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 and 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.

[2]

P. CardaliaguetR. 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.

[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.

[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.

[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.

[6]

F. Gensbittel and M. Oliu-Barton, Optimal strategies in repeated games with incomplete information, In revison.

[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.

[8]

N. Krasovskii and A. Subbotin, Game Theoretical Control Problems, Springer Verlag, 1988.

[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.

[10]

R. Laraki and J. Renault, Acyclic gambling games, 2017.

[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. 

[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.

[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.

[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.

[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.

[16]

R. Rockafellar, Convex Analysis, Princeton University Press, 1997.

[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.

[18]

S. Sorin, A First Course on Zero-Sum Repeated Games, Springer, 2002.

[19]

N. Vieille, Weak approachability, Mathematics of Operations Research, 17 (1992), 781-791.  doi: 10.1287/moor.17.4.781.

show all references

References:
[1]

R. Aumann and M. Maschler, Repeated Games with Incomplete Information, With the collaboration of Richard E. Stearns. MIT Press, Cambridge, MA, 1995.

[2]

P. CardaliaguetR. 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.

[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.

[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.

[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.

[6]

F. Gensbittel and M. Oliu-Barton, Optimal strategies in repeated games with incomplete information, In revison.

[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.

[8]

N. Krasovskii and A. Subbotin, Game Theoretical Control Problems, Springer Verlag, 1988.

[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.

[10]

R. Laraki and J. Renault, Acyclic gambling games, 2017.

[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. 

[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.

[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.

[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.

[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.

[16]

R. Rockafellar, Convex Analysis, Princeton University Press, 1997.

[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.

[18]

S. Sorin, A First Course on Zero-Sum Repeated Games, Springer, 2002.

[19]

N. Vieille, Weak approachability, Mathematics of Operations Research, 17 (1992), 781-791.  doi: 10.1287/moor.17.4.781.

Figure 1.  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 $
Figure 2.  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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