Nonzero-Sum Games of Optimal Stopping for Markov Processes

Optimal stopping games, often referred to as Dynkin games, are extensions of optimal stopping problems. Since the seminal paper of Dynkin [14], optimal stopping games have been studied extensively. Martingale methods for zero-sum games were studied by Kifer [32], Neveu [44], Stettner [55], Lepeltier and Maingueneau [41] and Ohtsubo [45]. The Markovian framework was initially studied by Frid [22], Gusein-Zade [26], Elbakidze [18] and Bismut [5]. Bensoussan and Friedman [2] and Friedman [20, 21] considered zero-sum optimal stopping games for diffusions and developed an analytic approach by relying on variational and quasi-variational inequalities. Ekström and Peskir [16] proved the existence of a value in two-player zero-sum optimal stopping games for right-continuous strong Markov processes and construct a Nash equilibrium point under the additional assumption that the underlying process is quasi-left continuous. Peskir in [51] and [52] extended these results further by deriving a semiharmonic characterisation of the value of the game without assuming that a Nash equilibrium exists a priori. In particular, a necessary and sufficient condition for the existence of a Nash equilibrium is that the value function coincides with the smallest superharmonic and the largest subharmonic function lying between the gain and the loss function which, in the case of absorbed Brownian motion in [0,1], is equivalent to ‘pulling a rope’ between ‘two obstacles’ (that is finding the shortest path between the graphs of two functions). Connections between zero-sum optimal stopping games and singular stochastic control problems were studied in [23, 31] and[4]. Cvitanic and Karatzas [11] showed that backward stochastic differential equations are connected with the value function of a zero-sum Dynkin game. Advances in this direction can be found in [27]. Various authors have also studied zero-sum optimal stopping games with randomised strategies. For further details one can refer to [39] and the references therein. Zero-sum optimal stopping games have been used extensively in the pricing of game contingent claims both in complete and incomplete markets (see for example [15, 17, 24, 25, 30, 33, 35, 36, 38] and [19]).

Literature on nonzero-sum optimal stopping games is mainly concerned with the existence of a Nash equilibrium. Initial studies in discrete time date back to Morimoto [42] wherein a fixed point theorem for monotone mappings is used to derive sufficient conditions for the existence of a Nash equilibrium point. Ohtsubo [46] derived equilibrium values via backward induction and in [47] the same author considers nonzero-sum games in which the lower gain process has a monotone structure, and gives sufficient conditions for a Nash equilibrium point to exist. Shmaya and Solan in [54] proved that every two player nonzero-sum game in discrete time admits an \(\varepsilon \)-equilibrium in randomised stopping times. In continuous time Bensoussan and Friedman [3] showed that, for diffusions, a Nash equilibrium exists if there exists a solution to a system of quasi-variational inequalities. However, the regularity and uniqueness of the solution remain open problems. Nagai [43] studies a nonzero-sum stopping game of symmetric Markov processes. A system of quasi-variational inequalities is introduced in terms of Dirichlet forms and the existence of extremal solutions of a system of quasi-variational inequalities is discussed. Nash equilibrium points of the stopping game are then obtained from these extremal solutions. Cattiaux and Lepeltier [8] study special right-processes, namely Hunt processes in the Ray topology, and they prove existence of a quasi-Markov Nash Equilbrium. The authors follow Nagai’s idea but use probabilistic tools rather than the theory of Dirichlet forms. Huang and Li in [29] prove the existence of a Nash equilibrium point for a class of nonzero-sum noncyclic stopping games using the martingale approach. Laraki and Solan [40] proved that every two-player nonzero-sum Dynkin game in continuous time admits an \(\varepsilon -\)equilibrium in randomised stopping times. Hamadène and Zhang in [28] prove existence of a Nash equilibrium using the martingale approach, for processes with positive jumps. One application of nonzero-sum optimal stopping games is seen in the study of game options in incomplete markets, via the consideration of utility-based arguments (see [34]). Nonzero-sum optimal stopping games have also been used to model the interaction between bondholders and shareholders in the study of convertible bonds, when corporate taxes are included and when the company is allowed to claim default (see [9]).

In this work we consider two player nonzero-sum games of optimal stopping for a general strong Markov process. The aim is to use probabilistic tools to study the optimal stopping problem of player one (resp. player two) when the stopping time of player two (resp. player one) is externally given. Although this work does not deal with the question of existence of mutually best responses (that is the existence of a Nash equilibrium) the results obtained can be exploited further in various examples, to show the existence of a pair of first entry times that will be a Nash equilibrium. Indeed, the results derived here will be used in a separate work (see [1]) to construct Nash equilibrium points for one dimensional regular diffusions and for a certain class of payoff functions.

This paper is organised as follows: In Sect. 2 we introduce the underlying setup and formulate the nonzero-sum optimal stopping game. In Sect. 3 we show that if the strategy chosen by player two (resp. player one) is \(\sigma _{D_2}\) (resp. \(\tau _{D_1}\)), the first entry time into a regular Borel subset \(D_2\) (resp. \(D_1\)) of the state space, then the value function of player one associated with \(\sigma _{D_2}\) (resp. the value function of player two associated with \(\tau _{D_1}\)), which we shall denote by \(V^1_{\sigma _{D_2}}\) (resp. \(V^2_{\tau _{D_1}}\)), is finely continuous. In Section 4 we shall use this regularity property of \(V^1_{\sigma _{D_2}}\) (resp. \(V^2_{\tau _{D_1}}\)) to construct an optimal stopping time for player one (resp. player two). In Sect. 5 we shall use the results obtained in Sects. 3 and 4 to provide a partial superharmonic characterisation for \(V^1_{\sigma _{D_2}}\) (resp. \(V^2_{\tau _{D_1}}\)). More precisely if \({D_2}\) (resp. \(D_1\)) is also a closed or finely closed subset of the state space then \(V^1_{\sigma _{D_2}}\) (resp. \(V^2_{\tau _{D_1}}\)) can be identified with the smallest finely continuous function that is superharmonic in \(D_2^c\) (resp. in \(D_1^c\)) and that majorises the lower payoff function. In Section 6 we shall consider stationary one-dimensional Markov processes and we shall assume that there exists a pair of stopping times \((\tau _{A_*},\sigma _{B_*})\) of the form \(\tau _{A_*}= \inf \{t\ge 0: X_t \le A_*\}\) and \(\sigma _{B_*} = \inf \{t \ge 0 : X_t \ge B_*\}\) where \(A_*< B_*\), that is a Nash equilibrium point. We first show that \(V^{1}_{\sigma _{B_*}}\) (resp. \(V^{1}_{\tau _{A_*}}\)) is continuous at \(A_*\) (resp. at \(B_*\)). Then for the special case of one dimensional regular diffusions we shall use the results obtained in Sect. 5 to show that \(V^{1}_{\sigma _{B_*}}\) (resp. \(V^{2}_{\tau _{A_*}}\)) is also smooth at \(A_*\) (resp. \(B_*\)) provided that the payoff functions are smooth. This is in line with the principle of smooth fit observed in standard optimal stopping problems (see for example [49] for further details).

In this section we shall formulate rigorously the nonzero-sum optimal stopping game. For this we shall first set up the underlying framework. This will be similar to the one presented by Ekström and Peskir (cf. [16, p. 3]). On a given filtered probability space \(\left( \Omega ,\mathcal {F},(\mathcal {F}_t)_{t\ge 0},\mathsf {P}_{x}\right) \) we define a strong Markov process \(X=(X_{t})_{t \ge 0}\) with values in a measurable space \((E,\mathcal {B})\), with E being a locally compact Hausdorff space with a countable base (note that since E has a countable base then it is a Polish space) and \(\mathcal {B}\) the Borel \(\sigma \)-algebra on E. We shall assume that \(\mathsf {P}_{x}\left( X_0=x\right) =1\), that the sample paths of X are right-continuous and that X is quasi-left-continuous (that is \(X_{\rho _{n}} \rightarrow X_{\rho }\) \(\mathsf {P}_{x}\)-a.s. whenever \(\rho _{n}\) and \(\rho \) are stopping times such that \(\rho _{n} \uparrow \rho \) \(\mathsf {P}_{x}\)-a.s.). All stopping times mentioned throughout this text are relative to the filtration \((\mathcal {F}_{t})_{t \ge 0}\) introduced above, which is also assumed to be right-continuous. This means that entry times in open and closed sets are stopping times. Moreover \(\mathcal {F}_0\) is assumed to contain all \(\mathsf {P}_{x}\)-null sets from \(\mathcal {{F}}^{X}_\infty = \sigma \left( X_{t}:t \ge 0\right) \), which further implies that the first entry times to Borel sets are stopping times. We shall also assume that the mapping \(x \mapsto \mathsf {P}_{x}(F)\) is (universally) measurable for each \(F \in \mathcal {F}\) so that the mapping \(x \mapsto \mathsf {E}_{x} [Z]\) is (universally) measurable for each (integrable) random variable Z.

Finally we shall assume that \(\Omega \) is the canonical space \(E^{\left[ 0,\infty \right) }\) with \(X_{t}(\omega ) = \omega (t)\) for \(\omega \in \Omega \). In this case the shift operator \(\theta _{t}:\Omega \rightarrow \Omega \) is well defined by \(\theta _{t}(\omega )(s) = \omega (t+s)\) for \(\omega \in \Omega \) and \(t,s \ge 0\).

The game is said to have a solution if there exists a pair of stopping times \((\tau _*,\sigma _{*})\) which is a Nash equilibrium point, that is \(\mathsf {M}_{x}^1(\tau ,\sigma _{*}) \le \mathsf {M}_{x}^1(\tau _*,\sigma _{*})\) and \(\mathsf {M}_{x}^2(\tau _*,\sigma ) \le \mathsf {M}_{x}^2(\tau _*,\sigma _{*})\) for all stopping times \(\tau , \sigma \). This means that none of the players will perform better if they change their strategy independent of each other. In this case \(V^{1}_{\sigma _{*}}(x) = \mathsf {M}_{x}^{1}(\tau _{*},\sigma _{*})\) is the payoff function of player one and \(V^{2}_{\tau _{*}}(x)=\mathsf {M}_{x}^{2}(\tau _{*},\sigma _{*})\) the payoff function of player two in this equilibrium. So \(V^{1}_{\sigma _*}\) and \(V^{2}_{\tau _*}\) can be called the value functions of the game (corresponding to \((\tau _*,\sigma _*))\). In general, as we shall see in Sect. 6, there might be other pairs of stopping times that form a Nash equilibrium point, which can lead to different value functions.

In this section we show that if the strategy chosen by player two (resp. player one) corresponds to the first entry time into a subset \(D_2\) (resp. \(D_1\)) of E, whose boundary \(\partial D_2\) (resp. \(\partial D_1\)) is regular, then \(V^{1}_{\sigma _{D_2}}\) (resp. \(V^{2}_{\tau _{D_1}})\) is continuous in the fine topology (i.e. finely continuous). For literature on the fine topology one can refer to [6, 10] and [13]. We first define the concept of a finely open set and a regular boundary of a Borel subset of E.

We now introduce preliminary results which are needed to prove the main theorem of this section.

The next lemma and theorem, which we shall exploit in this study, provide conditions for fine continuity. The proofs of these results can be found in [13].

We next state and prove the main result of this section, that is the fine continuity property of \(V^{1}_{\sigma _{D_2}}\) (resp. \(V^{2}_{\tau _{D_1}}\)).

We next present an example to show that if \(\partial D\) is not regular for D then \(V^{1}_{\sigma _{D}}\) may not be finely continuous.

The main result of this section is to show that if \(\sigma _{D_2}\) (resp. \(\tau _{D_1}\)) is externally given as the first entry time in \(D_2\) (resp. \(D_1\)), a set that is either closed or finely closed, and has a regular boundary, then the first entry time \(\tau _{*}^{\sigma _{D_2}} = \inf \{t \ge 0:X_t \in D^{\sigma _{D_2}}_1\}\) (resp. \(\sigma _{*}^{\tau _{D_1}} = \inf \{t \ge 0: X_t \in D^{\tau _{D_1}}_1 \}\)) where \(D^{\sigma _{D_2}}_1 = \{V^{1}_{\sigma _{D_2}} = G_1\}\) (resp. \(D^{\tau _{D_1}}_2 = \{V^{2}_{\tau _{D_1}} = G_2\}\)) solves the optimal stopping problem \(V_{\sigma _{D_{2}}}^{1}(x)=\sup _{\tau } \mathsf {M}_{x}^{1}(\tau ,\sigma _{D_{2}})\) (resp. \(V_{\tau _{D_{1}}}^{2}(x)=\sup _{\sigma } \mathsf {M}_{x}^{2}(\tau _{D_1},\sigma ))\). The proof of this result will be divided into several lemmas and propositions.

We now state and prove the main result of this section.

In this section we shall assume that the Markov process X takes values in \(\mathbb {R}\) and is such that Law\((X|\mathsf {P}_{x}) = \) Law\((X^{x}|\mathsf {P})\). We shall also assume that there exist points \(A_*\) and \(B_*\) satisfying \(-\infty< A_*< B_*< \infty \) such that (i.) for given \(D_2\) of the form \([B_*,\infty )\), the first entry time \(\tau _{A_*} = \inf \{t \ge 0:X_t \le A_*\}\) (as obtained from Theorem 4.4) is optimal for player one and (ii.) for given \(D_1\) of the form \((-\infty ,A_*]\), the first entry time \(\sigma _{B_*} = \inf \{t \ge 0:X_t \ge B_*\}\) (as obtained from Theorem 4.4) is optimal for player two.

So in this section we will assume the existence of a pair \((\tau _{A_*}, \sigma _{B_*})\) that is a Nash equilibrium.

6.2 The principle of double smooth fit

In this section we will consider the special case when X is a one dimensional regular diffusion process and we shall assume that \(V^{1}_{\sigma _{B_*}}\) and \(V^{2}_{\tau _{A_*}}\) are obtained from the double partial superharmonic characterisation as explained in Sect. 5. More precisely we shall assume that the functions u, v introduced in Theorem 5.3 (i.) coincide with those from Theorem 5.3 (ii.) so that a mutual response is assumed to exist. The aim is to use this characterisation to derive the so-called principle of double smooth fit. This principle is an extension of the principle of smooth fit observed in standard optimal stopping problems (see [49]). We note that in the case of more general strong Markov processes in \(\mathbb {R}\) this principle may break down. As observed in standard optimal stopping problems this may happen for example when the scale function of X is not differentiable (see [50]) or in the case of Poisson process (see [48]). Carr et. al in [7], for example, also showed that this principle breaks down in a CGMY model.

Remark 6.4

Examples of nonzero-sum optimal stopping games for one dimensional regular diffusion processes, for which the optimal stopping regions are of the threshold type are given in [1] and [12]¹. In particular the authors therein provide sufficient conditions for existence and uniqueness of Nash equilibria.

So suppose that X is a regular diffusion process with values in \(\mathbb {R}\). We shall also assume that the fine topology coincides with Euclidean topology so that fine continuity is equivalent to continuity in the classical sense. In this context we can define the scale function S of the process X, that is the mapping \(S:\mathbb {R} \rightarrow \mathbb {R}\) which is a strictly increasing continuous function satisfying

$$\begin{aligned} \mathsf {P}_{x}(\tau _{c}< \tau _{d}) = \frac{S(d)-S(x)}{S(d)-S(c)} \quad \text {and} \quad \mathsf {P}_{x}(\tau _{d}<\tau _{c}) = \frac{S(x)-S(c)}{S(d)-S(c)} \end{aligned}$$

(6.9)

for any \(c< x < d\) where \(\tau _{y} = \inf \{t \ge 0: X_{t} = y\}\) for \(y \in \mathbb {R}\). Since we are assuming that \(D_{2} = [B_{*},\infty )\) then for any given \(a,b \in (-\infty ,B_{*})\) such that \(a < b\) we have

$$\begin{aligned}&u(x) \ge \mathsf {E}_{x}[u(X_{\tau _{a,b} \wedge \sigma _{B_*}})] = \mathsf {E}_{x}[u(X_{\tau _{a,b}})] = u(a)\frac{S(b)-S(x)}{S(b)-S(a)} + u(b)\frac{S(x)-S(a)}{S(b)-S(a)}\nonumber \\ \end{aligned}$$

(6.10)

for all \(a< x < b\), where \(\tau _{a,b} = \inf \{t \ge 0:X_{t} \notin (a,b)\}\). The first inequality follows from the fact that u is superharmonic in \(D_{2}^{c}\) (recall Definition 5.1). This means that u is S-concave in every interval in \((-\infty ,B_{*})\) and as for concave functions this implies that the mapping

$$\begin{aligned} y \mapsto \frac{u(y)-u(x)}{S(y)-S(x)} \end{aligned}$$

(6.11)

is decreasing provided that \(y \ne x\). By symmetry we have that v is S-concave in every interval in \((A_{*},\infty )\) and that the mapping \(y \mapsto \frac{v(y)-v(x)}{S(y)-S(x)}\) is decreasing provided \(y \ne x\).

From the results for the Dirichlet problem (see for example [49, (7.1.2)-(7.1.3)]) one can show that

$$\begin{aligned}&\mathbb {L}_{X}u = 0 \text { and } \mathbb {L}_{X}v = 0 \text { in } C_{1} \cap C_{2} u\left| _{\partial C_{1}} \right. = G_{1} \text { and } u\left| _{\partial C_{2}} = H_{1} \right. \nonumber \\&v\left| _{\partial C_{1}} \right. = H_{2} \text { and } v\left| _{\partial C_{2}} \right. = G_{2} \nonumber \end{aligned}$$

where \(C_1 = D_1^c\) and \(C_2 = D_2^{c}\). The aim is to show that \(u'\left| _{A_*}\right. = G'_{1} \left| _{A_*}\right. \) and \(v'\left| _{B_*}\right. = G'_{2} \left| _{B_*}\right. \) These two conditions will be referred to as the principle of double smooth fit. Informally this principle states that the optimal stopping boundary points \(A_{*}\) and \(B_{*}\) must be selected in such a way that u and v are respectively smooth at these points. The proof of this result follows in a similar way as the proof of Theorem 2.3 in [50]. We shall first state the following lemma, the proof of which can be found in [50].

Lemma 6.5

Suppose that \(f,g:\mathbb {R}_{+} \rightarrow \mathbb {R}\) are two continuous functions such that \(f(0)=g(0)=0\), \(f(\varepsilon ) > 0\) whenever \(\varepsilon > 0\), and \(g(\delta ) > 0\) whenever \(\delta > 0\). Then for every \(\varepsilon _{n} \downarrow 0\) as \(n \rightarrow \infty \), there exists \(\varepsilon _{n_{k}} \downarrow 0\) and \(\delta _{k} \downarrow 0\) as \(k \rightarrow \infty \) such that \(\lim _{k \rightarrow \infty } \frac{f(\varepsilon _{n_{k}})}{g(\delta _{k})} = 1.\)

Proposition 6.6

Suppose that \(D_{1}\) is of the form \((-\infty ,A_{*}]\) and \(D_{2}\) of the form \([B_{*},\infty )\) for some points \(A_{*},B_{*}\) such that \( A_{*} < B_{*} \). Suppose that \(G_{1}\) is differentiable at \(A_{*}\) and \(G_{2}\) is differentiable at \(B_{*}\). If the scale function S of X is differentiable at \(A_{*}\) and \(B_{*}\), then \(u'(A_{*}) = G'_{1}(A_{*})\) and \(v'(B_{*}) = G'_{2}(B_{*})\).

Proof

We shall first consider the case when \(S'(A_{*}) \ne 0\). Since u is superharmonic in \(\left( -\infty ,B_{*}\right) \) we have that \(\mathsf {E}_{x}[u(X_{\rho \wedge \sigma _{B_*}})] \le u(x)\) for all stopping times \(\rho \) and for all \(x\in \mathbb {R}\). Define the exit time \(\tau _{\varepsilon }=\inf \left\{ t\ge 0:X_{t}\notin \left( A_{*}-\varepsilon ,A_{*}+\varepsilon \right) \right\} \) for \(\varepsilon >0\) such that \(A_{*}+\varepsilon < B_{*}.\) Then

$$\begin{aligned}&\mathsf {E}_{A_{*}}[u(X_{\tau _{\varepsilon } \wedge \sigma _{D_{2}}})] = \mathsf {E}_{A_{*}}[u(X_{\tau _{\varepsilon }})] \nonumber \\&\quad \le u(A_{*}) = u(A_{*})\mathsf {P}_{A_{*}}(X_{\tau _{\varepsilon }} = A_{*}+\varepsilon ) + G_{1}(A_{*}) \mathsf {P}_{A_{*}}(X_{\tau _{\varepsilon }} = A_{*}- \varepsilon )\qquad \end{aligned}$$

(6.12)

where the last equality follows from the fact that \(u(A_{*})= G_{1}(A_{*})\). On the other hand we have

$$\begin{aligned} \mathsf {E}_{A_{*}}[u(X_{\tau _{\varepsilon }})]= u(A_{*}+\varepsilon )\mathsf {P}_{A_{*}}(X_{\tau _{\varepsilon }} = A_{*}+\varepsilon ) + G_{1}(A_{*}-\varepsilon ) \mathsf {P}_{A_{*}}(X_{\tau _{\varepsilon }} = A_{*}- \varepsilon ). \end{aligned}$$

(6.13)

By combining (6.12) and (6.13) it follows that

$$\begin{aligned}&(u(A_{*}+\varepsilon ) - u(A_{*})) \mathsf {P}_{A_{*}}\left( X_{\tau _{\varepsilon }}=A_{*}+\varepsilon \right) \nonumber \\&\quad \le \left( G_{1}(A_{*})-G_{1}(A_{*}-\varepsilon )\right) \mathsf {P}_{A_{*}}\left( X_{\tau _{\varepsilon }}=A_{*}-\varepsilon \right) . \end{aligned}$$

(6.14)

Since we are assuming that S and \(G_{1}\) are differentiable at \(A_{*}\) and that \(S'(A_{*}) \ne 0\) we get, upon using the facts \(\mathsf {P}_{A_{*}}(X_{\tau _{\varepsilon }} = A_{*}-\varepsilon ) = \frac{S(A_{*}+\varepsilon )-S(A_{*})}{S(A_{*}+\varepsilon ) - S(A_{*}-\varepsilon )}\) and \(\mathsf {P}_{A_{*}}(X_{\tau _{\varepsilon }} = A_{*}+\varepsilon ) = \frac{S(A_{*})-S(A_{*}-\varepsilon )}{S(A_{*}+\varepsilon ) - S(A_{*}-\varepsilon )}\), that

$$\begin{aligned}&\frac{(u(A_{*}+\varepsilon ) - u(A_{*})) (S(A_{*})-S(A_{*}-\varepsilon ))}{S(A_{*}+\varepsilon ) - S(A_{*}-\varepsilon )} \\&\quad \le \frac{( G_{1}(A_{*})-G_{1}(A_{*}-\varepsilon ))(S(A_{*}+\varepsilon ) - S(A_{*}))}{S(A_{*}+\varepsilon )-S(A_{*}-\varepsilon )} \end{aligned}$$

which is equivalent to (since the scale function is increasing)

$$\begin{aligned}&\frac{u(A_{*}+\varepsilon ) - u(A_{*})}{\varepsilon } \le \frac{( G_{1}(A_{*})-G_{1}(A_{*}-\varepsilon ))}{\varepsilon } \frac{\frac{(S(A_{*}+\varepsilon ) - S(A_{*}))}{\varepsilon }}{\frac{(S(A_{*}) - S(A_{*}-\varepsilon ))}{\varepsilon }}. \end{aligned}$$

(6.15)

Taking the limit as \(\varepsilon \downarrow 0\) on both sides of (6.15) and using the fact that \(G_1\) and S are differentiable at \(A_*\) and that \(S'(A_*) \ne 0\) we get that \(u'(A_{*}+) = \lim _{\varepsilon \downarrow 0} \frac{u(A_*+ \varepsilon ) - u(A_*)}{\varepsilon } \le G_1'(A_*)\). On the other hand we have

$$\begin{aligned}&u(A_{*}+\varepsilon ) - u(A_{*}) \ge \frac{G_{1}\left( A_{*}+\varepsilon \right) -G_{1}\left( A_{*}\right) }{\varepsilon }. \end{aligned}$$

(6.16)

Taking limits on both sides of (6.16) as \(\varepsilon \downarrow 0\) we get that \(u'(A_*+) \ge G_{1}'(A_{*})\). So we can conclude that \(u'(A_*+) = G_1'(A_*)\). On the other hand, since \(u(A_{*}-\varepsilon ) = G_{1}(A_{*}-\varepsilon )\) for all \(\varepsilon >0\) sufficiently small we have that \(u'(A_*-) = \lim _{\varepsilon \downarrow 0} \frac{u(A_*- \varepsilon ) - u(A_*)}{\varepsilon } = G_{1}'(A_{*})\). So the result is proved in the case \(S'(A_{*}) \ne 0\). Now suppose that \(S'(A_{*}) = 0\). Since \(G_{1} \le u\), we have, for any \(\varepsilon ,\delta > 0\) sufficiently small, that

$$\begin{aligned}&\frac{G_{1}(A_{*}+\varepsilon ) - G_{1}(A_{*})}{S(A_{*}+\varepsilon )-S(A_{*})} \le \frac{u(A_{*}+\varepsilon )-u(A_{*})}{S(A_{*}+\varepsilon )-S(A_{*})} \le \frac{u(A_{*}-\delta )-u(A_{*})}{S(A_{*}-\delta )-S(A_{*})} \nonumber \\&\qquad = \frac{G_{1}(A_{*}-\delta )-G_{1}(A_{*})}{S(A_{*}-\delta )-S(A_{*})} \end{aligned}$$

(6.17)

where the second inequality follows from the fact that the mapping \(y \mapsto \frac{u(y)-u(x)}{S(y)-S(x)} \) is decreasing. Multiplying both sides of (6.17) by \(\frac{S(A_{*}+\varepsilon )- S(A_{*})}{\varepsilon }\) and using the fact that the scale function is increasing we get

$$\begin{aligned}&\frac{G_{1}(A_{*}+\varepsilon ) - G_{1}(A_{*})}{\varepsilon } \le \frac{u(A_{*}+\varepsilon )-u(A_{*})}{\varepsilon } \nonumber \\&\quad \le \frac{\frac{G_{1}(A_{*}-\delta )-G_{1}(A_{*})}{-\delta }}{\frac{S(A_{*}-\delta )-S(A_{*})}{-\delta }}\frac{S(A_{*}+\varepsilon )-S(A_{*})}{\varepsilon } \end{aligned}$$

(6.18)

Setting \(f(\varepsilon ) := \frac{S(A_{*}+\varepsilon ) - S(A_{*})}{\varepsilon }\) and \(g(\delta ) := \frac{S(A_{*}-\delta ) - S(A_{*})}{-\delta }\) in Lemma 6.5 we get that for any \(\varepsilon _{n} \downarrow 0\), there exists \(\varepsilon _{n_{k}} \downarrow 0\) and \(\delta _{k} \downarrow 0\) as \(k \rightarrow \infty \) such that

$$\begin{aligned} G_{1}'(A_{*})\le & {} \lim _{k \rightarrow \infty } \frac{u(A_{*}+\varepsilon _{n_{k}}) - u(A_{*})}{\varepsilon _{n_{k}}} \le \lim _{k \rightarrow \infty } \frac{f(\varepsilon _{n_k})}{g(\delta _{k})} \frac{G_{1}(A_{*}-\delta _k) - G_1(A_*)}{-\delta _k} \nonumber \\= & {} G'_{1}(A_{*}). \end{aligned}$$

(6.19)

From this it follows that that \(u'(A_*+) = G'_{1}(A_{*})\). On the other hand if we multiply (6.17) by \(\frac{S(A_{*}-\delta )- S(A_{*})}{-\delta }\) we get

$$\begin{aligned} \frac{\frac{G_{1}(A_{*}+\varepsilon )-G_{1}(A_{*})}{\varepsilon }}{\frac{S(A_{*}+\varepsilon )-S(A_{*})}{\varepsilon }}\frac{S(A_{*}-\delta )-S(A_{*})}{-\delta } \le \frac{u(A_{*}-\delta )-u(A_{*})}{-\delta } = \frac{G_{1}(A_{*}-\delta )-G_{1}(A_{*})}{-\delta } \end{aligned}$$

Interchanging \(\varepsilon \) and \(\delta \) in (6.20) and using Lemma 6.5 again with \(f(\varepsilon ):= \frac{S(A_{*}) - S(A_{*}-\varepsilon )}{\varepsilon }\) and \(g(\delta ):= \frac{S(A_{*}+\delta ) - S(A_{*})}{\delta }\) we get that \(\frac{d^{-}u(A_{*})}{dx} = G_{1}'(A_{*})\). This assertion together with \(\frac{d^{+}u(A_{*})}{dx} = G_{1}'(A_{*})\) proves the required result. By symmetry one can show that \(v'(B_{*})=G_{1}'(B_{*})\). \(\square \)

Figure 2a, b show that there can exist more than one Nash equilibrium point. In this example, if both players cooperate and decide to stop the process in the regions \(D_{1}\) and \(D_{2}\) given in Fig. 2a, then their expected gains are higher than those earned if they stop the process in the regions given in Fig. 2b. However this is not the case in the example presented in Fig. 2c, d, where it is evident that nothing is gained if the players cooperate. For a more detailed study on the existence and uniqueness of a Nash equilibrium in the case of absorbed Brownian motion in [0,1] and when the class of payoff functions \(G_i\) are of a similar form to the one presented in Fig. 2, the reader is referred to [1].

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