Abstract
Aumann has proved that common knowledge of substantive rationality implies the backwards induction solution in games of perfect information. Stalnaker has proved that it does not. Roughly speaking, a player is substantively rational if, for all vertices v, if the player were to reach vertex v, then the player would be rational at vertex v. It is shown here that the key difference between Aumann and Stalnaker lies in how they interpret this counterfactual. A formal model is presented that lets us capture this difference, in which both Aumann’s result and Stalnaker’s result are true (under appropriate assumptions).
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Notes
- 1.
Supported in part by NSF under grant IRI-96-25901.
- 2.
The model that I use to prove Stalnaker’s result is a variant of the model that Stalnaker (1996) used, designed to be as similar as possible to Aumann’s model so as to bring out the key differences. This, I believe, is essentially the model that Stalnaker had in mind at the round table.
- 3.
Again, I should stress that this is not exactly the model that Stalnaker uses in (1996), but it suffices for my purposes. I remark that in Halpern (1999), I use selection functions indexed by the players, so that player 1 may have a different selection function than player 2. I do not need this greater generality here, so I consider the simpler model where all players use the same selection function.
- 4.
There are certainly other reasonable properties we could require of the selection function. For example, we might want to require that if v is reached in some state in \(\mathcal{K}_{i}(\omega )\), then \(f(\omega,v) \in \mathcal{K}_{i}(\omega )\). I believe that it is worth trying to characterize the properties we expect the selection function should have, but this issue would take us too far afield here. (See Stalnaker RC, 1999, Counterfactual propositions in games, Unpublished manuscript, for further discussion of this point.) Note that F1–F3 are properties that seem reasonable for arbitrary games, not just games of perfect information.
- 5.
A game is nondegenerate if the payoffs are different at all the leaves.
- 6.
Actually, F4 says that player i considers at least as many strategies possible at ω as at f(ω, v). To capture the fact that player i’s beliefs about other players’ possible strategies do not change, we would need the opposite direction of F4 as well: if \(\omega ' \in \mathcal{K}_{i}(\omega )\) then there exists a state \(\omega '' \in \mathcal{K}_{i}(f(\omega,v))\) such that s(ω′) and s(ω″) agree on the subtree of \(\Gamma \) below v. I do not impose this requirement here simply because it turns out to be unnecessary for Aumann’s Theorem.
- 7.
Samet does not use selection functions to capture counterfactual reasoning, but hypothesis transformations, which map cells (in the information partition) to cells. However, as I have shown (Halpern 1999), we can capture what Samet is trying to do by using selection functions.
References
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Acknowledgements
I’d like to thank Robert Stalnaker for his many useful comments and criticisms of this paper.
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Halpern, J.Y. (2016). Substantive Rationality and Backward Induction. In: ArlĂł-Costa, H., Hendricks, V., van Benthem, J. (eds) Readings in Formal Epistemology. Springer Graduate Texts in Philosophy, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-20451-2_43
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