Discounted Stochastic Games With No Stationary Nash Equilibrium: Two Examples


  • Yehuda Levy

    1. Center for the Study of Rationality and Dept. of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel;
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    • Research supported in part by Israel Science Foundation Grant 1596/10. I am grateful to Abraham Neyman for going over the drafts of this paper and providing countless corrections and suggestions. I thank Eilon Solan for pointing out the work Simon (2003) of Bob Simon, with its inspirational techniques, to me. I also thank Bob Simon, Ilan Nehama, and Ziv Hellman for many helpful conversations, and the anonymous referees who performed extremely diligent proofreadings.


We present two examples of discounted stochastic games, each with a continuum of states, finitely many players, and actions, that possess no stationary equilibria. The first example has deterministic transitions—an assumption undertaken in most of the early applications of dynamics games in economics—and perfect information, and does not possess even stationary approximate equilibria or Markovian equilibria. The second example satisfies, in addition to stronger regularity assumptions, that all transitions are absolutely continuous with respect to a fixed measure—an assumption that has been widely used in more recent economic applications. This assumption has been undertaken in several positive results on the existence of stationary equilibria in special cases, and in particular, guarantees the existence of stationary approximate equilibria.