Backward Induction without Common Knowledge

Abstract
A large class of games is that of non-cooperative, extensive form games of perfect information. When the length of these games is finite, the method used to reach a solution is that of a backward induction. Working from the terminal nodes, dominated strategies are successively deleted and what remains is a unique equilibrium. Game theorists have generally assumed that the informational requirement needed to solve these games is that the players have common knowledge of rationality. This assumption, however, has given rise to several problems and paradoxes. Most notably, it has been shown that the common knowledge assumption makes the theory of the game inconsistent at some information set. The present paper shows that a) no common knowledge of rationality need be assumed for the backward induction solution to hold. Rather, it is sufficient that the players have a number of levels of knowledge proportional to the length of the game, and b) it is also necessary that the number of levels of knowledge is finite and proportional to the length of the game. For a higher number of levels of knowledge, inconsistencies arise.
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Cristina Bicchieri (1990). Paradoxes of Rationality. Midwest Studies in Philosophy 15 (1):65-79.
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Philip J. Reny (1988). Common Knowledge and Games with Perfect Information. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1988:363 - 369.
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