Value and Nash Equilibrium in Games of Optimal Stopping

We study games of optimal stopping (Dynkin games). A Dynkin game is a mathematical model involving several competing players, each of them interested in capturing the moments when certain stochastic processes are at an extremum. Actions of players are referred to as “stopping” (of an underlying process), and the outcome for every player depends on the stopping decisions of the other players. Our focus is on Dynkin games with asymmetric information. Asymmetry of information refers to the situation in which different players have different (possibly incomplete) knowledge of the underlying world. Observations of the underlying processes (or of a more general information flow) and of the actions of competitors allow the players to make optimal stopping choices. An important aspect of our framework is a possibility of randomising these choices: for example, in order to avoid revealing private information to competitors. We develop a general stochastic framework for studying Dynkin games with asymmetric information. In particular, we provide conditions for the existence of the value in such games. Separately, we study issues arising in games with mixed first-mover advantage, in which sometimes it is beneficial for the players to act as soon as possible, and sometimes to wait for another player to act.