5 research outputs found
The asymptotic value in finite stochastic games
We provide a direct, elementary proof for the existence of , where is the value of a -discounted
finite two-person zero-sum stochastic game
General limit value in zero-sum stochastic games
Bewley and Kohlberg (1976) and Mertens and Neyman (1981) have proved,
respectively, the existence of the asymptotic value and the uniform value in
zero-sum stochastic games with finite state space and finite action sets. In
their work, the total payoff in a stochastic game is defined either as a Cesaro
mean or an Abel mean of the stage payoffs. This paper presents two findings:
first, we generalize the result of Bewley and Kohlberg to a more general class
of payoff evaluations and we prove with a counterexample that this result is
tight. We also investigate the particular case of absorbing games. Second, for
the uniform approach of Mertens and Neyman, we provide another counterexample
to demonstrate that there is no natural way to generalize the result of Mertens
and Neyman to a wider class of payoff evaluations
Best-response Dynamics in Zero-sum Stochastic Games
We define and analyse three learning dynamics for two-player zero-sum discounted-payoff stochastic games. A continuous-time best-response dynamic in mixed strategies is proved to converge to the set of Nash equilibrium stationary strategies. Extending this, we introduce a fictitious-play-like process in a continuous-time embedding of a stochastic zero-sum game, which is again shown to converge to the set of Nash equilibrium strategies. Finally, we present a modified δ-converging best-response dynamic, in which the discount rate converges to 1, and the learned value converges to the asymptotic value of the zero-sum stochastic game. The critical feature of all the dynamic processes is a separation of adaption rates: beliefs about the value of states adapt more slowly than the strategies adapt, and in the case of the δ-converging dynamic the discount rate adapts more slowly than everything else
Advances in Zero-Sum Dynamic Games
International audienceThe survey presents recent results in the theory of two-person zero-sum repeated games and their connections with differential and continuous-time games. The emphasis is made on the following(1) A general model allows to deal simultaneously with stochastic and informational aspects.(2) All evaluations of the stage payoffs can be covered in the same framework (and not only the usual Cesàro and Abel means).(3) The model in discrete time can be seen and analyzed as a discretization of a continuous time game. Moreover, tools and ideas from repeated games are very fruitful for continuous time games and vice versa.(4) Numerous important conjectures have been answered (some in the negative).(5) New tools and original models have been proposed. As a consequence, the field (discrete versus continuous time, stochastic versus incomplete information models) has a much more unified structure, and research is extremely active