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

  • János Flesch (a1), Dries Vermeulen (a1) and Anna Zseleva (a2)


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 The approach by Dubins and Savage turns out to be essentially maximal: these two statements are no longer valid if we take any extension of their approach that considers all singleton plays.


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



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Aryal, G. and Stauber, R. (2014). A note on Kuhn’s theorem with ambiguity averse players. Econom. Lett. 125, 110114.
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. 627650.
Bingham, N. H. (2010). Finite additivity versus countable additivity. Electron. J. Hist. Probab. Statist. 6, no 1.
Capraro, V. and Scarsini, M. (2013). Existence of equilibria in countable games: An algebraic approach. Games Econom. Behavior 79, 163180.
Dubins, L. E. (1974). On Lebesgue-like extensions of finitely additive measures. Ann. Prob. 2, 456463.
Dubins, L. E. (1975). Finitely additive conditional probabilities, conglomerability and disintegrations. Ann. Prob. 3, 8999.
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..
Dunford, N. and Schwartz, J. T. (1964). Linear Operators, Part I: General Theory. New York, Interscience Publishers.
de Finetti, B. (1972). Probability, Induction and Statistics. New York, Wiley.
de Finetti, B. (1975). The Theory of Probability. Chichester, J. Wiley and Sons.
Flesch, J., Vermeulen, D. and Zseleva, A. (2017). Zero-sum games with charges. Games Econom. Behavior 102, 666686.
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, 208224.
Kechris, A. S. (1995). Classical Descriptive Set Theory. Berlin, Springer.
Kuhn, H. W. (1953). Extensive games and the problem of information. Ann. Math. Study 28, 193216.
Loś, J. and Marczewski, E. (1949). Extensions of measure. Fundam. Math. 36, 267276.
Maitra, A. and Sudderth, W. (1993). Finitely additive and measurable stochastic games. Internat. J. Game Theory 22, 201223.
Maitra, A. and Sudderth, W. (1998). Finitely additive stochastic games with Borel measurable payoffs. Internat. J. Game Theory 27, 257267.
Marinacci, M. (1997). Finitely additive and epsilon Nash equilibria. Internat. J. Game Theory 26, 315333.
Maschler, M., Solan, E. and Zamir, S. (2013). Game Theory. Cambridge University Press.
Muraviev, I., Riedel, F. and Sass, L. (2017). Kuhn’s theorem for extensive form Ellsberg games. J. Math. Econom. 68, 2641.
Purves, R. and Sudderth, W. (1976). Some finitely additive probability. Ann. Prob. 4, 259276.
Rao, K. P. S. B. and Rao, B. (1983). Theory of Charges: A Study of Finitely Additive Measures. New York, Academic Press.
Savage, L. J. (1972). The Foundations of Statistics. New York, Dover Publications.
Schirokauer, O. and Kadane, J. B. (2007). Uniform distributions on the natural numbers. J. Theoret. Prob. 20, 429441.
Sudderth, W. (2016). Finitely additive dynamic programming. Math. Operat. Res. 41, 92108.
Takahashi, M. (1969). A generalization of Kuhn’s theorem for an infinite game. J. Sci. Hiroshima Univ. Ser. A-I Math. 33, 237242.


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

  • János Flesch (a1), Dries Vermeulen (a1) and Anna Zseleva (a2)


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