arXiv:1412.6072v1 [cs.DM] 20 Nov 2014 Nested Family of Cyclic Games with k -total Effective Rewards Endre Boros * Khaled Elbassioni † Vladimir Gurvich † Kazuhisa Makino ‡ December 19, 2014 Abstract We consider Gillette’s two-person zero-sum stochastic games with per- fect information. For each k ∈ Z+ we introduce a payoff function, called the k-total reward. For k = 0 and 1 these payoffs are known as mean payoff and total reward, respectively. We restrict our attention to the determin- istic case, the so called cyclic games. For all k, we prove the existence of a saddle point which can be realized by pure stationary strategies. We also demonstrate that k-total reward games can be embedded into (k +1)-total reward games. In particular, all of these classes contain mean payoff cyclic games. Keywords: stochastic game with perfect information, cyclic games, two-person, zero-sum, mean payoff, total reward 1 Introduction We consider two-person zero-sum stochastic games with perfect informa- tion and, for each positive integer k we define an effective payoff func- tion, called the k-total reward, generalizing the classical mean payoffs [4] (k = 0), as well as the total rewards [19, 20] (k = 1). In this paper, we restrict ourselves by two-person zero-sum games with deterministic states, and the solution concept is Nash equilibrium, which is just a saddle point in the considered case. We call the considered family of games k-total reward BW-games, where B and W stand for the two players, Black, the minimizer and White, the maximizer. We denote by R the set of reals, by Z the set of integers, and by Z+ the set of nonnegative integers. For a subset S ⊆ Z+, let R S denote the ∗ MSIS Department and RUTCOR, Rutgers University, 100 Rockafellar Road, Livingston Campus Piscataway, NJ 08854, USA; ({boros,gurvich}@rutcor.rutgers.edu) † Masdar Institute of Science and Technology, P.O.Box 54224, Abu Dhabi, UAE; (kelbas- sioni@masdar.ac.ae) ‡ Research Institute for Mathematical Sciences (RIMS) Kyoto University, Kyoto 606-8502, Japan; (makino@kurims.kyoto-u.ac.jp) 1