The FedEx Problem (Working Paper)

Transcription

1 The FedEx Problem (Working Paper) Amos Fiat Kira Goldner Anna R. Karlin Elias Koutsoupias June 6, 216 Remember that Time is Money Abstract Benjamin Franklin in Adice to a Young Tradesman (1748) Consider the following setting: a customer has a package and is willing to pay up to some alue to ship it, but needs it to be shipped by some deadline d. Gien the joint prior distribution from which (, d) pairs are drawn, we characterize the auction that yields optimal reenue, contributing to the limited understanding of optimal auctions beyond single-parameter settings. Our work requires a new way of combining and ironing reenue cures which illustrate why randomization is necessary to achiee optimal reenue. Finally, we strengthen the emerging understanding that duality is useful for both the design and analysis of optimal auctions in multi-parameter settings. 1 Introduction Consider the pricing problem faced by FedEx. Each customer has a package to ship, a deadline d by which he needs his package to arrie, and a alue for a guarantee that the package will arrie by his deadline. FedEx can (and does) offer a number of different shipping options in order to extract more reenue from their customers. In this paper, we sole the optimal (reenue-maximizing) auction problem for the single-agent ersion of this problem. Our paper adds to the relatiely short list of multi-parameter settings for which a closed-form solution is known. This pricing problem is extremely natural and arises in numerous scenarios, whether it is Amazon.com proiding shipping options, Internet Serice Proiders offering bandwidth plans, Bitcoin miners setting a policy for transaction fees, or a myriad of other settings in which customers hae a sensitiity to time or some other feature of serice. In these settings, a seller can price discriminate or otherwise segment his market by delaying serice, or proiding lower quality/cheaper ersions of a product. It is important to understand how buyer deadline (or quality) constraints impact the design of auctions and what leerage they gie to the seller to extract more reenue. Tel Ai Uniersity. fiat@tau.ac.il. Uniersity of Washington. kgoldner@cs.washington.edu. Funded by the National Science Foundation under CCF grant Uniersity of Washington. karlin@cs.washington.edu. Funded by the National Science Foundation under CCF grant Uniersity of Oxford. elias@cs.ox.ac.uk. Funded by the European Research Council under the European Union s Seenth Framework Programme (FP7/27-213) / ERC grant agreement no (ALGAME). This research was done in part while the authors were isiting the Simons Institute for Theoretical Computer Science.

2 We consider a model in which a seller proides n different options for serice, and a customer is interested in buying an option that meets his quality demand of d. We use the running example of shipping packages by deadlines. A customer s utility for getting his package shipped by day t at a price of p is p if t d (i.e., it is receied by his deadline) and p otherwise. A customer s (, d) pair is his priate information. We study the Bayesian setting, where this pair (, d) is drawn from a prior distribution known to the seller. Related Work The FedEx problem is a ariant of price discrimination in which the customers are grouped by their deadline. Price discrimination offers different prices to users with the goal of improing reenue [?]. Alternatiely one can iew the FedEx problem as a multi-dimensional optimal auction problem. There are two ways to express the FedEx problem in this way. First, as a 2-dimensional (alue deadline) problem of arbitrary joint distribution in which the second ariable takes only integer alues in a bounded interal. Alternatiely, as a ery special case of the n-dimensional unit-demand problem with correlated alues (the customer buys a day among the n choices). There is an extensie body of literature on optimal auction design. The seminal work of Myerson [?] has completely settled the case of selling a single item to multiple bidders and extends directly to the more general framework of single-parameter settings. The most complicated part of his solution is his handling of distributions that are not regular by ironing them, that is, by replacing the reenue cures by their upper concae enelope. Myerson s ironing is done in quantile space. In this work, we also need to iron the reenue cures, but we need to do this in alue space. Extending Myerson s solution to the multidimensional case has been one of the most important open problems in Microeconomics. For the case of unit-demand agents, a beautiful sequence of papers [?,?,?,?,?,?] showed how to obtain approximately optimal auctions. For the case of finite type spaces, [?,?,?] are able to use linear and conex programming techniques to formulate and sole the optimal auction problem. This gies a black-box reduction from mechanism to algorithm design that yields a PTAS for reenue maximization in unit-demand settings. For the case of additie agents, additional recent breakthroughs [?,?,?,?,?,?] hae also resulted in approximately optimal mechanisms. But if we insist on optimal auctions for continuous probability distributions, no general solution is known for the multidimensional case een for the two-dimensional single-bidder case and it is ery possible that no such simple solution exists for the general case. One of the reasons that the multidimensional case is so complex is that optimal auctions are not necessarily deterministic [?,?,?,?,?,?,?,?]. The optimal auction for the FedEx problem also turns out to be randomized with possibly exponentially many different price leels. There are some releant results that sole special cases of the two-parameter setting. One of the earliest works is by Laffont, Maskin, and Rochet [?], who soled a distant ariant of the FedEx problem in which the utility of the bidder is expressed as a quadratic function of the two alues (, d) when the alues are uniformly distributed in [, 1]; unlike the FedEx problem the second parameter is drawn from a continuous probability distribution. Their solution was highly non-triial, which was an early indication that the multidimensional optimal auction problem may not be easy. This work was followed by McAfee and McMillan [?] who generalized this example to a class of distributions that hae the single-crossing property. These initial results were followed by more general results. In particular, Haghpanah and

3 Hartline [?] considered the multidimensional problem of selling a product with multiple quality leels and gae sufficient conditions under which the optimal auction is to sell only the highest quality (in the FedEx problem, this corresponds to haing a single price for eery day). They also gae sufficient conditions under which selling the grand bundle to an agent with additie preferences is optimal. Their work generalizes results from Armstrong [?]. Our approach is based on a duality framework. Two such frameworks were proposed. The first framework by Daskalakis, Deckelbaum, and Tzamos [?,?,?] reduces the problem to optimal transport theory. With its use, the authors gae optimal mechanisms for a number of two-item settings and gae necessary and sufficient conditions under which grand bundle selling is optimal. The second framework is by Giannakopoulos and Koutsoupias [?,?], which is based on expressing the problem as an optimization problem with linear partial differential inequalities, computing its dual, and using complementary slackness conditions to extract the optimal auction and proe its optimality. With its use, the authors gae the optimal auction for a large class of distributions for two items and for the uniform distribution for up to 6 items. Our solution of the FedEx problem follows the latter duality framework. For much more on both exact and approximate optimal mechanism design, see [?,?,?,?,?]. For background on duality in infinite linear and conex programs, see e.g., [?,?]. This Paper Our main result is a characterization of the reenue-optimal auction for this setting. For each deliery option of 1 through n days, the mechanism specifies a distribution of prices. The customer, knowing the distributions, specifies a deliery option of i days, and then a price is drawn randomly from day i s distribution and offered to the customer. We formulate the optimal auction problem as a continuous infinite linear program, take its dual, and determine a sufficient set of conditions for optimality. We then show how to construct a sequence of reenue-type cures Γ i ( ). Each such cure represents the optimal reenue on days i through n gien a price that might be set on day i. That is, Γ i (p) corresponds to the sum of (1) the reenue from selling day i deliery at a price of p to customers with a deadline of i and (2) the optimal reenue (when constrained by this choice on day i) from days i + 1 through n, where the constraint comes from the requirement that it is incentie compatible for a customer to report his true deadline rather than an earlier one. This cure also incorporates ironing so as to ensure incentie compatibility, and this ironing may lead to randomization oer prices. We use these cures to construct a solution to the primal and dual linear programs that satisfies conditions sufficient for optimality. Our result is one of relatiely few exact and explicit closed-form generalizations of [?] to multiparameter settings with arbitrary joint distributions, and contributes to recent breakthroughs in this space. Key take-aways are the following: 1. Our result requires a new way of combining and ironing reenue cures. In Myerson s optimal auction for irregular distributions, ironing ensures incentie compatibility and gies an upper bound on the optimal reenue. Myerson shows that this upper bound is in fact achieable using randomization. Similarly, our combined and ironed cures yield upper bounds on the reenue, and we show that these upper bounds can be realized with lotteries. In Myerson s setting, ironing is required to enforce incentie compatibility constraints among multiple bidders. In our setting, we need ironing een for one bidder because of the multiple parameters.

4 This may suggest that ironing is one of the biggest hurdles in extending Myerson s results to more general settings. 2. The optimal auction we obtain is constructed inductiely and, consequently, is relatiely simple to describe. Once the distribution of prices has been determined for deliery by day i, using Γ i+1 ( ), we show how to define the distribution of prices for deliery by day i + 1. The latter inoles randomizing oer up to 2 i prices. 3. The duality approach gies a closed-form allocation rule. Naie attempts to sole this problem, een for the case where there are only two or three possible deadlines, leads to a massie case analysis depending on the priors. Een in cases where the optimal auction is deterministic, setting these prices is not straightforward. For example, for three possible deadlines, the optimal deterministic mechanism can require 1, 2, or 3 distinct prices, and determining how many prices to use and how to set them seems non-triial. Our duality approach, howeer, leads to a unified allocation rule with no case analysis at all. This paper strengthens the emerging understanding that duality is useful for determining the structure of the optimal auction in non-triial settings and for analyzing the resulting auction. 2 Preliminaries As discussed aboe, the type of a customer is a (alue, deadline) pair. An auction takes as input a reported type t = (, d) and determines the shipping date in 1,..., n} and the price. We denote by a d () the probability that the package is shipped by day d, when the agent reports (, d), and by p d () the corresponding expected price (the expectation is taken oer the randomness in the mechanism). Our goal is to design an optimal auction for this setting. By the reelation principle, we can restrict our attention to incentie-compatible mechanisms. Denote by u(, d, d) the utility of the agent when his true type is (, d) but he reports a type of (, d ). That is, u(, d a d ( ) p d ( ) if d d, d) = p d ( ) otherwise. The incentie compatibility requirement is that u(, d) := u(, d, d) u(, d, d), d. (1) We also require indiidual rationality, i.e., u(, d) for all (, d). Without loss of generality, a d () is the probability that the package is shipped on day d, since any incentie-compatible mechanism which ships a package early can be conerted to one that always ships on the due date 1, while retaining incentie compatibility and without losing any reenue. For each fixed d, this implies the standard (single parameter) constraints [?], namely d, a d () is monotone weakly increasing and in [, 1]; (2) d, p d () = a d () 1 At least that s when the customer thinks it s being shipped. a d (x)dx and hence u(, d) = a d (x)dx. (3)

5 Clearly no agent would eer report d d, as this would result in negatie utility. Howeer, we do need to make sure that the agent has no incentie to under-report his deadline, and hence another IC constraint is that for all 2 d n: which is equialent to a d 1 (x)dx u(, d 1, d) u(, d, d) (4) a d (x)dx d s.t. 1 < d n. (5) We sometimes refer to this as the inter-day IC constraint. Since a d () is the probability of allocation on day d, gien report (, d), constraints (2), (3) and (5) are necessary and sufficient, by transitiity, to ensure that u(, d, d) u(, d, d) for all possible misreports (, d ). The prior We assume that the agent s (alue, deadline) comes from a known prior F. Let q i be the probability that the customer has a deadline i 1,..., n}, that is, q i = Pr (,d) F [d = i] and let F i ( ) be the marginal distribution function of alues for bidders with deadline i, that is, F i (x) = Pr (,d) F [ x d = i]. We assume that F i is atomless and strictly increasing, with density function defined on [, H]. Let f i () be the deriatie of F i (). The objectie Let ϕ i () = 1 F i() f i () be the irtual alue function for drawn from distribution F i. Applying the Myerson payment identity implies that the expected payment of a customer with deadline i is E Fi [p i ()] = E Fi [ϕ i ()a i ()]. Thus, we wish to choose monotone allocation rules a i (), for days 1 i n, so as to maximize E (,i) F [p i ()] = n q i E Fi [p i ()] = i=1 n q i E Fi [ϕ i ()a i ()] = i=1 n q i ϕ i ()f i ()a i ()d, i=1 subject to (2), (3) and (5).

6 A triial case and discussion Conditioned on the fact that a customer has a day d deadline, i.e., if we knew that his alue for serice was drawn from F d, the optimal pricing would be triial, since this is a single agent, single item auction. Thus, the optimal mechanism for such a customer is to set the price for serice by day d to the resere price r d for his prior 2. Moreoer, if it is the case that r d r d+1 for each d, where r d is the resere price for the distribution F d, the entire Fedex problem is triial, since then we could set r d as the price for serice on day d, all IC constraints would be satisfied, and we would be optimizing pointwise for each conditional distribution. In this paper, we do not make the assumption that the resere prices are weakly decreasing with the deadline, let alone the stronger assumption that the distribution F d stochastically dominates the distribution F d+1 for each d. There may be seeral reasons that these assumptions do not hold. For one, the prior F captures the result of random draws from a population consisting of a mixture of different types. Obiously any particular indiidual with deadline d is at least as happy with day d 1 serice as with day d serice, but two random indiiduals may hae completely uncorrelated needs, so if one them is of type (, d), and the other is of type (, d ), with d > d, it is not necessarily the case that. A second factor has to do with costs. It is likely that the cost that FedEx incurs for sending a package within d days is higher than the cost FedEx incurs for sending a package within d > d days, since in the latter case, for example, FedEx has more flexibility about which of many planes/trucks to put the package on, and een may be able to reduce the total number of plane/truck trips to a particular destination gien this flexibility. More generally, in other applications of this problem, the cost of proiding lower quality serice is lower than the cost of proiding higher quality serice. Thus, een if resere prices tend to decrease with d, all bets are off once we consider a customer s alue for deadline d conditioned on that alue being aboe the expected cost to FedEx of shipping a package by deadline d for each d. In this paper, we are not explicitly modeling the costs that FedEx incurs, the optimization problems that it faces, the online nature of the problem, or any limits on FedEx s ability to ship packages. These are interesting problems for future research. The discussion in the preceding few paragraphs is here merely to explain why the problem remains interesting and releant een when r d is below r d+1. 3 Warm-up: The case of n = 2 Suppose that the customer has a deadline of either one day or two days. By the taxation principle, the optimal mechanism is a menu, in this case a price p i (possibly selected by some randomized procedure) for haing the package shipped by (on) day i. Let R i () be the reenue cure for day i, that is R i () := (1 F i ()). Let r i := argmax (R i ()), the price at which expected reenue from a bidder with alue drawn from F i is maximized, and let Ri := R i(r i ) denote this maximum expected reenue. Since Ri is the optimal expected reenue from the agent [?], conditioned on haing a deadline of i, q 1 R1 + q 2R2 is an upper bound on the optimal expected reenue for the two-day FedEx problem. If r 1 r 2, then this optimum is indeed achieable by an IC mechanism: just set the day one shipping price p 1 to r 1 and the day two shipping price p 2 to r 2. 2 r d is defined properly at the beginning of Section 3.

7 But what if r 2 > r 1? In this case, the inter-day IC constraint (5) is iolated by this pricing (a customer with d = 2 will prefer to pretend his deadline is d = 1). Attempt #1: One alternatie is to consider the optimal single price mechanism (i.e., p 1 = p 2 = p). In this case, the optimal choice is clear: p := argmax [q 1 R 1 () + q 2 R 2 ()], (6) i.e., set the price that maximizes the combined reenue from both days. There are cases where this is optimal, e.g., if both F 1 and F 2 are regular. 3 Attempt #2: Another auction that retains incentie compatibility, and, in some cases, improes performance is to set the day one price p 1 to p and the day two price to p 2 := argmax p R 2 (). (7) Howeer, een if we fix p 1 = p, further optimization may be possible if F 2 is not regular. Attempt #3: Consider the concae hull of R 2 ( ), i.e., the ironed reenue cure. If R 2 () is maximized at r 2 > p and R 2 ( ) is ironed at p, then offering a lottery on day two with an expected price of p yields higher expected reenue than offering any deterministic day two price of p 2. As we shall see, for this case, this solution is actually optimal. (See Figure 1.) Howeer, if p > r 2, (which is possible if F 1 and F 2 are not regular, een if r 1 < r 2 ), then we will see that the optimal day one price is indeed aboe r 2, but not necessarily equal to p. Attempt #4: If p > r 2, set the day one price to p 1 := argmax r2 R 1 (). This should make sense: if we re going to set a day one price aboe r 2, we may as well set the day two price to r 2, but in that case, the day two cure should not influence the pricing for day one (except to set a lower bound for it). Admittedly, this sounds like a tedious case analysis, and extending this reasoning to three or more days gets much worse. Happily, though, there is a nice, and relatiely simple way to put all the aboe elements together to describe the solution, and then, as we shall see in Section 5, proe its optimality ia a clean duality proof. A solution for n = 2 Define R( ) to be the concae (ironed) reenue cure corresponding to reenue cure R( ) and let q 1 R 1 () + q 2 R 2 () r 2 R 12 () := (8) q 1 R 1 () + q 2 R 2 (r 2 ) > r 2 Note that because R 2 ( ) is the least concae upper bound of R 2 ( ) and by definition of r 2 that R 2 (r 2 ) = R 2 (r 2 ). The optimal solution is to set p 1 := argmax R 12 (), and then take p 2 := r 2 if r 2 p 1 and E(p 2 ) := p 1 otherwise, 3 A distribution F is regular if its irtual alue function is increasing in.

8 1 1 p p 1 p r 2 p Figure 1: A two-day case: Suppose that the optimal thing to do on day one is to offer a price of p. In the upper left, we see the corresponding allocation cure a 1 (). The bottom left graph shows the reenue cure R 2 ( ) for day two (the thin black cure) and the ironed ersion R 2 ( ) (the thick blue concae cure). Optimizing for day two subject to the inter-day IC constraint a 1(x)dx a 2(x)dx suggests that the most reenue we can get from a deadline d = 2 customer is R 2 (p) on day two, which can be done by offering the price of p with probability 1/3 and a price of p with probability 2/3 (since, in this example, p = (1/3)p + (2/3)p). This yields the pink allocation cure a 2 () shown in the upper right. The fact that these cures satisfy the inter-day IC constraint follows from the fact that the area of the two grey rectangles shown in the bottom right are equal.

9 where the randomized case is implemented ia the lottery as in the example of Figure 1. The key idea: R 12 () describes the best reenue we can get if we set a price of on day 1. Since r 2 is the optimal day two price, if we are going to set a price aboe r 2 for day one, we may as well be optimal for day two. On the other hand, if the day one price is going to be below r 2, we hae to be careful about the inter-day IC constraint (5), and ironing the day two reenue cure may be necessary. This is precisely what the definition of R 12 ( ) in (8) does for us. The asymmetry between day one and day two, specifically the fact that the day one cure is neer ironed, whereas the day two cure is, is a consequence of the inter-day IC constraint (5). We generalize this idea in the next section to sole the n-day problem. 4 An optimal allocation rule 4.1 Preliminaries Our goal is to choose monotone allocation rules a i (), for days 1 i n, so as to maximize n i=1 q i ϕ i()f i ()a i ()d. For a distribution f i ( ) on [, H] with irtual alue function ϕ i ( ) = 1 F i() f i (), define γ i () := q i ϕ i ()f i (). Then we aim to choose a i () to maximize n i=1 γ i()a i ()d. Let Γ i () = γ i(x)dx. Obsere that this function is the negatie of the reenue cure, that is, Γ i () = q i R i () = q i (1 F i ()). 4 Thus, Γ i () = Γ i (H) = and Γ i () for [, H]. Definition 1. For any function Γ, define ˆΓ( ) to be the lower conex enelope 5 of Γ( ). We say that ˆΓ( ) is ironed at if ˆΓ() Γ(). Since ˆΓ( ) is conex, it is continuously differentiable except at countably many points and its deriatie is monotone (weakly) increasing. Definition 2. Let ˆγ( ) be the deriatie of ˆΓ( ) and let γ( ) be the deriatie of Γ( ). Claim. The following facts are immediate from the definition of lower conex enelope (See Figure 2.): ˆΓ() Γ(). ˆΓ( min ) = Γ( min ) where min = argmin Γ(). (This implies that there is no ironed interal that crosses oer min.) ˆγ() is an increasing function of and hence its deriatie ˆγ () for all. If ˆΓ() is ironed in the interal [l, h], then ˆγ() is linear and ˆγ () = in (l, h). We next define the sequence of functions that we will need for the construction: 4 Γ i() = q i [xfi(x) (1 Fi(x)] dx. Integrating the first term by parts gies xfi(x) dx = Fi() Fi(x) dx. Combining this with the second term yields Γi() = qi(1 Fi()). 5 The lower conex enelope of function f(x) is the supremum oer conex functions g( ) such that g(x) f(x) for all x. Notice that the lower conex enelope of Γ( ) is the negatie of the ironed reenue cure R().

10 h H Figure 2: The black cure is Γ i (), and its lower conex enelope ˆΓ() is traced out by the thick light blue line. The cure is ironed in the interal [l, h] (among others), so in that interal, ˆΓ() is linear, and thus has second deriatie equal to. Definition 3. Let Γ n () := Γ n () and r n := argmin Γ n (). Inductiely, define, for i := n 1 down to 1, Γ i () + Γ i () := ˆΓ i+1 () < r i+1 Γ i () + ˆΓ and r i := argmin Γ i (). i+1 (r i+1 ) r i+1 The deriatie of Γ i ( ) is then γ i () := γ i () + ˆγ i+1 () < r i+1. γ i () r i+1 Rewriting this yields γ i () γ i () = ˆγ i+1 () < r i+1. (9) r i The allocation rule We define the allocation cures a i ( ) inductiely. We will show later that they are optimal. Each allocation cure is piecewise constant. For day one, set if < r 1, a 1 () = 1 otherwise. Suppose that a i 1 has been defined, for some i < n, with jumps at 1,..., k, and alues = β < β 1 β 2... β k = 1. That is, if < 1, a i 1 () = β j j < j+1 1 j < k. 1 k

11 Thus, we can write where Next we define a i (). a i 1 () = k (β j β j 1 )a i 1,j () j=1 a i 1,j () = if < j 1 j. 1 1 Figure 3: This figure shows an example allocation cure a i 1 () in purple, and illustrates some aspects of Definition 4. The cures Γ i () and ˆΓ i () are shown directly below the top figure. In this case, r i [ j+1, j+2 ), so j = j + 1. The bottom figure shows how a i,j () is constructed from a i 1,j (). Definition 4. Let j be the largest j such that j r i. For any j j, consider two cases:

12 ˆΓ i ( j ) = Γ i ( j ), i.e. ˆΓ i not ironed at j : In this case, define if < j a i,j () = 1 otherwise.. ˆΓ i ( j ) Γ i ( j ): In this case, let j := the largest < j such that ˆΓ i () = Γ i () i.e., not ironed, and j := the smallest > j such that ˆΓ i () = Γ i () i.e., not ironed. Let < δ < 1 such that j = δ j + (1 δ) j. Then ˆΓ i ( ) is linear between j and j : ˆΓ i ( j ) = δγ i ( j ) + (1 δ)γ i ( j ). Define if < j a i,j () = δ j < j. 1 otherwise. Finally, set a i () as follows: j j=1 (β j β j 1 )a ij () if < r i, a i () =. (1) 1 r i Remark: In order to continue the induction and define a i+1 () we need to rewrite a i () in terms of functions a i,j () that take only /1 alues. This is straightforward. Lemma 1. The allocation cures a i ( ), for 1 i n, are monotone increasing from to 1 and satisfy the inter-day IC constraints (5). Moreoer, each a i ( ) changes alue only at points where ˆΓ i ( ) is not ironed. Proof. That the allocation cures a i ( ) are weakly increasing, start out at, and end at 1 is immediate from the fact that they are conex combinations of the monotone allocation cures a ij ( ). Also, by construction, each a i ( ) changes alue only at points where ˆΓ i () is not ironed. So we hae only left to erify that a i 1 (x)dx a i (x)dx. From the discussion aboe, for r i, we hae a i 1 () = j j=1 (β j β j 1 )a i 1,j () and a i () = j j=1 (β j β j 1 )a i,j ()

13 since a i 1,j () = for r i and j > j. Thus, it suffices to show that for each j j and r i a i 1,j (x)dx a i,j (x)dx. If Γ i is not ironed at j, then this is an equality. Otherwise, for j, the left hand side is and the right hand side is nonnegatie. For j j the left hand side is ( j ), whereas the right hand side is δ( j ). Rearranging the inequality j = δ j + (1 δ) j δ j + (1 δ) implies that j δ( j ). This completes the proof that (5) holds. Notice that a i 1,j(x)dx = a i,j(x)dx for < j and > j, so a i 1 () = a i () unless Γ i is ironed at, or r i. We will use this fact in the proof of Claim 5.3 below. 5 Proof of optimality In this section, we proe that the allocation rules and pricing of the preious section are optimal. To this end, we formulate our problem as an (infinite) linear program. We discussed the objectie and constraints of the primal program in Section 2, and we hae already shown aboe that our allocation rules are feasible for the primal program. We then construct a dual program, and a feasible dual solution for which we can proe strong duality and hence optimality of our solution. 5.1 The linear programming formulation Recall the definitions from Section 2: The function γ i () is the deriatie of Γ i () = q iϕ i (x)f i (x) dx, where ϕ i () = 1 F i() f i () is the day i irtual alue function and q i is the fraction of bidders with deadline i. Similarly ˆγ i () is the deriatie of ˆΓ i (). We use [n] to denote the set of integers 1,..., n}. The Primal Variables: a i (), for all i [n], and all [, H]. Maximize n i=1 a i ()γ i ()d Subject to a i (x)dx a i+1 (x)dx i [n 1] [, H] (dual ariables α i ()) a i () 1 i [n] [, H] (dual ariables b i ()) a i() i [n] [, H] (dual ariables λ i ()) a i () i [n] [, H].

14 The Dual Variables: b i (), λ i (), for all i [n], and all [, H], α i (x) for i [n 1] and all x [, H]. Minimize [b 1 () + + b n ()] d Subject to b 1 () + λ 1() + b i () + λ i() + α i (x)dx b n () + λ n() α 1 (x)dx γ 1 () α i 1 (x)dx γ i () α n 1 (x)dx γ n () λ i (H) = [, H] (primal ar a 1 ()) [, H], i = 2,..., n 1 (primal ar a i ()) i [n] [, H] (primal ar a n ()) α i () [, H], i [n 1] b i (), λ i () 5.2 Conditions for strong duality: i [n] [, H]. As long as there are feasible primal and dual solutions satisfying the following conditions, strong duality holds. See Appendix A for a proof that these conditions are sufficient. b i () + λ i() + α i (x)dx b 1 () + λ 1() + b n () + λ n() a i (x)dx < a i () > λ i () continuous at i [n] (11) a i () < 1 b i () = i [n] (12) a i() > λ i () = i [n] (13) a i+1 (x)dx α i () = i [n 1] (14) α i 1 (x)dx > γ i () a i () = i = 2,..., n 1 (15) α 1 (x)dx > γ 1 () a 1 () = (16) α n 1 (x)dx > γ n () a n () = (17) We allow a i () R + }, otherwise we could not een encode a single-price auction The proof Theorem 2. The allocation cures presented in Subsection 4.2 are optimal, that is, obtain the maximum possible expected reenue. 6 In particular, a i() may hae (countably many) discontinuities, in which points a i() = + >. Howeer, in our proof of optimality a i() appears only as a factor of the product a i()λ i(). Eery time a i() = +, the corresponding dual alue of λ i() is by condition (13). See also Appendix A.

15 Proof. To proe the theorem, we erify that there is a setting of feasible dual ariables for which all the conditions for strong duality hold. To this end, set the ariables as follows: λ i () = Γ i () ˆΓ i () (18) < r i b i () = (19) ˆγ i () r i ˆγ i+1 α i () = i+1 (2) r i+1 From Claim 4.1, it follows that λ i (), α i () for all and i. Since r i is the minimum of ˆΓ i ( ), we hae ˆγ i (r i ) =. Moreoer, since ˆγ i ( ) is increasing, b i () for all and i. Taking the deriatie of (18), and using Equation (9), we obtain: γ i () λ ˆγ i () ˆγ i+1 () < r i+1 i() = (21) ˆγ i () r i+1 γ n () λ n() = ˆγ n () (22) Also, using (2) and the fact that ˆγ i+1 (r i+1 ) =, we get: H ˆγ i+1 () < r i+1 A i () := α i (x) dx = (23) r i+1 Condition (11) from Section 5.2 holds since Γ i () and ˆΓ i () are both continuous functions. The proofs of all remaining conditions for strong duality from Section 5.2 can be found below. Claim. Condition (12): For all i and, a i () < 1 = b i () =. Proof. If a i () < 1, then < r i, so by construction, b i () =. Claim. Condition (13): For all i and, a i () > = λ i() =. Proof. From Subsection 4.2, a i () > only for unironed alues of, at which λ i() =. Claim. Condition (14): For all i and, a i(x)dx < a i+1(x)dx = α i () =. Proof. As discussed at the end of the proof of Lemma 1, a i(x)dx = a i+1(x)dx unless Γ i+1 is ironed at, or r i. In both of these cases α i () = (by part 4 of Claim 4.1 and Definition 2, respectiely). Claim. Conditions (15)- (17) and dual feasibility: For all i and, a i () > = the corresponding dual constraint is tight, and the dual constraints are always feasible. Proof. Rearrange the dual constraint b i () + A i () A i 1 () + λ i () γ i() to b i () A i 1 () γ i () λ i() A i ().

16 Fact 1: For i [n 1], γ i () λ i () A i() = ˆγ i () for all. To see this use (21) and (23): γ i () λ i() = ˆγ i () ˆγ i+1 () < r i+1 A i () = ˆγ i () r i+1 ˆγ i+1 () Fact 2: For i 2,..., n}, b i () A i 1 () = ˆγ i () for all. < r i ˆγ i () < r i b i () = A i 1 () = ˆγ i () r i r i < r i+1 r i+1 Hence for i 2,..., n 1}, b i () A i 1 () = γ i () λ i () A i() for all. For i = n, since γ n = γ n, and γ n () λ n() = ˆγ n (). Combining this with Fact 2 aboe, we get that b n () A n 1 () + λ n() = γ n () for all. Finally, for i = 1, using Fact 1, for < r 1, we get which is true for < r 1. For r 1, we get b 1 () = ˆγ 1 () = γ 1 () λ 1() A 1 () b 1 () = γ 1 () = γ 1 () λ 1() A 1 (), so the dual constraint is tight when a 1 () > as this starts at r 1. The aboe claims proe that this dual solution satisfies feasibility and all complementary slackness and continuity conditions from Section 5.2 hold. A Proof of strong duality Theorem 3. Let a i ( ), b i ( ), λ i ( ), α i ( ) be functions feasible for the primal and dual, satisfying all the conditions from Section sec:cs. Then they are optimal. Proof. First, we proe weak duality. For any feasible primal and dual: i=1 = n b i () d n (1 b i () + [λ i () + α i ()]) d. (25) i=1 Applying primal feasibility, we see that this quantity is n i=1 (24) ( [ ] ) a i ()b i ()) a i()λ i () + a i (x) a i+1 (x)dx α i () d. (26) We rewrite this expression using the following.

17 Figure 4: This figure illustrates what some of the dual ariables might be for the case of two days when r 1 < r 2. The upper figure plots the functions ˆγ 12 () and ˆγ 2 (), and the lower figure shows b 1 () in dark grey, b 2 () in pink and A 1 () = α 1(x)dx in green. Note that up to r 2, the function A 1 () = ˆγ 2 ().

18 Applying integration by parts,using the facts that λ i ( ) is continuous (Condition (11)) and equal to at any point that a i () =,7 we get a i()λ i () d = a i ()λ i () since a i () = and λ i (H) =. Second, interchanging the order of integration, we get [a i (x) a i+1 (x)dx] α i () d = H + a i ()λ i() d = a i ()λ i() d, ( ) a i () α i (x) dx a i+1 () α i (x) dx d. Combining these shows that (26) equals ( H n [ a i () b i () + λ i() + α i (x) α i 1 (x) dx] ) d i=1 H a i ()γ i () d (27) i=1 where the last inequality is dual feasibility. (Note that α ( ) = α n ( ) =.) Comparing (24) and (27) yields weak duality, i.e., i b i() d i a i()γ i () d. If the conditions (11)-(17) hold, we also hae strong duality and hence optimality: To show that (25) = (26), obsere that (12) a i () < 1 implies that b i () = ; (13) a i () > implies that λ i() =. (14) (a i+1(x) a i (x))dx > implies that α i () = for i = 1,..., n 1. Finally, (27) is an equality rather than an inequality because of conditions (15)-(17). 7 a i() can be at only countably many points.