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