OPTIMAL STOPPING WITH PRIVATE INFORMATION
JOBMARKET PAPER
THOMAS KRUSE AND PHILIPP STRACK
Abstract. Many economic situations are modeled as stopping problems. Examples are timing of market entry decisions, search, and irreversible investment. We analyze a principal- agent problem where the principal and the agent have dierent preferences over stopping rules. The agent privately observes a signal that in uences his own and the principal's payo. Based on his observation the agent decides when to stop. In order to in uence the agents stopping decision the principal chooses a transfer that conditions only on the time the agent stopped. We derive a monotonicity condition such that all cut-o stopping rules can be imple- mented using such a transfer. The condition generalizes the single-crossing condition from static mechanism design to optimal stopping problems. We give conditions under which the transfer is unique and derive a closed form solution based on re ected processes. We prove that there always exists a cut-o rule that is optimal for the principal and, thus, can be implemented using a posted-price mechanism. An application of our result leads to a new purely stochastic representation formula for the value function in optimal stopping problems based on re ected processes. Finally, we apply the model to characterize the set of implementable policies in the context of job search and irreversible investment.
Keywords: Dynamic Mechanism Design, Optimal Stopping, Dynamic Implementability, Posted-Price Mechanism
JEL codes: D82, D86, C02
Thomas Kruse, Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53113 Bonn, Germany. Web: http://www.uni-bonn.de/∼tkruse/, Email: [email protected]; Philipp Strack, Institute for Microeconomics & Hausdor Center for Mathematics, University of Bonn, Lennestrasse 43, 53111 Bonn, Germany. Web: http://www.uni-bonn.de/∼pstrack/, Email: [email protected]. We would like to thank Stefan Ankirchner, Dirk Bergemann, Paul Heidhues, Daniel Krähmer, Benny Moldovanu, Sven Rady, Juuso Toikka and seminar audiences in Yale, Maastricht and Bonn for many helpful comments and suggestions. All remaining errors are our own. 1 1. Introduction
Many economic situations are modeled as stopping problems. Examples are job search, timing of market entry decisions, or irreversible investment decisions. In an optimal stopping problem information arrives over time and a decision maker decides at every point in time whether to stop or continue. The stopping decision is irreversible and, thus, stopping today implies forgoing the option to stop later with a potentially larger return. The literature on optimal stopping assumes that the decision maker observes all relevant information directly. There are, however, many economic environments modeled as opti- mal stopping problems where private information is a crucial feature. Important examples include the regulation of a rm that decides when to enter a market, or the design of unem- ployment bene ts when the worker privately observes his job oers. Consider for example a pharmaceutical rm that observes the stochastically changing demand for a new product and needs to decide on whether and when to enter the market for this product. If the rm starts selling the product, it pays an investment cost and receives the ow returns from selling the product at all future times. When timing its market entry decision, the rm solves an irreversible investment problem. As famously shown in the irreversible investment literature, it is optimal for the rm to delay investment beyond the point where the expected return of investing becomes positive (for an excellent introduction in the irreversible investment literature see Dixit and Pindyck 2008). While this investment strategy is optimal for the rm, it is typically wasteful from a social perspective: As the rm does not internalize the consumer surplus, it will enter the market too late. A question that naturally arises is how a regulator can improve the incentives of the rm if the demand is private information of the rm. The result we derive in this paper implies that the regulator can always implement a socially e cient market entry decision through a simple posted-price-like-mechanisms that conditions on when the rm enters. Our model is a dynamic principal-agent model. The agent privately observes a randomly changing signal and chooses a stopping rule. The principal observes the stopping decision of the agent, but not the signal. The principal commits to a transfer in order to in uence the agent's decision. Our main result is that under a single-crossing condition all cut-o rules can be implemented using transfers that condition only on the time the agent stopped. As the transfer does not depend on private information of the agent it can be implemented without communication. This feature resembles a posted-price mechanism and makes our mechanism especially appealing from an applied perspective. Finally, we show that the minimal stopping rule that is optimal for the principal is a cut-o rule and thus can be implemented.
2 We establish our result as follows. First, we derive a dynamic single-crossing condition that ensures the monotonicity of the continuation gain in the agent's signal at a every point in time. A consequence of this monotonicity is that the incentive to stop is increasing in the type. Thus, to implement a cut-o rule it su ces to provide incentives to the (marginal) cut- o type. Intuitively, taking future transfers as given, the transfer providing incentives to the marginal type today could be calculated recursively. But as future transfers are endogenous, this recursive approach is di cult and requires the calculation of a value function at every point in time. We circumvent all these problems attached to the recursive approach by directly constructing transfers using re ected stochastic processes. We de ne a re ected process as a Markov process that equals the original process as long as the latter stays below the cut-o. Once the original process exceeds the cut-o, the re ected process follows its own dynamics and is de ned to stay below the cut-o. We show that every cut-o rule can be implemented through a transfer that equals the agent's expected payo evaluated at the process re ected at the cut-o. To the best of our knowledge, we are the rst to use re ected processes in the context of mechanism design. We also believe that this paper contributes to the mathematical literature on optimal stopping as we are able to give a new, completely probabilistic su cient condition for optimal stopping rules. The approach of this paper diers from the standard mechanism-design one, since we do not rely on the revelation principle and direct mechanisms. We do not use direct mechanisms in order to bypass technical problems that a formalization of the cheap talk protocol between the principal and the agent would entail: the agent would have to communicate with the principal constantly and, hence, the space of communication strategies would be very rich. As optimal communication strategies are not necessarily Markovian, an optimization over those is a hard problem. The direct mechanism design approach, nevertheless, has been successfully used in general discrete-time settings in Bergemann and Välimäki (2010) for welfare maximization, in Pavan et al. (2012) for revenue maximization, or in continuous time to solve principal-agent problems in Sannikov (2008) and Williams (2011). Surprisingly, it turns out that for optimal stopping problems the focus on posted-price mechanisms is not restrictive from a welfare perspective. We show that a principal-optimal full information stopping rule ( rst best) is always implementable by posted-price mecha- nisms. Moreover, while direct mechanisms often require communication and a transfer at every point in time, our posted-price mechanism demands no communication and only a single transfer at the time of stopping. The paper proceeds as follows. Section 2 introduces the model. In Section 3 we show that all cut-o rules are implementable using posted-price mechanisms and derive the transfer. Furthermore Section 3 establishes that the minimal stopping rule that is optimal for the 3 principal is a cut-o rule. Section 4 presents applications to irreversible investment, search, and proves that the classical static implementability result is a special case of our dynamic result. Section 5 concludes.
2. The Model
We rst give an informal presentation of the model before we present all the precise mathematical assumptions. There is a risk-neutral principal and a risk-neutral agent, who depending on the stopping rule τ and the signal receive X a payo of τ agent: V (τ) = E h(t, Xt)dt + g(τ, Xτ ) ˆ0 τ principal: W (τ) = E β(t, Xt)dt + α(τ, Xτ ) . ˆ0 The signal X is a Markov process which is only observed by the agent. Depending on his past observation of the signal the agent can decide at every point in time between stopping and continuing. In order to in uence the agents decision the principal commits to a transfer π that only conditions on the time the agent stopped. The transfer π implements the stopping rule τ ? if τ ? is optimal for the agent given he receives the transfer π in addition to his payos g and h V (τ ?) + π(τ ?) ≥ V (τ) + π(τ) for all τ . The next sections state the precise assumptions of the model.
2.1. Evolution of the Private Signal. Time is continuous and indexed by t ∈ [0,T ] for some xed time horizon T < ∞.1 At every point in time t the agent privately observes a real valued signal Xt. The signal is adapted to the ltration F of some ltered probability space (Ω, P, F = {Ft}t∈[0,T ]). The ltered probability space supports a one-dimensional F-
Brownian motion Wt and a F-Poisson random measure N(dt dz) with compensator ν(dz) on R. We denote by N˜(dt dz) = N(dt dz) − ν(dz)vt the compensated Poisson random measure.
The time zero signal X0 is distributed according to the cumulative distribution function
F : R → [0, 1]. The signal observed by the agent Xt changes over time and follows the inhomogeneous Markovian jump diusion dynamics
˜ (2.1) dXt = µ(t, Xt)dt + σ(t, Xt)dWt + γ(t, Xt−, z)N(dtdz). ˆR The functions µ, σ : [0,T ] × R → R and γ : [0,T ] × R × R → R satisfy the usual assump- tions listed in Appendix A to ensure that the process exists and is well-behaved . Under these assumptions there exists a path-wise unique solution t,x to (2.1) for every initial (Xs )s≥t condition t,x . Xt = x 1We show later that discrete time models are a special case where we restrict the set of possible strategies. 4 We assume that jumps preserve the order of signals. At every point in time t a lower signal x ≤ x0 before a jump has a lower level after a jump
x + γ(t, x, z) ≤ x0 + γ(t, x0, z) ν(dz) − a.s. This implies a comparison result (c.f. Peng and Zhu 2006): The path of a signal starting at a lower level x ≤ x0 at time t is smaller than the path of a signal starting in x0 at all later times s > t (2.2) t,x t,x0 P Xs ≤ Xs − a.s. 2.2. Information, Strategies and Payos of the Agent. Based on his observations of the signal the agent decides when to stop. The agent can only stop at times T ⊆ [0,T ]. Note that we can embed discrete time in our model by only allowing the agent to stop at a nite number of points T = {1, 2,...,T }. Denote by T the set of F adapted stopping rules taking values in T. A stopping rule is a complete contingent plan which maps every history observed by the agent (t, (Xs)s≤t) into a binary stopping decision. The agent is risk-neutral and receives a ow payo h : [0,T ] × R → R and terminal payo g : [0,T ] × R → R.2 His value V : T → R of using the stopping rule τ equals τ V (τ) = E h(t, Xt)dt + g(τ, Xτ ) . ˆ0 The problem analyzed in the optimization theory literature is to nd an agent-optimal stop- ping rule, i.e. a stopping rule τ ? such that for all stopping rules τ ∈ T
V (τ ?) ≥ V (τ) .
To ensure that the optimization problem is well posed we assume that g and h are Lipschitz continuous and of polynomial growth in the x variable. Moreover we assume that g is twice dierentiable in the x variable and once in the t variable. Then the in nitesimal 1 (2.3) (Lg)(t, x) = lim (E [g(t + ∆,Xt+∆)|Xt = x] − g(t, x)) ∆&0 ∆ is well de ned for every t and every x and we have the identity 1 (Lg)(t, x) = g (t, x) + µ(t, x)g (t, x) + σ(t, x)2g (t, x) t x 2 xx
+ g(t, x + γ(t, x, z)) − g(t, x) − gx(t, x)γ(t, x, z)ν(dz). ˆR Finally we assume that the payos of the agent satisfy the following single-crossing condition
2Note that our model can easily extended to include a ow payo after stopping, i.e. payos of the form h i E τ T ˜ 0 h(t, Xt)dt + g(τ, Xτ ) + τ h(t, Xt)dt . ´ ´ 5 De nition 1 (Dynamic Single-Crossing). The mapping x 7→ h(t, x) + (Lg)(t, x) is non- increasing.
The dynamic single-crossing condition means that if after every history Ft the expected change in the payo of the agent if he waits an in nitesimal short time is decreasing in the current signal Xt = x t+∆ t 1 E lim ∆ h(s, Xs)ds + g(t + ∆,Xt+∆) − h(s, Xs)dt + g(t, Xt) | Ft ∆&0 ˆ0 ˆ0 1 t+∆ = lim E h(s, Xs)dt + g(t + ∆,Xt+∆) − g(t, Xt) | Xt = x ∆&0 ∆ ˆt = h(t, x) + (Lg)(t, x) . The dynamic single-crossing condition is a condition naturally satis ed in most optimal stopping problems. Examples in the economic literature are irreversible investment (Dixit and Pindyck 2008), learning in one armed bandit models, job and other search problems as well as American options.
2.3. Information, Strategies and Payos of the Principal. Similar to the agent the principal is risk neutral and wants to maximize the functional W : T → R which is the sum of a ow payo β : [0,T ] × R → R and terminal payo α : [0,T ] × R → R τ W (τ) = E β(t, Xt)dt + α(τ, Xτ ) . ˆ0 The payos α, β of the principal satisfy the same regularity assumptions as the payos of the agent g, h as well as the dynamic single-crossing condition. We call a stopping time τ ? principal-optimal if it maximizes the utility of the principal, W (τ ?) ≥ W (τ) for all τ ∈ T . Note that the principal optimal stopping time ( rst best) is the stopping time the principal would like to use if he observes the signal X. The principal however, does not observe the signal X. He only knows the distribution F of X0 and the probability measure P describing the evolution of X, i.e µ, σ and γ. As the signal X is unobservable the principal has no information about the realized ow payos of the agent or the agents expectation about future payos. The only information the principal observes is the realized stopping decision of the agent. In order to in uence this decision she commits to a transfer scheme π : [0,T ] → R that depends only on the time the agent stopped. As the transfer only conditions on publicly observable information it requires no communication between the principal and the agent and can be interpreted as a posted-price. We de ne the set of stopping rules that are implementable through a transfer 6 De nition 2 (Implementability). A stopping rule τ ? ∈ T is implemented by the transfer π if for all stopping rules τ ∈ T τ ? τ ? ? (2.4) E h(t, Xt)dt + g(τ ,Xτ ? ) + π(τ ) ≥ E h(t, Xt)dt + g(τ, Xτ ) + π(τ) . ˆ0 ˆ0 A transfer π implements the stopping rule τ ? if the stopping rule τ ? is optimal for the agent if he receives the transfer π in addition to his payos h and g. Implementability generalizes the notion of optimality for stopping rules, as a stopping rule is optimal if and only if it is implemented by a constant transfer.3
3. Implementable Stopping Rules
We introduce the notion of cut-o rules. Cut-o rules are stopping rules such that the
agent stops the rst time the signal Xt exceeds a time-dependent threshold b.
De nition 3 (Cut-O Rule). A stopping rule τ ∈ T is called a cut-o rule if there exists a function b : [0,T ] → R such that
τ = inf{t ≥ 0 | Xt ≥ b(t)}.
4 In this case we write τ = τb. If b is càdlàg we call τb a regular cut-o rule. In Section 3.2 we show that all regular cut-o rules are implementable. The associated transfer admits an explicit representation in terms of the re ected version of X which we introduce in Section 3.1. Under the assumption, that the agent stops the rst time it is optimal to stop we show in Section 3.3 that every implementable stopping rule, is a cut-o rule.
3.1. Re ected Processes. For a given barrier b : [0,T ] → R the re ected version of X is a process X˜ which evolves according to the same dynamics as X as long it is strictly below b but is pushed downwards any time it tries to exceed b.
De nition (Re ected Process). Let b : [0,T ] → R be a càdlàg function. A processes ˜ (Xt)0≤t≤T on (Ω, P, F) is called a re ected version of X with barrier b if it satis es the conditions ˜ ˜ 1) X is constrained to stay below b: For every t ∈ [0,T ] we have Xt ≤ b(t) a.s.
2) Until X hits the barrier both processes coincide: For every 0 ≤ t < τb we have ˜ Xt = Xt a.s.
3The de nition of implementability does not require a regularity assumption on the transfer π ensuring that the resulting optimization problem of the agent is well posed, as all transfers that lead to an ill-posed optimization problem implement no stopping rule and thus are irrelevant. 4A function b : [0,T ] → R is called càdlàg if it is right-continuous and has left-limits. 7 Random Walk Xt , Reflected RW X t Brownian Motion Xt , Reflected BM X t 18 18 17 17 16 16 15 15 14 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 t 0 t 10 20 30 40 10 20 30 40 Figure 3.1. Realization of a random walk (left) and a Brownian motion
(right) starting in X0 = 8 re ected at the constant barrier b(t) = 10.
˜ ˜ 3) X is always smaller than X: For every t ∈ [0,T ] we have Xt ≤ Xt a.s. 4) When hits the barrier, ˜ is at : ˜ a.s. X X b Xτb = b(τb)
Moreover, we assume that for every stopping rule τ ∈ T and every Fτ measurable random variable there exists a process ˜ τ,η such that η (Xt )τ≤t≤T 5) ˜ τ,η starts at time in : P ˜ τ,η X τ η [Xτ = η] = 1. ˜ 6) ˜ has the strong Markov Property: We have ˜ τ,Xτ ˜ a.s. for every X Xt = Xt t ∈ [τ, T ]. 7) A higher value x0 ≥ x of X˜ at time t leads to higher values of X˜ at all later times : ˜ t,x ˜ t,x0 a.s. s ≥ t Xs ≤ Xs
The next Proposition ensures the existence of a re ected version of X for a wide range of settings.
Proposition 4. Let b be a càdlàg function. a) If X has no jumps (i.e. γ = 0), then there exists a version of X re ected at b. b) If X is non decreasing and b non increasing, then a version of X re ected at b is given by ˜ τ,η τ,η Xs = min{Xs , b(s)}. Proposition 4 shows that re ected versions of the process exist for very large classes of en- vironments, especially Diusion processes. One assumption of Proposition 4 will be satis ed in all the examples we discuss later.
3.2. All Cut-O Rules are implementable. In this section we prove that every cut-o rule is implementable. For a given cut-o rule we de ne the transfer and explicitly verify that it implements the cut-o rule. The transfer equals the expected future change in payo evaluated at the process re ected at the cut-o. 8 Theorem 5. Let b : [0,T ] → R such that there exists a version X˜ of X re ected at b. Then
τb is implementable by the transfer T (3.1) E ˜ t,b(t) ˜ t,b(t) π(t) = h(s, Xs ) + (Lg)(s, Xs )ds . ˆt
Proof. Let π be given by Equation (3.1). To shorten notation we de ne z(t, x) = h(t, x) + (Lg)(t, x). By the single-crossing condition x 7→ z(t, x) is non-increasing. We need to show for all stopping times τ ∈ T
τ τb (3.2) E E h(t, Xt)dt + g(τ, Xτ ) + π(τ) ≤ h(t, Xt)dt + g(τb,Xτb )ds + π(τb) . ˆ0 ˆ0 As X is a Feller process and the generator of g is well de ned we can apply Dynkin's formula to get that (3.2) is equivalent to
τ τb E z(s, Xs)ds + π(τ) ≤ E z(s, Xs)ds + π(τb) . ˆ0 ˆ0
First, we prove that the stopping min{τ, τb} performs at least as well as τ. We readily compute τ E 1{τb<τ} z(s, Xs)ds + π(τ) ˆ0 τb τ T = E 1 z(s, X )ds + z(s, X )ds + z(s, X˜ τ,b(τ))ds . {τb<τ} ˆ s ˆ s ˆ s 0 τb τ
The comparison principle for the re ected process X˜ τb,b(τb) with the original process X yields
for every time s ≥ τb after the process hit the barrier that the re ected process is not greater ˜ τb,b(τb) ˜ than the original process Xs ≥ Xs . As the re ected process X is Markovian it follows ˜ τb,b(τb) τ ,b(τ ) that ˜ τ,Xτ ˜ b b for all times . By the comparison principle for re ected Xs = Xs s ≥ τ > τb
processes we have that for all times s ≥ τ > τb the re ected process started in (τ, b(τ)) is not τ ,b(τ ) τ,b(τ) ˜ τb,b(τb) smaller than the re ected process started in ˜ b b , i.e. ˜ ˜ τ,Xτ . These (τ, Xτ ) Xs ≥ Xs
two inequalities combined with the monotonicity of z yield that for all τb < τ τ τ z(s, X )ds ≤ z(s, X˜ τb,b(τb))ds ˆ s ˆ s τb τb T T ˜ τ,b(τ) ˜ τb,b(τb) z(s, Xs )ds ≤ z(s, Xs )ds . ˆτ ˆτ 9 Hence conditional on τb < τ the stopping rule τb gives the agent a higher expected payo than the stopping rule τ τ E 1{τb<τ} z(s, Xs)ds + π(τ) ˆ0 τb τ T ≤ E 1 z(s, X )ds + z(s, X˜ τb,b(τb))ds + z(s, X˜ τb,b(τb))ds {τb<τ} ˆ s ˆ s ˆ s 0 τb τ τb E = 1{τb<τ} z(s, Xs)ds + π(τb) . ˆ0
Consequently it is never optimal to continue after τb. The agent is at least as well of if he uses the stopping rule min{τ, τb} instead of τ τ E z(s, Xs)ds + π(τ) ˆ0 τ τ E E = 1{τb<τ} z(s, Xs)ds + π(τ) + 1{τb≥τ} z(s, Xs)ds + π(τ) ˆ0 ˆ0 τ∧τb ≤ E z(s, Xs)ds + π(τ ∧ τb) . ˆ0
Thus it su ces to consider stopping rules τ ≤ τb. We have τ E z(s, Xs)ds + π(τ) ˆ0 τ τb T = E z(s, X )ds + z(s, X˜ τ,b(τ))ds + z(s, X˜ τ,b(τ))ds . ˆ s ˆ s ˆ s 0 τ τb τ,b(τ) From the comparison principle for re ected processes follows ˜ τ,Xτ ˜ for all Xs ≤ Xs s ≥ τ. By the minimality property of re ected processes the original process X and the re ected process ˜ τ,Xτ coincide for all times before they hit the barrier, i.e. ˜ τ,Xτ for all . X Xs = Xs s < τb τ ,b(τ ) Moreover, the Markov property of the re ected process yields ˜ b b ˜ τ,Xτ for . Xs = Xs s ≥ τb
10 The monotonicity of z implies τ τ T E E ˜ τ,b(τ) z(s, Xs)ds + π(τ) = z(s, Xs)ds + z(s, Xs )ds ˆ0 ˆ0 ˆτ τ T E ˜ τ,Xτ ≤ z(s, Xs)ds + z(s, Xs )ds ˆ0 ˆτ τ τb T = E z(s, X )ds + z(s, X˜ τ,Xτ )ds + z(s, X˜ τ,b(τ))ds ˆ s ˆ s ˆ s 0 τ τb τ τb T = E z(s, X )ds + z(s, Xτ,Xτ )ds + z(s, X˜ τb,b(τb))ds ˆ s ˆ s ˆ s 0 τ τb τb E = h(t, Xt)dt + g(τb,Xτb ) + π(τb) . ˆ0
3.3. All Implementable Stopping Rules are Cut-o Rules. For a xed transfer π and π a xed time-t signal x = Xt the agent's continuation value w (t, x) is the increase in expected payo if the agent uses the optimal continuation strategy instead of stopping at time t τ π E t,x t,x w (t, x) = sup h(s, Xs )ds + g(τ, Xτ ) + π(τ) − g(t, x) − π(t). τ∈Tt,T ˆt The continuation value satis es the following properties.5
Lemma 6. The mapping x 7→ wπ(t, x) is non-increasing and Lipschitz continuous.
By de nition the continuation value is non-negative. If the continuation value is positive it is never optimal for the agent to stop. If the continuation value equals zero it is optimal to stop immediately but potentially there exists another strategy that stops later and is also optimal. Clearly if the agent is indierent strategies that randomize over stopping and continuing are also optimal.
De nition 7 (Minimal Stopping Rule). We call a stopping time τ which is implemented by the transfer π minimal if and only if the agent stops the rst time it is optimal to stop, i.e.
π τ = inf{t ≥ 0 | w (t, Xt) = 0}.
The restriction to minimal stopping rules allows us to excludes non-generic cases. The next proposition shows only cut-o rules can be implemented using a transfer π if we assume that the agent stops the rst time it is optimal for him.
5The monotonicity of x 7→ wπ(t, x) and the resulting form of the continuation region is shown in Jacka and Lynn (1992) for the case that X is a Itô diusion. 11 Proposition 8. Let the stopping rule τ be implemented by the transfer π and minimal then τ is a cut-o rule.
Proof. We x a point in time t ∈ [0,T ] and introduce the stopping region Dt = {x ∈ π 0 R|w (t, x) = 0}. Let x ∈ Dt and x ≥ x. Then Lemma 6 implies that the continuation value wπ(t, x0) at x0 is smaller than the continuation value wπ(t, x) at x and hence x0 is in the stopping region Dt as well. This implies that Dt is an interval which is unbounded on the π right. By Lemma 6 the function x → w (t, x) is continuous and hence Dt is closed. Therefore we have Dt = [b(t), ∞) for some b(t) ∈ R. This implies that τ is a cut-o rule with barrier b
τ = inf{t ≥ 0 | Xt ∈ Dt} = inf{t ≥ 0 | Xt ≥ b(t)}.
3.4. A Principal-Optimal Policy is Implementable. In this section we prove that a principal-optimal policy is implementable. A principal-optimal policy that maximizes the payo of the principal τ W (τ) = E β(t, Xt)dt + α(τ, Xτ ) . ˆ0 Recall that we made no assumptions on the payos a, b of the principal, apart from dier- entiability and the single-crossing assumption.
Theorem 9. There exists a principal-optimal stopping rule that is implementable.
Proof. Assume that the preferences of the principal and the agent are perfectly aligned V ≡ W . It follows from Proposition 8 that the minimal stopping rule implemented by the transfer of zero π ≡ 0 is a cut-o rule. As the preferences of the agent and the principal are the same it follows that the minimal principal-optimal stopping rule is a cut-o rule. By Theorem 5 all cut-o rules can be implemented and it follows that the minimal principal- optimal stopping rule is implementable. 3.5. Uniqueness of The Payment. In this section we derive conditions such that the payment implementing a cut-o rule τb is uniquely determined up to a constant.
Proposition 10. Let X be a regular continuous diusion (γ = 0 and σ bounded away from zero) and the barrier b be continuously dierentiable, then there exists up to an additive constant at most one continuously dierentiable transfer π : [0,T ] → R implementing a cut-o rule τb.
Proof. Let π denote a continuously dierentiable transfer implementing the stopping rule τb. We assume that π(T ) = 0 and show that π is uniquely determined. The value function is 12 de ned by τ (3.3) π E t,x t,x v (t, x) = sup h(s, Xs )ds + g(τ, Xτ ) + π(τ) . τ∈Tt,T ˆt Since g, h and π are su ciently smooth it follows that vπ is continuously dierentiable in t and twice continuously dierentiable in x for all x < b(t). Moreover, since τb is an optimal stopping time in (3.3) the value function is also given by
τb vπ(t, x) = E h(s, Xt,x)ds + g(τ ,Xt,x) + π(τ ) . s b τb b ˆt In particular, at every point x < b(t) below the barrier it is optimal to continue, which implies that
(3.4) Lvπ(t, x) + h(t, x) = 0. On the barrier it is optimal to stop, which yields
(3.5) vπ(t, b(t)) = g(t, b(t)) + π(t) for all t ∈ [0,T ]. Since the agent has to stop at time T we have for all x ∈ R
(3.6) vπ(T, x) = g(T, x). Moreover the smooth pasting principle implies that
π (3.7) ∂xv (t, b(t)) = ∂xg(t, b(t)). Observe that by Ladyzhenskaja et al. (1968; Theorem 5.3 in Chapter 4) there exists a unique solution to Equations (3.4),(3.6) and (3.7). As these three equations do not depend on the transfer π the value function does not depend on π neither. So for all continuously dierentiable transfers π with π(T ) = 0 the associated value functions coincide vπ = v. But π by Equation (3.5) π is uniquely determined by π(t) = v (t, b(t)) − g(t, b(t)).
4. Applications and Examples
4.1. Irreversible Investment and Regulation. In this section we present an application of our result to the classical irreversible investment literature. A pharmaceutical rm ob- serves demand for a new product X. The demand Xt is a geometric Brownian motion with drift
dXt = Xtµdt + XtσdWt , where 0 ≤ µ < r and σ > 0. If the rm starts selling the product at time t it pays continuous investment cost c : R+ → R+ and receives the ow returns from selling the product. The 13 total expected payo of the rm equals ∞ −rt −rτ Xτ V (τ) = E −c(τ) + e Xtdt = E −c(τ) + e . ˆτ r − µ When timing its market entry decision the rm solves an irreversible investment problem. As famously shown in the irreversible investment literature (see for example Dixit and Pindyck (2008)) it is optimal for the rm to delay investment beyond the point where the net present
value −rt Xt of investing becomes positive. While this behavior is optimal for the −c(t) + e r−µ rm it is wasteful from a social perspective. Assume that the consumers discount at the same rate have an additional gain Γ > 0 from the pharmaceutical such that the overall bene t of society equals ∞ −rt −rτ (1 + Γ)Xτ W (τ) = V (τ) + E e ΓXtdt = E −c(τ) + e . ˆτ r − µ As the rm does not internalize the bene t of the consumers it will enter the market too late.
Proposition 11. Let V be the optimal stopping time of the rm and τ = arg supτ V (τ) W be the socially optimal stopping time, then we have that τ = arg supτ W (τ) τ W < τ V .
A question that naturally arises is how a regulator could improve the incentives of the rm if demand is private information of the rm. By Theorem 1 we know that using a simple posted-price mechanisms the regulator can always implement the socially e cient market entry decision. In the next paragraph we will explicitly calculate the associated transfer π. De ne −rt x , the rm faces the maximization problem E The g(t, x) = e r−µ −c(t) V (τ) = [g(τ, Xτ )] . generator Lg is given by x µx (Lg)(t, x) = −c0(t) − re−rt + e−rt = −c0(t) − e−rtx . r − µ r − µ Clearly (Lg)(t, x) is decreasing in x and thus the dynamic single-crossing condition is satis- ed. The transfer is given by
T T E ˜ t,b(t) E 0 −rs ˜ t,b(t) π(t) = (Lg)(s, Xs )ds = −c (s) − e Xs ds ˆt ˆt
= (c(T ) − c(t)) − φb(t, b(t)) , 14 h i where E T −rs ˜ t,x is the expectation of the re ected process ˜.The function φb(t, x) = t e Xs ds X φ is the unique solution´ to the partial dierential equation
φx(t, x) = 0 for all x = b(t) x2σ2 φ (t, x) + φ (t, x)µx + φ (t, x) = e−rtx for all x < b(t) . t x xx 2 This allows one to easily calculate φ numerically.
4.2. Constant Processes and the Relation to Static Mechanism Design. In this sec- tion we show that for the special case of constant processes our setting reduces to the classical static mechanism design setting analyzed among others by Mirrlees (1971) and Guesnerie and Laont (1984). We will explicitly construct the re ected version X˜ of the process X. In this context the set of cut-o rules is equivalent to the set of monotone allocations and thus we recover the famous result that every monotone allocation is implementable. Consider the case without ow transfers h ≡ 0. Let the signal X be constant after time zero, i.e. Xt = X0 for all t ∈ [0,T ]. As the signal is constant after time zero the private information is one dimensional. Consequently the outcome of any (deterministic) stopping rule τ depends only on the initial signal X0. Henceforth we de ne τˆ : R → [0,T ] as a function
of the initial signal τˆ(x) = E [τ|X0 = x]. As the outcome τˆ(X0) depends only on the initial
signal X0 we can de ne a transfer πˆ : R → R that only conditions on X0 by πˆ(x) = π(τ(x)) . Implementability reduces to the following well known de nition of static Implementability (Guesnerie and Laont (1984), De nition 1)
De nition 12 (Static Implementability). An allocation rule τ : R → [0,T ] is implementable if there exists a transfer πˆ : R → R such that for all x, x0 ∈ X
g(ˆτ(x), x) +π ˆ(x) ≥ g(ˆτ(x0), x) +π ˆ(x0) .
As famously shown by Mirrlees (1971) and Guesnerie and Laont (1984) any monotone decreasing allocation rule is implementable (Guesnerie and Laont (1984), Theorem 2).6 Furthermore the transfers implementing τˆ are of the from x ∂ πˆ(x) = g(ˆτ(z), z)dz − g(ˆτ(x), x) + c , ˆx ∂x with some constant c ∈ R. In the following paragraph we show how our dynamic single-crossing condition, cut-o rules and the transfer simplify to the results well-known from static mechanism design. The
6Both Mirrlees and Guesnerie & Laont analyze a more general setting as they do not assume that the agent is risk-neutral or put dierently that his utility is linear in the transfer. 15 generator Lg equals 1 (Lg)(t, x) = lim E [g(t + ∆,Xt+∆) − g(t, x) | Xt = x] ∆&0 ∆ g(t + ∆, x) − g(t, x) ∂ = lim = g(t, x) . ∆&0 ∆ ∂t Thus the dynamic single-crossing condition (1) simpli es to ∂ decreasing in for all ∂t g(t, x) x t ∈ [0,T ] and all x ∈ R. As the process X is constant, for every barrier there exists a decreasing barrier that induces the same stopping time. Hence without loss of generality we restrict attention to decreasing barriers b. It is easy to check that for every decreasing barrier b a version of the process X re ected at b is given by
˜ t,x Xs = min{x, b(t)} . t,b(t) From the fact that the re ected version of the process X˜s started on the barrier at time t stays on the barrier at all future times s > t follows that it is deterministic and given by ˜ t,b(t) Xs = b(s). Hence the transfer implementing the stopping time τb equals T T ∂ T ∂ E ˜ t,b(t) E π(t) = (Lg)(s, Xs )ds = g(s, b(s))ds = g(s, b(s))ds . ˆt ˆt ∂t ˆt ∂t Finally we show that the transfer π(τ(x)) equals the transfer πˆ(x) up to a constant.
Proposition 13. Let τ :[x, x] → [0,T ] be measurable and non-increasing then πˆ(x) = π(τ(x)) for all x ∈ [x, x].
Proof. Recall that πˆ is given by x πˆ(x) = ∂xg(ˆτ(z), z)dz − g(ˆτ(x), x) + c , ˆx for some constant c ∈ R. We x x ∈ [x, x] and compute x πˆ(x) − c = ∂xg(ˆτ(z), z) − ∂xg(ˆτ(x), z)dz − g(ˆτ(x), x) ˆx x τ(z) = ∂t∂xg(s, z)dsdz − g(ˆτ(x), x). ˆx ˆτ(x) 16 Then Fubini's Theorem implies
τ(x) b(s) πˆ(x) − c = ∂x∂tg(s, z)dzds − g(τ(x), x) ˆτ(x) ˆx τ(x) = ∂tg(s, b(s)) − ∂tg(s, x)ds − g(τ(x), x) ˆτ(x) τ(x) = ∂tg(s, b(s))ds − g(τ(x), x). ˆτ(x) Choosing T yields c = τ(x) ∂tg(s, b(s))ds + g(τ(x), x) πˆ(x) = π(τ(x)). ´ 4.3. Monotone Increasing Processes and Decreasing Cut-O. In this section we restrict attention to increasing processes and stopping times with decreasing cut-o b. It is easy to check that for every decreasing barrier b a version of the process X re ected at b is given by ˜ t,x Xs = min{x, b(t)} . t,b(t) From the fact that the re ected version of the process X˜s started on the barrier at time t stays on the barrier at all future times s > t follows that it is deterministic and given by ˜ t,b(t) Xs = b(s). Hence the transfer implementing the stopping time τb equals T T T (4.1) E ˜ t,b(t) E π(t) = (Lg)(s, Xs )ds = (Lg)(s, b(s))ds = (Lg)(s, b(s))ds . ˆt ˆt ˆt 4.4. Search with Recall. In the next section we apply our result to (job) search with
recall. Oers arrive according to a Poisson process (Nt)t∈[0,T ] with constant rate λ > 0.
Oers (wi)i∈N are identically and independently distributed according to the distribution R R. s,t is the best oer the agent got in the past, where we assume that the agent G : → Xt started with an oer of x ∈ R at time s
s,x Xt = max{wi | i ≤ Nt − 1} ∨ z . We assume that the agent receives a nal payo g(t, x) = e−rtx and a ow payo of zero. The generator of g equals ∞ (Lg)(t, x) = −rx + λ (1 − G(z))dz . ˆx As G(z) ≤ 1 the generator is monotone decreasing in x and thus the single-crossing condition is satis ed. By de nition the process X is monotone increasing and by equation (4.1) the transfer that implements a monotone cut-o b is given by T T ∞ π(t) = (Lg)(s, b(s))ds = −rb(s) + λ (1 − G(z))dzds . ˆt ˆt ˆb(s) 17 4.5. Application to Optimal Stopping. As an immediate consequence of Theorem (5) we establish a link between re ected processes and optimal stopping rules. If a stopping rule τ ? is implemented by a constant transfer, then τ ? is optimal in the stopping problem τ (4.2) sup E h(t, Xt)dt + g(τ, Xτ ) . τ∈T ˆ0 This observation leads to the following purely probabilistic su cient condition for optimal stopping rules.
Corollary 14. Let b : [0,T ] → R be a barrier such that the re ected process satis es T E ˜ t,b(t) ˜ t,b(t) h(s, Xs ) + (Lg)(s, Xs )ds = 0 ˆt for all t ∈ [0,T ]. Then the associated cut-o rule τb solves the optimal stopping problem (4.2).
Conclusion
We have shown under weak assumptions that in optimal stopping settings a stopping rule that is optimal for the principal can be implemented by a simple posted-price mechanism. The new approach we introduced is a forward construction of transfers. The transfer is the expected payo evaluated at the process re ected at the time-dependent cut-o one wants to implement. It is an interesting question for future research whether a similar approach can be used in environments where non cut-o rules are of interest. Two such situations arise naturally. First, with multiple agents, the socially-e cient cut-o depends not only on time but also on the signals of the other agents. We think that a generalization of the method developed in this paper can be used to implement the principal-optimal behavior in this multi-agent settings. As the cut-o depends on the other agents' signals, one needs to de ne processes re ected at stochastic barriers. If one de nes payments as expected payos evaluated at the process re ected at the stochastic cut-o, we conjecture that our implementability result (Theorem 5) holds also for those generalized stochastic cut-os. The payment, however, will depend on the other agents' signals and, thus, the mechanism will require constant communication between the principal and the agents. A second, more challenging, problem is revenue maximization. For special cases the rev- enue maximizing policy depends only on the initial and the current signal. In those cases one might be able to use the method developed here for revenue maximization by oering at time zero a menu of posted-price mechanisms. Those extensions illustrate that posted-price transfers without communication are often insu cient in more general dynamic setups. We hope, however, that due to its simplicity 18 and the explicit characterization of transfers, our result will prove useful to introduce private information into the wide range of economic situations that are modeled as single-agent optimal stopping problems.
Appendix
Appendix A. We impose the following assumptions on the coe cients in (2.1). We assume thatµ, σ and γ are Lipschitz continuous, i.e. there exist positive constants L and L(z) such that
|µ(t, x) − µ(t, y)| + |σ(t, x) − σ(t, y)| ≤ L|x − y| |γ(t, x, z) − γ(t, y, z)| ≤ L(z)|x − y| for all t ∈ [0,T ] and x, y ∈ R. Moreover, we assume that these functions are of linear growth, i.e. there exist positive constants K and K(z) such that
|µ(t, x)| + |σ(t, x)| ≤ K(1 + |x|) |γ(t, x, z)| ≤ K(z)(1 + |x|) for all t ∈ [0,T ] and x ∈ R. Furthermore the constants L(z) and K(z) satisfy
K(z)p + L(z)pν(dz) < ∞ ˆR for all p ≥ 2.
Proof of Lemma 6. Let us rst show monotonicity of x 7→ wπ(t, x). By Dynkin's formula the continuation value satis es τ π E t,x t,x w (t, x) = sup h(s, Xs ) + Lg(s, Xs )ds + π(τ) − π(t) τ∈Tt,T ˆt for all (t, x) ∈ [0,T ] × R. The dynamic single-crossing condition and Equation (2.2) imply that the ow payos given that X started in a higher level x0 ≥ x are smaller than the ow payos if it started in x
t,x0 t,x0 t,x t,x h(s, Xs ) + Lg(s, Xs ) ≤ h(s, Xs ) + Lg(s, Xs ) for all s ≥ t. Hence, we have wπ(t, x0) ≤ wπ(t, x). For the Lipschitz continuity x t ∈ [0,T ] and x, y ∈ R. We have
|wπ(t, x) − wπ(t, y)| τ E t,x t,y t,x t,y ≤ sup h(s, Xs ) − h(s, Xs ) ds + g(τ, Xτ ) − g(τ, Xτ ) + |g(t, x) − g(t, y)| . τ∈Tt,T ˆt 19 Let C denote the Lipschitz constants of h and g, i.e.
|h(t, x) − h(t, y)| + |g(t, x) − g(t, y)| ≤ C|x − y| Then we have " # π π E t,x t,y |w (t, x) − w (t, y)| ≤ 2C sup Xs − Xs + |x − y| . s∈[t,T ]
By Theorem 3.2 in Kunita there exists a constant ˜ such that E t,x t,y C sups∈[t,T ] |Xs − Xs | ≤ ˜ C|x − y|, which yields the claim.
Appendix B.
Proof of Proposition 4. For case b) it is straightforward to verify that ˜ τ,η τ,η Xs = min{Xs , b(s)} satis es all the conditions from the de nition of a re ected process. For case a) we show that the solution to a re ected stochastic dierential equation (RSDE) is a re ected version of X. A solution to a RSDE with initial condition (t, η) is a pair of processes (X,˜ l) which satis es the equation s s ˜ ˜ ˜ Xs = η + µ(r, Xr)dr + σ(r, Xr−)dWr − ls , ˆt ˆt ˜ ˜ Moreover, X is constrained to stay below the barrier Xs ≤ b(s) and the process l is non- decreasing and minimal in the sense that it only increases on the barrier, i.e. T t b(s) − ˜ ˜ Xsdls = 0. Existence of (X, l) follows from Slominski and Wojciechowski (2010;´ Theorem 3.4). Conditions (3.1), (3.1) and (3.1) are satis ed by de nition. Furthermore pathwise uniqueness holds, which implies condition (3.1). Note that the solution to the unre ected
SDE (2.1) solves the RSDE for s < τb. As the solution to the RSDE is unique (3.1) follows. ˜ i i ˜ 0,ξi 0,ξi It remains to verify conditions (3.1) and (3.1). For ξ1, ξ2 ∈ R we write (X , l ) = (X , l ), ( ) and introduce the process ˜ 1 ˜ 2. Applying the Meyer-Itô formula Protter i = 1, 2 ∆s = Xs − Xs (2005; Theorem 71, Chapter 4) to the function x 7→ max(0, x)2 yields s 2 2 c max(0, ∆s) = max(0, ∆0) + 2 1{∆r>0}∆rd∆r ˆ0 s X + 1 dh∆ic + max(0, ∆ )2 − max(0, ∆ )2, ˆ {∆r>0} r r r− 0 0
For t > 0 performing the same computations with the expectation taken conditional to Ft 2 yields for ξ1, ξ2 ∈ L (Ft) E ˜ 1 ˜ 2 2 2 [max(0, Xs − Xs ) |Ft] ≤ C max(0, ξ1 − ξ2) for all s ∈ [t, T ]. If ξ1 ≤ ξ2 this directly yields (3.1). Claim (3.1) follows by the same argument.
21 References
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