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On the equivalence of mixed and behavior strategies in finitely additive decision problems

Part of: Game theory Mathematical economics

Published online by Cambridge University Press:  01 October 2019

János Flesch ,
Dries Vermeulen  and
Anna Zseleva
János Flesch*
Affiliation:
Maastricht University
Dries Vermeulen*
Affiliation:
Maastricht University
Anna Zseleva*
Affiliation:
Tel Aviv University
*
*Postal address: School of Business and Economics, Department of Quantitative Economics, Maastricht University, PO Box 616, 6200 MD Maastricht, The Netherlands.
*Postal address: School of Business and Economics, Department of Quantitative Economics, Maastricht University, PO Box 616, 6200 MD Maastricht, The Netherlands.
****Postal address: School of Mathematical Sciences, Tel Aviv University, 6997800 Tel Aviv, Israel.
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Abstract

We consider decision problems with arbitrary action spaces, deterministic transitions, and infinite time horizon. In the usual setup when probability measures are countably additive, a general version of Kuhn’s theorem implies under fairly general conditions that for every mixed strategy of the decision maker there exists an equivalent behavior strategy. We examine to what extent this remains valid when probability measures are only assumed to be finitely additive. Under the classical approach of Dubins and Savage (2014), we prove the following statements: (1) If the action space is finite, every mixed strategy has an equivalent behavior strategy. (2) Even if the action space is infinite, at least one optimal mixed strategy has an equivalent behavior strategy. The approach by Dubins and Savage turns out to be essentially maximal: these two statements are no longer valid if we take any extension of their approach that considers all singleton plays.

Keywords

Mixed strategybehavior strategyfinitely additive probability measureequivalent strategyKuhn’s theorem

MSC classification

Primary: 91B06: Decision theory
Secondary: 91A18: Games in extensive form
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 3 , September 2019 , pp. 810 - 829
Copyright
© Applied Probability Trust 2019 

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Footnotes

Support from the Basic Research Program of the National Research University Higher School of Economics is gratefully acknowledged.

References

Aryal, G. and Stauber, R. (2014). A note on Kuhn’s theorem with ambiguity averse players. Econom. Lett. 125, 110–114.CrossRefGoogle Scholar
Aumann, R. J. (1964). Mixed and behavior strategies in infinite extensive games. In Advances in Game Theory, Annals of Mathematics Studies vol. 52, eds Dresher, M., Shapley, L. S. and Tucker, A. W.. Princeton University Press, pp. 627–650.Google Scholar
Bingham, N. H. (2010). Finite additivity versus countable additivity. Electron. J. Hist. Probab. Statist. 6, no 1.Google Scholar
Capraro, V. and Scarsini, M. (2013). Existence of equilibria in countable games: An algebraic approach. Games Econom. Behavior 79, 163–180.CrossRefGoogle Scholar
Dubins, L. E. (1974). On Lebesgue-like extensions of finitely additive measures. Ann. Prob. 2, 456–463.CrossRefGoogle Scholar
Dubins, L. E. (1975). Finitely additive conditional probabilities, conglomerability and disintegrations. Ann. Prob. 3, 89–99.CrossRefGoogle Scholar
Dubins, L. E. and Savage, L. J. (2014). How to Gamble if you Must: Inequalities for Stochastic Processes. New York, Dover Publications. Edited and updated by Sudderth, W. D. and Gilat, D..Google Scholar
Dunford, N. and Schwartz, J. T. (1964). Linear Operators, Part I: General Theory. New York, Interscience Publishers.Google Scholar
de Finetti, B. (1972). Probability, Induction and Statistics. New York, Wiley.Google Scholar
de Finetti, B. (1975). The Theory of Probability. Chichester, J. Wiley and Sons.Google Scholar
Flesch, J., Vermeulen, D. and Zseleva, A. (2017). Zero-sum games with charges. Games Econom. Behavior 102, 666–686.CrossRefGoogle Scholar
Harris, J. H., Stinchcombe, M. B. and Zame, W. R. (2005). Nearly compact and continuous normal form games: Characterizations and equilibrium existence. Games Econom. Behavior 50, 208–224.CrossRefGoogle Scholar
Kechris, A. S. (1995). Classical Descriptive Set Theory. Berlin, Springer.CrossRefGoogle Scholar
Kuhn, H. W. (1953). Extensive games and the problem of information. Ann. Math. Study 28, 193–216.Google Scholar
Loś, J. and Marczewski, E. (1949). Extensions of measure. Fundam. Math. 36, 267–276.CrossRefGoogle Scholar
Maitra, A. and Sudderth, W. (1993). Finitely additive and measurable stochastic games. Internat. J. Game Theory 22, 201–223.CrossRefGoogle Scholar
Maitra, A. and Sudderth, W. (1998). Finitely additive stochastic games with Borel measurable payoffs. Internat. J. Game Theory 27, 257–267.CrossRefGoogle Scholar
Marinacci, M. (1997). Finitely additive and epsilon Nash equilibria. Internat. J. Game Theory 26, 315–333.CrossRefGoogle Scholar
Maschler, M., Solan, E. and Zamir, S. (2013). Game Theory. Cambridge University Press.CrossRefGoogle Scholar
Muraviev, I., Riedel, F. and Sass, L. (2017). Kuhn’s theorem for extensive form Ellsberg games. J. Math. Econom. 68, 26–41.CrossRefGoogle Scholar
Purves, R. and Sudderth, W. (1976). Some finitely additive probability. Ann. Prob. 4, 259–276.CrossRefGoogle Scholar
Rao, K. P. S. B. and Rao, B. (1983). Theory of Charges: A Study of Finitely Additive Measures. New York, Academic Press.Google Scholar
Savage, L. J. (1972). The Foundations of Statistics. New York, Dover Publications.Google Scholar
Schirokauer, O. and Kadane, J. B. (2007). Uniform distributions on the natural numbers. J. Theoret. Prob. 20, 429–441.CrossRefGoogle Scholar
Sudderth, W. (2016). Finitely additive dynamic programming. Math. Operat. Res. 41, 92–108.CrossRefGoogle Scholar
Takahashi, M. (1969). A generalization of Kuhn’s theorem for an infinite game. J. Sci. Hiroshima Univ. Ser. A-I Math. 33, 237–242.CrossRefGoogle Scholar