Abstract
I study games with countably many players, each of whom has finitely many pure strategies. The following are constructed: (i) a game that has a strong \(\epsilon \) equilibrium for all \(\epsilon >0\) but does not have a Nash equilibrium, and (ii) a symmetric game in which Nash equilibria exist, but all of them are asymmetric. Some additional results about infinite symmetric games are also derived.
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Notes
A more detailed discussion on the merits of the countable player set model can be found in Voorneveld (2010).
See Billingsley (1995), p. 287.
In Flesch et al. (2014) a discussion about this reasoning appears in the context of multi-stage games.
Solan and Vielle use the term “perfect \(\epsilon \) equilibrium.”
The proof of this fact is essentially the same as the proof of Nash equilibrium non-existence in this game.
This is a direct consequence of the Borel–Cantelli Lemma; see Billingsley (1995), pp. 59–60.
A clear case in point is provided by models of intergenerational equity [see, e.g., Asheim (2010)], in which there are infinitely many generations, and the central issue is how to treat them fairly.
Relatedly, there is a significant difference between invariance under finite and arbitrary permutations in the intergenerational equity setting; see Lauwers (1998).
To the best of my knowledge, a proof of this result has not been previously published; however, since it boils down to a straightforward application of Kakutani’s fixed-point theorem, I choose not to include it in the paper.
A G such that N is finite is called a finite game.
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Acknowledgments
Insightful comments by Igal Milchtaich, Ron Peretz, Amnon Schreiber, Igor Ulanovsky, and anonymous referees are gratefully acknowledged. A special thanks goes to an Associate Editor, whose comments resulted in a serious improvement of the paper.
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Rachmilevitch, S. Symmetry and approximate equilibria in games with countably many players. Int J Game Theory 45, 709–717 (2016). https://doi.org/10.1007/s00182-015-0479-5
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DOI: https://doi.org/10.1007/s00182-015-0479-5