Grappling With the Centipede: Defence of Backward Induction for BI-Terminating Games

Economics and Philosophy 14 (01):95- (1998)
Abstract
According to a standard objection to the use of backward induction in extensive-form games with perfect information, backward induction (BI) can only work if the players are confident that each player is resiliently rational - disposed to act rationally at each possible node that the game can reach, even at the nodes that will certainly never be reached in actual play - and also confident that these beliefs in the players’ future resilient rationality are robust, i.e. that they would be kept come what may, whatever evidence of irrationality would by then transpire concerning past performance of the players. Since both resiliency and robustness assumptions are extremely strong and their appropriateness as idealizations is quite problematic, it has been argued (by Binmore, Reny, Bicchieri, Pettit and Sugden, among others) that BI is an indefensible procedure. Therefore, we need not be worried that BI can be used to justify seemingly counter-intuitive game solutions. I show, however, that there is a restricted class of extensive-form games in which BI solutions can be defended without assuming resiliency or robustness. For these ”BI-terminating games” (= games in which BI moves always terminate the play, at each choice node), to defend BI solutions, it is enough to make confidence-in-rationality assumptions concerning actual play; stipulations about various counterfactual developments are unnecessary. For this class of games, then, the standard objection to BI is inapplicable. At the same time, however, it will transpire that the class in question contains some well-known games, such as the Centipede in its different versions, in which BI recommends a seemingly unreasonable behaviour.
Keywords backward induction  decision theory  centipede  Robert Aumann  Ken Binmore  Philip Renyi  rationality  game theory  extensive-form games  Robert Stalnaker
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Cristina Bicchieri (1988). Backward Induction Without Common Knowledge. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1988:329 - 343.
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