Dynkin games with incomplete and asymmetric information

Abstract

We study Nash equilibria for a two-player zero-sum optimal stopping game with incomplete and asymmetric information. In our 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 Nash equilibria 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.

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

“…Players assess the game by looking at the expected payoff as in (2). Finally, we remark that under a number of (restrictive) technical assumptions and with infinite horizon [10] and [11] show the existence of a value (actually of a saddle point) in a smaller class of strategies. In [10] both players use (F X t )-stopping times, with no need for additional randomisation.…”

“…Both these papers attack the problem via a characterisation of the value as the viscosity solution of a certain variational inequality (of a type which is rather new in the literature and is inspired to similar results in the context of differential games; see, e.g., Cardaliaguet and Rainer [9]). Free boundary methods in connection with randomised stopping times are instead used in De Angelis et al [11], where players have asymmetric information regarding the drift of a linear diffusion underlying the game, and in Ekström et al [18], where the two players estimate the drift parameter according to two different models. The methods used in those papers cannot be extended to the non-Markovian framework of our paper.…”

“…Our second example generalises the set-ups of [10] and [11] and reduces to those cases when J = 2, the time horizon is infinite and the payoff processes are (particular) time-homogeneous functions of a (particular) one-dimensional diffusion. Here the underlying dynamics of the game is a diffusion, whose drift depends on the realisation of an independent multinomial random variable J ∈ {1, .…”

“…, where (F X t ) is generated by the sample paths of the process X and it is augmented by the P-null sets (notice that F X t F t ). In [11] instead, the first player (minimiser) observes the true value of J . In that case…”

“…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.…”