Agreeing on Decisions After All

The Decision Agreement Principle, originally promulgated by Cave and Bacharach (independently), asserts that if like-minded decision makers commonly know each other’s decisions, then the decisions are the same. Subsequently, Moses and Nachum discovered a flaw in the reasoning underlying the principle. Here we provide a careful, coherent, and correct formulation of the principle, which avoids the Moses-Nachum flaw, and enables us to understand just when the principle applies, and when it doesn’t. 1. A Detective Story A murder has been committed. To increase the chances of conviction, the chief of police puts two detectives on the case, with strict instructions to work independently, to exchange no information. The two, Alice and Bob, went to the same police school; so given the same clues, they would reach the same conclusion. But as they work independently, they presumably do not get the same clues. At the end of thirty days, each is to decide whom to arrest. On the night before the thirtieth day, they happen to meet in the locker room at headquarters, and get to talking about the case. True to their instructions, they exchange no substantive information, no clues; but both are self-confident individuals, and feel that there is no harm in discussing whom they plan to arrest. When they leave the locker room, it is common knowledge between them whom Alice will arrest, and whom Bob will arrest. Conclusion: They arrest the same people; and this, in spite of knowing nothing about each other’s clues.