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We investigate zero-sum turn-based two-player stochastic games in which theobjective of one player is to maximize the amount of rewards obtained during aplay, while the other aims at minimizing it. We focus on games in which theminimizer plays in a fair way. We believe that these kinds of games enjoyinteresting applications in software verification, where the maximizer playsthe role of a system intending to maximize the number of "milestones" achieved,and the minimizer represents the behavior of some uncooperative but yet fairenvironment. Normally, to study total reward properties, games are requested tobe stopping (i.e., they reach a terminal state with probability 1). We relaxthe property to request that the game is stopping only under a fair minimizingplayer. We prove that these games are determined, i.e., each state of the gamehas a value defined. Furthermore, we show that both players have memoryless anddeterministic optimal strategies, and the game value can be computed byapproximating the greatest-fixed point of a set of functional equations. Weimplemented our approach in a prototype tool, and evaluated it on anillustrating example and an Unmanned Aerial Vehicle case study.

Playing Against Fair Adversaries in Stochastic Games with Total Rewards