Discounting Infinite Games But How and Why?

In a recent paper de Alfaro, Henzinger and Majumdar [Luca de Alfaro, Thomas A. Henzinger, and Rupak Majumdar. Discounting the future in systems theory. In ICALP 2003, volume 2719 of LNCS, pages 1022-1037. Springer, 2003] observed that discounting successive payments, the procedure that is employed in the classical stochastic game theory since the seminal paper of Shapley [L.S. Shapley. Stochastic games. Proceedings Nat. Acad. of Science USA, 39:1095-1100, 1953], is also pertinent in the context of much more recent theory of stochastic parity games [L. de Alfaro and R. Majumdar. Quantitative solution to omega-regular games. In STOC'01, pages 675-683. ACM Press, 2001. final version to appear in Journal of Computer and System Sciences, L. de Alfaro, T.A. Henzinger, and O. Kupferman. Concurrent reachability games. In FOCS'98, pages 564-575. IEEE Computer Society Press, 1998, L. de Alfaro and T.A. Henzinger. Concurrent ω-regular games. In LICS'00, pages 142-154. IEEE Computer Society Press, 2000] which were proposed as a tool for verification of probabilistic systems. We show that, surprisingly perhaps, the particular discounting used in [Luca de Alfaro, Thomas A. Henzinger, and Rupak Majumdar. Discounting the future in systems theory. In ICALP 2003, volume 2719 of LNCS, pages 1022-1037. Springer, 2003] is in fact very close to the original ideas of Shapley. This observation allows to realize that the specific discounting of [Luca de Alfaro, Thomas A. Henzinger, and Rupak Majumdar. Discounting the future in systems theory. In ICALP 2003, volume 2719 of LNCS, pages 1022-1037. Springer, 2003] suffers in fact from some needless restrictions. We advocate that dropping the constraints imposed in [Luca de Alfaro, Thomas A. Henzinger, and Rupak Majumdar. Discounting the future in systems theory. In ICALP 2003, volume 2719 of LNCS, pages 1022-1037. Springer, 2003] leads to a more general and elegant theory that includes parity and mean payoff games as particular limit cases.