Knowledge, probability, and adversaries
Proceedings of the eighth annual ACM Symposium on Principles of distributed computing
Knowledge and common knowledge in a distributed environment
PODC '84 Proceedings of the third annual ACM symposium on Principles of distributed computing
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Bacharach and Cave independently generalized Aumann's celebrated agreement theorem to the case of decision functions. Roughly speaking, they showed that once two like-minded agents reach common knowledge of the actions each of them intends to perform, they will perform identical actions. This theorem is proved for decision functions that satisfy a condition that Bacharach calls the sure thing condition, which is closely related to Savage's sure thing principle. The assumption that any reasonable decision function should satisfy the sure thing condition seems to have been widely accepted as being natural and intuitive. By taking a closer look at the meaning of the sure thing condition in this context, we argue that the technical definition of the sure thing condition does not capture the intuition behind Savage's sure thing principle very well. It seems to involve nontrivial hidden assumptions, whose appropriateness in the case of non-probabilistic decision functions is questionable. Similar trouble is found with the technical definition of the like-mindedness of two agents. Alternative definitions of the sure thing principle and like-mindedness are suggested, and it is shown that the agreement theorem does not hold with respect to these definitions. In particular, it is shown that the agreement theorem does not apply to a particularly appealing example attributed to Bacharach. Conditions that do guarantee the agreement theorem for decision functions are presented. Finally, we consider similar issues that arise in the case of communication among more than two agents, as studied by Parikh and Krasucki.