Abstract
A generalization of mixed strategy equilibrium is proposed, where mixed strategies need only be finitely additive and payoff functions are not required to be integrable or bounded. This notion of best-response equilibrium is based on an extension of the idea that an equilibrium strategy is supported in the player’s set of best-response actions, but is applicable also when no best-response actions exist. It yields simple, natural equilibria in a number of well-known games where other kinds of mixed equilibrium are complicated, not compelling or do not exist.
This is a preview of subscription content, log in via an institution to check access.
Access this article
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
Notes
This contrasts with the usual definition of mixed strategy, where the domain is some pre-specified measurable structure on \(S_{i}\), for example, the collection of all Borel sets. A conceptual problem with the latter approach is that, unless \(S_{i}\) is finite, the choice of measurable structure is arguably arbitrary, as it is not indicated by the game itself. Yet choosing it is necessary for defining the mixed extension of the game, where players use mixed strategies rather than actions. In the framework presented here, there is no mixed extension.
An example is \(f(x,\;y) = x\sin 1/y:(0,\;1)^{2} \to {\mathbb{R}}\). With \(\mu_{1} = \mu_{2} = \delta_{{0^{ + } }}\) and the algebra \({\mathcal{I} }\) (both are defined at the end of Sect. 3), \(\int f\,d\mu = \int \int f\,d\mu_{1}\, d\mu_{2} = 0\) but the other iterated integral does not exist.
Recall that the asterisk denotes outer measure.
An open problem is to find a two-player game with bounded payoff functions that does not have a best-response equilibrium, or to prove that such a game does not exist.
The collection of all sets \(A\) satisfying the first or second condition is easily seen to be an algebra. If neither condition holds, \(A\) is not measurable. However, by Proposition 2, there are extensions of \(\sigma\) that render all sets measurable. Such an extension is the function \(A \mapsto \mathop {\lim }\nolimits_{n \to \infty } \delta_{{s^{n} }} (A)\), where \({\text{lim}}\) refers to some fixed Banach limit (so that it exists for every bounded sequence).
More generally, for any \(S \subseteq {\mathbb{R}}\), the collection \(\left\{ {A \cap S \mid A \in {\mathcal{I}}} \right\}\) is an algebra on \(S\), which may also be denoted by \({\mathcal{I}}\) if the meaning is clear from the context.
For an alternative solution to the problem of nonexistence of equilibrium, which employs a set-valued solution concept, see Milchtaich (2019).
It is easy to see that a legitimate equilibrium \(\sigma\) remains so if one (or more) of the algebras \({\mathcal{A}}_{i}\) is replaced by a subalgebra, to which the strategy \(\sigma_{i}\) is restricted. Note that this is the opposite of the situation for best-response equilibria, which are preserved by extensions rather than restrictions.
References
Bhaskara Rao KPS, Bhaskara Rao M (1983) Theory of charges: a study of finitely additive measures. Academic Press, London
Dasgupta P, Maskin E (1986) The existence of equilibrium in discontinuous economic games, I: theory. Rev Econ Stud 53:1–26
Dastidar KG (2011) Existence of Bertrand equilibrium revisited. Int J Econ Theory 7:331–350
de Finetti B (1974) Theory of probability. Wiley, New York
Dubins LE, Savage LJ (2014) How to gamble if you must: inequalities for stochastic processes. Edited and updated by Sudderth WD, Gilat D. Dover, New York
Dunford N, Schwartz JT (1988) Linear operators. Part I. Wiley, New York
Eaton BC, Lipsey RG (1975) The principle of minimum differentiation reconsidered: some new developments in the theory of spatial competition. Rev Econ Stud 42:27–49
Flesch J, Vermeulen D, Zseleva A (2021) Legitimate equilibrium. Int J Game Theory 50:787–800
Harris CJ, Stinchcombe MB, Zame WR (2005) Nearly compact and continuous normal form games: characterizations and equilibrium existence. Games Econ Behav 50:208–224
Hoernig SH (2007) Bertrand games and sharing rules. Econ Theory 31:573–585
Karlin S (1950) Operator treatment of minmax principle. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games I, Annals of Mathematics Studies, vol 24. Princeton University Press, Princeton, pp 133–154
Marinacci M (1997) Finitely additive and epsilon Nash equilibria. Int J Game Theory 26:315–333
Milchtaich I (2019) Polyequilibrium. Games Econ Behav 113:339–355. https://www.sciencedirect.com/science/article/pii/S0899825618301556
Osborne MJ, Pitchik C (1986) The nature of equilibrium in a location model. Int Econ Rev 27:223–237
Savage, LJ (1954) The foundations of statistics. Wiley, New York (Revised and enlarged edition, Dover, New York, 1972)
Shaked A (1982) Existence and computation of mixed strategy Nash equilibrium for 3-firms location problem. J Ind Econ 31:93–96
Sion M, Wolfe P (1957) On a game without a value. In: Dresher M, Tucker AW, Wolfe P (eds) Contributions to the theory of games III, Annals of Mathematics Studies, vol 39. Princeton University Press, Princeton, pp 299–306
Vasquez MA (2017) Optimistic equilibria in finitely additive mixed strategies. Mimeo
Ville J (1938) Sur la théorie général des jeux où intervient l’habilité des joueurs. In: Traité du Calcul des Probabilités et de ses Applications by Borel É et al., vol 2, Paris, pp 105–113
von Neumann J, Morgenstern O (1953) Theory of games and economic behavior, 3rd edn. Princeton University Press, Princeton
Yanovskaya EB (1970) The solution of infinite zero-sum two-person games with finitely additive strategies. Theory Probab Appl 15:153–158
Acknowledgements
I am grateful to János Flesch and Miklós Pintér for stimulating discussions concerning the subject matter. I also thank two anonymous referees for their detailed and helpful reports.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Bernhard von Stengel.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Milchtaich, I. Best-response equilibrium: an equilibrium in finitely additive mixed strategies. Int J Game Theory 52, 1317–1334 (2023). https://doi.org/10.1007/s00182-023-00871-2
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00182-023-00871-2