We investigate two-player infinitely repeated games where the discount factor is less than but close to unity. Monitoring is private and players cannot communicate. We require no condition concerning the accuracy of players' monitoring technology. We show the folk theorem for the prisoners' dilemma with conditional independence. We also investigate more general games where players' private signals are correlated only through an unobservable macro shock. We show that efficiency is sustainable for generic private signal structures when the size of the set of private signals is sufficiently large. Finally, we show that cartel collusion is sustainable in price-setting duopoly.

The authors examine discounted repeated games where players privately observe different signals. A leading example is secret price cutting; a firm cannot directly observe rival firms' price cutting but its own sales can imperfectly indicate what is going on. The characterization of equilibria in this class of games has been an open question. The authors construct equilibria where players voluntarily communicate what they have observed and prove folk theorems. Their results thus provide a theoretical support for the conventional wisdom that communication facilitates collusion.

The authors investigate the implementation of social choice functions that map to lotteries over alternatives. They require virtual implementation in iteratively undominated strategies. Under very weak domain restrictions, they show that if there are three or more players, any social choice function may be so implemented. The literature on implementation in Nash equilibrium and its refinements is compromised by its reliance on game forms with unnatural features (for example, "integer games") or "modulo" constructions with mixed strategies arbitrarily excluded. In contrast, the authors' results employ finite (consequently "well-behaved") mechanisms and allow for mixed strategies.

WE ARE MOST GRATEFUL to Glazer and Rosenthal for the attention they have paid to our work (Abreu and Matsushima (1992a, b, c)). In the process of criticizing our mechanism, they have provided an elegant exposition of it which usefully supplements our own efforts. The criticisms themselves we feel are misplaced, and based on a close reading but narrow interpretation of our results. Glazer and Rosenthal seek to show that in our mechanisms .. . the iterative removal of strictly dominated strategies is (or indeed ought to be) controversial. In terms of conventional decision theory the iterative logic is impeccable. When iterative deletion leads to a unique profile, that profile is the unique rationalizable profile, the unique Nash equilibrium, and so on. Of course, the logic of iterative dominance entails common knowledge of rationality. The theoretical coherence of this assumption has been questioned in the context of backward programming in extensive form games (see Luce and Raiffa (1957, pp. 80-81), Rosenthal (1981), Basu (1988), Reny (1992a, 1992b), Bonanno (1991), Binmore (1987a, 1987b), and Gul (1989) among others). These paradoxes of common knowledge of rationality have been highlighted in stylized examples such as the finitely repeated prisoners' dilemma and Rosenthal's centipede which in fact has been a seminal inspiration for this recent literature.2 But our mechanisms are simultaneous. There is no opportunity to demonstrate irrationality, strategic or otherwise, and therefore no scope to rationalize iteratively dominated behavior. From a decision theory perspective, the Glazer and Rosenthal critique might therefore be viewed as an elaborate revival of the (indefensible) claim that players in a one-short prisoners' dilemma will choose not to confess because they are both better off doing so than by both playing their dominant strategies. Within the standard game theory paradigm there is really nothing more to be said. Before turning to issues of bounded rationality, we note that the Glazer and Rosenthal critique is essentially premised on the existence of a countervailing focal point. But what is a focal point? The notion is notoriously vague, and we are aware of no widely accepted definition.3 This is not to deny that in very simple and particular games, certain behavior may be focal. Frequently, focalness is identified with a Pareto dominating equilibrium, and this is the point of view Glazer and Rosenthal seem to adopt.4 This is implicit in their acknowledgement that ... . when the social choice function does satisfy Pareto optimality, our objection loses much of its immediate