Efficient sequential assignment with incomplete information

Games and Economic Behavior 68 (2010) 144–154

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Efficient sequential assignment with incomplete information ✩ Alex Gershkov ∗ , Benny Moldovanu Department of Economics, University of Bonn, Lennestr. 37, 53113 Bonn, Germany

a r t i c l e

i n f o

Article history: Received 7 May 2008 Available online 21 June 2009 JEL classification: C7 D7 D8 Keywords: Dynamic mechanism design Sequential assignment

a b s t r a c t We study the welfare maximizing assignment of several heterogeneous, commonly ranked objects to impatient agents with privately known characteristics who arrive sequentially according to a Poisson or renewal process. There is a deadline after which no more objects can be allocated. We first show that the dynamically efficient allocation, characterized by Albright [Albright, S.C., 1974. Optimal sequential assignments with random arrival times. Manage. Sci. 21 (1), 60–67], is implementable by the dynamic version of VCG mechanism. We then obtain several properties of the welfare maximizing policy using stochastic dominance measures of increased variability and majorization arguments. We also propose redistribution mechanisms that 1) implement the efficient allocation, 2) satisfy individual rationality, 3) never run a budget deficit, 4) may run a budget surplus that vanishes asymptotically. © 2009 Elsevier Inc. All rights reserved.

1. Introduction

We study efficient, individually rational and budget-balanced schemes for the following dynamic mechanism design problem: the designer wants to assign a fixed, finite set of heterogeneous objects to a sequence of randomly arriving agents with privately known characteristics. Monetary transfers are feasible. The objects are substitutes, and each agent derives utility from at most one object. Moreover, all agents share a common ranking over the available objects, and values for objects have a multiplicative structure involving the agents’ types and objects’ qualities. We assume that there is a deadline by which all objects must be sold. In Appendix B we analyze a scenario with an infinite horizon and discounting (in both scenarios, time is a continuous variable). Examples of such settings include the dynamic allocation of limited resources among incoming projects, the allocation of limited research facilities among research units, the assignment of dormitory rooms to potential tenants, and the allocation of available positions to arriving candidates (it is often the case that positions in an organization’s hierarchy cannot be re-assigned without very high penalties). The yield-management literature has analyzed the simpler model of allocating identical objects (e.g., seats on an aeroplane, or hotel rooms) from the point of view of revenue-maximization.1 Our model also shares several common features with the classical job search models.2 The main difference is that in the search litera-

✩ Some of the present results previously appeared in a discussion paper entitled “The Dynamic Assignment of Heterogeneous Objects: A Mechanism Design Approach”. We wish to thank David Parkes, an associated editor and two referees for many helpful remarks. We are grateful for financial support from the German Science Foundation, and from the Max Planck Research Prize. Corresponding author. E-mail addresses: [email protected] (A. Gershkov), [email protected] (B. Moldovanu). 1 See McAfee and te Velde (2008), and Gershkov and Moldovanu (forthcoming-a) for several references to that large literature. 2 For extensive surveys of the search literature see Lippman and McCall (1981), and Mortensen (1986).

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145

ture it is usually assumed that stream of the job offers is generated by a non-strategic player, without private information. Hence, implementation issues do not arise there.3 The assumption of non-strategic arrivals is an important limitation of the model. But, many of the model’s features do appear in practical situations of interest. For example, customers in duty-free or central railway station shops are impatient, “short-lived” and exhibit non-strategic arrivals. Shopping in “mega-markets” situated outside population agglomerations (thus requiring long travel) present similar characteristics. Perishable items (such as flowers, foods, etc. . . ) sold in such shops have obvious deadlines attached. Similarly, on every single evening hotels face “last-minute” non-strategic travelers in need of a room, and the hotels’ allocation/pricing policies do indeed change early in the evening as a function of their vacancy rates. Compared to a static setting, the new trade-off is between an assignment today and the valuable option of assigning the object in the future, possibly to an agent who values it more. Since the arrival process of agents is stochastic, the “future” on which the option value depends may never materialize (if there is a deadline) or it may be farther away in time, and thus discounted. The efficient dynamic allocation under complete information has been characterized – via a system of differential equations – by Albright (1974).4 This policy is described by partitions of the set of possible agent types: an arriving agent gets the best available object if his type lies in the highest interval of the partition, the second best available object if his type lies in the second highest interval, and so on. These intervals may depend on the point in time of the arrival. We derive associated menus of prices (one menu for each point in time, and for each subset of remaining objects) that implement the efficient allocation, and show that these menus have an appealing recursive structure: each agent who is assigned an object has to pay the value he displaces in terms of the chosen allocation – thus we obtain here a dynamic version of the Vickrey–Clarke–Groves mechanism. Dolan (1978) used a dynamic version of the Vickrey–Clarke–Groves mechanism in order to achieve welfare maximization in queues with random arrivals and with incomplete information about the agents’ characteristics. Dynamic extensions of VCG schemes (for much more general situations than those considered here) have recently attracted a lot of interest – see for example Athey and Segal (2007), Bergemann and Välimäki (2006), and Parkes and Singh (2003). Gershkov and Moldovanu (forthcoming-b) analyze the limitations encountered in a framework where the designer needs also to learn about the distribution of agents’ characteristics. Although agents have private values in their setting, that model exhibits phenomena typical of interdependent valuations, and the efficient policy cannot always be implemented. Gershkov and Moldovanu (forthcoming-a) derive the revenue maximizing policy in the sequential assignment model with heterogeneous objects, and relate it to classical results from the Management Science literature. We next show that a decrease in the second order stochastic sense in the distribution of agents’ types (which implies an increase in variability) leads to an increase in expected welfare in the dynamic assignment problem. The proof of this result uses several simple insights from majorization theory.5 Majorization by a vector of weighted sums of order statistics plays also the main role in our next result that assesses the welfare loss due to the sequential nature of the allocation process versus the scenario where the allocation can be executed after all arrivals and types have been observed. In general, it is not enough to focus on the physical allocation in order to ensure efficiency since one also has to describe what happens to the monetary payments. In some frameworks these accrue to a third-party (e.g., a seller in an auction) and physical efficiency is thus what matters. In other settings, the implementation of the efficient physical allocation generates a deficit or a surplus. If there is no third party, full efficiency calls then for budget-balancedness. It is well known that in the static setup no VCG mechanism can generally be both individually rational and budget balanced. Guo and Conitzer (2009) and Moulin (2009) analyze surplus minimizing VCG schemes in the static environment, and derive bounds on the ensuing efficiency loss. For the study of efficient, individually rational and budget-balanced mechanisms in our frameworks, we distinguish between two scenarios: 1) Both physical assignments and monetary payments must take place upon the agents’ arrival; 2) Payments can be postponed to later stages. While in the second scenario budget balancedness can be always reached, in the first scenario budget balancedness can only be reached asymptotically, as the arrival rate (or the time up to the deadline) increases without bound. The rest of the paper is organized as follows: in Section 2 we present the complete information continuous-time model of sequential assignment, and we present a theorem, due to Albright (1974) that characterizes the dynamic welfare maximization policy. In Section 3 we introduce incomplete information, and, for a setting where all objects perish after a deadline, we compute a dynamic VCG mechanism implementing the efficient allocation. We also consider the effect of changes in the distribution of agents’ types, and assess the welfare loss due to the sequential nature of the allocation process. Section 4

3

Other differences are: the job search model corresponds to our one object case, and sampling has an explicit cost. Derman et al. (1972) introduced the basic assignment model in a framework with a finite number of periods (time is discrete), and one arrival per period. An early paper that uses optimal stopping theory to characterize the efficient assignment of a single object to randomly arriving agents in continuous time is Elfving (1967). The dynamically efficient allocation policy can be explicitly computed if the distribution of agents’ types is exponential, while this is not often the case for general distributions. In a model with identical objects, McAfee and te Velde (2008) compute the dynamic welfare maximizing policy for a Pareto distribution of agents’ values, and show that it coincides then with the revenue optimizing policy. For a general comparison of the welfare-maximizing and revenue-maximizing policies see Gershkov and Moldovanu (forthcoming-a). 5 See Hardy et al. (1934, p. 45). 4

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focuses on budget balancedness, and we construct mechanisms that redistribute the raised revenue from the allocation process. Section 5 concludes. Proofs are relegated to Appendix A. Appendix B deals with an infinite horizon model with exponential discounting. For general renewal arrival processes we show that increased variability in inter-arrival times leads to higher expected welfare. 2. The complete information model There are n items (or objects). Each item i is characterized by a “quality” q i . Each agent j is characterized by a “type” x j . Agents arrive according to a (possibly non-homogeneous) Poisson process with intensity λ(t ), and each can only be served upon arrival (i.e., agents are impatient). After an item is assigned, it cannot be reallocated in the future. For some results we relax the Poisson assumption, and we allow for a more general renewal stochastic process to describe arrivals. An agent with type x j who obtains an item with characteristic q i and who is requested to pay a price p enjoys a utility of q i x j − p . If an item of quality q i is assigned to an agent with type x j at time t, then the utility for the designer is given by r (t )qi x j where r is a piecewise continuous, non-negative, non-increasing discount function which satisfies r (0) = 1. The items’ types 0  qn  qn−1  · · ·  q1 are assumed to be a-priori known constants, while the agents types become known to the designer upon arrival. Agents’ types are represented by independent and identically distributed random variables X i with common c.d.f. F that has a support [0, Υ ] with Υ  ∞. We assume that each X i has a finite mean, denoted by μ, and a finite variance. We denote by Πt the set of items available at t and by kt the number of objects available at t. Albright (1974) characterized the allocation policy that maximizes the total expected welfare from the designer’s point of view. His main result is: Theorem 1 (Albright (1974)). There exist n unique functions yn (t )  yn−1 (t )  · · ·  y 1 (t ), where y 0 ≡ ∞, ∀t, which do not depend on the q’s such that: 1. If an agent with type x arrives at a time t, it is optimal to assign to that agent the jth highest element of Πt if x ∈ [ y j (t ), y j −1 (t )), and not to assign any object if x < ykt (t ). 2. For each k, the function yk (t ) satisfies: (a) limt →∞ r (t ) yk (t ) = 0. y d[r (t ) yk (t )] = −λ(t )r (t ) y k−1 (1 − F (x)) dx  0. (b) dt k

3. The designer’s expected utility starting from time t is given by [r (t )

kt

q y (t )], i =1 (i ) i

where q(i ) is the ith highest element of Πt .

The surprising element in the above result is that the cutoff curves y j (t ) do not depend on the items’ characteristics. Since selling one object is equivalent to exchanging one of the currently available items with an item having a quality equal to zero, the above observation implies that after any sale at time t, the kt − 1 curves that determine the optimal allocation from time t on, coincide with the kt − 1 highest curves that were relevant for the decision at time t. 3. The incomplete information model In this section we introduce incomplete information to the above framework under the assumption that there is a deadline T after which all objects perish. While the items’ types 0  qn  qn−1  · · ·  q1 are assumed to be known constants, the agents’ types are assumed to be private information. The discount rate satisfies:

r (t ) =



1 if 0  t  T . 0 if t > T

It is then obvious that the dynamically efficient policy needs to satisfy

y 1 ( T ) = y 2 ( T ) = · · · = y n ( T ) = 0. 3.1. The dynamic VCG mechanism In order to implement the efficient allocation, we can restrict attention, without loss of generality, to direct mechanisms where every agent reports his characteristic x upon arrival, and where the mechanism specifies an allocation (which item, if any, the agent gets) and a payment. The needed allocation rule is clear: the agent who arrives at time t gets the object with the ith highest characteristic if and only if he reports a type x ∈ [ y i (t ), y i −1 (t )), where y i (t ) is given by Theorem 1. We calculate now the expected externality that each agent imposes on future agents. From Theorem 1(3) it follows that kt q y (t )]. Therefore, the the expected welfare starting from time t if the set Πt of object is still available is given by [ i = 1 (i ) i agent who gets the object of quality q( j ) at time t imposes an externality on future agents given by kt  (q(i ) − q(i +1) ) y i (t ). i= j

(1)

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If the agent arriving at period t faces the menu of prices where the price of the object of quality q( j ) is given by (1), then x ∈ [ y j (t ), y j −1 (t )) is given by his utility from reporting type 

q( j ) x −

kt 

(q(i ) − q(i +1) ) y i (t ).

i= j

Note that if x = y j (t ), then this menu of prices makes the agent indifferent between the objects q( j ) and q( j +1) . If x > y j (t ), the agent prefers q( j ) over q( j +1) and the opposite occurs if x < y j (t ). This observation allows us to the state the following result: Theorem 2. The complete information, dynamically efficient allocation policy is implementable also under incomplete information. In the supporting payment scheme an agent who gets the object of quality q( j ) at time t pays kt  (q(i ) − q(i +1) ) y i (t ). i= j

As usual for VCG schemes, this is not the unique payment that does the job. Adding to all prices an arbitrary function S (t ) which is independent of the objects’ quality still implements the efficient allocation. 3.2. Welfare analysis For the main results in this section we need a well-known concept, due to Hardy et al. (1934). Definition 1. For any n-tuple γ = (γ1 , γ2 , . . . , γn ) let γ( j ) denote the jth largest coordinate (so that γ(n)  γ(n−1)  · · ·  γ(1) ). Let α = (α1 , α2 , . . . , αn ) and β = (β1 , β2 , . . . , βn ) be two n-tuples. We say that α is majorized by β and we write α ≺ β if the following system of n − 1 inequalities and one equality is satisfied:

α(1)  β(1) , α(1) + α(2)  β(1) + β(2) , ···  ···

α(1) + α(2) + · · · + α(n−1)  β(1) + β(2) + β(n−1) , α(1) + α(2) + · · · + α(n) = β(1) + β(2) + · · · + β(n) . We say that

α is weakly sub-majorized by β and we write α ≺ w β if all relations above hold with weak inequality.

Our next result shows that an increase in the variability of the distribution of the agents’ values (while keeping a constant mean) increases expected welfare. We want to emphasize that this result holds even if all available objects are identical! Theorem 3. Consider two distributions of agents’ types F and G such that μ F = μG = μ and such that F second-order stochastically dominates G (in particular F has a lower variance than G). Then it holds that:

k

k

1. ∀k, t, i =1 y iF (t )  i =1 y iG (t ). 2. For any time t and for any set of available objects at t , Πt = ∅, the expected welfare in the efficient dynamic allocation under F is lower than that under G . Proof. See Appendix A.

2

The proof of the above theorem proceeds by showing that the efficient cutoffs of one distribution are majorized by those of the other.6 For a constant arrival rate, the system of differential equations that characterizes the efficient dynamic allocation can be solved explicitly for any number of objects if the distribution of the agents’ types is exponential, while this is rarely the case for other distributions. Together with the above result, that solution can be used to bound the optimal policy and the

6 This theorem has a counterpart for the cutoffs zi (t ) appearing in the next subsection for the scenario where allocations can be delayed till the deadline. The proof uses then a recent result about majorization of mean order statistics due to De La Cal and Caracamo (2006).

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associated welfare for large, non-parametric classes of distributions that are often used in applications, such as new better (worse) than used in expectation distributions.7 3.2.1. The welfare loss versus a delayed allocation Assume that at time t there are n objects left, with qualities q1  q2  · · ·  qn . Instead of the original formulation, consider now the scenario where the allocation decision to all subsequently arriving agents can be made at time T (the deadline) where their precise number is known, and where their types can be extracted via the usual, static VCG payments. At time T a welfare maximizing designer will allocate the object with the highest quality to the agent with the highest type, the object with the second highest quality to the agent  with the second highest type, and so on. . . (assortative matching). n This means that expected welfare at time t is given by i =1 q i z i (t ), where z i (t ) represents the expected type of an agent who arrives after t, and who get assigned to the object with the ith highest quality. In order to calculate the zi (t ) terms, let X (i ,l) denote the ith order statistic out of l copies of X , and denote by μi ,l its expectation (note that μl.l is the expectation of the maximum, or highest order statistic). Let Prl (t ) = e −λ( T −t ) 9

λl ( T −t )l be the probability that there will be l arrivals, l  1, after time t.8 Given assortative matching l!

at time T , we obtain :

zi (t ) =

∞ 

Prl (t )μl−i +1,l = e −λ( T −t )

l =i

∞ l  λ ( T − t )l μl−i +1,l , l!

i = 1, 2, . . . , n .

l =i

The next result establishes a relation between the n-vector of optimal dynamic cutoffs { y i (t )}ni=1 and the n-vector of n corresponding static expected types { zi (t )}ni=1 . Intuitively, i =1 q i [ y i (t ) − z i (t )] measures the welfare loss due to the sequentiality constraint. Theorem 4. For any period t , and for any n, the vector { y i (t )}ni=1 of optimal cutoffs in the welfare maximizing sequential allocation of n objects to agents arriving according to a Poisson process with parameter λ is weakly sub-majorized by the vector { zi (t )}ni=1 . Moreover, n n limn→∞ i =1 y i (t ) = limn→∞ i =1 zi (t ) = λ( T − t )μ, where μ is the mean of the distribution of agents’ types. Proof. See Appendix A.

2

If the number of homogeneous objects n is large, sequentiality does not cause any welfare loss since any arriving agent should get an object. The limit expression is intuitive: for each arrival, expected welfare is given by the average type μ, and λ( T − t ) is the expected number of arrivals in the interval [t , T ]. 4. Budget balancedness The above analysis was concerned solely with physical allocational efficiency. The employed mechanisms were also individually rational. We assume below that there are no alternative resources to subsidize the allocation process, and hence we require the total payment to be non-negative. But, an inability to redistribute the raised money among agents reduces their welfare and hence prevents reaching a fully efficient outcome. Roughly speaking, in an allocatively efficient mechanism, the payment of a buyer is tied to the externality he imposes on other agents, and hence it depends on the values of others. A redistribution of this payment to others is therefore bound to affect their incentives. In our dynamic setting, budget balancedness is easier to achieve due to the sequential nature of the allocation process: the amount each agent pays depends on the imposed expected externality, and hence it is not affected by the realized reports of the agents that should get this amount as a refund. In other words, the “future” acts here as a non-strategic agent that can receive transfers without affecting her incentives. We propose below mechanisms that reallocate the gathered money among the agents without distorting their incentives.10 We consider two mechanisms: 1) An Online mechanism, where monetary transfers pertaining to a specific transaction need to be completed together with the physical allocation; 2) An Offline mechanism, where payments can be postponed to later stages. For simplicity, we restrict attention to a homogeneous, Poisson arrival process with parameter λ. It is possible that the mechanisms we propose look somewhat strange because of the underlying goal of efficiency: if revenue is to be maximized, then refunds are of course not likely.

−

x

−

x

Note that F second order stochastically dominates G (x) = 1 − e μ (is second-order stochastically dominated by G (x) = 1 − e μ ) is equivalent to F being NBUE (NWUE). Any distribution with an increasing failure (or hazard) rate is NBUE, while any distribution with a decreasing failure rate is NWUE. 8 We assume here for simplicity that arrivals follow a homogeneous Poisson process with rate λ. The argument is easily extended to non-homogeneous processes. 9 Note that an object remains unassigned if there are not sufficient arrivals, yielding a zero reward. 10 We restrict attention here to the more interesting deadline scenario. The results can be easily extended to the infinite horizon case. 7

A. Gershkov, B. Moldovanu / Games and Economic Behavior 68 (2010) 144–154

149

4.1. Online mechanisms Denote by R (t ) the budget surplus accumulated by time t when using a mechanism that implements the efficient physical allocation, and that never runs a deficit (thus R (t )  0, ∀t ). While it is impossible here to precisely obtain ex-post budget balancedness, we describe below a scheme where the expected surplus vanishes asymptotically, as λ → ∞. Definition 2 (Online redistribution mechanism). A scheme that implements the efficient allocation such that the typeindependent part of the payment scheme for the agent arriving at t is S (t ) = − R (t ) will be called a dynamically efficient online redistribution mechanism (DEON hereafter). That is, each agent’s payment consists of two parts: first, a non-negative fee for the object he gets, which is determined by the report and by the efficient cutoffs; second, a refund equal to the payment of the previously arrived agent. Clearly, the DEON mechanism never runs a budget deficit, but it may run a surplus if, after the last sale, no agent arrives till the deadline. The next result shows that the expected surplus goes to zero if the rate of the Poisson process goes to infinity, that is if λ → ∞. Note that increasing λ has two opposite effects on the expected surplus: On the one hand, both prices and the expected surplus in case no agent arrives after the last sale go up. On the other, the probability of no arrival after the last sale goes down. The next theorem shows that the second effect dominates, that is the expected surplus at the last period goes to zero.11 Theorem 5. For any distribution of values, and for any deadline T , limλ→∞ E R ( T ) = 0. Proof. See Appendix A.

2

Implicitly, the assumption of non-strategic arrivals plays a significant role in the DEON mechanism. If an agent can split her identity, then she will find it optimal to arrive “again” immediately after a sale, and to claim type zero, say, which allows her to collect the payment from the last sale. Note though that analogous manipulations also afflict static budget balanced mechanisms. For example, in the well-known static AGV mechanism, an agent would benefit from creating and controlling additional dummy agents since monetary receipts are evenly distributed.12 The specific mechanism exhibited above (and the offline mechanism described below) is of course a theoretical construct, tailored to the assumptions made here. One can envisage variations that go some – but not all – of the way in the direction of a balanced budget and that could be used in practice, e.g. the available budget surplus need not be observable to outsiders; refunds may be made randomly, etc. . . 4.2. Offline mechanism In this subsection we show how to construct a mechanism that satisfies ex-post budget balancedness for any λ and T by relaxing the “online” requirement that all monetary transfers need to be implemented upon arrival. There are many examples where the physical part of the allocation is a matter of urgency to the agents, but where there is a possibility to delay final monetary payoffs until more information arrives (e.g., electricity or heat is continuously supplied in large apartment houses, but the precise cost allocation among tenants is done – at least in Germany – at the end of the year and is a function of later information such as vacancy rates, average temperature, fuel costs, etc. . . ). It is of practical interest to determine where such offline schemes are feasible and less prone to moral hazard problems. We sketch here the case where there is one object, with quality q = 1, but the generalization to the case with several heterogeneous objects is straightforward. The designer can now observe the number of agents that eventually arrive, and may ex-post refund the entire fee to the buyer who paid it if nobody else shows up. But, in order not to distort that buyer’s incentives, he has to pay a higher net price if additional agents subsequently arrive. Consider a time t where the object is still available, and the direct mechanism where: 1) the arriving agent at t does not get the object and pays nothing if his reported type is below y 1 (t ); 2) the arriving agent at t gets the object and pays P (t ) if his reported type is above y 1 (t ). Moreover, at time T , the designer distributes the raised revenue equally among all agents that arrived after the sale. Obviously, this mechanism is budget-balanced. Denote by Prl (t ) the probability that exactly l additional agents arrive after time t. In order to implement the efficient allocation, type y 1 (t ) should be indifferent ∞ P (t ) Pr (t ) between getting the object, which yields utility of y 1 (t ) − P (t ) + l=0 1+ll , and not getting the object, which generates utility of zero. Noting that

P (t ) =

11 12

∞

Prl (t ) l=0 1+l

=

1−e −λ( T −t ) λ( T −t )

for the Poisson arrival process, yields

λ( T − t ) y 1 (t ) . λ( T − t ) + e −λ(T −t ) − 1

Since λ governs the number of arrivals per unit of time, a similar result holds if λ is constant but T → ∞. For problems created by false identities in VCG mechanisms see also Yokoo et al. (2004).

(2)

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A. Gershkov, B. Moldovanu / Games and Economic Behavior 68 (2010) 144–154

Although the proposed mechanism is interim individually rational, it is not individually rational ex-post. It is impossible to attain ex-post individual rationality if the designer insists on an efficient allocation and on budget balancedness. To see this, observe that if a buyer is the only arriving agent, he should ultimately pay zero (by budget-balancedness); since, by efficiency, the type y 1 (t ) should be indifferent between buying and not, we obtain that sometimes this type needs to pay strictly more than his valuation. It can be easily shown that limt → T P (t ) = 2μ. Thus, the payment P (t ) does not “explode” as one gets near the deadline.13 5. Conclusion Traditionally, the mechanism design literature has focused on information and incentives issues in static models, whereas the Operations Research literature has looked at elegant dynamic models that were lacking strategic aspects. There seems to be an increased recent interest in combining these bodies of knowledge, and we believe that this is a very fruitful avenue. In our present context, we see the need for several extensions that relax some of the made assumptions: a) allow for several heterogeneous objects that are not-ranked, or for other, even more complex demand structures; b) allow for strategic arrivals of agents; c) analyze the dynamic competition among several designers. Appendix A First, we prove a lemma that is used in the proof of Theorem 3. Lemma (nα1 , α2 , . . . , αn ) and β = (β1 , β2 , . . . , βn ) be two n-tuples such that n 1. Let α = if i =1 q i α(i )  i =1 q i β(i ) for any constants qn  qn−1  · · ·  q 1 .

n

n

i =1

αi =

n

i =1 βi .

Then α ≺ β if and only

n

Proof of Lemma 1. ⇐ Assume that i =1 q i α(i )  i =1 q i β(i ) for any qn  qn−1  · · ·  q 1 . For each k = 1, 2, . . . , n − 1 consider qk = (qk1 , qk2 , . . . , qnk ) where qki = 1 for i = 1, 2, . . . , k, and qki = 0 for i = k + 1, k + 2, . . . , n. Then, for each k we obtain n 

qki α(i ) 

i =1

n 

qki β(i )

k 

⇔

i =1

α(i) 

i =1

k 

β( i )

i =1

and thus α ≺ β . ⇒ Assume α ≺ β and let qn  qn−1  · · ·  q1 . Then we have the following chain: n 

qi [β(i ) − α(i ) ] = qn

i =1

n n −1   [β(i ) − α(i ) ] + (qi − qn )[β(i ) − α(i ) ] i =1

= qn

i =1

n 

n −1  [β(i ) − α(i ) ]

i =1

i =1

[β(i ) − α(i ) ] + (qn−1 − qn )

n −2  + (qi − qn−1 )[β(i ) − α(i ) ] i =1

= ···  j  n n −1    [β(i ) − α(i ) ] + (q j − q j +1 ) [β(i ) − α(i ) ]  0. = qn i =1

− α( i ) ] = [β − α ]  0 by majorization. 2 ( i ) ( i ) i =1

The last inequality follows since: 1. definition; 3. ∀ j ,

j =1

n

j

i =1 [β(i )

n

i =1

i =1 βi

−

n

i =1

αi = 0 by definition; 2. ∀ j , q j − q j−1  0 by

Proof of Theorem 3. 1. By Theorem 1 we know that

k

d(

i =1

dt

y iF (t ))

∞ = −λ(t )





1 − F (x) dx;

ykF (t )

13 On the one hand, an agent that buys the good near the deadline is almost sure to get the fee back, thus he will be willing to pay a high amount for it. On the other, the fee is calculated to generate indifference between buying or not for an agent with a very low type. We are grateful to a referee who suggested this observation.

A. Gershkov, B. Moldovanu / Games and Economic Behavior 68 (2010) 144–154

k

d(

i =1

y iG (t ))

dt

∞ = −λ(t )





1 − G (x) dx.

ykG (t )

∞

∞

Define first: H F (s) = s (1 − F (x)) dx and H G (s) = H F (0) = H G (0) = μ. By SSD, for any s  0 it holds that

s

s F (x) dx 

0

s ⇔

G (x) dx

151

0

⇔



 1 − F (x) dx 

s

(1 − G (x)) dx. These are both positive, decreasing functions with

s



0

0

∞

∞



1 − F (x) dx 

s



1 − G (x) dx



1 − G (x) dx

s

∞

⇔

H F (s)  H G (s)

∞

where the second line follows because 0 (1 − F (x)) dx = μ F = μG = 0 (1 − G (x)) dx. Thus, the curve H F is always below H G . Consider now y 1F (t ) and y 1G (t ). These are, respectively, the solutions to the differential equations:

y  = −λ(t ) H F ( y )

y  = −λ(t ) H G ( y )

and

with boundary condition y ( T ) = 0. Integrating the above equations from t to T , and using the boundary condition, we get the integral equations:

T y ( T ) − y (t ) = −

 λ(s) H F y (s) ds

T ⇔

y (t ) =

t

t

T

T

y ( T ) − y (t ) = −

 λ(s) H G y (s) ds

⇔

y (t ) =

t

 λ(s) H F y (s) ds and

 λ(s) H G y (s) ds.

t

Because H F is always below H G and because these are decreasing functions, we obtain y 1F (t )  y 1G (t ). Consider now y 1F (t ) + y 2F (t ) and y 1G (t ) + y 2G (t ). These functions satisfy the differential equations:

∞



y = −λ(t )



 1 − F (x) dx and y  = −λ(t )

y − y 1F (t )

∞





1 − F (x) dx

y − y 1G (t )

with boundary condition y ( T ) = 0. Integrating from t to T yields the equations:

T y (t ) =



∞



λ(s) t

T    1 − F (x) dx ds = λ(s) H F ( y (s) − y 1F (s) ds, t

y (s)− y 1F (s)

T y (t ) =



∞



λ(s) t

T    1 − G (x) dx ds = λ(s) H G ( y (s) − y 1G (s) ds. t

y (s)− y 1G (s)

We have:

∀t ,













H F y (t ) − y 1F (t )  H F y (t ) − y 1G (t )  H G y (t ) − y 1G (t )

where the first inequality follows because y 1F (t )  y 1G (t ) ⇔ y (t ) − y 1F (t )  y (t ) − y 1G (t ) and because the function H F is decreasing, and the second inequality follows because H F is always below H G . This yields y 1F (t ) + y 2F (t )  y 1G (t ) + y 2G (t ), as required. The rest of the proof follows analogously. k F 2. The expected welfare terms from time t on if there are k objects left are given by i =1 q (i ) y i (t ) and by

k

G i =1 q (i ) y i (t ), respectively. By point G ≺ w ( y 1 (t ), y 2G (t ), . . . , ykG (t )) := ykG (t ). kG

vector z(t ) such that z(t ) ≺ y

1, we know that for each k and for each t , ykF (t ) = ( y 1F (t ), y 2F (t ), . . . , ykF (t ))

By Result 12.5(b) in Pecaric et al. (1992), for each k and each t there exists a k-

(t ) and such that zi (t )  ykF , ∀i . We obtain then: i

152

A. Gershkov, B. Moldovanu / Games and Economic Behavior 68 (2010) 144–154

k 

∀k, t ,

q(i ) y iF (t ) 

k 

i =1

q(i ) zi (t ) 

i =1

k 

q(i ) y iG (t )

i =1

where the last inequality follows from Lemma 1 since qn  qn−1  · · ·  q1 . 2

α ≺ β if and only if

n

i =1 q i

α( i ) 

n

i =1 q i β(i )

for any constants

Proof of Theorem 4. Consider a situation with m  n identical objects available at time t. For any particular realization of arrivals and agents types from period m t on, welfare in the case where the decision can be delayed until the entire realization has been observed is given by i =1 z i (t ). Consider now the original sequential assignment problem where agents arrive according to a Poisson process, and allocations must be made upon arrival. By Theorem 1, the expected total welfare is then m given by i =1 y i (t ). It is obvious that, for any realization of arrivals and types, this total reward cannot be larger than that obtained in the previous scenario where allocations can be delayed since mistakes may have occurred due to the constraints of sequentiality, i.e., an agent with a lower type was served although there was another arrival with a higher type.14 This m m y ( t )  yields i =1 i i =1 z i (t ), ∀m  n, as required. The second part follows by considering the limit when the number of identical objects tends to infinity. In that limit, all arriving agents should be served, and there is no welfare loss due to the sequentiality constraint. By Theorem 1 we know that

n

d[

i =1

y i (s)]

ds

∞ = −λ





1 − F (x) dx.

yn (s)

Integrating the above expression between t and T and recalling that n 

T  ∞ y i (t ) = λ

i =1

t



n

i =1

y i ( T ) = 0 yields:

  1 − F (x) dx ds.

yn (s)

Taking the limit with respect to n, we finally obtain:

lim

n→∞

n 

T  ∞ y i (t ) = lim λ



n→∞

i =1

t

yn (s)

T  ∞



=λ t

  1 − F (x) dx ds

  1 − F (x) dx ds = λ( T − t )μ

0

where the second line follows since limn→∞ yn (t ) = 0 uniformly.

2

Proof of Theorem 5. In the DEON mechanism, if an agent arrives at time t and reports type xt ∈ [ y j (t ), y j −1 (t )), while the previous agent arrived at τ < t and reported a type xτ ∈ [ yl (τ ), yl−1 (τ )) then the time t agent’s payment is given by kτ kt   (q(i ) − q(i +1) ) y i (t ) − (q(i ) − q(i +1) ) y i (τ ). i= j

i =l

Since the probability that no agent arrives between periods s and T is e −λ( T −s) , the expected surplus of the mechanism is given by

E R (T ) =

T  Πs ⊆Π 0

Prs (Πs )

ks ks   (q(i ,Πs ) − q(i +1,Πs ) ) y i (s) g i ,Πs (s)e −λ(T −s) ds j =1 i = j

where Prτ (Πs ) is the probability that at time τ the set of the objects still available is Πs , q( j ,Πs ) is the jth highest quality out of Πs , and g i ,Πs (s) is the density that at time s an agent arrives with a type that leads to the allocation of the jth highest quality out of Πs . Let k0 = 1. That is, initially there is only one object available of quality q, normalized to be one. The expected surplus of the DEON mechanism is given by

14

In other words, any allocation obtained in the sequential process can be replicated when delayed allocations are possible.

A. Gershkov, B. Moldovanu / Games and Economic Behavior 68 (2010) 144–154

T





t  y 1 (t )λ 1 − F y 1 (t ) e −λ 0 [1− F ( y 1 (z))] dz e −λ( T −t ) dt =

0

T

y 1 (t )λ[1 − F ( y 1 (t ))]e λ

153

t 0

F ( y 1 ( z)) dz

eλT

dt .

0

Since [1 − F ( y 1 (t ))]  1 it is sufficient to show that

T

λ y 1 (t )

lim

λ→∞ 0

e

λ( T −t +

t

0 [1− F ( y 1 ( z))] dz)

dt = 0.

The claim will follow by showing that, for any t ∈ [0, T ], the integrand goes to 0 when λ → ∞. If t = T , the result follows immediately since y ( T ) = 0. Assume then that t < T , and recall that

∞



y 1 (t ) = −λ





1 − F (x) dx.

y 1 (t )

Integrating both sides between t and T yields

T ∞ y 1 (t ) = λ





1 − F (x) dx ds  λ( T − t )μ

t y 1 (s)

where the last inequality follows from

t

∞ 0

[1 − F (x)] dx = μ. In addition, for any t < T , there exists a constant A > 0 such

that T − t + 0 [1 − F ( y 1 ( z))] dz > A. Therefore, it is sufficient to show that limλ→∞ rule to this ratio provides the required result. If k0 = 2 the expected surplus of the DEON mechanism is

T E R (T ) =

T

λ((q1 − q2 ) y 1 (t ) + q2 y 2 (t ))[1 − F ( y 1 (t ))] e

0

T T + q1

λ( T −t +

0 [1− F ( y 2 ( z))] dz)

T T + q2



t s e λ( T −t + s [1− F ( y 1 (z))] dz+ 0 [1− F ( y 2 (z))] dz)

λ2 y 1 (t )[1 − F ( y 1 (s))][1 − F ( y 1 (t ))] 

0 s



s t e λ( T −t + 0 [1− F ( y 2 (z))] dz+ s [1− F ( y 2 (z))] dz)

eλ A

= 0. Twice applying L’Hospital’s

y 2 (t )λ[ F ( y 1 (t )) − F ( y 2 (t ))] 

0

λ2 y 1 (t )[1 − F ( y 1 (t ))][ F ( y 1 (s)) − F ( y 2 (s))] 

0 s

dt + q2

t

λ2 ( T −t )μ

t e λ( T −t + 0 [1− F ( y 2 (z))] dz)

dt

dt ds

dt .

Similarly to the above argument, each element in the summation goes to 0 as λ goes to infinity, and the result follows.

2

Appendix B. The dynamic efficient allocation with an infinite horizon and discounting In this section we assume that r (t ) = e −αt . Given this specification, the arrival process can be more general, and it is assumed here to be a renewal process with general inter-arrival distribution B (instead of a Poisson process where the inter-arrival distribution is exponential).15 The complete-information efficient dynamic assignment – which turns out to be stationary – is characterized in the following theorem: Theorem 6. (See Albright (1974).) Assume that r (t ) = e −αt . The efficient allocation curves are constants (i.e., independent of time) yn  yn−1  · · ·  y 1 . These constants do not depend on the q’s, and are given by the implicit recursion:



 B (α ) ( y k + y k −1 + · · · + y 1 ) = 1 − F (x) dx, 1 − B (α ) ∞

1kn

yk

where  B is the Laplace-transform of the inter-arrival distribution B.

15 The derived controlled stochastic process is semi-Markov since the Markov property is preserved only at decision points, but not between them. See Puterman (2005) for solution approaches to such problems by an uniformization procedure.

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A. Gershkov, B. Moldovanu / Games and Economic Behavior 68 (2010) 144–154

In addition to an analog of Theorem 3 about the benefits of increased variability in the agents’ types, we can also prove a comparative-statics result about the benefits of variability in arrival times. Interestingly, this next result holds for a stochastic order that is much weaker than second-order stochastic dominance. Definition 3. (See Shaked and Shanthikumar (2007, p. 233).) Let X , Y be two non-negative random variables. Then X is said to be smaller than Y in the Laplace transform order, denoted by X  Lt Y , if







E e −s X  E e −sY



for all s > 0.

Theorem 7. Consider two inter-arrival distributions B and E such that B  Lt E. Then, for any fixed distribution of agents’ characteristics F , it holds that:

k

k

1. ∀k, i =1 y iB  i =1 y iE . 2. For any t and for any Πt = ∅, the expected welfare in the efficient dynamic allocation under B is lower than that under E .  B (α ) 1− B (α )

∞

 E (α ) 1− E (α )

∞

(1 − F (x)) dx where  B and  E are the respective Laplace B (α )   E (α ). This transforms. By the definition of the Laplace transform, and by the assumption B  Lt E, we know that  Proof. 1. Let H B (s) =

s

(1 − F (x)) dx, H E (s) =

s

yields:

 B (α )   E (α )

⇔

  B (α ) E (α )   1 − B (α ) 1 − E (α )

⇔

H B (s)  H E (s).

x x The first equivalence follows because the function 1− is increasing on the interval [0, 1) with limx→1 1− = ∞, and because x x Laplace transforms take values in the interval [0, 1]. Thus, we obtained that the decreasing function H B (s) is always below the decreasing function H E (s). Consider first y 1B and y 1E . These are, respectively, the solutions to the equations s = H B (s) and s = H E (s). The rest of the proof continues analogously to the proof of Theorem 3. 2

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