**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."