# American Institute of Mathematical Sciences

October  2019, 6(4): 259-275. doi: 10.3934/jdg.2019018

## Asymptotically optimal strategies in repeated games with incomplete information and vanishing weights

 Université Paris-Dauphine, PSL Research University, CNRS, CEREMADE, 75016 Paris, France

Received  November 2018 Revised  September 2019 Published  October 2019

We construct asymptotically optimal strategies in two-player zero-sum repeated games with incomplete information on both sides in which stages have vanishing weights. Our construction, inspired in Heuer (IJGT 1992), proves the convergence of the values for these games, thus extending the results established by Mertens and Zamir (IJGT 1971) for $n$-stage games and discounted games to the case of arbitrary vanishing weights.

Citation: Miquel Oliu-Barton. Asymptotically optimal strategies in repeated games with incomplete information and vanishing weights. Journal of Dynamics & Games, 2019, 6 (4) : 259-275. doi: 10.3934/jdg.2019018
##### References:
 [1] R. Aumann and M. Maschler, Repeated Games with Incomplete Information, With the collaboration of Richard E. Stearns. MIT Press, Cambridge, MA, 1995.  Google Scholar [2] P. Cardaliaguet, R. Laraki and S. Sorin, A continuous time approach for the asymptotic value in two-person zero-sum repeated games, SIAM Journal on Control and Optimization, 50 (2012), 1573-1596.  doi: 10.1137/110839473.  Google Scholar [3] B. De Meyer, Repeated games and partial differential equations, Mathematics of Operations Research, 21 (1996), 209-236.  doi: 10.1287/moor.21.1.209.  Google Scholar [4] B. De Meyer, Repeated games, duality and the Central Limit theorem, Mathematics of Operations Research, 21 (1996), 237-251.  doi: 10.1287/moor.21.1.237.  Google Scholar [5] B. De Meyer and A. Marino, Duality and optimal strategies in the finitely repeated zero-sum games with incomplete information on both sides, Cahiers de la MSE, 27. Google Scholar [6] F. Gensbittel and M. Oliu-Barton, Optimal strategies in repeated games with incomplete information, In revison. Google Scholar [7] M. Heuer, Asymptotically optimal strategies in repeated games with incomplete information, International Journal of Game Theory, 20 (1992), 377-392.  doi: 10.1007/BF01271132.  Google Scholar [8] N. Krasovskii and A. Subbotin, Game Theoretical Control Problems, Springer Verlag, 1988.  Google Scholar [9] R. Laraki, Variational inequalities, system of functional equations and incomplete information repeated games, SIAM Journal on Control and Optimization, 40 (2001), 516-524.  doi: 10.1137/S0363012900366601.  Google Scholar [10] R. Laraki and J. Renault, Acyclic gambling games, 2017. Google Scholar [11] P. Maldonado and M. Oliu-Barton, A strategy-based proof of the existence of the value in zero-sum differential games, Morfismos, 18 (2014), 31-44.   Google Scholar [12] J.-F. Mertens and S. Zamir, The value of two-person zero-sum repeated games with lack of information on both sides, International Journal of Game Theory, 1 (1971), 39-64.  doi: 10.1007/BF01753433.  Google Scholar [13] J.-F. Mertens and S. Zamir, Incomplete information games with transcendental values, Mathematics of Operations Research, 6 (1981), 313-318.  doi: 10.1287/moor.6.2.313.  Google Scholar [14] M. Oliu-Barton, Differential games with asymmetric and correlated information, Dynamic Games and Applications, 5 (2015), 378-396.  doi: 10.1007/s13235-014-0131-1.  Google Scholar [15] M. Oliu-Barton, Splitting game: Uniform value and optimal strategies, Dynamic Games and Applications, 8 (2018), 157-179.  doi: 10.1007/s13235-017-0216-8.  Google Scholar [16] R. Rockafellar, Convex Analysis, Princeton University Press, 1997.  Google Scholar [17] D. Rosenberg and S. Sorin, An operator approach to zero-sum repeated games, Israel Journal of Mathematics, 121 (2001), 221-246.  doi: 10.1007/BF02802505.  Google Scholar [18] S. Sorin, A First Course on Zero-Sum Repeated Games, Springer, 2002.  Google Scholar [19] N. Vieille, Weak approachability, Mathematics of Operations Research, 17 (1992), 781-791.  doi: 10.1287/moor.17.4.781.  Google Scholar

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##### References:
 [1] R. Aumann and M. Maschler, Repeated Games with Incomplete Information, With the collaboration of Richard E. Stearns. MIT Press, Cambridge, MA, 1995.  Google Scholar [2] P. Cardaliaguet, R. Laraki and S. Sorin, A continuous time approach for the asymptotic value in two-person zero-sum repeated games, SIAM Journal on Control and Optimization, 50 (2012), 1573-1596.  doi: 10.1137/110839473.  Google Scholar [3] B. De Meyer, Repeated games and partial differential equations, Mathematics of Operations Research, 21 (1996), 209-236.  doi: 10.1287/moor.21.1.209.  Google Scholar [4] B. De Meyer, Repeated games, duality and the Central Limit theorem, Mathematics of Operations Research, 21 (1996), 237-251.  doi: 10.1287/moor.21.1.237.  Google Scholar [5] B. De Meyer and A. Marino, Duality and optimal strategies in the finitely repeated zero-sum games with incomplete information on both sides, Cahiers de la MSE, 27. Google Scholar [6] F. Gensbittel and M. Oliu-Barton, Optimal strategies in repeated games with incomplete information, In revison. Google Scholar [7] M. Heuer, Asymptotically optimal strategies in repeated games with incomplete information, International Journal of Game Theory, 20 (1992), 377-392.  doi: 10.1007/BF01271132.  Google Scholar [8] N. Krasovskii and A. Subbotin, Game Theoretical Control Problems, Springer Verlag, 1988.  Google Scholar [9] R. Laraki, Variational inequalities, system of functional equations and incomplete information repeated games, SIAM Journal on Control and Optimization, 40 (2001), 516-524.  doi: 10.1137/S0363012900366601.  Google Scholar [10] R. Laraki and J. Renault, Acyclic gambling games, 2017. Google Scholar [11] P. Maldonado and M. Oliu-Barton, A strategy-based proof of the existence of the value in zero-sum differential games, Morfismos, 18 (2014), 31-44.   Google Scholar [12] J.-F. Mertens and S. Zamir, The value of two-person zero-sum repeated games with lack of information on both sides, International Journal of Game Theory, 1 (1971), 39-64.  doi: 10.1007/BF01753433.  Google Scholar [13] J.-F. Mertens and S. Zamir, Incomplete information games with transcendental values, Mathematics of Operations Research, 6 (1981), 313-318.  doi: 10.1287/moor.6.2.313.  Google Scholar [14] M. Oliu-Barton, Differential games with asymmetric and correlated information, Dynamic Games and Applications, 5 (2015), 378-396.  doi: 10.1007/s13235-014-0131-1.  Google Scholar [15] M. Oliu-Barton, Splitting game: Uniform value and optimal strategies, Dynamic Games and Applications, 8 (2018), 157-179.  doi: 10.1007/s13235-017-0216-8.  Google Scholar [16] R. Rockafellar, Convex Analysis, Princeton University Press, 1997.  Google Scholar [17] D. Rosenberg and S. Sorin, An operator approach to zero-sum repeated games, Israel Journal of Mathematics, 121 (2001), 221-246.  doi: 10.1007/BF02802505.  Google Scholar [18] S. Sorin, A First Course on Zero-Sum Repeated Games, Springer, 2002.  Google Scholar [19] N. Vieille, Weak approachability, Mathematics of Operations Research, 17 (1992), 781-791.  doi: 10.1287/moor.17.4.781.  Google Scholar
Duality between the hyperplanes above $v_1(\, \cdot \, , Q)$ (resp. supporting $v_1(\, \cdot \, , Q)$ at $p\in \Delta(K)$) and the set $B(Q)$ (resp. $B(p, Q)$). Here, $z \in B(Q)$ corresponds to the hyperplane $p'\mapsto \langle z, p'\rangle$ and $x\in B(p, Q)$ corresponds to $p'\mapsto \langle x, p'\rangle$
Illustration of the strategy at stage $m$, in the case $v(\pi_m)<u(\pi_m)$ where player $2$ needs to use his private information. In the figure, $R = \{r, r'\}$ and $\alpha_m = ( \alpha, \alpha')\in \Delta(R)$. The vectors $z_m(r)$ and $z_m(r')$ belong, respectively, to $B(p_m(r), Q_m(r))$ and $B(p_m(r), Q_m(r'))$. The construction is trivial in the case $v(\pi_m)\geq u(\pi_m)$, for it is enough to take $z_m(r) = z_m(r') = z_m$
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