When all is done but not (yet) said: Dynamic rationality in extensive games

The jury is still out concerning the epistemic conditions for backward induction, the “oldest idea in game theory” ([2, p. 635]). Aumann [2] and Stalnaker [31] take contradictory positions in the debate: Aumann claims that common ‘knowledge’ of ‘rationality’ in a game of perfect information entails the backward-induction solution; Stalnaker that it does not.1 Of course there is nothing wrong with any of their relevant formal proofs, but rather, as pointed out by Halpern [22], there are differences between their interpretations of the notions of knowledge, belief, strategy and rationality. Moreover, as pointed out by Binmore [14, 15], Bonanno [17], Bicchieri [13], Reny [26], Brandenburger [18] and others, the reasoning underlying the backward induction method seems to give rise to a fundamental paradox (the so-called “BI paradox”): in order to even start the reasoning, a player assumes that (common knowledge, 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! In this paper we use as a foundation the relatively standard and well-understood setting of Conditional Doxastic Logic (CDL, [16, 5, 7, 6]), and its “dynamic” version (obtained by adding to CDL operators for truthful public announcements [!φ]ψ): the logic PAL-CDL, introduced by Johan van Benthem [11]. In fact, we consider a slight ex-