Blackwell Optimal Strategies in Priority mean-Payoff Games
نویسندگان
چکیده
One of the recurring themes in the theory of stochastic games is the interplay between discounted games and mean-payoff games. This culminates in the seminal paper of Mertens and Neyman [12] showing that mean-payoff games have a value and this value is the limit of the values of discounted games when the discount factor tends to 1. Note however that optimal strategies in both games are very different. As shown by Shapley [13] discounted stochastic games admit memoryless optimal strategies. On the other hand mean-payoff games do not have optimal strategies, they have only ε-optimal strategies and to play optimally players need an unbounded memory. The connections between discounted and mean-payoff games become much tighter when we consider perfect information stochastic games (games where players play in turns). As discovered by Blackwell [3], if the discount factor is close to 1 then optimal memoryless deterministic strategies in discounted games are also optimal for mean-payoff games (but not the other way round). Thus both games are related not only by their values but also through their optimal strategies. Blackwell’s result extends easily to two-player perfect information stochastic games. What happens if instead of mean-payoff games we consider parity games – a class of games more directly relevant to computer science [9]? In particular, are parity games related to discounted games? It is well known that deterministic mean-payoff games and parity games are related, see [2]. The first insight that there is some link between parity games and discounted games is due to de Alfaro at al. [1]. It turns out that parity games are related to multi-discounted games with multiple discount factors that depend on the state. This should be compared with discounted games with a unique, state independent, discount factor which are used in the study of mean-payoff games. Like in the classical theory of stochastic games, we examine what happens when the discount factors tend to 1, the idea is that in the limit we want to obtain parity games. Note that if we have several state dependent discount factors λ1, . . .λk then there are two possibilities to approach 1: • we can study the iterated limit limλ1→1 . . . limλk→1 when discount factors tend to 1 one after another (i.e. first we go to 1 with the discount factor λk associated with some group of states, when the limit is reached then we go to 1 with the next discount factor λk−1 etc.,
منابع مشابه
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ورودعنوان ژورنال:
- Int. J. Found. Comput. Sci.
دوره 23 شماره
صفحات -
تاریخ انتشار 2012