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Time-inconsistent stopping, myopic adjustment & equilibrium stability: with a mean-variance application

For a discrete time Markov chain and in line with Strotz’ consistent planning we develop a framework for problems of optimal stopping that are timeinconsistent due to the consideration of a non-linear function of an expected reward. We consider pure andmixed stopping strategies and a (subgame perfect Nash) equilibrium. We provide different necessary and sufficient equilibrium conditions including a verification theorem. Using a fixed point argument we provide equilibrium existence results. We adapt and study the notion of the myopic adjustment process and introduce different kinds of equilibrium stability. We show that neither existence nor uniqueness of equilibria should generally be expected. The developed theory is applied to a mean-variance problem and a variance problem.


Introduction
Consider a stochastic process X on a state space E and the problem of finding a stopping time τ that maximizes J τ (x) := E x (f (X τ )) + g (E x (h(X τ ))) , X 0 = x where f, h : E → R and g : R → R.
(1. 1) This problem is in general time-inconsistent in the sense that if a stopping rule is optimal for a particular initial value x then it is generally not optimal for x ′ = x; the reason being that g may be non-linear. Note that we may formulate both mean-variance and variance stopping problems as special cases of (1), see Section 6. The consistent planning approach to time-inconsistent problems pioneered by Strotz and Selten [44,45,47] corresponds -in a stopping problem context -to viewing (1) from the perspective of a person who decides when to stop X but whose preferences, due to the time-inconsistency, change as X evolves; and therefore (1) is viewed as an intrapersonal non-cooperative stopping game. The approach is formalized by formulating an appropriate mathematical definition of a subgame perfect Nash equilibrium. We refer to e.g. [8,13,15,38] for more comprehensive interpretations of the equilibrium approach to time-inconsistent problems.
The present paper is structured as follows. In Section 1.1 we motivate the study of timeinconsistent stopping by formulating three types of problems that are studied in finance and economics and review some of the related literature. In Section 2 we define mixed and pure stopping strategies and the equilibrium. In Section 2.1 we show that the definition of equilibrium coincides with standard optimality when the problem is time-consistent (i.e. when g = 0 in (1)). In Section 3 we derive several results with necessary and sufficient equilibrium conditions including a verification theorem. In Section 4 we provide a fixed point problem characterization of equilibrium and related equilibrium existence results. In Section 5 we adapt and study the notion of a myopic adjustment process. We also define and study different notions equilibrium stability. In Sections 6.1 and 6.2 the developed theory is applied to mean-variance and variance optimization. In Section 6.1.2 we show that an equilibrium does not necessarily exist and if it does then it is not necessarily unique. In Section 7 we discuss the framework of the present paper in relation to the literature. The appendix contains some technical results.

Time-inconsistency in economics & related literature
In order to motivate the study of time-inconsistent stopping problems in general we here present three simple examples which correspond to time-inconsistent problems commonly studied in finance and economics. Similar presentations are contained in [13,15] while [8,38] present these problems in a regular stochastic control framework. Note that of the three kinds of problems described in this section only mean-variance optimization can directly be studied within the framework of the present paper. In [15] we develop a general framework for the equilibrium approach to time-inconsistent stopping problems of the type in the present paper for a one-dimensional diffusion. We remark that timeinconsistent problems can also be studied using the pre-commitment approach and the dynamic optimality approach. In the context of the present paper the pre-commitment approach corresponds to maximizing (1) for a particular x. The dynamic optimality approach was invented in [41,42] and corresponds to choosing a strategy that is optimal with respect to all present states.
Mean-variance optimization: In a stopping problem context the mean-variance problem can be motivated with the following example. Suppose an investor wants to sell an asset whose price follows a stochastic process X. Suppose the investor wants, for any particular x, to use a selling strategy, i.e. a stopping time τ , that maximizes for a fixed parameter γ > 0 corresponding to risk aversion. The interpretation is that the investor wants a large expected payoff but is averse to risk measured in terms of selling price variance. A mean-variance stopping problem is studied in Section 6.1. In [15, Section 4.2] a mean-variance stopping problem for a geometric Brownian motion is studied using the equilibrium approach. In [4] a mean-standard deviation and mean-variance stopping problem for a discrete time Markov chain is studied using the equilibrium approach. In [41] a mean-variance stopping problem for a geometric Brownian motion is studied using a precommitment approach and the dynamic optimality approach. We note that there is a large literature on mean-variance optimization especially for regular stochastic control (often corresponding to dynamic asset portfolio selection), see e.g. [5,6,9,15,16,18,26,35,36,37,42,43,49,50,52,55].
Endogenous habit formation: An example of this kind of problem is a version of the asset selling problem introduced above corresponding to where F (·, x) is a utility function parametrized by x. The interpretation is that the current price of the asset determines the utility function of the investor. In [13] we develop a general framework for the equilibrium approach to time-inconsistent stopping problems of the endogenous habit formation type for a continous time Markov process and in [13,Example 5.8.] an endogenous habit formation asset selling problem is studied. Endogenous habit formation is also studied in e.g. [7,19,21,54].
Non-exponential discounting: An example of this kind of problem is the version of the asset selling problem corresponding to where δ(·) is a non-exponential discounting function (i.e a non-increasing function taking values in [0, 1] with δ(0) = 1). Non-exponential discounting stopping problems are studied in [3,28,32,33]. They can also be studied within the continuous time framework of [13]. Non-exponential discounting is also studied in e.g. [1,2]. Now we mention some other related problems studied in the recent literature. A timeinconsistent stopping problem under model ambiguity is studied using the equilibrium approach in [30]. Conditional optimal stopping is studied using the equilibrium approach in [39]. Precommitment and naive strategies for optimal exit times for gambling are studied in [27]. Optimal stopping under probability distortion is studied with a precommitment approach in [51]. In [29] a general framework for naive and equilibrium strategies for time-inconsistent stopping problems for a diffusion is developed and applied to probability distortion. The principle of smooth pasting for a particular problem is considered in [48]. [20] studies a framework under general preference structure. A version of the classical dividend problem with a time-inconsistent restriction is studied in [14] using both the precommitment and consistent planning approach. Further references to the literature are found throughout the paper. A recent survey of time-inconsistent stochastic control is [53].

Problem formulation
We consider the time-inconsistent stopping problem (1) for a discrete time strong timehomogeneous Markov chain X = {X n }, n ∈ N 0 , taking values in a finite state space E with N elements. We also consider a stochastic process {Y n }, n ∈ N 0 , where each Y n is uniformly distributed on [0, 1] and independent of X and of every Y k , k = n. We denote by P x the measure under which X 0 = x ∈ E a.s. The associated expectations are denoted by E x .
As a notational convenience we consider an ordering of the state space E and identify N -dimensional vectors, for instance p, with functions on the state space, i.e.
Definition 2.1 (Mixed stopping strategies). A vector p ∈ [0, 1] N is said to be a mixed (Markov) stopping strategy and τ p := min{n ≥ 0 : Y n ≤ p Xn } is said to be a mixed (Markov) strategy (profile) stopping time.
Note that τ p is a stopping time with respect to the filtration σ(X 0 , ..., X n , Y 0 , ..., Y n ), n ∈ N 0 . Definition 2.2 (Pure stopping strategies). A mixed stopping strategy p is said to be a pure stopping strategy if p ∈ {0, 1} N .
. For a mixed stopping strategy p and any n the conditional probability of stopping X at n, before having observed Y n , given that X has not been stopped before n, is p Xn . In this sense a stopping strategy p corresponds to using the random variable Y n as a randomization device for the stopping decision made at n. Note that the randomization device can be interpreted as flipping a biased coin at each n and stopping at n if the outcome is, say, heads, where the probability of heads is p y if the observed state is X n = y. For a pure stopping strategy the conditional probability of stopping X at n given that X has not been stopped before n is either one or zero; and in this sense the decision to stop or not at n depends only on the payoff relevant quantity X n without randomization. For a more thorough description of the game theory terms used in this section in another time-inconsistent stopping context see [13].
Remark 2.4. Note that p is a complete specification of the strategies of all players in the game and that p is in this sense a strategy profile, although we refer to p as a stopping strategy to be more in line with the existing literature.
Since the distribution of τ p is determined by p we typically perform the analysis of the present paper from the viewpoint of stopping strategies p. Hence, instead of J τp (x) we write J p (x), cf. (1).

Definition 2.5 (Equilibrium).
A stopping strategyp is said to be a (subgame perfect Nash) equilibrium if The equilibrium is said to be pure ifp is pure. Ifp is an equilibrium then τp is said to be the (corresponding) equilibrium stopping time and Jp is said to be the (corresponding) equilibrium value function.
Remark 2.6 (Interpretation). The interpretation of the expression in the right hand side of (EqI) -or equivalently of K(x, q,p) see (EqII)-(EqIV) below -is that it is the value obtained at x when stopping at x with probability q given that the strategyp is used subsequently. The interpretation of the expression in the left hand side of (EqI)-(EqIV) is that it is the value obtained at x when using the strategyp given thatp is used subsequently. The interpretation of an equilibriump is therefore that, for each x, there is the possibility to deviate fromp at x in the sense of using any other biased coin -cf. q in (EqI) -to determine whether to stop X at the present time or not; but that such a deviation is never preferred to usingp x given thatp is used at all subsequent dates.
This paper is devoted to the question of how to find equilibria as defined above. Throughout the paper we suppose the following assumptions hold: Assumption 2.8. X is an absorbing Markov chain; that is, for each X 0 = x ∈ E, there is (at least) one absorbing state in E that X reaches with positive probability in a finite number of steps.
We use the convention where the limits exist due to Assumption 2.8, and the notation which we note implies that We remark that if p x = 0 for each x ∈ E then τ p = ∞ a.s. meaning that X is never stopped. However, in this case X eventually reaches an absorbing state by Assumption 2.8 and therefore stops in this sense. We remark the related fact that φ p and ψ p are independent of the value of p x whenever x is an absorbing state. We also use the notation We now provide three equivalent equilibrium definitions that will be used in the sequel.
Proposition 2.9. Each one of the following conditions is equivalent to the equilibrium condition (EqI): Proof. Use the notation (2) and (2) to see that the first result holds. The second and third results follow from the first result and the observation that if we set q =p x in (EqII) then equality is attained, cf. Lemma A.1.
Remark 2.10. It should be possible to relax Assumptions 2.7 and 2.8 at the cost of increasing the amount of technical details. We have chosen not to do so in order to focus on the main ideas and not overburden the presentation.

The time-consistent case
If we consider a standard stopping problem, corresponding to maximization for (1) with g = 0, then an equilibrium stopping time has the desirable property of being characterized as an -in the usual sense -optimal stopping time: Then, τp is an equilibrium stopping time for (1) if and only if τp is an optimal stopping time for (1).
Proof. g = 0 implies that the equilibrium condition (EqI) can be written as or equivalently as

We find that Jp(x) is an equilibrium value function if and only if
is the optimal value function of the problem (1) with g = 0 (by well-known results from the general theory of optimal stopping, see e.g. [46]).

Necessary and sufficient equilibrium conditions
It is instructive to note that the right side of the equality in (EqIII), or equivalently (EqIV), is for each fixed x andp an elementary optimization problem of a function [0, 1] → R, cf. (2); which in particular can be written as where c 1 , ..., c 4 are constants (depending on x andp). Using this observation we immediately obtain: Then, a necessary condition for a stopping strategŷ p to be an equilibrium is that, for each x ∈ E, the following inequalities hold and (at least) one of them holds with equality: • Suppose g in (1) is twice differentiable. Then, a necessary condition for a stopping strategyp to be an equilibrium is that for each x ∈ E withp x ∈ (0, 1) (if such points exist) holds that, Proof. Inequalities (3.1) and (3.1) are obtained by setting q = 1 and q = 0 in (EqII), respectively. Inequality (3.1) is trivial. If none of (3.1)-(3.1) holds with equality thenp x cannot satisfy (EqIV); to see this use e.g. that (3.1) is essentially a first order condition for the maximization in (EqIV). Hence, the first result holds. The second result is proved similarly; it is essentially a second order condition for the maximization in (EqIV).
We now provide an equilibrium verification theorem.
whereq x is the maximal constant in the set of maximizers in (3.2).  .2), is an equilibrium whose equilibrium value function is given by Proof. Note that if x is an absorbing state then the expression to be maximized in (3.2) is independent of q and henceq x = 1. The result follows directy from Lemma A.3 and Proposition 2.9.
In the rest of this section we suppose g in (1) is either convex or concave. We remark that the mean-variance problem studied in Section 6.1 uses a convex g while the variance problem studied in Section 6.2 uses a concave g. Proof. If, for each x, the necessary conditions (3.1) and (3.1) hold and one of them holds with equality thenp is a equilibrium since (EqIII) is then satisfied, for each x, with q = 0 or q = 1; to see this use that convexity of g implies (Lemma A.4) that q → K(x, q,p) is convex, (3.6) and that it generally holds (Lemma A.1) that Let us show the reverse implication: Ifp is an equilibrium then trivially (3.1) and (3.1) hold. Moreover, by (3) holds that the maximum in (EqIII) is attained by either q = 0 or q = 1. Recall that ifp is an equilibrium then (EqIII) holds. Now note that (i) if q = 0 is the maximizer in (EqIII) then (3.1) holds with equality, and (ii) if q = 1 is the maximizer in (EqIII) then (3.1) holds equality. • if, for a stopping strategyp, (3.1) holds with equality for each x ∈ E, thenp is an equilibrium.
Proof. Let us prove the first result: Theorem 3.1 implies that ifp is an equilibrium then, for each x ∈ E, (3.1), (3.1) and (3.1) hold and (at least) one of them holds with equality. To see that the other implication is true use concavity of q → K(x, q,p) (Lemma A.4) and basic optimization theory to see that if, for each x ∈ E, (3.1), (3.1) and (3.1) hold and (at least) one of them holds with equality then the equilibrium condition (EqIII) must be satisfied (use also the general observation (3)). The second results is proved similarly.
Theorem 3.7. Suppose g in (1) is strictly convex and an equilibriump exists. Then, an equivalent pure equilibrium exists and such a pure equilibrium can be obtained by changing eachp x ∈ (0, 1) (in case they exist) to 1.
Proof. Suppose y ∈ E is such thatp y ∈ (0, 1). Then has a maximum at q =p y . Since this function is convex (Lemma A.4) and has an interior maximum (by definition of equilibrium and sincep y ∈ (0, 1)) it must be a constant function. In particular, using that g is strictly convex and that is linear, we find that is constant i.e. h(y) = E y ψp(X 1 ) , but then it also follows that f (y) = E y φp(X 1 ) . Now writep Fix a state x ∈ E. Clearly, τp ≤ τp a.s. Using the above and the Markov property we obtain It can similarly be shown that It can now be directly verified thatp is an equilibrium. It it also easy to see that Jp(x) = Jp(x). Now, the claim holds by a trivial induction.
Remark 3.8. The previous theorem implies that in the strictly convex case we just have to check at most the 2 N pure strategies to check whether equilibria exist.

A fixed point problem characterization and existence results
In this section we derive equilibrium existence results which rely on the observation that an equilibrium is the solution to a certain fixed point problem. K(x, q, p). Proof. Follows directly from Proposition 2.9. Proof. The assumed convexity for the set of maximizers for each particular x in (4.1) implies that Γ(p) will be a hyperrectangle and thus a convex set. Since a maximizer in (4.1) necessarily exists follows that Γ(p) is non-empty.
Using the analogous result for ψ p and Assumption 2.7 we find that for any fixed q and x it holds that lim k→∞ K(x, q, k p) = K(x, q, p).
Hence, using also that q → K(x, q, p) is continous (for any fixed x and p), it is easy to see that: For any two sequences { k p} and { k q} on [0, 1] N , with lim n→∞ k p = p and lim n→∞ k q = q, that satisfy k q ∈ Γ( k p) for all k, holds q ∈ Γ(p). We also note that: The set [0, 1] N is non-empty, compact and convex. From (4), (4) and (4)   Remark 4.5. In [31] time-inconsistent regular stochastic control in continuous time is studied and an equilibrium existence result is proved using fixed point arguments similar to those used here. In [8] time-inconsistent stochastic control in discrete time is studied and it is noted that an equilibrium can be viewed as the fixed point of a particular mapping.

The myopic adjustment process and equilibrium stability
An iteration of the type p k+1 ∈ Γ(p k ) where p 0 ∈ [0, 1] N (see Definition 4.1) corresponds to what in economics is known as a myopic adjustment process for decisions in repeated interactive situations, see e.g. [10,22,34]. The interpretation here is that every agent in the game adjusts his decision at each k-step under the (myopic) assumption that all other agents will stay with their strategy. Since Γ(p k ) is in general a set of vectors, i.e. a set of stopping strategies, it holds that this iteration is not uniquely defined. We thus definē Γ(p k ) as the largest (in Euclidean norm, or, equivalently, element-wise) vector in Γ(p k ) and consider the myopic adjustment process This corresponds to the interpretation that there is preference in the myopic adjustment for a higher probability of stopping over a smaller one when they give the same value (in the maximization in (4.1)). Note that the myopic adjustment process can be tried as a constructive algorithm for finding equilibria.
The following are now natural questions: 1. Supposep is an equilibrium and that we perturbp slightly by considering a stopping strategyp ǫ ∈ B(p; ǫ) for some small ǫ > 0, where B(p; ǫ) denotes a ball with radius ǫ centered atp. In which circumstances does then the myopic adjustment process (5) with p 0 =p ǫ converge top?
2. In which circumstances does the myopic adjustment process (5) converge to an equilibriump for any initial value p 0 ?
In the rest of this section we try to shed some light on these questions by defining and investigating different notions of equilibrium stability.
1. An equilibriump is said to be strongly locally stable if there exists a constant ǫ such that for everyp ǫ ∈ B(p; ǫ) there exists an equivalent equilibrium p ∈P such that for every x it holds that Definition 5.
2. An equilibriump is said to be locally stable if for some ǫ > 0 and everŷ p ǫ ∈ B(p; ǫ) the myopic adjustment process with p 0 =p ǫ converges to an equivalent equilibrium p ∈P. An equilibrium is said to be unstable if it is not locally stable.
It is easy to see that a strongly locally stable equilibrium is locally stable.
Remark 5.3. The interpretation of a strongly locally stable equilibrium is that the best response (at each x) to a small deviation from the equilibrium is to return to an equivalent equilibrium immediately; i.e., if a small deviation from a strongly locally stable equilibrium occurs then the myopic adjustment process converges in one step to an equivalent equilibrium. The interpretation of a locally stable equilibrium is that if a small deviation from the equilibrium occurs then the equilibrium (or more precisely an equivalent equilibrium) will eventually be restored under the myopic adjustment process.
We obtain: Theorem 5.4. Suppose g is strictly convex and that an equilibriump exists. Then,p is strongly locally stable.
Proof. In this proof let us use the notatioñ p =Γ(p ǫ ).
Now, if we can show thatp is an equilibrium equivalent top for some small ǫ > 0 then we are done.
Fix an arbitrary state x ∈ E. Since q → K(x, q,p) is a convex function it follows that exactly one of the following cases holds: Summarizing the three cases, for a sufficiently small ǫ we conclude thatp satisfies We now show thatp is an equilibrium equivalent top. Indeed, note that the condition argmax q∈[0,1] K(x, q,p) = [0, 1] means that q → K(x, q,p) is a constant function. Hence, using the exact same arguments as in the proof of Theorem 3.7, we see that Jp = Jp, proving the claim.
Theorem 5.4 implies that the equilibria in the mean-variance problems studied in Section 6.1 below are locally stable.
Definition 5.5. An equilibriump is said to be globally stable if the myopic adjustment process converges to an equivalent equilibrium p ∈P for any starting value p 0 .
Obviously, a globally stable equilibrium is a unique equilibrium and a globally stable equilibrium is also locally stable. However, a globally stable equilibrium is not generally strongly locally stable.
A strictly convex function g makes the problem of checking strong stability a finite problem as only pure stopping strategies have to be considered. With this notion, we may analyze the structure using the notion of directed graphs: The vertices are the pure strategies p ∈ {0, 1} N and there is a directed edge from p to q if q =Γ(p). Now, the problem of studying strong stability boils down to checking whether this directed graph is acyclic.
We remark that Example 6.2 is a problem with a convex g and two equilibria and hence convexity of g is not a sufficient condition for global stability.
We immediately obtain the following (trivial) result: Theorem 5.6. Suppose g is strictly convex. Then: Either the myopic procedure does not converge but runs in cycles or it terminates in at most 2 N + 1 steps.

Applications
In this section we apply the developed theory to mean-variance and variance optimization problems.

A mean-variance problem
The mean-variance stopping problem -see Section 1.1 for a motivation -is attained in the framework of the present paper when The convexity of g implies that if an equilibrium exists then a pure version of that equilibrium exists, cf. Theorem 3.7, and that it is moreover strongly locally stable, cf. Theorem 5.4.

Equilibrium strategies of threshold-type
In [15,Section 4.2] it was shown that the equilibrium for the mean-variance problem for a geometric Brownian motion corresponds, in case it exists, to using a particular threshold stopping strategy. In this section we first study a threshold strategy ansatz to find an equilibrium stopping time for the mean-variance problem assuming only that X is a skip free Markov chain absorbed in x 1 and x N on some state space We furthermore assume x 1 = 0 (this makes some expressions shorter and can be easily relaxed). Second, we use this ansatz to a more particular problem.
An (upper) threshold stopping time is easily seen to be attained by the stopping strategy To see this note e.g. that p being pure implies that one of these conditions necessarily holds. This implies that p in (6.1.1) is an equilibrium if and only: To see this use the threshold structure of p and that x 1 and x N are absorbing. Now consider the function Since p in (6.1.1) is an equilibrium if and only if (6.1.1) and (6.1.1) hold it follows that p in (6.1.1) is an equilibrium if and only if: is determined by the recurrence relation P r(i, b) = 1 2 P r(i+1, b)+ 1 2 P r(i−1, b) for i ∈ {2, ..., b − 1} with boundary conditions P r(1, b) = 0 and P r(b, b) = 1, yielding Basic probability calculations now yield Using (6.1) and x 1 = 0 this implies that Analogously, Note also that −f (x i ) − g(h(x i )) = −x i . The observations above yield, with some calculations Using this explicit formula we can -for any N , γ and further specification of the state space E -check if (6.1.1) is satisfied for someb ∈ {2, ..., N − 1}, in which case thisb corresponds to an equilibrium. Moreover, if such ab exists then the observations above imply that the corresponding equilibrium value function is wherep denotes the threshold strategy, cf. (6.1.1), corresponding tob.
Let us now consider a specific example. Suppose γ = 0.07 and that The state space E is depicted in Figure 1. For this example we conclude from (6.1.1) and the second picture in Figure 1 that the threshold strategy (6.1.1) with b =b = 16 is an equilibrium. Figure 1 also depicts the corresponding equilibrium value function together with the value for the strategy of always stopping immediately.
Remark 6.1. The equilibrium value function in Figure 1 looks very similar to the equilibrium value function for the geometric Brownian motion mean-variance stopping problem depicted in [15, Figure 2].

Counterexamples to uniqueness and existence
In this section we show that one should not in general expect an equilibrium to be unique, not only in the trivial sense that more than one equilibrium strategy may exist, but also in the sense that these may correspond to different equilibrium value functions. We also show that one should not in general expect an equilibrium to exist.
Example 6.2 (Two different equilibria). Consider the mean-variance problem -i.e. f, g and h as defined in (6.1) -for some γ > 2 and the Markov chain X defined by: Let us show thatp = (1, 1) T , i.e. the strategy corresponding to always stopping immediately, is an equilibrium. Clearly, φp(x i ) + g(ψp(x i )) = f (x i ) + g(h(x i )) = x i for each i. Hence (3.1) holds with equality for each i. Simple calculations give that Hence, (3.1) holds for i = 1. Moreover, (3.1) holds also for x 2 since this is an absorbing state. It thus follows from Corollary 3.4 thatp is an equilibrium. The corresponding equilibrium value is easily found to be Jp = (1, 2) T . With similar calculations it can be shown that alsop = (0, 1) T , i.e. waiting until the value 2 is reached, is an equilibrium with corresponding equilibrium value function Jp = (2, 2) T . Example 6.3 (No equilibrium). Consider the mean-variance problem with γ = 1 for the skip free Markov chain X defined by: Since g is convex -implying that if an equilibrium exists then a pure equilibrium exists, cf. Theorem 3.7 -and x 1 and x N are absorbing it follows that verifying that the strategies (i) -(iv) below are not equilibrium strategies corresponds to verifying that this problem has no equilibrium. From the calculations below and the convexity of g it is easy to see that a myopic adjustment process for this problem does not converge but instead -regardless of the initial value p 0 -runs in a cycle according to (i) → (ii) → (iii) → (iv) → (i) and so on as long as it is allowed to run.
5210. Hence, deviating by not stopping is optimal at x = 2 and this is therefore not an equilibrium (cf. e.g. Theorem 3.1).
Hence, this is not an equilibrium.
Hence, this is not an equilibrium.

A variance problem
The variance problem is defined by setting J τ (X τ ) = Var x (X τ ) in (1) which in our framework is attained when The equilibrium approach to the variance stopping problem for a geometric Brownian motion was studied in [15,Section 4.1]. Optimal variance problems are also studied in e.g. [11,12,24,25,40].
From Corollary 4.4 and the concavity of g follows that an equilibrium always exists for the variance problem. Let us now consider a symmetric random walk X on the state space where x 0 is absorbing and x M is reflecting, for some natural number M . Note that the number of states is N = M + 1. Let us try the ansatz that the equilibrium is of the kind p = (1, 0, ..., 0, p) T (6.8) for some p ∈ [0, 1] to be determined. Similarly to Section 6.1 we find that Using that p is the probability of stopping at x M , and also (6.2), (6.2) and that x 0 = 0 is absorbing, we find This implies that .
Since x M is reflecting it follows that It is similarly found that Putting everything together yields , for all i,

Similar calculations yield
, for all i, Using the findings above it is easy to verify: (i) condition (3.1) is satisfied for all i and all p (note that this corresponds to the fact that the variance is always non-negative), (ii) condition is an equilibrium. Simple calculations imply that (6.2) corresponds to The corresponding equilibrium value function is , for all i.
The equilibrium value function in the case M = 100 is depicted in Figure 2.
We remark that for mixed equilibria one should not in general expect equilibrium stability. This can be made rigorous in this example: If we considerp ǫ by just changingp M top ǫ M = 1 M +1 + ǫ, the myopic adjustment process can be found explicitly using the calculations above. Indeed, for small enough ǫ, it holds that Therefore, in case M ≥ 3, iterating this, we see that the myopic adjustment process cannot converge to the fixed pointp. Hencep is not locally stable.

Discussion and relation to the literature
The definitions of pure and mixed strategies as well as the equilibrium definition of the present paper are in line with the definitions of [4] which studies mean-standard deviation and mean-variance stopping in a discrete time framework. A pure stopping strategy for continous time is in [13,15] defined as the entry time into a set in the state space. The stopping strategies considered in [28,29,30,33] are of the same type. Noticing that a pure stopping strategy p corresponds to τ p = min{n ≥ 0 : X n ∈ {x ∈ E : p x = 1}}, we see that our definition is in line with the literature. A mixed stopping strategy for continuous time is in [15] defined as the first stopping time of an X-associated Cox process, which is a natural continous time interpretation of the definition of a mixed stopping strategy in the present paper; see [15, Section 2.1] for further arguments. The continuous time mixed equilibrium in [15] corresponds to a first order condition whose interpretation is inline with the present paper in the sense that a stopping strategy is an equilibrium if it is at no x desirable to deviate from the equilibrium by using an alternative probability for stopping at x. The equilibrium definition in [13] is analogous but in a framework considering only pure strategies. Further comparisons of definitions in the literature on time-inconsistent stopping is found in e.g. [13,15].
In [33] a non-exponential discounting stopping problem in discrete time for stopping strategies that correspond to pure strategies in the sense of the present paper is studied and a method for finding equilibria similar to the myopic adjustment process, there called a fixed-point iteration, is used to establish equilibrium existence. The equilibrium definition of [33] differs from that of the present paper in the sense that it is not possible to deviate at x from a proposed equilibrium stopping strategy if it suggest stopping at x, and hence the strategy of always stopping immediately is necessarily an equilibrium , see also the discussion in [15, Section 2.1]. Similar frameworks and approaches to finding equilibria for time-inconsistent stopping problems in continuous time is studied in [28,29,30]. We also note that a similar iteration approach is used to finding equilibria for a portfolio selection problem in [17,Example 4.].

A. Technical results
In this section we derive properties for the function φ p defined in (2). Analogous results are of course true for ψ p defined in (2).
Lemma A.1. For each p and x holds that, Proof. Follows from the definitions of p and τ p .
Lemma A.2. For each p and x, the following identities hold: where we use the convention l j=k := 1 for l < k. Moreover, the functions are, for each fixed x ∈ E, continuous.
Proof. Using the notation (2), Fubini's Theorem and Assumption 2.8 we immediately obtain the identities. Recall that φ p is independent of the choice of p x for each absorbing state x ∈ E. The continuity follows from majorized convergence.
Lemma A.3. Consider a function φ : E → R and a vector p ∈ [0, 1] N with p x > 0 for each absorbing state x ∈ E and suppose Proof. Note that τ p < ∞ a.s. Repeated substitution in (A.3) and the Markov property give, with a slight abuse of notation, Now use Lemma A.2.
Lemma A.4. If g : R → R is a convex function then q → (c 1 q + c 2 (1 − q) + g(c 3 q + c 4 (1 − q)) is a convex function, for any constants c 1 , ..., c 4 . The analogous result holds in the case g is concave.
Proof. Follows directly from the definition convexity/concavity.