Stochastic Switching Games

TL;DR

This paper models competitive stochastic switching games where two players control market regimes influenced by a stochastic factor, deriving equilibrium strategies and analyzing long-term market dynamics through approximation schemes.

Contribution

It introduces threshold-type Feedback Nash Equilibria for nonzero-sum stochastic switching games, linking them to optimal switching and timing game frameworks.

Findings

Recurrent equilibrium market organization with Ornstein-Uhlenbeck process.

Absorbing equilibrium where one player gains permanent advantage with Geometric Brownian Motion.

Explicit computations and comparative statics of market equilibria.

Abstract

We study nonzero-sum stochastic switching games. Two players compete for market dominance through controlling (via timing options) the discrete-state market regime $M$. Switching decisions are driven by a continuous stochastic factor $X$ that modulates instantaneous revenue rates and switching costs. This generates a competitive feedback between the short-term fluctuations due to $X$ and the medium-term advantages based on $M$. We construct threshold-type Feedback Nash Equilibria which characterize stationary strategies describing long-run dynamic equilibrium market organization. Two sequential approximation schemes link the switching equilibrium to (i) constrained optimal switching, (ii) multi-stage timing games. We provide illustrations using an Ornstein-Uhlenbeck $X$ that leads to a recurrent equilibrium $M^\ast$ and a Geometric Brownian Motion $X$ that makes $M^\ast$ eventually "absorbed" as one player eventually gains permanent advantage. Explicit computations and comparative statics regarding the emergent macroscopic market equilibrium are also provided.