Abstract
The variation of a martingale \(p_{0}^{k}=p_{0},\ldots,p_{k}\) of probabilities on a finite (or countable) set X is denoted \(V(p_{0}^{k})\) and defined by
It is shown that \(V(p_{0}^{k})\leq\sqrt{2kH(p_{0})}\), where H(p) is the entropy function H(p)=−∑ x p(x)logp(x), and log stands for the natural logarithm. Therefore, if d is the number of elements of X, then \(V(p_{0}^{k})\leq\sqrt{2k\log d}\). It is shown that the order of magnitude of the bound \(\sqrt{2k\log d}\) is tight for d≤2k: there is C>0 such that for all k and d≤2k, there is a martingale \(p_{0}^{k}=p_{0},\ldots,p_{k}\) of probabilities on a set X with d elements, and with variation \(V(p_{0}^{k})\geq C\sqrt{2k\log d}\). An application of the first result to game theory is that the difference between v k and lim j v j , where v k is the value of the k-stage repeated game with incomplete information on one side with d states, is bounded by \(\|G\|\sqrt{2k^{-1}\log d}\) (where ∥G∥ is the maximal absolute value of a stage payoff). Furthermore, it is shown that the order of magnitude of this game theory bound is tight.
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Notes
This book is based on reports by Robert J. Aumann and Michael Maschler which appeared in the 1960s in Report of the U.S. Arms Control and Disarmament Agency. See “Game theoretic aspects of gradual disarmament” (1966, ST–80, Chap. V, pp. V1–V55), “Repeated games with incomplete information: a survey of recent results” (1967, ST–116, Chap. III, pp. 287–403), and “Repeated games with incomplete information: the zero-sum extensive case” (1968, ST–143, Chap. III, pp. 37–116).
I wish to thank Benjamin Weiss for raising the question of the tightness of the factor \(\sqrt{\log d}\) in the bound and demonstrating for each positive ℓ the existence of a simple martingale of probabilities \(p_{0}^{\ell}\) on a set with 2ℓ elements and with variation ℓ. Specifically, starting with the uniform probability, in each stage half of the nonzero probabilities (each half equally likely) move to zero, and the other half double their probabilities. Therefore, for each fixed α>0, there is a positive constant 0<C(α) (→ α→0+0) such that for k and d with \(\alpha\leq\frac{\log_{2} d}{k} \leq1\), \(V(k,d)\geq[\log_{2} d]\geq C(\alpha)\sqrt{k\log d}\).
I wish to thank Stanislaw Kwapien for suggesting a proof that avoids the information-theoretic techniques and communicating a simple analytical proof of the above displayed version of Pinsker’s inequality.
References
Aumann, R.J., Maschler, M.: Repeated Games with Incomplete Information. MIT Press, Cambridge (1995), with the collaboration of R. Stearns
Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley Series in Telecommunications. Wiley, New York (1991)
Mertens, J.-F., Zamir, S.: The value of two-person zero-sum repeated games with lack of information on both sides. Int. J. Game Theory 1, 39–64 (1971)
Mertens, J.-F., Zamir, S.: The maximal variation of a bounded martingale. Isr. J. Math. 27, 252–276 (1977)
Zamir, S.: On the relation between finitely and infinitely repeated games with incomplete information. Int. J. Game Theory 1, 179–198 (1972)
Acknowledgements
This research was supported in part by Israel Science Foundation grants 1123/06 and 1596/10.
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Neyman, A. The Maximal Variation of Martingales of Probabilities and Repeated Games with Incomplete Information. J Theor Probab 26, 557–567 (2013). https://doi.org/10.1007/s10959-012-0447-y
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DOI: https://doi.org/10.1007/s10959-012-0447-y