Playing with ghosts in a Dynkin game

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Abstract

We study a class of two-player optimal stopping games (Dynkin games) of preemption type, with uncertainty about the existence of competitors. The set-up is well-suited to model, for example, real options in the context of investors who do not want to publicly reveal their interest in a certain business opportunity. We show that if the underlying process is a Rd-valued, continuous, strong Markov process, and the stopping payoff is a continuous function (with mild integrability properties) there exists a Nash equilibrium in randomised stopping times for the game. Moreover, the equilibrium strategies and the expected payoffs of the two players are computed explicitly in terms of the corresponding one-player game. To the best of our knowledge this is the first paper to address this version of Dynkin games.

Introduction

Stopping games have received huge attention in the stochastic control literature since their inception, dating back to work by Dynkin in [10]. In the standard modern formulation of the two-player game (due to Neveau [22]) the players have gains/losses depending on a stochastic process X, which they both observe. Their aim is to maximise gains (or minimise losses) by finding stopping rules that allow for a Nash equilibrium in the game. Both players know the structure of the game and have full information on the specifications of the process X.

Many real-world applications call for incomplete and/or asymmetric information about the game structure and/or the underlying stochastic process. In particular, in this paper we are interested in determining equilibria for two-player Dynkin games in which each player is uncertain about the existence of a competitor. Before describing our contribution in further detail we spend a few words on the existing literature in order to contextualise the problem.

It is difficult to provide a detailed literature review that would do justice to the numerous contributions on Dynkin games in the standard framework (i.e., full and symmetric information) and it falls outside the scope of our introduction. For the case of zero-sum games one may for example refer to [20] and [12] for general treatments of the martingale and the Markovian set-ups, respectively, and to [17] for a reduction of a financial game option into a stopping game. Other seminal contributions to this literature can be found in [3], [24] and [26] among others. For the case of nonzero-sum games one may refer to [2] for connections between such games and variational inequalities, to [15] for martingale methods for existence of a saddle-point in a general set-up, and to [1] and [8] for sufficient conditions for the existence (and uniqueness) of Nash equilibria in hitting times to thresholds for one dimensional diffusions.

In our paper we will be dealing with a class of problems that share similarities with nonzero-sum Dynkin games. However we depart from the standard set-up by including the key feature of uncertainty about competition. In this respect we draw from the literature on games with incomplete/asymmetric information, whose main common denominator is the need of the players to hide their information from the competitors. Mathematically this translates into the use of randomised stopping times; the latter can be informally understood as stopping rules which prescribe to stop according to some ‘intensity’; for example, in a discrete-time setting, it means that stopping may occur at each time with some probability. The other key feature of this type of games is the need to account for the dynamic evolution of the players’ beliefs concerning those parameters that they cannot fully observe (in other words, the players update their private views on the ‘state of the world’ by observing what happens during the game, and one needs to keep track of such updates). We incorporate beliefs in our game by adding a state variable, denoted by Π, whose dynamic is constructed starting from the randomised stopping strategies of the two players (see Section 4).

The literature on Dynkin games with asymmetric/incomplete information has started to gain traction in recent years. The first contribution that we are aware of is by Grün [14]. In [14], a zero-sum stopping game with asymmetric information about the payoff functions is considered. In a setting where one of the players has the informational advantage of knowing the payoff functions and the other player only knows the distribution of possible payoff functions, a value of the game is obtained (with randomised stopping times) and characterised as a viscosity solution of a nonlinear variational problem. Later on, in [13], authors use methods inspired by dynamic programming to study a zero-sum stopping game in which the players have access to different filtrations generated by the dynamics of two different processes. Explicitly solvable examples in this literature are still rare and the first one is given in [6] (we provide another one in Section 6 for our game). Authors in [6] study a zero-sum stopping game with asymmetric information about a drift parameter of the underlying diffusion process. An explicit Nash equilibrium is obtained in which the uninformed player uses a normal stopping time (pure strategy) and the informed one uses a randomised stopping time (mixed strategy).

For completeness, but without elaborating further, we also mention that there exists a vast literature on stochastic differential games with asymmetric information and the interested reader may look into [4] and [5] and references therein.

In a strategic context of agents playing hide and seek, it seems natural to ask what happens if players cannot be certain about the existence of competition. Several real-world situations fall under this category as, for example, (a) investors who do not want to reveal their interest for specific business opportunities (so-called real options), (b) potential house buyers who are not aware of how many other offers will be put forward (and how quickly), (c) buyers of depletable assets/goods (e.g., cheap flight tickets), etc. The feature of uncertain competition has indeed been addressed in a static setting of auction theory, see, e.g. [16] and [21], and more recently [11]. However, to the best of our knowledge there are no studies featuring uncertain competition in a dynamic setting and in particular there seem to be no contributions to the theory of Dynkin games. With this paper we aim at filling that gap and encourage further research in that direction.

Here, we study a combined stopping/preemption game between two players who are interested in the same asset. The player who stops first receives the full payoff, defined in terms of an underlying Markov process X representing the asset. The process X is required to be strong Markov, continuous and taking values in Rd. The game is a nonzero-sum Dynkin game but, in contrast with the classical set-up, in our model the players face uncertain competition, i.e. each player is uncertain as to whether the other player exists or not.

At the start of the game each player estimates the probability of competition. That is, Player 1 believes she has competition with probability p1 and Player 2 believes she has competition with probability p2. As the game evolves, both players adjust their beliefs according to the dynamic of their own belief process Πi, with i=1,2. Such adjustment is based on a combination of two key elements: (i) the observation of the underlying asset X and (ii) the lack of action from the other player. Intuitively, if the payoff associated with the current asset value becomes large, this is appealing for both players; therefore, from the point of view of Player 1, the fact that Player 2 has not stopped yet, suggests that Player 2 may not exist at all. (This simple heuristics also motivates the title of our paper.)

Within this context the use of randomised stopping times stems from two observations. On the one hand, it allows the players to hide their participation in the game in order to ‘fool’ their opponent. On the other hand, it is intuitively clear that, due to the preemption feature of the game, it would be impossible to reach a non-trivial equilibrium using pure stopping times (with respect to the filtration generated by the asset). Indeed, if Player 1 picks a stopping time τ, then Player 2 would possibly stop just before τ so as to receive the full payoff.

In this paper, with no loss of generality we consider p1p2. Then we prove that there exists a Nash equilibrium in terms of strategies whose character completely depends on the initial belief of Player 1. Here we only describe the main ideas around the structure of the equilibrium but we emphasise that, at a deeper level, we find several remarkable properties of the players’ optimal strategies which will be described in fuller detail in Section 5.3 (as they need a more extensive mathematical discussion).

It turns out that the state space of the (d+1)-dimensional process (Π1,X) divides into three disjoint regions: the no-action region (C), the action region (C) and the stopping region (S). If (Π1,X) starts in the no-action region (note that Π01=p1), then the equilibrium consists of no action (from either player) until the boundary C of the region is reached (notice that no action results in Πi being constant for i=1,2); then Player 2 employs a randomised strategy in such a way that the process (Π1,X) reflects along the boundary C towards the interior of C (as an effect of decreasing Π1); at the same time Player 1 will follow a (equilibrium) strategy consisting of stopping with a fraction p1p2 of the ‘stopping intensity’ of Player 2. If, instead, the initial beliefs are such that (Π1,X) starts in C, then Player 2 employs a strategy consisting of stopping immediately with a certain probability. That makes the process (Π1,X) jump strictly into the interior of the no-action region C. After this initial jump the reflection strategy described above is employed. Also in this case Player 1 will follow the (equilibrium) strategy of stopping with a fraction p1p2 of the ‘stopping intensity’ of Player 2. Finally, if (Π1,X) starts in the stopping region S, then the equilibrium strategy consists of immediate stopping for both players. In this case the players split the payoff evenly.

Remarkably, the boundaries of the no-action region and of the action-region can be specified explicitly in terms of the corresponding one-player game with no competition. This allows to construct explicitly randomised stopping times, at equilibrium, in all those games whose one-player counterpart is explicitly solvable. (For instance the literature on optimal stopping of one-dimensional diffusions is rich with such examples, see e.g. [23].) In particular, in Section 6 we provide the explicit solution of a real option game with uncertain competition as an illustration of our results.

As explained above, our work is the first one to address uncertain competition in a dynamic setting. We find a surprisingly explicit yet rich structure of the equilibrium strategies, which allows for a collection of interesting considerations (see details in Section 5.3). In short, we observe that the most active player is the one who has the largest initial belief about the existence of the opponent, and that she is the one who benefits the most from randomisation; we call this feature the ‘benefit of wariness’. Also, the strategy adopted by Player 2 when the game starts with (Π01,X0)C is somewhat surprising: in the stopping literature there is no analogue for the region C and in the singular control literature (which is also related to the present work) one should not expect jumps strictly in the interior of the no-action region C (at least in absence of fixed costs of control). Finally, it is possible to draw a parallel between the strategies that we construct and a concept of entry time to ‘randomised’ sets. This is also rather surprising because, from the beginning, we look for Nash equilibria without any restrictions on the randomised stopping times.

The rest of the paper is organised as follows. The game is set in Section 2, where we also define randomised stopping times and recall the concept of Nash equilibrium in our context. In Section 3 we derive a number of properties of the two players’ expected payoffs, which are needed for the subsequent analysis. Section 4 is devoted to the construction of the belief processes, the specification of the sets C¯ (which corresponds to C plus a portion of its boundary), C and S, and the construction of a suitably reflected belief process. The existence and the main additional facts around the Nash equilibrium are derived in Section 5. We conclude the paper with a fully solved example in Section 6.

Section snippets

Set-up

Let (Ω,P,F) be a probability space hosting the following:

  • (a)

    a continuous, Rd-valued, strong Markov process X which is regular (it can reach any open set in finite time with positive probability, for any value of the initial point X0=x);

  • (b)

    two Bernoulli distributed random variables θi, i=1,2;

  • (c)

    two Uniform (0,1)-distributed random variables Ui, i=1,2.

Furthermore, we assume that these processes and random variables are mutually independent, and that P(θi=1)=1P(θi=0)=pi(0,1].

We denote by FX={FtX}0t<

Some useful observations on the game’s payoffs

In this section we make a few general observations that provide some intuition for the structure of the equilibrium payoffs of the two players.

Adjusted beliefs

In order to find an equilibrium for the game we study the evolution (during the game) of the players’ beliefs regarding the existence of their opponent. In other words, if γT2R is generated by Γ2A, then Player 1 dynamically evaluates the conditional probability of Player 2 being active as Πt1P(θ1=1|FtX,γˆ>t)=P(θ1=1|FtX)P(γˆ>t|FtX,θ1=1)P(γˆ>t|FtX)=p1P(γ>t|FtX)1p1+p1P(γ>t|FtX)=p1(1Γt2)1p1Γt2 provided p1(0,1), where we recall that γˆ=γ1{θ1=1}+1{θ1=0} as in (5). Here we used independence of

Construction of Nash equilibria

With no loss of generality, we assume that p2 dominates p1, i.e. 0p1p21. We first comment on the two extreme cases p1=0 and p1=1.

If p1=1, then also p2=1 (since p1p2), so both players are certain that the other player is active. In this case it is clear that immediate stopping, i.e. (τ,γ)(0,0), provides a Nash equilibrium, and the corresponding equilibrium payoff for each player is g(x)2.

If p1=0, then Player 1 is certain that Player 2 is not active, and will consequently play the optimal

Real options with unknown competition

In this section we consider an example of a real option with uncertain competition (for the classical case of real options with no competition, see e.g. [9], and for a related problem of real option pricing under incomplete information about the competition, see [18]). For this, let g(x)=(xK)+, p1=p2p(0,1) and dXt=μXtdt+σXtdWtfor some constants K>0, μ<r and σ>0. Here X represents the present value of future revenues from entering a certain business opportunity, K is the sunk cost for

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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  • Acknowledgements: T. De Angelis gratefully acknowledges support by the EPSRC, United Kingdom grant EP/R021201/1 and E. Ekström by the Knut and Alice Wallenberg Foundation, Sweden . Parts of this work were carried out during T. De Angelis’ visits to Uppsala University and E. Ekström’s visits to University of Leeds. We thank both institutions for their hospitality.

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