Strategic manipulation in Bayesian dialogues

In a Bayesian dialogue two individuals report their Bayesian updated belief about a certain event back and forth, at each step taking into account the additional information contained in the updated belief announced by the other at the previous step. Such a process, which operates through a reduction of the set of possible states of the world, converges to a commonly known posterior belief, which can be interpreted as a dynamic foundation for Aumann’s agreement result. Certainly, if two individuals have diverging interests, truthfully reporting one’s Bayesian updated belief at every step might not be optimal. This observation could lead to the intuition that always truthfully reporting one’s Bayesian updated belief were the best that two individuals could do if they had perfectly coinciding interests and these were in line with coming to know the truth. This article provides an example which shows this intuition to be wrong. In this example, at some step of the process, one individual has an incentive to deviate from truthfully reporting his Bayesian updated belief. However, not in order to hide the truth, but to help it come out at the end: to prevent the process from settling into a commonly known belief—the “Aumann conditions”—on a certain subset of the set of possible states of the world (in which the process then would be blocked), and this way make it converge to a subset of the set of possible states of the world on which it will be commonly known whether the event in question has occurred or not. The strategic movement described in this example is similar to a conversational implicature: the correct interpretation of the deviation from truthfully reporting the Bayesian updated belief thrives on it being common knowledge that the announced probability cannot possibly be the speaker’s Bayesian updated belief at this step. Finally, the argument is embedded in a game-theoretic model.