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# REPUTATION WITH LONG RUN PLAYERS

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1. Introduction and Related Literature2. The Model2.1. Examples3. One-Sided Reputation3.1. The Main One-Sided Reputation Result3.2. Uniformly Learnable Types3.3. Examples with no Reputation Effects4. Two-Sided Reputation and a War of Attrition4.1. The War of Attrition4.2. The Main Two Sided Reputation ResultAppendix A. Omitted ProofsA.1. Omitted Steps in the Proof of Corollary 2A.2. Omitted Steps in the Proof of Theorem 2ReferencesREPUTATION WITH LONG RUN PLAYERSALP E. ATAKAN AND MEHMET EKMEKCIAbstract.Previous work shows that reputation results may fail in repeated games with long-runplayers with equal d iscount factors. Attention is restricted to extensive-form stage games ofperfect information. One and two-sided reputation results are provided for repeated gameswith two long-run players with equal discount factors where the first mover advantage ismaximal. If one of the players is a Stackelberg type with positive probability, then thatplayer receives the highest payoff, that is part of an individ ually rational payoff profile, inany perfect equilibria, as agents become patient. If both players are Stackelberg types withpositive probability, then perfect equilibrium payoffs converge to a unique payoff vector;and the equilibrium play converges to the unique equilibrium of a continuous time war ofattrition. All results generalize to simultaneous move stage games, if the stage game is agame of strictly conflicting interest.Keywords: Repeated Games, Reputation, Equal Discount Factor, Long-run Players, Warof Attrition.JEL Classification Numbers: C73, D83.1. Introduction and Related LiteratureThis paper proves one and two-sided reputation results when two players with equaldiscount factors play a repeated game where th e fi rst mover advantage is maximal. The stagegame, which is repeated in each period, is an extensive-form game of perfect information.Date: First draft, March, 2008. This revision, October, 2008.We would like to thank Martin Cripps, Eddie Dekel and Christoph Kuzmics for helpful discussions; ourco-editor Larry Samuelson and three referees for detailed comments; and Umberto Garfagnini for excellentresearch assistance.12 ATAKAN AND EKMEKCIA Stackelberg strategy is a player’s optimal repeated game strategy, if the player couldpublicly commit to this strategy ex-ante; and a Stackelberg type is a commitment type thatonly plays the Stackelberg s tr ategy. Th e first mover advantage is maximal for player 1 ifthe repeated game Stackelberg strategy delivers player 1 his highest payoff, that is partof an individually rational payoff p rofile, whenever player 2 best responds. Our first mainresult shows that if there is one-sided incomplete information and player 1 is a Stackelbergtype with positive probability, then player 1 receives his highest possible payoff, in anyequilibrium, as the discount factor converges to one, and the probability of being any othercommitment type converges to zero. Th is one-sided reputation result extends to arbitraryprobability d istributions over other commitment types if these other types are uniformlylearnable. Our second main result (two-sided reputation result) establishes that if there isincomplete information about both players ’ types and each player is a Stackelberg type withpositive probability, then all equilibrium paths of play resemble the unique equilibr ium ofan appropriately defined continuous time war of attrition, as the time between repetitionsof the stage game shrinks to zero. Also, all equilibrium payoffs converge to the uniqueequilibrium payoff of the war of attrition.A one-sided reputation result was first established for finitely repeated games by Krepsand Wilson (1982) and Milgrom and Roberts (1982); and extended to infinitely repeatedgames by Fudenberg and Levine (1989). However, most reputation results in the literatureare for repeated games where a long-run player, that is possibly a Stackelberg type, facesa sequence of short-run players (as in Fudenberg and Levine (1989, 1992)); or for repeatedgames where the player building the reputation is infinitely more patient than his rival andso the rival is essentially a s hort-run player, at the limit (for example, see Schmidt (1993b),Celantani, Fud enberg, Levine, and Pesendorfer (1996), Battigalli and Watson (1997) orEvans and Thomas (1997)). Also, previous research has shown that reputation results arefragile in infinitely repeated games where long-run players with equal discount factors playa simultaneous-move stage game. In particular, one-sided reputation results obtain onlyif the stage game is a strictly conflicting interest game (Cripps, Dekel, an d PesendorferREPUTATION 3(2005)), or if there is a strictly dominant action in the s tage game (Chan (2000)).1Forother simultaneous move games, such as the common interest game, a folk theorem byCripps and Thomas (1997) shows that any individually rational and feasible payoff can besustained in perfect equilibria of the infinitely repeated game, if the players are s ufficientlypatient (also see the analysis in Chan (2000)).Almost all th e recent work on reputation has focused on simultaneous move stage games.In sharp contrast, we restrict attention to extensive-form stage games of perfect in formation.The stage games we allow for include common interest games, the battle of the sexes, th echain store game as well as all str ictly conflicting interest games (see section 2.1). For theclass of games we consider, without incomplete information, the folk theorem of Fudenbergand Maskin (1986) applies, under a full dimensionality condition (see Wen (2002) or Mailathand Samuelson (2006)). Also, if the normal form representation of the extensive formgame we consider is played simultaneously in each period, then under one-sided incompleteinformation, a folk theorem applies for a subset of the class of games we consider (see Crippsand Thomas (1997)). Consequently, our one-sided reputation r esult covers a significantlylarger class of games than those covered by p revious reputation results.Our two-sided reputation result is motivated by the approach in Kreps and Wilson (1982)and is closely related to previous work by Abreu and Gul (2000) and Abreu and Pearce(2007). Abreu and Gul (2000) show that in a two player bargaining game, as the frequencyof offers increases, the equilibr ia of the (two-sided) incomplete information game convergesto th e unique equilibrium of a continuous time war of attrition. Their two-sided reputationresult

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