An anti-folk theorem for finite past equilibria in repeated games with private monitoring

We prove an anti-folk theorem for repeated games with private monitoring. We assume that the strategies have a finite past (they are measurable with respect to finite partitions of past histories), that each period players' preferences over actions are modified by smooth idiosyncratic shocks, and that the monitoring is sufficiently connected. In all repeated game equilibria, each period play is an equilibrium of the stage game. When the monitoring is approximately connected, and equilibrium strategies have a uniformly bounded past, then each period play is an approximate equilibrium of the stage game.Repeated games, anti-folk theorem, private monitoring

Generalization of a Result on "Regression, Short and Long"

This paper is concerned with the problem of combining a data set that identifies the conditional distribution P (y|x) with one that identifies the conditional distribution P (z|x), in order to identify the regressions E (y|x, middot) identical with [E (y|x, z = j), j element of Z] when the conditional distribution P (y|x, z) is unknown. Cross and Manski (2002) studied this problem and showed that the identification region of E (y|x, middot) can be precisely calculated, when y has finite support. Here we generalize Cross and Manski's result showing that the identification region can be precisely calculated also in the case in which y has infinite support.

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

Repeated games with incomplete information on one side

This paper studies repeated games with incomplete information on one side and equal discount factors for both players. The payoffs of the informed player I depend on one of two possible states of the world, which is known to her. The payoffs of the uninformed player U do not depend on the state of the world (that is, U knows his payoffs), but player I's behavior makes knowledge of the state of interest to player U. We define a finitely revealing equilibrium as a Bayesian perfect equilibrium where player I reveals information in a bounded number of periods. We define an ICR profile as a strategy profile in which (a) after each history the players have individually rational payoffs and (b) no type of player I wants to mimic the behavior of the other type. We show that when the players are patient, all Nash equilibrium payoffs in the repeated game can be approximated by payoffs in finitely revealing equilibria, which themselves approximate the set of all ICR payoffs. We provide a geometric characterization of the set of equilibrium payoffs, which can be used for computations.Repeated games, incomplete information, discounting

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.