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We extend Kreps and Wilson's concept of sequential equilibrium to games with infinite sets of signals and actions. A strategy profile is a conditional ε ‐equilibrium if, for any of a player's positive probability signal events, his conditional expected utility is within ε of the best that he can achieve by deviating. With topologies on action sets, a conditional ε ‐equilibrium is full if strategies give every open set of actions positive probability. Such full conditional ε ‐equilibria need not be subgame perfect, so we consider a non‐topological approach. Perfect conditional ε ‐equilibria are defined by testing conditional ε ‐rationality along nets of small perturbations of the players' strategies and of nature's probability function that, for any action and for almost any state, make this action and state eventually (in the net) always have positive probability. Every perfect conditional ε ‐equilibrium is a subgame perfect ε ‐equilibrium, and, in finite games, limits of perfect conditional ε ‐equilibria as ε → 0 are sequential equilibrium strategy profiles. But limit strategies need not exist in infinite games so we consider instead the limit distributions over outcomes. We call such outcome distributions perfect conditional equilibrium distributions and establish their existence for a large class of regular projective games. Nature's perturbations can produce equilibria that seem unintuitive and so we augment the game with a net of permissible perturbations.
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Record-Keeping and Cooperation in Large Societies
Abstract We introduce a new model of repeated games in large populations with random matching, overlapping generations, and limited records of past play. We prove that steady-state equilibria exist under general conditions on records. When the updating of a player’s record can depend on the actions of both players in a match, any strictly individually rational action can be supported in a steady-state equilibrium. When record updates can depend only on a player’s own actions, fewer actions can be supported. Here, we focus on the prisoner’s dilemma and restrict attention to strict equilibria that are coordination-proof, meaning that matched partners never play a Pareto-dominated Nash equilibrium in the one-shot game induced by their records and expected continuation payoffs. Such equilibria can support full cooperation if the stage game is either “strictly supermodular and mild” or “strongly supermodular,” and otherwise permit no cooperation at all. The presence of “supercooperator” records, where a player cooperates against any opponent, is crucial for supporting any cooperation when the stage game is “severe.”
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- Award ID(s):
- 1951056
- PAR ID:
- 10282687
- Date Published:
- Journal Name:
- The Review of Economic Studies
- ISSN:
- 0034-6527
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/restud/rdab016
