A Dynkin game with asymmetric information

Abstract

Abstract. We study a Dynkin game with asymmetric information. The game has a random expiry time, which is exponentially distributed and independent of the underlying process. The players have asymmetric information on the expiry time, namely only one of the players is able to observe its occurrence. We propose a set of conditions under which we solve the saddle point equilibrium and study the implications of the information asymmetry. Results are illustrated with an explicit example.

Recursive construction of a Nash equilibrium in a two-player nonzero-sum stopping game with asymmetric information

“…Recently, there is a growing attention towards stopping games with asymmetric information. Lempa et al [22] considered a stopping game with a finite-horizon which is an exponentially distributed random variable, where only one player is exposed to the value of this random variable. Esmaeeli et al [9] and Grün [11] presented two models in which the asymmetry of information is modelled as in the classical work of Aumann and Maschler [3].…”

On the value of non-Markovian Dynkin games with partial and asymmetric information

“…Here we consider a non-Markovian extension of the the framework of [30], where the time horizon of the game is exponentially distributed and independent of the payoff processes. On a probability space (Ω, F, P) we have a filtration (G t ) t∈[0,T ] , augmented with P-null sets, and a positive random variable θ which is independent of G T .…”

“…Our framework encompasses most (virtually all) examples of zero-sum Dynkin games (in continuous time) with partial/asymmetric information that we could find in the literature (see, e.g., De Angelis et al [10], [11], Gensbittel and Grün [21], Grün [22] and Lempa and Matomäki [30]) and we give a detailed account of this fact in Section 3 (notice that [10] and [30] obtain equilibria for the game in pure strategies, i.e., using stopping times, but in very special examples). Broadly speaking, all those papers' solution methods hinge on variational inequalities and PDEs and share two key features: (i) a specific structure of the information flow in the game and (ii) the Markovianity assumption.…”

Optimal stopping games in models with various information flows

“…Gensbittel and Grün [31] considered a simpler version of such a game in a model in which the dynamics of the underlying process are modelled by continuous-time Markov chains. The asymmetry of information in other models for such optimal stopping games was described in Lempa and Matomäki [44] by a random time horizon which is independent of the underlying process, in Ekström, Glover, and Leniec [17] by heterogeneous beliefs about the drift of the underlying diffusion process, in Esmaeeli, Imkeller, and Nzengang [22] by a random variable which is not necessarily independent of the underlying process, and in De Angelis, Ekström, and Glover [13] by a Bernoulli random variable affecting the drift of the underlying process only at the initial time (see also De Angelis, Gensbittel, and Villeneuve [14] for a similar problem where both players have partial information). In our model, the asymmetry of information is described by a continuous-time Markov chain which is independent of the standard Brownian motion driving the underlying process.…”