Half empty, half full and the possibility of agreeing to disagree

Aumann (1976) derives his famous we cannot agree to disagree result under the assumption of rational Bayesian learning. Motivated by psychological evidence against this assumption, we develop formal models of optimistically, resp. pessimistically, biased Bayesian learning within the framework of Choquet expected utility theory. As a key feature of our approach the posterior subjective beliefs do, in general, not converge to "true" probabilities. Moreover, the posteriors of different people can converge to different beliefs even if these people receive the same information. As our main contribution we show that people may well agree to disagree if their Bayesian learning is psychologically biased in our sense. Remarkably, this finding holds regardless of whether people with identical priors apply the same psychologically biased Bayesian learning rule or not. A simple example about the possibility of ex-post trading in a financial asset illustrates our formal findings. Finally, our analysis settles a discussion in the no-trade literature (cf. Dow, Madrigal, and Werlang 1990, Halevy 1998) in that it clarifies that ex-post trade between agents with common priors and identical learning rules is only possible under asymmetric information.