# 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
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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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