Arbitrage-Free Equilibria of Noncooperative Games

by Robert F. Nau (Fuqua School of Business Working Paper)

Abstract: A refinement of subjective correlated equilibrium is proposed: "arbitrage-free equilibrium." This solution concept is derived from the requirement that the outcome of a noncooperative game should not present arbitrage opportunities to an outside observer when the players publicly accept small side-gambles consistent with their preferences, whether they are risk neutral or risk averse. An arbitrage-free equilibrium is a correlated equilibrium in which the common prior assumption applies to the players' risk neutral probabilities (products of probabilities and relative marginal utilities for money) rather than their true probabilities. The players' true probabilities may therefore be discordant, but nonetheless they satisfy a strong consistency condition given the marginal utilities. This reinterpretation of the common prior assumption guarantees that, when the players are risk averse, their equilibrium expected payoffs are Pareto efficient, a condition which is generally not satisfied by completely mixed Nash or objective correlated equilibria among risk averse players. The knowledge requirements for arbitrage-free correlated equilibria are also weaker than those of Nash or objective correlated equilibria in the sense that we need not observe the player's true probabilities or utilities for outcomes: the parameters of the game which are common knowledge are merely the small monetary side-gambles that the players are willing to accept. These results generalize earlier work by Nau and McCardle (1990) concerning the arbitrage-free properties of objective correlated equilibria for risk neutral players.

Key Words: Correlated equilibrium, joint coherence, arbitrage, common knowledge, common prior assumption, risk neutral probabilities, separation of probability and utility