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

Read more

Summary

INTRODUCTION

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.

STOCHASTIC GAME MODEL
An Informal Description of the Construction
Construction
Observations and Characterization of Equilibria
Nonexistence of Stationary Equilibria
Markovian Strategies
Nonexistence of Markovian Equilibrium in Example I
The Idea of The Construction
Additional Notations and Conventions
Construction from Kohlberg and Mertens’ Game
The Normal-Form Game
The Stochastic Game
Necessary Components of Construction
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call