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

Jing Fu and Frank Page

LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library

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.

Keywords: m-tuples of Caratheodory functions; upper Caratheodory correspondences; the 3M property; continuum valued upper Caratheodory sub-correspondences; weak star upper semicontinuous measurable selection valued correspondences; approximate Caratheodory selections; discounted stochastic games; stationary Markov perfect equilibria (search for similar items in EconPapers)
JEL-codes: C70 (search for similar items in EconPapers)
Pages: 26 pages
Date: 2022-04-12
New Economics Papers: this item is included in nep-gth
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