Keep 'hoping' for rationality: a solution to the backward induction paradox

The jury is still out concerning the epistemic conditions for backward induction, the “oldest idea in game theory” (Aumann, 1995, p. 635). Aumann (1995) and Stalnaker (1996) take conflicting positions in the debate: the former claims that common “knowledge” of “rationality” in a game of perfect information entails the backwardinduction solution; the latter that it does not. Of course there is nothing wrong with any of their relevant formal proofs, but rather, as pointed out by Halpern (2001), there are differences between their interpretations of the notions of knowledge, belief, strategy and rationality. Moreover, as pointed out by Binmore (1987; 1996), Bonanno (1991), Bicchieri (1989), Reny (1992), Brandenburger (2007) and others, the reasoning underlying the backward induction method seems to give rise to a fundamental paradox: in order even to start the reasoning, a player assumes that (common knowledge of, or some form of common belief in) “rationality” holds at all the last decision nodes (and so the obviously irrational leaves are eliminated); but then, in the next reasoning step (going backward along the tree), some of these (last) decision nodes are eliminated, as being incompatible with (common belief in) “rationality”! Hence, the assumption behind the previous reasoning step is now undermined: the reasoning player can now see, that if those decision nodes that are now declared “irrational” were ever to be reached, then the only way that this could happen is if (common belief in) “rationality” failed. Hence, she was wrong to assume (common belief in) “rationality” when she was reasoning about the choices made at those last decision nodes. This whole line of arguing seems to undermine itself!