A Robust Folk Theorem for the Prisoners' Dilemma

We prove the folk theorem for the Prisoner's dilemma using strategies that are robust to private monitoring. From this follows a limit folk theorem : when players are patient and monitoring is sufficiently accurate, (but private and possibly independent) any feasible individually rational payoff can be obtained in sequential equilibrium. The strategies used can be implemented by finite (randomizing) automata.

Hierarchies of belief and interim rationalizability

In games with incomplete information, conventional hierarchies of belief are incomplete as descriptions of the players' information for the purposes of determining a player's behavior. We show by example that this is true for a variety of solution concepts. We then investigate what is essential about a player's information to identify behavior. We specialize to two player games and the solution concept of interim rationalizability. We construct the universal type space for rationalizability and characterize the types in terms of their beliefs. Infinite hierarchies of beliefs over conditional beliefs , which we call Delta-hierarchies, are what turn out to matter. We show that any two types in any two type spaces have the same rationalizable sets in all games if and only if they have the same Delta-hierarchies.Interim rationalizability, belief hierarchies

HIERARCHIES OF BELIEF AND INTERIM RATIONALIZABILITY

In games with incomplete information, conventional hierarchies of belief are incomplete as descriptions of the players’ information for the purposes of determining a player’s behavior. We show by example that this is true for a variety of solution concepts. We then investigate what is essential about a player’s information to identify rationalizable behavior in any game. We do this by constructing the universal type space for rationalizability and characterizing the types in terms of their beliefs. Infinite hierarchies of beliefs over conditional beliefs, what we call delta-hierarchies, are what turn out to matter. We show that any two types in any two type spaces have the same rationalizable sets in all games if and only if they have the same delta-hierarchies.

Evolution of Preferences

We model, using evolutionary game theory, the implications of endogenous determination of preferences over the outcomes of any given two-player normal form game, G. We consider a large population randomly and repeatedly matched to play G. Each individual has a preference relation over the outcomes of G which may be different than the "true" payoff function in G, and makes optimal choices given her preferences. The evolution of preferences is driven by the payoffs in G that each player obtains. We define stable outcomes (of G) as arising from the stable points of the evolutionary process described above. In our most general model players know the distribution of preferences in the population and observe their opponents' preferences with probability p. They then play a (Bayesian) Nash equilibrium of the resulting game of incomplete information. In the case in which players can perfectly observe their opponents' preferences, i.e., p=1, (where the game is actually one of complete information) an outcome is stable only if it is efficient. Also, an efficient outcome which arises from a strict Nash equilibrium is stable. We also characterize, for 2×2 games, both the stable outcomes and the stable distributions of preferences in the population. When preferences are unobservable, i.e., p=0, we show that stability in our model of evolution of preferences coincides with the notion of neutrally stable strategy (NSS). Finally, we consider robustness of these results. The necessity and sufficiency results are robust to slight changes in p, except for the sufficiency of NSS when p=0: There are in fact (Pareto-inferior) risk-dominant strict equilibria that are not stable for any p>0.Evolution of preferences, observability

When is Reputation Bad?

In traditional reputation theory, reputation is good for the long-run player. In “Bad Reputation,” Ely and Valimaki give an example in which reputation is unambiguously bad. This paper characterizes a more general class of games in which that insight holds, and presents some examples to illustrate when the bad reputation effect does and does not play a role. The key properties are that participation is optional for the short-run players, and that every action of the long-run player that makes the short-run players want to participate has a chance of being interpreted as a signal that the long-run player is “bad. ” We also broaden the set of commitment types, allowing many types, including the “Stackelberg type” used to prove positive results on reputation. Although reputation need not be bad if the probability of the Stackelberg type is too high, the relative probability of the Stackelberg type can be high when all commitment types are unlikely.