OPTIMAL DYNAMIC MECHANISM DESIGN WITH DEADLINES

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1 OPTIMAL DYNAMIC MECHANISM DESIGN WITH DEADLINES KONRAD MIERENDORFF Abstract. A seller maximizes revenue from selling an object in a ynamic environment, with buyers that iffer in their patience: Each buyer has a privately known ealine for buying an a privately known valuation. First, we erive the optimal mechanism, neglecting the incentive constraint for the ealine. The ealine of the winner etermines the time of the allocation an therefore also the amount of information available to the seller when he ecies to whether to allocate to a buyer. Depening on the shape of the markup that the seller uses, this can lea to a violation of the neglecte incentive constraint. We give sufficient conitions on the type istribution uner which the neglecte constraint is fulfille or violate. Secon, for the case that the constraint cannot be neglecte, we consier a moel with two perios an two buyers. Here, the optimal mechanism is implemente by a fixe price in perio one an an asymmetric auction in perio two. The asymmetry, which is introuce to prevent the patient type of the first buyer from buying in perio one leas to pooling of ealines at the top of the type space. Keywors: Dynamic Mechanism Design, Multiimensional Signals, Revenue Maximization, Dealines JEL-Coes: D44, D82 1. Introuction In many situations, sellers face a changing population of heterogeneous buyers. Buyers arrive at ifferent points in time. Some are impatient an want to buy immeiately, others are patient an willing to wait. Patient buyers can act strategically an use their flexibility with respect to the time of a purchase in orer to get better prices. Typical examples are online auctions, the sale of flight tickets, hotel reservations, or the sale of houses. To capture heterogeneity in the egree of patience, we assume that buyers have iiosyncratic ealines. A ealine can be viewe as an extreme form of time preferences, as in the case of a traveler who nees to buy tickets before a certain ate in orer to coorinate with other travel arrangements. Dealines may also be impose by thir parties. Consier for example a company that nees to buy an input from a seller in orer to Date: July 14, 215 (first raft June 22, 29). I greatly benefite from many conversations with Benny Molovanu an Philippe Jehiel. I wish to thank the eitor Alessanro Pavan, an associate eitor an two anonymous referees for suggestions that greatly improve the paper an Yeon-Koo Che, Olivier Compte, Jacob Goeree, Paul Heihues, Alexey Kushnir, Luke Linsay, Susanne Ohlenorf, Dezsö Szalay, Alexaner Westkamp, an seminar participants at Cologne, Bonn, Mannheim, Zürich, Paris, CORE, the Hausorff Center Workshop Information an Dynamic Mechanism Design, ESEM 29, Yonsei, EARIE 211 an the Arne Rye Lectures on Dynamic Mechanism Design for helpful comments an iscussions. I starte this project while visiting Paris School of Economics, which I woul like to thank for its hospitality. Financial support of the Bonn Grauate School of Economics, the German Acaemic Exchange Service an the European Research Council (ERC Avance Investigator Grant, ESEI ) is gratefully acknowlege. Contact: Department of Economics, Chair of Organizational Design, University of Zürich, Blümlisalpstrasse 1, CH-86 Zürich, konra.mierenorff@econ.uzh.ch, phone: +41 ()

2 2 KONRAD MIERENDORFF enter a contractual relationship with a thir party. This input coul be a physical object, an option contract, a license, a patent, etc. It is conceivable that the thir party sets a ealine, after which the contractual relationship is no longer available. Therefore, the input is worthless for the company if it is purchase after the ealine. This paper analyzes the implications of private information about patience (ealines) on the revenue maximizing mechanism. We consier the allocation of a single object over a finite time horizon with ranomly arriving buyers an inepenent private values. To focus on the effects of private information about time preferences, we assume that arrival times are observable for the seller. 1 Consumption is assume to take place at the en of the time horizon (e.g., when the plane takes off an not when the ticket is sol), so that we can ignore iscounting as long as all iscount rates are ientical. If we relax the incentive constraint for the ealine, revenue maximization is equivalent to maximizing expecte virtual surplus, subject to monotonicity with respect to the valuation. Virtual surplus maximization is a straightforwar ynamic programming problem which yiels the relaxe solution. Since it is costless to elay an allocation an waiting for more buyers to arrive can improve revenue, the object is allocate only at the ealine of the winner. We show that the relaxe solution can be implemente by a payment rule that applies a markup to the critical virtual valuation of the winning buyer. 2 The latter is the lowest virtual valuation with which a buyer can win, given the types of all competing buyers an the seller s expectation of future arrivals. It can be interprete as the virtual opportunity cost of awaring to the winning buyer. The first contribution of the paper is a regularity conition uner which the relaxe solution satisfies the neglecte incentive constraint for the ealine. 3 We show a martingale property: Increasing the ealine leas to a mean-preserving sprea of the virtual opportunity. Intuitively, a later ealine allows the seller to gather more information before allocating to a buyer, which increases the ispersion of the virtual opportunity cost. The martingale property is important, because for most istributions, the seller s markup is non-linear which inuces enogenous risk-preferences in the buyers. If the markup is convex, buyers become enogenously risk-averse with respect to the seller s virtual opportunity cost. If it is concave, buyers become enogenously risk-loving. Moreover, buyers can choose the ispersion of the virtual opportunity cost by choosing the reporte ealine. Therefore, the shape of the markup etermines whether a buyer prefers to report an earlier ealine (low ispersion) or a later ealine (high ispersion). A concave markup inuces a buyer to report the latest possible ealine. He will thus report his ealine truthfully, because over-reporting woul lea to an allocation after the true ealine. Formulate in terms of the istribution function, the regularity conition requires convexity of the inverse hazar rate of the valuation. Conversely, the relaxe solution is not implementable if the inverse hazar rate is concave. 1 The case of unobservable arrivals is consiere in Pai an Vohra (213). 2 Payoff equivalence implies that our results apply for any implementation of the relaxe solution. 3 For simplicity we consier the case that the valuation an ealine of a buyer are inepenent. Section 3.4 iscusses the correlate case.

3 OPTIMAL DYNAMIC MECHANISM DESIGN WITH DEADLINES 3 Unfortunately, examples for both the concave an the convex case aboun an we cannot conclue that for most commonly use istributions, the moel is regular. The secon contribution of the paper is to erive an optimal mechanism in the irregular case. We consier a special case with two buyers who arrive in two ifferent perios. The mechanism esign problem becomes multi-imensional, but the fact that the secon imension is a ealine provies some structure. The ealine enters the buyers preferences only as a constraint. As long as a buyer gets the object before his ealine, his utility is unaffecte by the ealine. This allows to formulate the seller s problem as virtual surplus maximization subject to the constraint that the utility of a buyer is non-ecreasing in the ealine. Formally, this constraint is equivalent to a type-epenent participation constraint: Patient/strategic buyers have the outsie option to buy before their ealines. This paper is the first so solve an optimal auction problem with a type-epenent participation constraint. The optimal control techniques employe here an the solution resemble Jullien (2), who stuies a principal-agent problem with a type-epenent (exogenous) participation constraint, but there are several notable ifferences. First, an auction moel involves an aitional feasibility constraint for the allocation rule. This is incorporate in the control problem using a characterization of asymmetric reuce form auctions (Mierenorff, 211). Secon, in an auction moel, the optimal allocation rule may be iscontinuous (at the reserve price). Thir, we explicitly show how ironing is use to hanle bunching in the valuation imension because the stanar assumption of a monotone virtual valuation oes not imply monotonicity in the presence of type-epenent participation constraints. Forth, in contrast to Jullien (2), the type epenent participation constraint is etermine enogenously by the mechanism because the outsie option of the patient buyer is given by the utility of an impatient buyer. We show that in the 2x2 moel, uner a mil regularity conition, the enogenous participation constraint only bins for the highest type, an then solve the moel using two separate mechanism esign problems. The technical reasons for the restriction to the 2x2 moel, as well as possible generalizations an applications in other settings are iscusse in Section 5. In the optimal mechanism, the seller posts a fixe price in the first perio. If buyer one oes not accept, the seller waits for the secon perio an conucts an auction. If buyer one is patient, both buyers will participate in the auction. The seller has two instruments to prevent the patient/strategic type of buyer one from choosing the fixe price. He can increase the fixe price, an he can istort the auction format in the secon perio in favor of buyer one. Both instruments increase the expecte payoff from the auction compare to the fixe price an thereby reuce the incentive for the patient/strategic type to eviate. We show that it is optimal for the seller to use both instruments. The optimal allocation rule is thus biase against impatient buyers an rewars patient buyers with a higher chance of winning. The most striking implication is that for high valuations, the mechanism no longer separates buyers with ifferent ealines: Buyers with low valuations have to wait until their ealine before they make a purchase. Buyers with higher valuations, on the other han, o not benefit from waiting an may buy earlier.

4 4 KONRAD MIERENDORFF 1.1. Relate Literature. The literature on ynamic revenue maximization can be broaly ivie into two strans. On the one han, there are moels where all buyers are impatient an therefore non-strategic with respect to the purchase time. 4 This is a stanar assumption in the classic revenue management literature. 5 On the other han, there are moels in which all buyers are assume to be patient an strategic. Most of this literature aopts a generalization of the stanar framework with one-imensional private information. (See Boar an Skrzypacz (215), for a moel with persistent types an Pavan et al. (214), for a general moel with arrival of new information in each perio.) In these moels, a buyer can influence the timing of the allocation by mimicking buyers with ifferent valuations. Moving to the multi-imensional moel allows us to stuy the timing ecision of a buyer inepenently of the incentives to reveal the valuation. We show that the incentives to reveal the ealine are influence by the markup, which reflects the seller s esire to reuce information rents arising from the private valuation. This interaction rives our new insights about the incentives to reveal time preferences an the consequences for revenue-maximization, which are obscure in a one-imensional moel. The moel analyze in this paper has also been stuie by Pai an Vohra (213), who allow for private information about the arrival time an o not make restrictions on the number of perios or objects. 6 Pai an Vohra focus on sufficient conitions for incentive compatibility of the relaxe solution. They observe that monotonicity of the winning probability of a buyer, in the ealine an the arrival time, is sufficient for incentive compatibility of the relaxe solution. For the arrival time, they show that monotonicity is guarantee if the conitional hazar rate of the valuation is monotone in the arrival time. For the ealine, however, no sufficient conition for monotonicity of the allocation rule is available. In Appenix C.5 in the Supplemental Material, we iscuss this further an show why monotonicity generally fails unless the support of the valuation istribution shifts with the ealine. In contrast to Pai an Vohra, we o not try to erive a sufficient conition via monotonicity of the winning probability. Instea, we show that the ealine of a buyer affects the amount of information available to the seller when he has to ecie whether the buyer is aware or not. The enogenous risk preferences in the buyers, arising from the seller s esire to reuce information rents, leas to our sufficient conition in terms of the shape of the hazar rate of the valuation. This paper is also relate to a literature on static mechanism esign with two-imensional private information, in which the secon imension is for example a buget constraint, a minimal capacity, or a quality constraint. 7 In these moels, the secon imension of 4 See for example Das Varma an Vettas (21); Vulcano et al. (22); Gershkov an Molovanu (29a); Dizar et al. (211). 5 See Elmaghraby an Keskinocak (23) for a survey. McAfee an te Vele (27) survey airline pricing. Su (27) stuies a moel with patient buyers. 6 Our sufficient conition also hols with many objects an two time perios. 7 See Beaury et al. (29) for an analysis of optimal taxation; Blackorby an Szalay (28) an Szalay (29) for regulation; Iyengar an Kumar (28) an Dizar et al. (211) for auction moels with capacitate biers; Che an Gale (2), Malakhov an Vohra (25) an Pai an Vohra (214) for moels with buget constraine buyers.

5 OPTIMAL DYNAMIC MECHANISM DESIGN WITH DEADLINES 5 private information has a special structure similar to the ealine in our moel, which makes the analysis tractable. 8 With the exception of Szalay (29), who stuies a singleagent moel, this literature typically makes assumptions on the correlation between the two imensions which guarantee that the relaxe solution is incentive compatible. We a to this literature by eriving the optimal mechanism in the irregular case. This paper iffers from the literature on ynamic screening in which buyers learn about their valuations over time (Courty an Li, 2; Eso an Szentes, 27; Pavan et al., 214). This literature stuies the case where agents receive (one-imensional) private information in each perio but types evolve over time. 9 Instea, we consier a moel with persistent types where private information is two-imensional, which allows for richer time preferences. Several papers have consiere efficient ynamic mechanism esign (see Parkes an Singh, 23; Bergemann an Välimäki, 21; Athey an Segal, 213). The incentive problems analyze in the present paper isappear for value-maximization because the seller oes not use a markup an the relaxe solution is always incentive compatible. Organization of the Paper. Section 2 escribes the moel an formulates the seller s problem. Section 3 presents the relaxe solution an conitions for incentive compatibility, formal proofs are in Appenix A. Section 4 presents the general solution for the irregular case. The formal erivation is in Appenix B. Section 5 conclues. Appenix C in the Supplemental Material in contains some omitte proofs, iscussions, an an extension of Section 3 to multiple objects. 2. The Moel Consier a seller who wants to maximize the expecte revenue from selling an inivisible object within T < time perios. In the case of an airline ticket, T is the time when the plane takes off. The seller s valuation (or prouction cost) is normalize to zero. In each perio, a ranom number of buyers N t arrives. The set of buyers who arrive in perio t is enote I t. I t := t τ=1 I τ an N t := I t escribe the arrivals until perio t. A buyer i I t is characterize by his arrival time a i = t, his valuation v i [, v], where v >, an his ealine i {t,..., T }. The object cannot be sol to a buyer before his arrival time. Utility is quasi-linear. If buyer i has to make a total payment of y i, then his total payoff is v i y i if he gets the object in perios a i,..., i, an y i otherwise. 1 8 The moels of Rochet an Choné (1998), Jehiel et al. (1999), an others, in which all imensions are symmetric, rarely have explicit solutions (see Armstrong, 1996, for an exception). 9 See also Battaglini (25), Nocke an Peitz (27), Möller an Watanabe (21), Deb (211) an references therein. For surveys of the literature on ynamic mechanism esign see Bergemann an Sai (211), an Gershkov an Molovanu (212a). Gershkov an Molovanu (29b, 212b, 21) have stuie ynamic mechanism esign problems in which the seller learns about future buyers type istributions from current buyers types. We abstract from learning by the seller an assume that types are uncorrelate. 1 Note that the ealine is part of the preferences of a buyer. An alternative interpretation woul be that the ealine is the time when the buyer will exit the market. This interpretation restricts the class of mechanism that the seller can use, because payments woul have to take place before the ealine of a buyer. It turns out, however, that the optimal mechanism oes not require payments after the ealine. Therefore, both interpretations of the ealine lea to the same results.

6 6 KONRAD MIERENDORFF Buyers are risk-neutral an maximize expecte payoff. Neither the buyers nor the seller iscount future payoffs. 11 The numbers of arrivals in ifferent perios an the types of ifferent buyers are inepenently istribute. Moreover, to focus on the novel insights that arise ue to the ynamic structure of the moel, we assume that conitional on the arrival time, the ealine an the valuation of a buyer are inepenent. 12 ν t (N t ) enotes the probability that N t buyers arrive in perio t. To exclue uninteresting cases, we assume that in each perio, there is a positive probability of new arrivals ( t : ν t () < 1). For a given arrival time a, the probability that the ealine of a buyer equals is enote ρ a (). The valuation has istribution function F a (v) an ensity f a (v). 13 Information realizes over time. In perio t, the numbers of future buyers N t+1,..., N T, an their types are not known to anyboy. In particular, the ecision to sell a unit in perio t cannot be base on this information. Upon arrival, each buyer privately observes his valuation an his ealine. In orer to focus on the incentive issues of private information about ealines, we assume that the seller observes arrivals. 14 ν t ( ), ρ a ( ) an F a ( ) are commonly known. Finally, we assume that for all a, f a (v) is continuous in v an strictly positive for all v [, v], continuously ifferentiable in v for v (, v), an that f 1 (.) can be extene continuously to [, v]. To avoi aitional technicalities, the following stanar regularity conition is impose throughout the paper. Assumption 1. For all a {1,... T }, the virtual valuation J a (v) := v 1 Fa(v) f a(v) increasing in v. is strictly For some results in Section 4, we will assume that the monopoly profit from selling to a single buyer in the first perio is concave: Assumption 2. v(1 F 1 (v)) is concave. Assumption 2 is equivalent to the assumption that J 1 (v)f 1 (v) is increasing Allocation an Payment Rules. In orer to escribe allocation an payment rules in this ynamic setting we efine a state as s t = (H t, k t ). H t = (a i, v i, i ) i I t enotes the history of buyer types that have arrive so far, an k t inicates whether the object is still available (k t = 1), or has alreay been allocate (k t = ). The history of buyer types excluing i is enote Ht i. 11 If only payments are iscounte an all agents have a common iscount factor, the results o not change. The assumption that the value of consumption is not iscounte is natural in the settings iscusse in the introuction were consumption takes place at time T, inepenent of the time of the allocation (e.g. of a ticket or a reservation). In other applications, iscounting may be more natural. 12 In Section 3.4, we will iscuss the consequences of correlations between the ealine an the valuation. 13 To simplify notation, we assume that buyers with the same arrival perio are ex-ante ientical. All results carry over to the case of ex-ante asymmetric biers. 14 See Pai an Vohra (213) for a iscussion of private information about arrival times.

7 OPTIMAL DYNAMIC MECHANISM DESIGN WITH DEADLINES 7 An allocation rule efines a winning probability x i (s t ) [, 1] for each state s t, an for each buyer i I t. An allocation rule must satisfy the feasibility constraint t, s t : x i (s t ) k t. (F) i I t The probability that the object is not allocate in state s t is enote by x (s t ) = k t i I t x i (s t ). An allocation rule allocates only at the ealine if x i (s t ) = for all t, s t an all i I t with i t. A payment rule efines a payment y i (s t ) for each buyer i I t in every state s t. 15 It will be without loss for the seller s maximal profit to consier only symmetric allocation an payment rules, i.e., rules that o not iscriminate between buyers with the same arrival times Mechanisms. The seller s goal is to esign a mechanism that has a Bayes-Nashequilibrium which maximizes expecte revenue. In general, a mechanism can be any game form with T stages, such that only buyers from I t are active in stage t. We assume that the mechanism esigner has full commitment power an can choose to conceal any information about the first t stages from the buyers that arrive in stages t + 1,..., T. 17 By the revelation principle, the seller can restrict attention to incentive compatible an iniviually rational irect mechanisms, in which no information is reveale. 18 Furthermore, we impose symmetry as iscusse above. Definition 1. A irect mechanism consists of message spaces S 1 = [, v] {1,..., T },..., S T = [, v] {T }, an symmetric allocation an payment rules (x, y). For a given irect mechanism (x, y), consier a buyer i I a who reports (v, ) S a. If all other buyers (past, current an future) report their types truthfully, the interim winning probability that is, the probability that this buyer wins in perio t a, is given by q t a(v, ) := E [x i (s t ) (a i, v i, i ) = (a, v, )]. The interim expecte payment is given by [ T ] p a (v, ) := E y i (s τ, k τ+1 ) (a i, v i, i ) = (a, v, ). τ=a 15 Here, we implicitly assume that the payment in perio t only epens on st. Since we will consier Bayesian incentive compatibility an interim participation constraints in what follows, this is without loss of generality. 16 Formally, an allocation rule (payment rule) is calle symmetric if for all t, all states s t, an all i, j I t such that a i = a j: x i(s t) = x j(σ i,j(h t), k t) (y i(s t) = y j(σ i,j(h t), k t)), where σ i,j enotes the permutation that interchanges the i th an the j th element of its argument. 17 This assumption yiels an upper boun on the revenue that can be achieve. We will see that this boun can also be achieve in a perioic ex-post equilibrium, i.e., if buyers observe all information from past an current stages. 18 The stanar revelation principle hols because the seller observes arrival times. Note that without this assumption, each buyer coul mimic all types with an arrival time greater or equal than his own arrival time. Therefore, the neste range conition is satisfie an the revelation principle hols (see Green an Laffont, 1986; Bull an Watson, 27).

8 8 KONRAD MIERENDORFF Note that we aggregate payments from ifferent perios in this efinition. 19 (q, p) is calle the reuce form of (x, y). The interim expecte utility from participating in a mechanism (x, y), for a buyer with true type (v, ) who reports (v, ), is given by [ ] U a (v,, v, ) := qa(v τ, ) v p a (v, ). (2.1) The expecte utility from truth-telling is abbreviate U a (v, ) := U a (v,, v, ). τ=a Definition 2. (i) A irect mechanism (x, y) is (Bayesian) incentive compatible if for all a {1,..., T }, v, v [, v], an, {a,..., T }, U a (v, ) U a (v,, v, ). (IC) (ii) A irect mechanism (x, y) is iniviually rational if for all 1 a T, an all v [, v], U a (v, ). (IR) 2.3. Characterization of Incentive Compatibility. In this section we erive a basic characterization of incentive compatibility. 2 We restrict attention to mechanisms that allocate only at the ealine. The following Lemma shows that in the absence iscounting, this is without loss. Lemma 1. Let (x, y) be a irect mechanism that satisfies (IC) an (IR). Then, there exists an alternative mechanism (ˆx, ŷ) such that ˆx allocates only at the ealine, (ˆx, ŷ) satisfies (IC) an (IR), an (x, y) an (ˆx, ŷ) yiel the same expecte revenue. Proof. The proof can be foun in Appenix C.1 in the Supplemental Material The iea behin the Lemma is that elaying an allocation until the reporte ealine of a buyer oes not change his payoff if he reports the true or an earlier ealine. At the same time, a buyer who reporte a later ealine than his true ealine may be worse off, if his allocation is elaye to the reporte ealine. Therefore, a moification of a given mechanism that moves all allocations to the reporte ealines of the respective buyers relaxes incentive constraints. Hence, the seller oes not loose revenue by allocating only at the ealine. In light of Lemma 1, we will only consier mechanisms that only allocate at the ealine an write q a (v, ) instea of q a(v, ) in what follows. The following characterization of incentive compatibility buils on Myerson s characterization for one-imensional private information. In aition, we have to ensure that buyers o not have an incentive to misreport the ealine. If a mechanism only allocates at the ealine, we only have to rule out ownwar eviations. The characterization requires that the expecte utility from participating with a truthful report is weakly increasing in the ealine. For the lowest type v =, we have to strengthen this conition an require 19 Explicit expressions for q t a an p a can be foun in Appenix C.3 in the Supplemental Material. 2 Similar characterizations have been use in the previous literature on two-imensional incentive problems. See for example Pai an Vohra (213).

9 OPTIMAL DYNAMIC MECHANISM DESIGN WITH DEADLINES 9 that the utility is inepenent of the ealine. The following theorem shows that this one-imensional constraint, together with Myerson s characterization is sufficient to rule out simultaneous misreports of the ealine an the valuation. Theorem 1. Let (x, y) be a irect mechanism that allocates only at the ealine, with reuce form (q, p). Then (x, y) is incentive compatible if an only if for all 1 a T, an all v, v [, v] : v > v q a (v, ) q a (v, ), U a (v, ) = U a (, ) + ˆ v q a (s, )s, (M) (PE) U a (v, ) U a (v, + 1), if < T, (ICD ) an U a (, ) = U a (, + 1), if < T. (ICD u ) Proof. The proof can be foun in Appenix C.1 in the Supplemental Material Sufficiency is implie by the special structure of preferences. The ealine is a constraint that oes not affect a buyer s payoff as long as the object is aware before the ealine. Therefore, a buyer who reports < i enjoys the same expecte payoff as a buyer whose true ealine is an who reports his ealine truthfully. For such a buyer, however, (M) an (PE) ensure that a truthful report of the valuation is optimal. Therefore, buyer i oes not have an incentive to misreport both imensions simultaneously. Formally, the ownwar incentive constraint for the ealine resembles an enogenous, type-epenent participation constraint. A patient/strategic buyer with arrival time a an ealine > a has the outsie option to report <. He only participates voluntarily with a truthful report, if his payoff with = excees the payoff of his best outsie option The Seller s Problem. By the revelation principle an Lemma 1, the seller s problem is to choose an incentive compatible an iniviually rational irect mechanism that allocates only at the ealine, to maximize T {E [N a ] E [p a (v, )]}. a=1 Using (2.1) an (PE) to substitute the payment rule, integrating by parts an setting U a (, ) = for all 1 a T, the objective of the seller can be rearrange to { } T T E [N a ] ρ a () E [q a (v, )J a (v)]. a=1 =a Next, we substitute q a (v, ), an bring the seller s maximization problem into a recursive form. The resulting ynamic program is enote R: V T (s T ) := max x i (s T )J ai (v i ), (R) x(s T ) i I T : i =T

10 1 KONRAD MIERENDORFF t < T : V t (s t ) := max x(s t) i I t : i =t x i (s t )J ai (v i ) + x (s t )E st+1 [V t+1 (s t+1 ) s t, k t+1 = 1], where the reuce form of the optimal policy x must satisfy (M), (ICD ), an (PE) where we set U a (, ). As is common in one-imensional auction problems, the seller chooses a policy that maximizes the expecte virtual valuation of the winning buyer. 3. The Relaxe Solution Once the seller s objective function has been transforme into virtual value form as in (R), the stanar approach to maximize revenue is to relax all remaining constraints except for participation an feasibility constraints. The solution to this problem is the relaxe solution x rlx. Assumption 1 ensures that x rlx satisfies the monotonicity constraint (M). Therefore x rlx is an optimal allocation rule if the ealines of all buyers are public. In this section we first efine a payment rule y rlx, that implements the allocation rule x rlx uner the assumption that ealines are public, but valuations are private. The efinition uses critical virtual valuations. Next, we erive the martingale property, which leas to Lemma 2. Finally we will show how this property can be use to erive sufficient conitions for incentive compatibility of the relaxe solution with respect to the ealine, which leas to our first main result in Theorem 3. Throughout this section will use the properties of one particular payment rule y rlx. Note, however, that payoff equivalence implies that the results of this exercise, in particular Theorem 3, also apply to all other payment rules that implement x rlx, as long as U a (, ) = U a (, + 1) is satisfie for all a an A Payment Rule that Implements the Relaxe Solution with Public Dealines. We efine a payment rule in which the transfer of a bier is zero if he oes not win the object. The winning bier has to make a payment that is equal to the lowest valuation with which he can win the object for given arrivals an types of competing biers. To efine this payment rule formally, fix a buyer i with arrival time a an suppose that the object is still available in the arrival perio (k a = 1). We efine the critical virtual valuation for all a an all continuations H i J a (H i ) := J a ( inf { v x rlx i of H i a as 21 (( H i, (a i, v i, i ) = (a, v, ) ), k ) = 1 }). (3.1) J a (H i ) is calle the critical virtual valuation because it is the lowest virtual valuation with which buyer i with ealine wins against the other buyers that arrive until perio, for a given history H i. With this notation we can rewrite xrlx as follows: 22 x rlx 1, if k ai = 1, t = i, an J a (v i ) J ai (H i i (s t ) := i ),, otherwise. 21 H i is a continuation of Ha i if Ĩa \ {i} = Ia \ {i} an for all j Ia \ {i}, (ãi, ṽi, i) = (a i, v i, i). 22 For simplicity, we ignore ties. The subsequent analysis is vali for any eterministic tie-breaking rule. With ranom tie-breaking we woul have to conition payments on the realize allocation ecision in aition to the state. This woul only complicate the notation without changing the results.

11 OPTIMAL DYNAMIC MECHANISM DESIGN WITH DEADLINES 11 While rewriting x rlx in this way is tautological, (3.1) serves well to efine a simple payment rule y rlx : ( ) yi rlx Ja 1 J ai (H i (s t ) := i ), if x rlx i (s t ) = 1, (3.2) otherwise. Theorem 2. If ealines are public information, then (x rlx, y rlx ) is incentive compatible an maximizes the seller s revenue. Moreover, truthfully reporting the valuation is a weakly ominant strategy. Proof. With the payment rule y rlx, the payment of a losing buyer is zero. The winner pays the lowest valuation with which he coul have obtaine the object for a given history of arrivals until perio i. Thus, truth-telling is a weakly ominant strategy if the ealine is public an buyers only report their valuations. (x rlx, y rlx ) is optimal because any mechanism that implements x rlx an satisfies U a (, ) = U a (, + 1) for all a an, yiels the same expecte revenue, which follows from payoff equivalence The Martingale Property. The next step is to ientify conitions uner which the mechanism (x rlx, y rlx ) satisfies the incentive constraint for the ealine (ICD ). The critical virtual valuation can be interprete as the virtual opportunity cost of an allocation to a buyer. This can be easily seen in the following example. The example also illustrates a property that is crucial for the subsequent analysis: the sequence of critical virtual valuations for ifferent ealines ( J ai (H i )) is a martingale. =a i,...,t Example. Let T = 2, an ν 1 (1) = ν 2 (1) = 1, that is, exactly one buyer arrives in each perio. If the first buyer has ealine 1 = 1, then he is aware the object if his virtual valuation is greater than the option value of waiting, which equals E v2 [max{, J 2 (v 2 )}]. Hence, the critical virtual valuation of buyer one for 1 = 1 is J 1 (H 1 1 ) = E v 2 [max{, J 2 (v 2 )}]. If 1 = 2, the object will not be allocate in perio one because there is no buyer who has reache his ealine. In perio two, the virtual opportunity cost of allocating to buyer one is max {, J 2 (v 2 )}. Therefore, the critical virtual valuation of buyer one for 2 = 1 is J 1 (H 1 2 ) = max{, J 2(v 2 )}. Clearly, (J 1 (H1 1 ), J 1(H2 1 )) is a martingale because E [ J 1 (H2 1 ) H1 1 ] = J 1 (H 1 1 ). To erive the martingale property more generally, the basic efinition of the critical virtual valuation by (3.1) is not very useful. A simpler expression can be obtaine by efining a score π i for each buyer, which epens only on the buyer s own type. Mierenorff (213) shows that a score can be efine such that buyer types with ifferent arrival times an ealines become irectly comparable. 23 In particular, in each perio, the allocation 23 That paper consiers value-maximization in a more general allocation problem. Replacing values by virtual values, however, oes not change the result we use here.

12 12 KONRAD MIERENDORFF rule in the relaxe solution will pick the buyer with the highest (positive) score among all buyers that have arrive so far. If the ealine of this buyer is the current perio, the object will be allocate to him. If his ealine is in the future, or if the highest score is negative, the object is not allocate in the current perio. Formally, we will have: } x rlx 1, if k ai = 1, t = i, an π i max {, max j I i π j, i (s t ) = (3.3), otherwise. This structure also implies that if the object is still available in perio t, then the ealine of the buyer with the highest positive score cannot be in the past. If it was in the past, then he woul have gotten the object at his ealine because he ha the highest score in that perio alreay. To efine the score, we introuce an artificial state ŝ (π) for a given perio. In this state, the object is still available (ˆk = 1), an there is a single artificial buyer (Î = {1}). The buyer has arrival time â 1 = 1, virtual valuation J 1 (ˆv 1 ) = π, an his ealine is ˆ 1 = T. 24 If V i (ŝ i ()) J a (v i ), the score of buyer i is efine implicitly by J a (v i ) = V i (ŝ i (π i )). (3.4) If V i (ŝ i ()) > J a (v i ), we set π i = 1. Mierenorff (213) shows that the relaxe solution coincies with the solution efine by (3.3) an (3.4). The expecte revenue in the artificial state ŝ i () is the same as the expecte revenue in a state where no buyer is currently available, because the artificial buyer in ŝ i () has virtual valuation zero. If V i (ŝ i ()) > J a (v i ) the seller woul therefore never allocate to buyer i in terms of virtual surplus, he is better off without buyer i. We assign an arbitrarily chosen negative score ( 1) to a buyer with such a low valuation. In the opposite case the score is non-negative. The score π i is then efine such that i s virtual value is equal to the continuation value of the seller s problem in the artificial state with a single buyer who has a virtual valuation equal to the score π i an ealine T. In other wors, (3.4) implies that in perio i, the seller is inifferent, in terms of virtual surplus, between allocating to buyer i, or switching to the artificial state ŝ i (π i ). We have thus transforme buyer i s valuation into a score that escribes the equivalent (in terms of seller-revenue) virtual valuation of a buyer with ealine T. (3.3) has further implications. Since the buyer with the highest score cannot loose against any buyer who has alreay arrive, V t (H t, 1) only epens on the type of the buyer with the highest score. Moreover, Mierenorff (213) shows that in any state where the object is still available, the seller s continuation value is the same as the continuation value in the artificial state ŝ t ( max {, maxj I t π j }). Formally, for all t an Ht such that x rlx (H t, 1) = 1 for all t < t, 24 We can choose â1 arbitrarily. If â 1 1, then Jâ1 (ˆv 1) = π. It is important, however, that the ealine is T.

13 OPTIMAL DYNAMIC MECHANISM DESIGN WITH DEADLINES 13 ( { })) V t (H t, 1) = V t ŝ t (max, max π j. (3.5) j I t Equippe with these properties of the relaxe solution, we can now stuy the relationship between critical virtual valuations for ifferent ealines. Consier a buyer i who arrives in a perio where the object is still available. If i s ealine is, he wins in perio if { } π i max, max π j. j I Using (3.4), this yiels ( { })) J ai (v i ) = V (ŝ (π i )) V ŝ (max, max π j, j I \{i} because the continuation value in the artificial state is weakly increasing in the artificial buyer s virtual valuation. Therefore, we have ( { })) ( ) J ai H i = V ŝ (max, max π j. (3.6) j I \{i} We can (3.6) for 1 to get J ai (H i 1 ) = V 1 [ = E = E (ŝ 1 ( max {, max j I 1 \{i} π j V (s ) s 1 = ŝ 1 (max ( [V (ŝ max = E [ J ai (H i ) H 1] i. {, max j I \{i} π j })) {, max j I 1 \{i} π j })) H i 1 ] }) ] H i 1 To obtain the secon line, we have use to efinition }) of V 1 an the fact that x rlx = 1 in the artificial state ŝ 1 (max {, max j I 1 \{i} π j, which hols because the ealine of the artificial buyer is T. Note that in the secon line, we take the expectation of V (s ) where s is the state in which the set of buyers is given by the artificial buyer from ŝ 1, joine with the new arrivals from perio. To obtain the thir line we have use (3.5) which applies here because the ealines of all buyers in state s are greater or equal than. To obtain the last line, we have use (3.6) for. To summarize, we have shown that the sequence of ranom variables ( J ai (H i )) is a martingale. The following =a,...,t lemma slightly strengthens this result an shows strict secon-orer stochastic ominance of critical virtual valuations for ifferent ealines It is well known in the literature on ynamic programming, that the stoppe sequence in an optimal stopping problem is a martingale. The martingale property of the critical virtual valuation, however, oes not follow from this result. For a more etaile iscussion of the two properties, see Appenix C.4 in the Supplemental Material.

14 14 KONRAD MIERENDORFF Lemma 2. For all states s a with k a = 1, an all i I a, ( J a (H i (with respect to ( H i ) ): for all {a + 1,..., T }, =a,...,t [ J a (H i ) H i ] 1 = J a (H i 1 ). E H i Furthermore, if J a (Ha i ) < v, then for all, {a,..., T }, < [ J a (H i ) ] [ sa SSD J a (H i ) ] sa, where SSD enotes strict secon-orer stochastic ominance. )) =a,...,t is a martingale Proof. The proof of strict secon-orer stochastic ominance can be foun in Appenix A. The martingale property implies that while the critical virtual valuation becomes more variable if the ealine increases, its expectation is inepenent of the ealine, exactly as in the example. Intuitively, competition by other buyers oes not become more or less intense if the ealine changes. The ealine only etermines which cohorts of buyers are incorporate in the option value an for which cohorts a buyer competes against the realize virtual valuations Incentive Compatibility of the Relaxe Solution. The final step in this section is to use Lemma 2 to erive sufficient conitions uner which x rlx satisfies or violates (ICD ), respectively. The main iea can be illustrate by consiering (ICD ) for a buyer with the highest possible valuation v i = v. For any ealine, we have ( [ ]) U a (v, ) = v E yi rlx (s ) k a = 1 Prob [k a = 1]. A buyer v i = v wins with probability one whenever the object is still available at his arrival time. Therefore, the expecte utilities for ifferent ealines an > can iffer only in the expecte payments. The payment is efine in (3.2) by applying a markup given by J 1 a to the virtual opportunity cost of the seller. If the markup is linear, the buyer is be risk-neutral with respect to the virtual opportunity cost of the seller an hence inifferent between reporting ifferent ealines. If the markup is non-linear, however, the buyer becomes enogenously risk-averse or risk-loving. For example, if J a is weakly convex, then J 1 a is weakly concave. By Jensen s inequality an the law of iterate expectations, the expecte payment is higher for than for because the critical virtual valuation for the earlier ealine is less variable. If J a is concave, we get the opposite observation. The following theorem generalizes the insight to all valuations v [, v]. For the case of v < v, the expecte payoff also epens on the winning probability which requires a ifferent proof which uses secon-orer stochastic ominance. Theorem 3. (i) If J a (v) is weakly convex for all a, then (x rlx, y rlx ) is incentive compatible an maximizes the seller s revenue if both the valuation an the ealine are private information. (ii) If J a (v) is strictly concave for some a, then (x rlx, y rlx ) is not incentive compatible if the ealine an the valuation are private information.

15 OPTIMAL DYNAMIC MECHANISM DESIGN WITH DEADLINES 15 Proof. The proof can be foun in Appenix A. There are two important elements in this payment rule that rive Theorem 3. First, the arrival of new information over time leas to in crease ispersion in the critical virtual valuation. Secon, the seller s esire to minimize information rents for the valuation leas to a markup that is typically non-linear. These two elements, arrival of information, an the esire to extract information rents interact, an istort the incentives to report the ealine truthfully. If the seller maximize value instea of revenue, the markup woul isappear an the relaxe solution woul always be incentive compatible. 26 Moreover, if there was no information revelation over time, but the seller maximize revenue, the relaxe solution woul again be always incentive compatible. 27 Remark 1 (Ex-Post Incentive Compatibility). Note that in Lemma 2, strict secon-orer stochastic ominance is shown conitional on the state in the arrival perio. This implies that the incentive compatibility result of Theorem 3.i also hols if buyers can conition their reports on the state at their arrival time. In other wors, uner the conitions of Theorem 3.i, the relaxe solution is perioic ex-post incentive compatible. 28 This shows that the optimal solution oes not rely on the seller s ability to conceal information. Remark 2 (Convex an Concave Virtual Valuations). In static mechanism esign, the regularity conition that guarantees incentive compatibility of the relaxe solution (Assumption 1) is satisfie by a large class of istributions (see Ewerhart, 213). In the present moel, the picture is less clear. Strict concavity (weak convexity) of the virtual valuation is equivalent to 1 F (v) (f(v)) 2 (f(v)f (v) 2(f (v)) 2 ) ( ) < f (v). This implies that all istributions with an increasing ensity that is not too convex have strictly concave virtual valuations. Conversely, ecreasing ensities that are not too concave imply weak convexity of the virtual valuation. For example, the virtual valuation is concave if the ensity is linear an increasing (f(v) = 1 k + 2kv, k (, 1]), or a power of v (f(v) = (k + 1)v k, k > ), but we also have concavity for hump- an U-shape ensities (f(v) = 3 2 6(v 1 2 )2 an f(v) = 12(v 1 2 )2 ). The virtual valuation is linear or convex if the ensity is linear an ecreasing (f(v) = (k + 1)v k, k ) or a power of 1 v (f(v) = (1 + k)(1 v) k ). These examples show clearly that both the regular an the irregular case are economically relevant. We will therefore stuy the optimal mechanism in the irregular case in Section This also follows from general existence results for efficient ynamic mechanisms (Parkes an Singh, 23; Bergemann an Välimäki, 21). 27 In this (hypothetical) case, the seller has perfect foresight an we are essentially back in the static moel of Myerson (1981). 28 This hybri concept requires ex-post incentive compatibility with respect to all information that is realize in the current perio an Bayesian incentive compatibility for all information that realizes in the future (see Bergemann an Välimäki, 21).

16 16 KONRAD MIERENDORFF 3.4. Generalizations. The martingale property of J a (H i ) oes not epen on the assumption that the valuation an the ealine are conitionally inepenent. Appenix C.2 in the Supplemental Material shows the martingale property for the case of two perios (T = 2), a finite number of units (K > 1) an buyers with unit eman. We can therefore generalize Theorem 3 to allow for correlations an multiple units. 29 For the case that the valuation an the ealine of a buyer are not inepenent, the sufficient conitions in Theorem 3 have to be augmente. If the istribution of the valuation becomes weaker for later ealines (in the hazar-rate orer), then the seller will set prices less aggressively if a buyer has a later ealine. This makes it more likely that the relaxe solution is incentive compatible if the ealine is private. Conversely, if the istribution of valuations becomes stronger for later ealines, the markups use in the relaxe solution increase with the ealine, which makes it less likely that the relaxe solution is incentive compatible. 3 Combining the hazar-rate orer with the previous convexity/concavity conitions, we the following sufficient conitions for incentive compatibility of the relaxe solution. Theorem (3 ). Suppose that K = 1 an T <, or that T = 2 an K <. Suppose that both the valuation an the ealine are private information. Then (x rlx, y rlx ) (i) is incentive compatible an maximizes the seller s revenue if for all a < T (a) J a (v ) J a (v ) for all v [, v], an (b) J a (v ) or J a (v ) is weakly convex as a function of v. (ii) violates (ICD ) if for some a < T (a) J a (v ) J a (v ) for all v [, v], an (b) J a (v ) or J a (v ) is strictly concave as a function of v. 4. The General Solution In this section, we analyze how a bining incentive constraint for the ealine istorts the optimal allocation rule. The most important ifference in comparison to the relaxe solution is that the bining incentive constraint leas to bunching of ealines. In the irregular case, the analysis of the seller s problem is significantly more complex. We obtain an explicit solution for the case of two perios (T = 2) with eterministic arrival of one buyer in each perio. We will assume that the profit of a monopolist who is selling to the first buyer is concave, which is capture by Assumption 2. This assumption ensures that the optimal mechanism oes not use lotteries in the first perio. While Assumption 2 is neee for a complete solution, a main property of the optimal solution, namely that ealines are not separate for high valuations, is robust. 29 I conjecture that Lemma 2 generalizes to the case of many objects (with unit eman) an more than two time perios. This woul imply an immeiate generalization of Theorem 3. The main obstacle for a proof of the conjecture is that the formulation of the optimal allocation rule in terms of a score is not easily generalizable beyon the case of one object. 3 The observation is not new. It can also be foun in the previous literature on static moels with two-imensional private information. See Section 1.1 an Footnote 7.

17 OPTIMAL DYNAMIC MECHANISM DESIGN WITH DEADLINES 17 For the 2x2 moel we simplify notation. The ealine of the first buyer is enote by {1, 2}, an the probability of = 1 by ρ. x 1 (v 1, = 1) enotes the winning probability of buyer if his ealine is one, an x i (v 1,, v 2 ) enotes the conitional winning probability of buyer i in perio two, for a given type profile, conitional on the event that the object has not been allocate in the first perio. Note that an allocation rule x is feasible if an only if for all v 1, v 2 [, v], {1, 2}, an i {1, 2}: x 1 (v 1, 1), x i (v 1,, v 2 ) [, 1] an 31 x 1 (v 1, 2, v 2 ) + x 2 (v 1, 2, v 2 ) 1. (F) We write interim winning probabilities of buyer one as q 1 (v 1, 1) = x 1 (v 1, 1), an q 1 (v 1, 2) = ˆ v x 1 (v 1, 2, v 2 )f 2 (v 2 )v 2. (4.1) The interim winning probability of buyer two, conitional on the ealine of buyer one, an the event that the object has not been allocate in perio one, is given by 32 q 2 (v 2 = 1) := an q 2 (v 2 = 2) := ˆ v ˆ v x 2 (v 1, 1, v 2 ) (1 x 1(v 1, 1))f 1 (v 1 ) v (1 x 1(s, 1))f 1 (s)s v 1, x 2 (v 1, 2, v 2 )f 1 (v 1 )v 1. (4.2) With this notation, the seller s objective can be written as follows: ρπ 1 [q 1 (, 1), q 2 ( 1)] + (1 ρ) π 2 [q 1 (, 2), q 2 ( 2)]. (4.3) The first part of the objective, π 1, is the expecte profit conitional on = 1, which oes not epen on the allocation rule for = 2: ˆ v [ ˆ v ] π 1 [q 1 (, 1), q 2 ( 1)] = q 1 (v 1, 1)J 1 (v 1 ) + (1 q 1 (v 1, 1)) q 2 (v 2 1)J 2 (v 2 )f 2 (v 2 )v 2 f 1 (v 1 )v 1. (4.4) Similarly, π 2, the expecte profit conitional on = 2, oes not epen on the allocation rule for = 1: π 2 [q 1 (, 2), q 2 ( 2)] = ˆ v q 1 (v, 2) J 1 (v) f 1 (v) + q 2 (v 2) J 2 (v) f 2 (v) v. (4.5) The seller maximizes (4.3) subject to the constraint that q is the reuce form of a feasible allocation rule an incentive constraints: For all {1, 2}, an all v, v [, v]: an U 1 (v, 1) = v > v q 1 (v, ) q 1 (v, ), (M 1 ) ˆ v q 1 (s, 1)s ˆ v q 1 (s, 2)s = U 1 (v, 2) (ICD 1 ) Given that virtual valuations are assume to be increasing, the monotonicity constraint on q 2 can be relaxe an is therefore omitte. As before, the seller maximizes virtual 31 We o not have to impose a constraint on the sum x1(v 1, 1) + x 2(v 1, 1, v 2) because x 2(v 1, 1, v 2) is the winning probability of buyer two conitional on the event that the object has not been allocate in the first perio. 32 With these efinitions, we have q2(v 2) = ρ ( v (1 x1(v1, 1))f1(v1)v1 ) q 2(v 2 = 1)+(1 ρ) q 2(v 2 = 2).