Abstract
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.
Highlights
The question of the existence of stationary equilibria in discounted stochastic games with uncountable state spaces has received much attention
We present two examples of discounted stochastic games, each with a con5 tinuum of states, finitely many players and actions, that possess no stationary equilibria
The purpose of this paper is to show that such games need not possess equilibria in stationary20strategies, neither in the framework of deterministic transitions
Summary
The question of the existence of stationary equilibria in discounted stochastic games with uncountable state spaces has received much attention. As existence results for stationary equilibria in general classes proved to be elusive, it became common to assume additional continuity conditions on the transitions; in particular, many works have undertaken the assumption which we term the absolute continuity condition, ACC, which stipulates that all transition measures are absolutely continuous w.r.t. some fixed measure on the state space. The first example is of a discounted stochastic games with uncountable state space and deterministic transitions and that does not possess ε-equilibria in stationary strategies.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have