### Robert Nau

## Imprecise Probabilities in Non-cooperative Games

### Abstract

Game-theoretic solution concepts such as Nash equilibrium are commonly used to model strategic behavior in terms of precise probability distributions over outcomes. However, there are many potential sources of imprecision in beliefs about the outcome of a game: incomplete knowledge of payoff functions, non-uniqueness of equilibria, heterogeneity of prior probabilities, unobservable background risk, and distortions of revealed beliefs due to risk aversion, among others. This paper presents a unified approach for dealing with these issues, in which the typical solution of a game is a convex set of probability distributions that, unlike Nash equilibria, may be correlated between players. In the most general case, where players are risk averse, the probabilities do not represent beliefs alone. Rather they must be interpreted as products of subjective probabilities and relative marginal utilities for money.

### Keywords

coherence, previsions, lower and upper probabilities, correlated equilibrium, risk neutral probabilities, risk neutral equilibrium

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### Authors’ addresses

Fuqua School of Business

Duke University

Durham, NC 27708-0120

USA

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