A Dynkin Game with Independent Processes and Incomplete Information

TL;DR

This paper studies a two-player Dynkin game with independent, unobservable processes, revealing the existence of infinitely many equilibria with infinite payoffs and characterizing conditions for finite payoffs.

Contribution

It introduces a model of Dynkin games with incomplete information and independent processes, analyzing equilibrium existence and payoff properties.

Findings

Infinite Nash equilibria with infinite expected payoffs.

Finite expected payoffs only occur with immediate stopping by at least one player.

Results apply to both pure and mixed strategies.

Abstract

We analyze a two-player, nonzero-sum Dynkin game of stopping with incomplete information. We assume that each player observes his own Brownian motion, which is not only independent of the other player's Brownian motion but also not observable by the other player. The player who stops first receives a payoff that depends on the stopping position. Under appropriate growth conditions on the reward function, we show that there are infinitely many Nash equilibria in which both players attain infinite expected payoffs. In contrast, the only equilibrium with finite expected payoffs mandates immediate stopping by at least one of the players. Our results hold in the settings of both pure and mixed strategies.