A "Reputation" Refinement without Equilibrium

THE ECONOMIC LITERATURE concerning agents' reputations has grown steadily since the seminal work of Kreps, Milgrom, Roberts, and Wilson.2Early work focused on how incomplete information leads to equilibria that are vastly different (but more intuitive) than those possible in the complete information game.Recently, however, game theorists have been studying how incomplete information might refine the set of equilibria.3One important class of games is that in which a single long-run agent plays a simultaneous move (stage) game with a sequence of opponents, each of whom plays only once, yet observes all previous play.Fudenberg and Levine (1989) study the reputation of the long-run player in this type of game.They argue that the "'most reasonable' equilibrium is the one which the long-run player most prefers."Their intuition is sustained when one perturbs the game with the "Stackelberg strategy."Fudenberg and Levine show that in the perturbed game the equilibrium payoffs of the long-run player are bounded below by a number that converges to the "Stackelberg payoff."Fudenberg and Levine (and the others who have developed reputation models) take the notion of Nash equilibrium as fundamental in the analysis.However, it would seem as though our intuition about reputations relies little on equilibrium concepts.This leads to two questions.First, can meaningful reputations develop apart from equilibria?Second, if so, under what circumstances can reputations develop?As I will demonstrate, equilibrium concepts are not required in order for players to establish significant reputations.I study (following Fudenberg and Levine (1989)) games in which a long-run player faces a sequence of short-run opponents.Like Fudenberg and Levine, I consider perturbations of the game involving the Stackelberg strategy.However, whereas they focus on equilibria, I will only require that players "best-respond" to their beliefs.Whenever the conjectures of the short-run players are "generally comparable" (e.g.contained in a compact set), I obtain the same refinement as do Fudenberg and Levine, but without an equilibrium assumption.What is important for reputations is that the beliefs of the short-run agents not be too dispersed in a sense to be made precise.Players can thus establish meaningful reputations from within the loose confines of individual rationality.This paper borrows heavily from the work of Fudenberg and Levine (1989).In fact, their statistical result (their Lemma 1), which establishes the potential gain of building a reputation, requires no notion of equilibrium.It does require that the short-run agents hold the same belief, which is implied by equilibrium.I simply invoke their lemma in a more general setting (in which short-run players may hold different beliefs) and study the type of beliefs which allow it to refine the set of rational outcomes.Note that a similar style of research has been followed on another front as well.Cho (1991) extends the Coase conjecture to a nonequilibrium setting.Cho's work is similar to mine in that we both take as fundamental a rationalizability notion.Our analyses require additional restrictions, however, and it is the nature of these restrictions in which our 11 am grateful to David Kreps, Marco LiCalzi, two referees, and the editor for comments.This is Chapter 2 of my Ph.D.