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Lexicographic Probabilities and Equilibrium Refinements

Lawrence Blume
Adam Brandenburger
Eddie Dekel

Abstract

This paper develops a decision-theoretic approach to normal-form refinements of Nash equilibrium and, in particular, provides characterizations of (normal-form) perfect equilibrium (Selten (1975)) and proper equilibrium (Myerson (1978)). The approach relies on a theory of single-person decision making that is a non-Archimedean version of subjective expected utility theory. According to this theory, each player in a game possesses, in addition to a strategy space and a utility function on outcomes, a vector of probability distributions, called a lexicographic probability system (LPS), on the strategies chosen by the other players. These probability distributions can be interpreted as the player's first-order and higher order theories as to how the game will be played, and are used lexicographically in determining an optimal strategy. We define an equilibirum concept, called lexicographic Nash equilibrium, that extends the notion of Nash equilibrium in that it dictates not only a strategy for each player but also an LPS on the strategies chosen by the other players. Perfect and proper equilibria are described as lexicogrpahic Nash equilibria by placing various restrictions on the LPS's possessed by the players.