Discounting Infinite Games But How and Why?

In a recent paper de Alfaro, Henzinger and Majumdar [8] observed that discounting successive payments, the procedure that is employed in the classical stochastic game theory since the seminal paper of Shapley [16], is also pertinent in the context of much more recent theory of stochastic parity games [7,6,5] which were proposed as a tool for verification of probabilistic systems. We show that, surprisingly perhaps, the particular discounting used in [8] is in fact very close to the original ideas of Shapley. This observation allows to realize that the specific discounting of [8] suffers in fact from some needless restrictions. We advocate that dropping the constraints imposed in [8] leads to a more general and elegant theory that includes parity and mean payoff games as particular limit cases.