# A nested family of $$\varvec{k}$$ k -total effective rewards for positional games

## Author

Listed:
• Endre Boros

(Rutgers University)

• Khaled Elbassioni

() (Masdar Institute of Science and Technology)

(Rutgers University
National Research University, Higher School of Economics)

• Kazuhisa Makino

(Research Institute for Mathematical Sciences (RIMS) Kyoto University)

## Abstract

Abstract We consider Gillette’s two-person zero-sum stochastic games with perfect information. For each $$k \in \mathbb {N}=\{0,1,\ldots \}$$ k ∈ N = { 0 , 1 , … } we introduce an effective reward function, called k-total. For $$k = 0$$ k = 0 and 1 this function is known as mean payoff and total reward, respectively. We restrict our attention to the deterministic case. For all k, we prove the existence of a saddle point which can be realized by uniformly optimal pure stationary strategies. We also demonstrate that k-total reward games can be embedded into $$(k+1)$$ ( k + 1 ) -total reward games.

## Suggested Citation

• Endre Boros & Khaled Elbassioni & Vladimir Gurvich & Kazuhisa Makino, 2017. "A nested family of $$\varvec{k}$$ k -total effective rewards for positional games," International Journal of Game Theory, Springer;Game Theory Society, vol. 46(1), pages 263-293, March.
• Handle: RePEc:spr:jogath:v:46:y:2017:i:1:d:10.1007_s00182-016-0532-z
DOI: 10.1007/s00182-016-0532-z
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## References listed on IDEAS

as
1. Endre Boros & Khaled Elbassioni & Vladimir Gurvich & Kazuhisa Makino, 2013. "On Canonical Forms for Zero-Sum Stochastic Mean Payoff Games," Dynamic Games and Applications, Springer, vol. 3(2), pages 128-161, June.
2. N. N. Pisaruk, 1999. "Mean Cost Cyclical Games," Mathematics of Operations Research, INFORMS, vol. 24(4), pages 817-828, November.
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### Keywords

Stochastic game with perfect information; Cyclic games; Two-person; Zero-sum; Mean payoff; Total reward;

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