The Incoherence of Agreeing to Disagree

by Robert F. Nau (Theory and Decision 39, 219-239, 1995)

Abstract: The agreeing-to-disagree theorem of Aumann and the no-expected-gain-from-trade theorem of Milgrom and Stokey are reformulated under an operational definition of Bayesian rationality. Common knowledge of beliefs and preferences is achieved through transactions in a contingent claims market, and mutual expectations of Bayesian rationality are defined by the condition of joint coherence, i.e., the collective avoidance of arbitrage opportunities. The existence of a common prior distribution and the impossibility of agreeing to disagree follow from the joint coherence requirement, but the prior must be interpreted as a ``risk-neutral'' distribution: a product of probabilities and marginal utilities for money. The failure of heterogenous information to create disagreements or incentives to trade is shown to be an artifact of overlooking the potential role of trade in constructing the initial state of common knowledge.

Key words: arbitrage, joint coherence, common knowledge, subjective probability, revising probabilities, consensus, rational expectations

Comments: This paper further explores the implications of the no-arbitrage definition of mutually expected rationality, also called joint coherence, that was originally introduced in the paper "Coherent Behavior in Noncooperative Games." The reinterpretation of the common prior assumption in terms of risk neutral probabilities is also discussed in the papers "Arbitrage, Rationality, and Equilibrium" and "Arbitrage-Free Correlated Equilibria."