Abstract
We introduce the dependent splitting game, a zero-sum stochastic game in which the players jointly control a martingale. This game models the transmission of information in repeated games with incomplete information on both sides, in the dependent case: The state variable represents the martingale of posterior beliefs. We establish the existence of the value for any fixed, general evaluation of the stage payoffs, as a function of the initial state. We then prove the convergence of the value functions, as the evaluation vanishes, to the unique solution of the Mertens–Zamir system of equations is established. From this result, we derive the convergence of the values of repeated games with incomplete information on both sides, in the dependent case, to the same function, as the evaluation vanishes. Finally, we establish a surprising result: Unlike repeated games with incomplete information on both sides, the splitting game has a uniform value. Moreover, we exhibit a couple of optimal stationary strategies for which the stage payoff and the state remain constant.
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Notes
See Sect. 3.4.
For any countable set X, we denote by \(\Delta (X)\) the set of probability measures over X, which is the set of sequences \((a_x)_{x\in X}\) such that \(\sum _{x\in X} a_x=1\) and \(a_x\ge 0\), for all \(x\in X\).
Laraki [5] considers a more abstract setting in which K and L are not supposed to be finite sets, but rather some convex, compact metric spaces.
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Acknowledgements
This work started as part of my Ph.D. under the supervision of Sylvain Sorin. I am very much indebted to him for motivating this research and his insightful guidance. The comments from Rida Laraki, Guillaume Vigeral and Fabien Gensbittel, the careful reading and suggestions from the anonymous referees, and the editors’ support have been remarkable. I am really grateful to all of them. Finally, I also gratefully acknowledge the support of the French National Research Agency, under Grant ANR CIGNE (ANR-15-CE38-0007-01).
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Oliu-Barton, M. The Splitting Game: Value and Optimal Strategies. Dyn Games Appl 8, 157–179 (2018). https://doi.org/10.1007/s13235-017-0216-8
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DOI: https://doi.org/10.1007/s13235-017-0216-8