Abstract
In theory of play, a player needs to reason about other players’ types that could conceivably explain how play has reached a particular node of the extensive form game tree. Notions of rationalizability are relevant for such reasoning. We present a logical description of such player types and show that the associated type space is constructible (by a Turing machine).
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- 1.
By automata, we refer only to finite state devices here, though probablistic polynomial time Turing machines are a natural class to consider as well [13].
- 2.
A surprise move by an opponent is perhaps much harder for an automaton to digest than for a human player.
- 3.
This observation can be formalized by defining an indistinguishability relation \(\sim _i\) on player \(i\) nodes, but we do not pursue this here, since we do not attempt an axiomatization of the type space.
- 4.
Note that \( B _i\) is a “model changing” operator and thus ‘dynamic’, in the spirit of [8].
- 5.
We refer the reader to [8] for a more detailed justification.
- 6.
A formal characterization of this process as a recursive function on the tree is in progress, but there are many technical challenges.
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Acknowledgments
A part of this article was written when I was visiting Peking University. I thank Yanjing Wang for stimulating discussions and warm hospitality at Beijing. I also thank Johan van Benthem for thoughtful and encouraging remarks that greatly helped in the formulation.
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Ramanujam, R. (2014). Logical Player Types for a Theory of Play. In: Baltag, A., Smets, S. (eds) Johan van Benthem on Logic and Information Dynamics. Outstanding Contributions to Logic, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-06025-5_18
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