# Recursive games: uniform value, Tauberian theorem and the Mertens conjecture “ $$Maxmin=\lim v_n=\lim v_{\uplambda }$$ M a x m i n = lim v n = lim v λ ”

## Author

Listed:
• Xiaoxi Li

() (CNRS, IMJ-PRG, UMR 7586, Sorbonne Universités, UPMC Univ.)

• Xavier Venel

() (CES, Université Paris 1 Panthéon Sorbonne)

## Abstract

Abstract We study two-player zero-sum recursive games with a countable state space and finite action spaces at each state. When the family of n-stage values $$\{v_n,n\ge 1\}$$ { v n , n ≥ 1 } is totally bounded for the uniform norm, we prove the existence of the uniform value. Together with a result in Rosenberg and Vieille (Math Oper Res 39:23–35, 2000), we obtain a uniform Tauberian theorem for recursive game: $$(v_n)$$ ( v n ) converges uniformly if and only if $$(v_{\uplambda })$$ ( v λ ) converges uniformly. We apply our main result to finite recursive games with signals (where players observe only signals on the state and on past actions). When the maximizer is more informed than the minimizer, we prove the Mertens conjecture $$Maxmin=\lim _{n\rightarrow \infty } v_n=\lim _{{\uplambda }\rightarrow 0}v_{\uplambda }$$ M a x m i n = lim n → ∞ v n = lim λ → 0 v λ . Finally, we deduce the existence of the uniform value in finite recursive game with symmetric information.

## Suggested Citation

• Xiaoxi Li & Xavier Venel, 2016. "Recursive games: uniform value, Tauberian theorem and the Mertens conjecture “ $$Maxmin=\lim v_n=\lim v_{\uplambda }$$ M a x m i n = lim v n = lim v λ ”," International Journal of Game Theory, Springer;Game Theory Society, vol. 45(1), pages 155-189, March.
• Handle: RePEc:spr:jogath:v:45:y:2016:i:1:d:10.1007_s00182-015-0496-4
DOI: 10.1007/s00182-015-0496-4
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## References listed on IDEAS

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1. Lehrer Ehud & Monderer Dov, 1994. "Discounting versus Averaging in Dynamic Programming," Games and Economic Behavior, Elsevier, vol. 6(1), pages 97-113, January.
2. Monderer, Dov & Sorin, Sylvain, 1993. "Asymptotic Properties in Dynamic Programming," International Journal of Game Theory, Springer;Game Theory Society, vol. 22(1), pages 1-11.
3. repec:dau:papers:123456789/6231 is not listed on IDEAS
4. Nicolas Vieille & Dinah Rosenberg, 2000. "The Maxmin of Recursive Games with Incomplete Information on one Side," Post-Print hal-00481429, HAL.
5. Mertens,Jean-FranÃ§ois & Sorin,Sylvain & Zamir,Shmuel, 2015. "Repeated Games," Cambridge Books, Cambridge University Press, number 9781107030206.
• Mertens,Jean-FranÃ§ois & Sorin,Sylvain & Zamir,Shmuel, 2015. "Repeated Games," Cambridge Books, Cambridge University Press, number 9781107662636.
6. VIEILLE, Nicolas & ROSENBERG, Dinah & SOLAN, Eilon, 2002. "Stochastic games with a single controller and incomplete information," HEC Research Papers Series 754, HEC Paris.
7. Eilon Solan & Nicolas Vieille, 2000. "Uniform Value in Recursive Games," Discussion Papers 1293, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
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1. repec:spr:dyngam:v:8:y:2018:i:2:d:10.1007_s13235-017-0227-5 is not listed on IDEAS

### Keywords

Stochastic games; Recursive games; Asymptotic value; Uniform value; Tauberian theorem; Maxmin;

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