Dynkin games with incomplete and asymmetric information

TL;DR

This paper analyzes a two-player zero-sum stopping game with incomplete and asymmetric information, deriving the value and optimal strategies through quasi-variational inequalities and providing explicit solutions in a linear payoff case.

Contribution

It introduces a novel framework for Dynkin games with asymmetric information, reducing the problem to a Markovian setting and solving it via quasi-variational inequalities.

Findings

Explicit solution for linear payoff case

Verification theorem for optimal strategies

Reduction to quasi-variational inequalities

Abstract

We study the value and the optimal strategies for a two-player zero-sum optimal stopping game with incomplete and asymmetric information. In our Bayesian set-up, the drift of the underlying diffusion process is unknown to one player (incomplete information feature), but known to the other one (asymmetric information feature). We formulate the problem and reduce it to a fully Markovian setup where the uninformed player optimises over stopping times and the informed one uses randomised stopping times in order to hide their informational advantage. Then we provide a general verification result which allows us to find the value of the game and players' optimal strategies by solving suitable quasi-variational inequalities with some non-standard constraints. Finally, we study an example with linear payoffs, in which an explicit solution of the corresponding quasi-variational inequalities can be obtained.