Joint Coherence in Games of Incomplete Information

Abstract

Decisions are often made under conditions of uncertainty about the actions of supposedly-rational competitors. The modeling of optimal behavior under such conditions is the subject of noncooperative game theory, of which a cornerstone is Harsanyi's formulation of games of incomplete information. In an incomplete-information game, uncertainty may surround the attributes as well as the strategic intentions of opposing players. Harsanyi develops the concept of a Bayesian equilibrium, which is a Nash equilibrium of a game in which the players' reciprocal beliefs about each others' attributes are consistent with a common prior distribution, as though they had been jointly drawn at random from populations with commonly-known proportions of types. The relation of such game-theoretic solution concepts to subjective probability theory and nonstrategic decision analysis has been controversial, as reflected in critiques by Kadane and Larkey and responses from Harsanyi, Shubik, and others, which have appeared in this journal. This paper shows that the Bayesian equilibrium concept and common prior assumption can be reconciled with a subjective view of probability by (i) supposing that players elicit each others' probabilities and utilities through the acceptance of gambles, and (ii) invoking a multi-agent extension of de Finetti's axiom of coherence (no arbitrage opportunities, a.k.a. “Dutch books”). However, the Nash property of statistical independence between players is weakened, and the probability distributions characterizing a solution of the game admit novel interpretations.