In the literature on Bayesian dialogues, this assumption is not explicitly grounded in some other theory or model. In a Bayesian dialogue, individuals by assumption always truthfully report their Bayesian updated belief. There is continued interest in Bayesian dialogues as, for instance, recent work by Polemarchakis ( 2016) and Di Tillio et al. Nielsen ( 1984) extends on Aumann’s result and the dynamic scenario studied by Geankoplos and Polemarchakis by using the more general framework of sigma algebras, instead of partitions, for modeling knowledge. Geanakoplos and Polemarchakis refer to such a process as one of indirect communication, as opposed to a process of direct communication, by which they refer to a scenario in which the two individuals directly reveal to each other the information that they have received according to their information partition. While Bacharach works with the more general framework of conditional distributions over the set of all possible states of the world, Geanakoplos and Polemarchakis study a dynamic process of belief revision that stays closer to Aumman’s original model: in their model, two individuals who receive private information in the form of a finite information partition communicate their Bayesian posterior belief about a certain event back and forth (thereby making their posterior at that step common knowledge) until their respective posteriors at the next step will already be common knowledge (without the announcement needed), and as a consequence-this is Aumann’s result-will be equal. Aumann ( 1976) suggests that such a process, which he illustrates by an example, can be seen as a dynamic foundation for his “agreement result.”Īumann’s ( 1976) “agreement result” says the following: If two individuals impose the same prior probability on the set of all possible states of the world Ω and if, after realization of the true state of the world (in virtue of the common knowledge of the prior probability and individuals’ information partitions), it is common knowledge that the posterior probability that individual 1 attributes to a certain event A is q 1 and that the posterior probability that individual 2 attributes to A is q 2, then: q 1 = q 2, that is, the two posteriors have to be equal.īoth Bacharach ( 1979) as well as Geanakoplos and Polemarchakis ( 1982) provide dynamic foundations for Aumman’s result, in the sense that the result holds in a nonvacuous way at the absorbing state of the respective process. In a “Bayesian dialogue” (Bacharach 1979 Geanakoplos and Polemarchakis 1982), two individuals, who assign the same prior distribution to some random variable and then receive private information about that distribution, communicate their posterior distribution back and forth-thereby successively updating their posterior distribution in a Bayesian-rational way-until the process has reached an absorbing state, in which the two individuals will have reached consensus about the posterior distribution. Finally, the argument is embedded in a game-theoretic model. 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. 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. In this example, at some step of the process, one individual has an incentive to deviate from truthfully reporting his Bayesian updated belief. This article provides an example which shows this intuition to be wrong. 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. Certainly, if two individuals have diverging interests, truthfully reporting one’s Bayesian updated belief at every step might not be optimal. 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. 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.
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