Abstract
As demonstrated by the email game of Rubinstein (1989), the predictions of the standard equilibrium models of game theory are sensitive to assumptions about the fine details of the higher order beliefs. This paper shows that models of bounded depth of reasoning based on level-k thinking or cognitive hierarchy make predictions that are independent of the tail assumptions on the higher order beliefs. The framework developed here provides a language that makes it possible to identify general conditions on depth of reasoning, instead of committing to a particular model such as level-k thinking or cognitive hierarchy.
Highlights
One of the assumptions maintained in the standard equilibrium analysis of game theory is that agents have unlimited reasoning ability—they are able to perform arbitrarily complicated iterative deductions in order to predict their opponent’s behavior
The assumption of common knowledge of rationality entails everyone being rational, everyone knowing that their opponent is rational, everyone knowing that their opponent knows that they are rational, and so on ad infinitum
At some finite point in time, an email gets lost and this leaves the generals in a situation of “almost common knowledge,” but not “common knowledge”, i.e., they both know that the enemy is weak, they both know that they both know this, and so on, but only finitely many times; see Section 2 for a formal description of the game
Summary
One of the assumptions maintained in the standard equilibrium analysis of game theory is that agents have unlimited reasoning ability—they are able to perform arbitrarily complicated iterative deductions in order to predict their opponent’s behavior. Rubinstein finds it hard to believe that the generals will try to outguess each other and fail to coordinate even in cases when the number of messages they exchange is very large and intuitively close to the simple game of the first scenario. He finds this discontinuity of behavior with respect to higher order beliefs counterintuitive and writes: The sharp contrast between our intuition and the game-theoretic analysis is what makes this example paradoxical. The notion of cognitive type space developed here allows for studying richer forms of dependency between the bound and the belief, which offers a new direction for the analysis of models of bounded depth of reasoning and its applications to various economic settings
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