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.