Dynamic Games and Applications

, Volume 8, Issue 2, pp 401–422 | Cite as

Tauberian Theorem for Value Functions

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Abstract

For two-person dynamic zero-sum games (both discrete and continuous settings), we investigate the limit of value functions of finite horizon games with long-run average cost as the time horizon tends to infinity and the limit of value functions of \(\lambda \)-discounted games as the discount tends to zero. We prove that the Dynamic Programming Principle for value functions directly leads to the Tauberian theorem—that the existence of a uniform limit of the value functions for one of the families implies that the other one also uniformly converges to the same limit. No assumptions on strategies are necessary. To this end, we consider a mapping that takes each payoff to the corresponding value function and preserves the sub- and superoptimality principles (the Dynamic Programming Principle). With their aid, we obtain certain inequalities on asymptotics of sub- and supersolutions, which lead to the Tauberian theorem. In particular, we consider the case of differential games without relying on the existence of the saddle point; a very simple stochastic game model is also considered.

Keywords

Dynamic Programming Principle Zero-sum games Abel mean Cesaro mean Differential games 

Mathematics Subject Classification

91A25 49L20 91A05 91A23 40E05 

Notes

Acknowledgements

I would like to express my gratitude to Ya.V. Salii for the translation. I am grateful to an anonymous referees for helpful comments. This study was supported by the Russian Foundation for Basic Research, project no. 16-01-00505.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Krasovskii Institute of Mathematics and MechanicsUral Branch, Russian Academy of SciencesYekaterinburgRussia
  2. 2.Chair of Applied Mathematics, Institute of Mathematics and Computer ScienceUral Federal UniversityYekaterinburgRussia

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