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The Splitting Game: Value and Optimal Strategies

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  • Miquel Oliu-Barton

    (Université Paris-Dauphine, PSL Research University, CNRS, CEREMADE)

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.

Suggested Citation

  • Miquel Oliu-Barton, 2018. "The Splitting Game: Value and Optimal Strategies," Dynamic Games and Applications, Springer, vol. 8(1), pages 157-179, March.
    • Handle: RePEc:spr:dyngam:v:8:y:2018:i:1:d:10.1007_s13235-017-0216-8
    DOI: 10.1007/s13235-017-0216-8
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    References listed on IDEAS

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    1. Robert J. Aumann, 1995. "Repeated Games with Incomplete Information," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262011476, December.
    2. Mertens, J.-F. & Zamir, S., 1980. "Minmax and maxmin of repeated games with incomplete information," LIDAM Reprints CORE 433, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Rida Laraki, 2002. "The splitting game and applications," International Journal of Game Theory, Springer;Game Theory Society, vol. 30(3), pages 359-376.
    4. Heuer, M, 1992. "Asymptotically Optimal Strategies in Repeated Games with Incomplete Information," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(4), pages 377-392.
    5. Pierre Cardaliaguet & Rida Laraki & Sylvain Sorin, 2012. "A Continuous Time Approach for the Asymptotic Value in Two-Person Zero-Sum Repeated Games," Post-Print hal-00609476, HAL.
    6. MERTENS, Jean-François & ZAMIR, Shmuel, 1971. "The value of two-person zero-sum repeated games with lack of information on both sides," LIDAM Reprints CORE 154, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. repec:dau:papers:123456789/6775 is not listed on IDEAS
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    Cited by:

    1. Frédéric Koessler & Marie Laclau & Tristan Tomala, 2022. "Interactive Information Design," Mathematics of Operations Research, INFORMS, vol. 47(1), pages 153-175, February.
    2. Fabien Gensbittel & Miquel Oliu-Barton, 2020. "Optimal Strategies in Zero-Sum Repeated Games with Incomplete Information: The Dependent Case," Dynamic Games and Applications, Springer, vol. 10(4), pages 819-835, December.
    3. Rida Laraki & Jérôme Renault, 2020. "Acyclic Gambling Games," Mathematics of Operations Research, INFORMS, vol. 45(4), pages 1237-1257, November.

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