Backward induction in games: an attempt at logical reconstruction

Backward induction has been the standard method of solving finite extensive-form games with perfect information, notwithstanding the fact that this procedure leads to counter-intuitive results in various games (iterated prisoner's dilemma, centipede, chain store, etc.). However, beginning in the late eighties, the method of backward induction became an object of criticism. It is claimed (most notably, by Reny 1988, 1989, Binmore 1987, Bicchieri 1989, and Pettit & Sugden 1989) that the assumptions needed for its defence are quite implausible, if not incoherent. It is therefore natural to ask for the justification of backward induction: Can one show that rational players who know the structure of the game, have trust in each other's practical rationality and reason correctly, will act in accordance in backward induction? Several researchers have tried a justification of this kind, but the argument presented in Robert Aumann's paper from 1995 is perhaps the most well-known and influential attempt to provide such a justification. Clausing (1999) provides a sustained discussion of the justification problem for backward induction. It is an excellent work and the criticism I will present below does not detract from this evaluation: the issues discussed by the author are complex and it is difficult to get everything right. Furthermore, I hope that the criticism to be presented may be instructive; Even though it has not been Clausing's intention, his logical reconstruction of Aumann's defence of backward induction allows us to see very clearly what is wrong with that argument. It also provides us with * This paper was presented at workshops in Lund and in Uppsala, in the Fall of 1999. I am indebted to the participants for their useful comments. The work on the paper was supported by a generous research grant from The Bank of Sweden Tercentenary Foundation