WHAT HAPPENED TO THE THREE-LEGGED CENTIPEDE? Graciela Kuechle Witten/Herdecke, Germany Abstract. In the last two decades, several frameworks have been proposed to analyze the question of whether common knowledge of rationality is sufficient to justify the play of backward induction in games of perfect information. Three strands of literature have addressed this issue: the literature on equilibrium refinements, the literature on knowledge-based epistemology and the literature on interactive epistemology. This paper surveys seminal frameworks within the first two strands of research and assesses the extent to which they provide a satisfactory solution to the problem. These approaches are illustrated using a three-legged version of Rosenthal’s (1981) Centipede Game, which is the classical example in the literature. The paper argues that some of these frameworks provide sensible answers to the riddle of backward induction or at least, succeed in bringing the paradox to another level. The paper also points at consistency problems in the body of refinements of Nash equilibrium revealed by the surveyed literature. Keywords: Backward Induction; Centipede Game; Common Knowledge; Equilibrium Refinements; Rationality 1. Introduction In 1987, Binmore claimed that rational players who have common knowledge of rationality would not necessarily play the backward induction (BI) solution in games of perfect information (PI). Illustrating his argument with the n-legged Centipede Game (Rosenthal, 1981), Binmore argued that the usual assumption behind BI, namely that deviations are causally inconsequential, need not hold. In these games, deviations could provide valuable information and trigger further deviations, invalidating the logic of equilibrium. Yet the usual BI argument precludes this type of hypothetical reasoning. To allow for deliberation at unexpected nodes and regain the backward induction algorithm, Binmore