Consider a string of prisoner’s dilemmas, or a centipede game. Backward induction implies strategies that are intuitively unsatisfying – in the prisoner’s dilemma string, the only subgame perfect strategy is to always defect. There is no way to play some sort of strategic, tit-for-tat play. Pettit and Sugden, philosophers both, propose a resolution. If I, during stage n, play “cooperate”, then I have played an irrational strategy. As soon as this happens, the belief of the other player in common rationality must break down, since I have showed that either I am irrational or that I think the other player is irrational. The authors propose a strategy of equilibrium play that relies on this contradiction which is different from “always defect”. They also show that the assumption of common knowledge breaks down this solution.
The idea is interesting, but I tend to find descriptive arguments of the type in this paper (published in Journal of Philosophy) confusing, particularly given the complications inherent in common beliefs. Aumann (1995) is a well-known rejoinder, which I will cover here shortly.
http://www.princeton.edu/~ppettit/papers/BackwardInduction_JournalofPhilosophy_1989.pdf
A Fine Theorem 