Agreeing to disagree and dilation

We consider Aumann's famous result on “agreeing to disagree” in the context of imprecise probabilities. Our primary aim is to reveal a connection between the possibility of agreeing to disagree and the interesting and anomalous phenomenon known as dilation. For such a purpose it is convenient to use Geanakoplos and Polemarchakis' communication setting, where agents repeatedly announce and update credences until no new information is conveyed by the announcements. We show that for agents who share the same set of priors and update by conditioning on every prior, once the procedure of communicating credences stops, it is impossible to agree to disagree on the lower or upper probability of a hypothesis unless a certain dilation occurs. With some common topological assumptions, the result entails that it is impossible to agree not to have the same set of posterior probability values unless dilation is present. This result may be used to generate sufficient conditions for guaranteed full agreement for some important models of imprecise priors, and we illustrate the potential with an agreement result involving density ratio classes. We also provide a formulation of our results in terms of “dilation-averse” agents who ignore information about the value of a dilating partition but otherwise update by full Bayesian conditioning.