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Discounted stochastic games, the 3M property and stationary Markov perfect equilibria

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  • Fu, Jing
  • Page, Frank

Abstract

We show that all discounted stochastic games DSGs satisfying the usual assumptions have Nash payoff selection correspondences having fixed points. Our fixed point result is surprising because it is well known that Nash payoff selection correspondences are badly behaved, being in general neither convex valued nor closed valued in the appropriate topologies (in this case the weak star topologies). Here we show that because all DSGs satisfying the usual assumptions have upper Caratheodory (uC) Nash (equilibrium) correspondences containing uC Nash sub-correspondences having the 3M property (defined here), these uC Nash sub-correspondences are continuum valued and therefore induce interval-valued uC player payoff sub-correspondences - and therefore, Caratheodory approximable uC player payoff sub-correspondences. Finally, because these uC player payoff sub-correspondences are Caratheodory approximable, their induced Nash payoff selection sub-correspondences have fixed points - implying that the DSGs to which they belong have stationary Markov perfect equilibria.

Suggested Citation

  • Fu, Jing & Page, Frank, 2022. "Discounted stochastic games, the 3M property and stationary Markov perfect equilibria," LSE Research Online Documents on Economics 118865, London School of Economics and Political Science, LSE Library.
    • Handle: RePEc:ehl:lserod:118865
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    References listed on IDEAS

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    1. He, Wei & Sun, Yeneng, 2017. "Stationary Markov perfect equilibria in discounted stochastic games," Journal of Economic Theory, Elsevier, vol. 169(C), pages 35-61.
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    3. Anna Jaśkiewicz & Andrzej S. Nowak, 2016. "Stationary Almost Markov Perfect Equilibria in Discounted Stochastic Games," Mathematics of Operations Research, INFORMS, vol. 41(2), pages 430-441, May.
    4. Wei He & Yeneng Sun, 2013. "Stationary Markov Perfect Equilibria in Discounted Stochastic Games," Papers 1311.1562, arXiv.org, revised Jan 2017.
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