “individual rationality”. Given the unique pattern how the set of perceived short-run players evolves, we can derive feasible payoffs, as well as each player's least possible equilibrium payoff by induction on the set of perceived short-run

Main results

We now present our study in characterizing the set SPE payoffs in the repeated game with myopic perception when the discount factor is sufficiently close to one. Given Proposition 2, we know that the limiting set of SPE payoffs in Gδ(S) must be a subset of feasible and weakly individually rational payoffs in Gδ(S). Closely resembling the classical folk theorem, our next proposition establishes that any feasible and strictly individually rational payoff of Gδ(S) for all SN can be supported

Concluding remarks

In this paper, we explore the implications of myopic perception on equilibrium outcomes in repeated games. The idea is based on the question whether common perception on players' time preferences can persist in a repeated game. It turns a repeated game into a stochastic/dynamic game. However, our model is different from most existing stochastic model as transition depends on the strategy profile itself. Because of its relatively simply dynamics, we can solve this problem by induction on the set

Acknowledgements

We would like to thank Nageeb Ali, Drew Fudenberg, George Mailath, David Miller, Larry Samuelson, Edward Schlee, Eilon Solan, John Wooders, an editor, and two anonymous referees for their comments and suggestions. Usual disclaimer applies. This research is supported by the Spanish Ministerio de Economía y Competitividad under the project ECO2015-67519-P 8 (MINECO/FEDER), and the Departamento de Educación, Política Lingüística y Cultura del Gobierno Vasco (grupo de investigación IT568-13 e

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