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# Subgame maxmin strategies in zero-sum stochastic games with tolerance levels

## Author

Listed:
• Flesch, Janos

(QE / Mathematical economics and game the)

• Herings, P. Jean-Jacques

(General Economics 1 (Micro))

• Maes, Jasmine

(General Economics 1 (Micro))

(General Economics 1 (Micro))

## Abstract

We study subgame φ-maxmin strategies in two-player zero-sum stochastic games with finite action spaces and a countable state space. Here φ denotes the tolerance function, a function which assigns a non-negative tolerated error level to every subgame. Subgame φ-maxmin strategies are strategies of the maximizing player that guarantee the lower value in every subgame within the subgame-dependent tolerance level as given by φ. First, we provide necessary and sufficient conditions for a strategy to be a subgame φ-maxmin strategy. As a special case we obtain a characterization for subgame maxmin strategies, i.e. strategies that exactly guarantee the lower value at every subgame. Secondly, we present sufficient conditions for the existence of a subgame φ-maxmin strategy. Finally, we show the possibly surprising result that the existence of subgame φ-maxmin strategies for every positive tolerance function φ is equivalent to the existence of a subgame maxmin strategy.

## Suggested Citation

• Flesch, Janos & Herings, P. Jean-Jacques & Maes, Jasmine & Predtetchinski, Arkadi, 2018. "Subgame maxmin strategies in zero-sum stochastic games with tolerance levels," Research Memorandum 020, Maastricht University, Graduate School of Business and Economics (GSBE).
• Handle: RePEc:unm:umagsb:2018020
as

File URL: https://cris.maastrichtuniversity.nl/portal/files/27587143/content

## References listed on IDEAS

as
1. R. Laraki & A. Maitra & W. Sudderth, 2013. "Two-Person Zero-Sum Stochastic Games with Semicontinuous Payoff," Dynamic Games and Applications, Springer, vol. 3(2), pages 162-171, June.
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4. Drew Fudenberg & David Levine, 2008. "Subgame–Perfect Equilibria of Finite– and Infinite–Horizon Games," World Scientific Book Chapters,in: A Long-Run Collaboration On Long-Run Games, chapter 1, pages 3-20 World Scientific Publishing Co. Pte. Ltd..
5. Ayala Mashiah-Yaakovi, 2015. "Correlated Equilibria in Stochastic Games with Borel Measurable Payoffs," Dynamic Games and Applications, Springer, vol. 5(1), pages 120-135, March.
6. Eilon Solan & Nicolas Vieille, 2000. "Uniform Value in Recursive Games," Discussion Papers 1293, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
7. János Flesch & Arkadi Predtetchinski, 2016. "On refinements of subgame perfect $$\epsilon$$ ϵ -equilibrium," International Journal of Game Theory, Springer;Game Theory Society, vol. 45(3), pages 523-542, August.
8. Abate, Alessandro & Redig, Frank & Tkachev, Ilya, 2014. "On the effect of perturbation of conditional probabilities in total variation," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 1-8.
9. Radner, Roy, 1980. "Collusive behavior in noncooperative epsilon-equilibria of oligopolies with long but finite lives," Journal of Economic Theory, Elsevier, vol. 22(2), pages 136-154, April.
Full references (including those not matched with items on IDEAS)

### Keywords

stochastic games; zero-sum games; subgame φ-maxmin strategies;

### JEL classification:

• C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games

### NEP fields

This paper has been announced in the following NEP Reports:

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