Understanding the Brandenburger-Keisler Belief Paradox

In their textbook, Osborne and Rubinstein describe game theory as “a bag of analytical tools designed to help us understand the phenomena that we observe when decision-makers interact” ([8] page 1). They go on to say that one of the basic assumptions of game theory is that when agents make decisions, they take into account “their knowledge or expectations of other decision-makers’ behavior (they reason strategically).” In other words, when agents involved in a multiagent interactive situation are making decisions about what action to perform next, that decision is influenced by what actions they expect the other agents will perform. This assumption leads very naturally to questions about what agents believe about the other agents’ beliefs. This observation has prompted a number of game theorists to propose that the basic models of game theory (extensive games forms and normal game forms) be extended to include a representation of the agents’ beliefs (see [1, 9, 3, 4, 7] for a discussion of the relevant literature). Essentially the idea is that when describing a strategic interactive situation part of that description should include the agents’ beliefs about the relevant ground (non-epistemic) facts, beliefs about the other agents’ beliefs about these ground facts, beliefs about the other agents’ beliefs about the other agents’ beliefs about these ground facts, and so on. Early on in 1967, John Harsanyi [6] developed an elegant formal model which can be used to represent the epistemic state of the agent in a game theoretic situation. The