Abstract
When the strict rationality underlying the Nash equilibria in game theory is relaxed, one arrives at the quantal response equilibria introduced by McKelvey and Palfrey. Here, the players are assigned parameters measuring their degree of rationality, and the resulting equilibria are Gibbs type distribution. This brings us into the realm of the exponential families studied in information geometry, with an additional structure arising from the relations between the players. Tuning these rationality parameters leads to a simple geometric proof of the Nash existence theorem that only employs intersection properties of submanifolds of Euclidean spaces and dispenses with the Brouwer fixed point theorem on which the classical proofs depend. Also, in this geometric framework, we can develop very efficient computational tools for studying examples. The method can also be applied when additional parameters are involved, like the capacity of an information channel.
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Notes
- 1.
All essentials features of our setting and reasoning will show themselves already for the case where \(n=2\) and \(m_i=2\) for all players, that is, for the simplest game where we have only two players each of which can choose between two possible actions or moves. Nevertheless, for the sake of generality, we shall consider arbitrary values of n and the \(m_i\)s.
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Jost, J., Bertschinger, N., Olbrich, E., Wolpert, D. (2018). Information Geometry and Game Theory. In: Ay, N., Gibilisco, P., Matúš, F. (eds) Information Geometry and Its Applications . IGAIA IV 2016. Springer Proceedings in Mathematics & Statistics, vol 252. Springer, Cham. https://doi.org/10.1007/978-3-319-97798-0_2
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