Some problems in discounted stochastic games [védés előtt]

Abstract

This thesis addresses three problems in the theory of iscounted stochastic games, all motivated by the effects of time-dependent dis-counting. First, it investigates the Nash equilibrium of finite stochastic games with generalised discounting. By employing the framework of gener-alised continuous games, it is shown that every finite stochastic game with generalised discounting admits a Nash equilibrium. Moreover, an example is provided to illustrate that a stationary Nash equilibrium does not necessarily exist in these stochastic games. Second, the thesis examines zero-sum stochastic games with sepa-rable discounting. Using the concept of supergames, it is demonstrated that every zero-sum finite stochastic game with separable discounting admits a value. Furthermore, it is established that, under certain conditions, this result can be extended to zero-sum countable stochas-tic games with separable discounting. In addition, three models of zero-sum infinite stochastic games with separable discounting - Borel, Suslin, and Nowak - are considered, and the existence of a value is established for each case. Finally, the thesis investigates finitely additive Markov decision processes under separable and ripple discounting. In the case of sep-arable discounting, it is shown that the player always possesses an optimal Markov strategy. In contrast, under ripple discounting, only the existence of an optimal behavioural strategy can be guaranteed.