Fixed Point Solution to Stochastic Priority Games

We define and examine a class of two-player stochastic games that we call priority games. The priority games contain as proper subclasses the parity games studied in computer science [4] and the games with the limsup and liminf payoff mappings. We show that the value of the priority game can be expressed as an appropriate nested fixed point of the value mapping of the one-day game. This extends the result of de Alfaro and Majumdar [4], where the authors proved that the value of the stochastic parity game can be expressed as the nested fixed point of the one-day value mapping. The difference between our paper and [4] is two-fold. The value of the parity game is obtained by applying the least and the greatest fixed points to the value mapping of the one-day game. However, in general, the greatest and the least fixed-points are not sufficient in order to obtain the value of the priority game. To cope with this problem we introduce the notion of the nearest fixed point of a monotone bounded nonexpansive mapping. Our main result is that the value of the priority game can be obtained as the nested nearest fixed point of the value mapping of the one-day game. The second point that makes our proof different from [4] is that our proof is inductive. We give a game interpretation for the nested fixed point formula where some variables are free (not bounded by the fixed point operator). Thus instead of proving the main result in one big step as in [4] we can limit ourselves to the case when just one fixed point is added to the nested fixed point formula.