A Zero-Sum Stochastic Game with Compact Action Sets and no Asymptotic Value
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We give an example of a zero-sum stochastic game with four states, compact action sets for each player, and continuous payoff and transition functions, such that the discounted value does not converge as the discount factor tends to 0, and the value of the n-stage game does not converge as n goes to infinity.
KeywordsZero sum stochastic games Asymptotic behavior Uniform value Compact action sets
This research was supported by grant ANR-10-BLAN 0112 (France).
This paper owes a lot to Sylvain Sorin. I pleasantly remember countless discussions about compact games and why they should have an asymptotic value or not, as well as devising with him a number of “almost proofs” of convergence. This was decisive to understand the right direction to go to stumble upon this counterexample.
I also would like to thank Jérome Bolte for being the first to warn me about non-semialgebraic functions, Jérôme Renault for raising several interesting questions while I was writing this paper, as well as Andrzej S. Nowak for useful references.
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