On Optimal Auctions for Mixing Exclusive and …...their own expected surplus. Online auctions...

MANAGEMENT SCIENCEArticles in Advance, pp. 1–24

http://pubsonline.informs.org/journal/mnsc ISSN 0025-1909 (print), ISSN 1526-5501 (online)

On Optimal Auctions for Mixing Exclusive and Shared Matchingin PlatformsHemant K. Bhargava,a Gergely Csapó,b Rudolf Müllerb

aGraduate School of Management, University of California Davis, Davis, California 95616; b School of Business and Economics, MaastrichtUniversity, Maastricht, Limburg 6200 MD, NetherlandsContact: [email protected], http://orcid.org/0000-0002-9268-2234 (HKB); [email protected] (GC);[email protected] (RM)

Received: November 1, 2017Revised: September 6, 2018Accepted: January 3, 2019Published Online in Articles in Advance:October 17, 2019

https://doi.org/10.1287/mnsc.2019.3309

Copyright: © 2019 INFORMS

Abstract. Platforms create value bymatching participants on alternate sides of themarketplace.Although many platforms practice one-to-one matching (e.g., Uber), others can conductand monetize one-to-many simultaneous matches (e.g., lead-marketing platforms). Bothformats involve one dimension of private information, a participant’s valuation for ex-clusive or shared allocation, respectively. This paper studies the problem of designing anauction format for platforms that mix the modes rather than limit to one and, therefore,involve both dimensions of information. We focus on incentive-compatible auctions (i.e.,where truthful bidding is optimal) because of ease of participation and implementation.We formulate the problem to find the revenue-maximizing incentive-compatible auctionas a mathematical program. Although hard to solve, the mathematical program leads toheuristic auction designs that are simple to implement, provide good revenue, and havespeedy performance, all critical in practice. It also enables creation of upper bounds on the(unknown) optimal auction revenue, which are useful benchmarks for our proposedauction designs. By demonstrating a tight gap for our proposed two-dimensional reserve-price-basedmechanism,we prove that it has excellent revenue performance and places lowinformation and computational burden on the platform and participants.

History: Accepted by Chris Forman, information systems.Funding: H. Bhargava thanks Google, Inc. for financial support through a research excellence gift forresearch in platforms. R. Müller thanks Province of Limburg, Netherlands, for financial support[Grant 2014-02207].

Keywords: information systems • IT policy and management • electronic markets and auctions • platforms • marketing • pricing • auctions

1. IntroductionTechnology-enabled platform marketplaces facilitatemultiple groups of trading partners (say, shoppersand merchants) to congregate, discover, and transactwith each other (Choudary et al. 2016). Because plat-forms focus on enabling value creation and exchangerather than value production itself, a vital functionis to match shoppers and merchants (Evans andSchmalensee 2016).1 For many platforms, matchingis the primary or only source of revenue. An exampleis the lead-marketing firm BuyerLink.com (formerlyReply.com), which matches merchants and shoppersfor automobiles, real estate, and insurance products.The platforms of interest in this paper operate at largescale and high speed, often conducting hundreds orthousands of matches per minute, and each matchneeds to be decided instantaneously when the op-portunity occurs. For instance, an online advertisingplatform must display ads as the user scrolls or switchesbetween pages. For these reasons, many platformsimplement real-time auctions that allow participantsto express preferences or prices and in which the

platform chooses matches to maximize some businessobjective.Many platforms pair each shopper exclusivelywith

one merchant (examples are CreditKarma and Uber).For such one-to-one matching auctions, the theoret-ical and technological infrastructure is relatively wellstudied (Milgrom 2004, Varian 2009). Myerson (1981)provides a thorough analysis and design of the revenue-maximizing 1:1 auction under standard conditionswith rules for allocations, prices, and the optimalreserve price (see Section 2). Some other platformsconduct 1:kmatching; that is, the same shopper couldbe simultaneously matched and monetized to k mer-chants. This is common in lead-marketing platforms,such as BuyerLink, with which (taking an automobilecontext) a single web userwho has displayed intent topurchase a car is matched, typically, with three dealerswho must share (and compete for) the opportunity totrade with the shopper. For such 1:k auctions, also,Myerson (1981) provides a thorough analysis for therevenue-maximizing auction and rules for the opti-mal reserve price.

1

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http://orcid.org/0000-0002-9268-2234

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https://doi.org/10.1287/mnsc.2019.3309

This paper examines the opportunity that manyplatforms have for mixing the 1:1 (exclusive) and 1:k(shared) auction formats, with k being a predefinedand small integer (e.g., k ! 3 in the case of Buyer-Link).2 In amixed-format auction,merchants can seekboth exclusive and shared matches by communicat-ing two separate bids, and the platform chooses boththe format and allocations after observing the bids.Over time, the platform can then produce a mix ofexclusive and shared matches. For platforms that use1:k auctions (with fixed k), the mixed format is nat-urally superior. It may also apply to platforms thatpresently use 1:1 matching but with which the busi-ness context allows for kmatches (e.g., CreditKarma).The underlying idea is that merchants have highervalue for exclusively being matched with a shopperthan for sharing the match with other merchants, but asharedmatch creates a higher total value than the highestindividual match. Because of this, the preferred format(with respect to some objective, such as platform reve-nue) depends on differences between the two valuations,how these differences are distributed across merchants,and what valuations are drawn in different instances.

Merchants participating in a mixed-format matchingplatform can seek both a shared match and, for a higherprice, an exclusive one-to-one match. Higher value forexclusive purchase can be fueled by the threat of com-petition, by a sense of privilege, or by special customerpreferences (e.g., luxury goods). For a platform thatfaces many and diverse matching opportunities (e.g.,the automobile segment of BuyerLink’s lead-marketingbusiness covers leads for 300 models across about35,000 zip codes), picking the mode of matching basedon instance-specific information could produce higheraggregate revenue than an ex ante commitment. Ex-ample 1 illustrates the benefit of this hybrid mode ofmatching as well as the challenge in designing rulesthat allow for both modes of matching.

Example 1. Consider a shopper in zip code 60173 whohas expressed interest in a BMWmini. The platform hasfive dealers in this area who are interested in such shop-pers, and it can connect the shopper simultaneouslywith up to three dealers. Dealers’ reservation valuesfor shared and exclusive purchase under two alternatescenarios are listed here.

Under these values, the surplus-maximizing solu-tion is a shared allocation (to dealers 1 [D1], 3, and 4

with total surplus 35) in scenario 1 and exclusiveallocation (to dealer 3 with surplus 38) in scenario 2.However, typically, the platform cannot capture thismaximum surplus as revenue because valuations areprivate to bidders, and they pick bids to maximizetheir own expected surplus. Online auctions generallycreate incentives for truthful bidding of valuations byemploying some variant of Vickrey pricing in whichwinners pay the highest losing bid. If the platformlimited itself to a single mode of matching across allscenarios, then, with Vickrey pricing, it would gen-erate a revenue of (1) 20 and 29 in scenarios 1 and 2,respectively, under 1:1 matching (winner pays thesecond-highest bid) and (2) 27 in both scenarios under1:k matching (every winner pays the highest losingbid of nine).Although these allocations maximize total surplus

and incentivize truthful bidding, the limitation to 1:1matching misses some revenue opportunity in sce-nario 1, and the limitation to 1:k matching misses arevenue opportunity in scenario 2. There are three well-known ways for the platform to enhance its revenue:(1) by using the revenue-maximizing auction withinalways-shared allocations, (2) by using the revenue-maximizing auction within always-exclusive allocations,and (3) by a mixed auction that decides the allocationformat based on bids and uses Vickrey–Clarke–Groves(VCG) prices (which we discuss in detail in Section4.2). We explain and illustrate all these approaches inSection 4, but none of them yields the optimal revenuefor the platform.What, then, is the revenue-maximizing auction design

in this setting? It is evident that we need a design thatintelligently switches ex interim between shared andexclusive matches. Unlike always 1:1 or always 1:kmatching, with which the revenue-maximizing auc-tion design is well understood, the correspondingproblem for mixed-format auctions is notoriously hardto solve when the auction leverages two-dimensionalprivate information (merchants’ valuations for sharedand exclusive allocation). Only a few papers have studiedmixed-format auctions, and with rather modest goalson account of the complexity of analyzing them(Section 2 provides additional details). Yet mixing thetwo formats is valuable because a well-designed auctionmechanism can generate higher revenue than the best-posted-price design in this setting (Deng and Pekec2013). Our primary objective and contributions in

Valuation D1 D2 D3 D4 D5 D1 D2 D3 D4 D5

Shared purchase (si) 10 8 14 11 9 10 8 14 11 9Exclusive purchase 25 20 20 15 9 29 20 38 15 9Exclusivity margin (mi) 15 12 6 4 0 19 12 24 4 0

Scenario 1 Scenario 2

Note. Cells with valuations in bold-face indicate the winning bidders.

Bhargava, Csapó, and Müller: Mixing Exclusive and Shared Matching in Platforms2 Management Science, Articles in Advance, pp. 1–24, © 2019 INFORMS

this paper are to formally study mixed-format auctions,develop useful theoretical results about them, andpresent a heuristic design that we demonstrate hashigh performance and is relatively easy to implementin practice.

We develop our reserve mechanism (RM) byextending the optimal reserve-price approach, whichunderpins the design of optimal 1:1 and 1:k matchingauctions. RM’s allocation rule relies on reserve pricesthat are configured based on the prior information(distributions of shared and exclusive valuations),and it picks the mode of matching (one versus mul-tiple winners) based on observed bids. RM belongs tothe class of affine maximizers, which form a gener-alization of VCG mechanisms. We provide a theo-retical and empirical comparison of RM against othercandidate designs mentioned, as well as against theunknown revenue of the optimal mechanism.

Our theoretical and practical progress in solving thetwo-dimensional auction-design problem involves athree-step approach. First, we formulate the search foran optimal mechanism in the space of mechanisms as amixed-integer mathematical program (see Section 3.2).Second,wedevelop amethod for generating specialized,parameterized mechanisms that can be easily imple-mentedandwhoseparameters canbe efficiently computed.These candidate solutions are presented in Section 4.Third, we develop techniques for estimating tight boundsfor the original problem (Section 5). We employ thesebounds to demonstrate the revenue performance ofthe proposed specialized mechanisms (Section 6) andto show that our proposed design (RM, a reserve-price mechanism that uses bid data to pick a one-to-one or one-to-many mode of matching) outperformsalternative and naive mechanisms (such as the opti-mal auction under an only-exclusive [OE] or only-shared [OS] policy or with VCG prices). Furthermore,a notable aspect of our work is to show that thesimplest auction designs in this setting, namely OEor OS, produce revenue within a constant factor ap-proximation of the optimal revenue. Finding simplemechanisms that provide constant approximationsconstitute an important line of research in mechanismdesign because it can explain salient features of opti-mal auctions and shed light on the practical success ofsimplemechanisms in complicated environments (see,e.g., Hartline 2012). Our results require mild condi-tions and that distributions of shared valuations andexclusivemargins are independent andhaveamonotonehazard rate.

Before proceeding further, we make a quick noteon terminology. Heretofore, we described two-sidedplatforms as matching shoppers on one side withmerchants on the other. As the paper moves forwardto discuss thematchingmechanism—an auction—it isuseful to adapt standard auction terminology. Hence

a merchant becomes an agent who participates as abidder in the auction, and a shopper is the item beingauctioned.

2. Related LiteratureMyerson (1981) showed how to design a revenue-maximizing auction for an item when private informa-tion is given by a single parameter. His approachprovides a closed-form characterization of the optimalmechanism under mild assumptions for most of thecases. In particular, if an agent’s valuation is (to theplatform) a random variable v with distribution D(v)with monotone nondecreasing hazard rate (MHR)and density d(v), then the optimal mechanism is adirect auction in which the highest bid wins as longas it meets the reserve price rD. The optimal reserveprice is determined as

rD ! v : φD(v) ! v − 1 −D(v)d(v)⏟ ⏞⏞⏟

virtual valuation φD(v)

! 0

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (1)

That is, the optimal reserve price is the value v at whichvirtual valuation φD(v) for distributionD equals zero.3

The payment is the minimum bid needed for this agentto win. MHR ensures the uniqueness of the optimal re-serve price. Distributions in this class have a tail notfatter than that of the exponential distribution and in-clude the normal, exponential, some parameteriza-tion of gamma, Pareto, and uniform distributions.The Myerson (1981) apparatus extends to auctions

that allocate k ≥ 1 identical items as long as privateinformation about agent type is described by a singleparameter (i.e., winners experience no value penaltywhen winning shared rather than exclusively). Incontrast, for settings with multidimensional types,the optimal mechanism-design problem quickly be-comes intractable, and there has not been a generalframework so far to treat these problems. Some re-searchers use linear programs to compute approxi-mations (e.g., Cai et al. 2011), but the solutions arenonpractical because they must be described by anexplicit table of inputs (bids) and outputs (allocationsand prices) that is exponential in the number of biddersand items. Furthermore, the optimal mechanism isusually tailored in a complexmanner to specific detailsof the distribution on agent preferences. Thus, even ifone finds the optimal mechanism, it is generally toocomplicated to implement in practice.There are different attempts to circumvent the in-

herent complexity in multidimensional mechanismdesign. One approach is to assume that apart froma single parameter of private information, the rest ofthe parameters are public information. This approach

Bhargava, Csapó, and Müller: Mixing Exclusive and Shared Matching in PlatformsManagement Science, Articles in Advance, pp. 1–24, © 2019 INFORMS 3

captures relevant information while maintaining thecomputational advantageof a single-dimensionalmodel.This path was chosen by Deng and Pekec (2013), whoassumed that the exclusivity margin is correlated withvalue for shared allocation, thereby leaving the latter assingle-dimensional private information.4 Because ofthe single-dimensional assumption, they can applythe techniques of Myerson (1981) and derive the opti-mal mechanism. However, for the problem discussedin this paper, their approach is unsuitable because iteliminates the cases in which the ability to dynami-cally choose between shared and exclusive allocations(versuspredetermined exclusive- or shared-only) ismostrelevant (i.e., when exclusivity margins are heteroge-neous, distributed differently across problem instances,and not correlated with shared valuations). In contrast,ourmodel keepsmultidimensionality, and then, becausethe Myerson (1981) framework does not work anylonger, we develop an innovative solution techniquein which the Deng and Pekec (2013) framework ishelpful in measuring the quality of our solution.

Salek and Kempe (2008) and Pei et al. (2014) studythe problem of selling a digital good with an unlimitedsupply of copies to bidders whose value for the good isdecreasing in the number of bidders obtaining it. Theyassume that the function according towhich the valuationdepreciates is public information, making their settingsingle dimensional. Salek and Kempe (2008) providethe revenue-maximizing Bayes–Nash implementablemechanism based on the Myerson (1981) techniques,and Pei et al. (2014) adapt the prior-free “single-sample”mechanism from Dhangwatnotai et al. (2015) thatyields a constant approximation for that setting. Theirtheorems hinge on the assumption that types are sin-gle dimensional and independently distributed accord-ing to monotone hazard rate distributions. Moreover,besides the fact that the single-sample mechanism isnot deterministic, it does not even extend directly toour setting because applying reserve prices combinedwith the VCG mechanism (see Vickrey 1961, Clarke1971, Groves 1973) is not incentive compatible in gen-eral, demonstrated by Example 2.

Sayedi et al. (2018) study an extension of generalizedsecond price auctions for sponsored search advertisingpatented by Yahoo. This auction takes two bids asinput: one bid for being displayed among other adsand a second for exclusive display. Because truthtelling is not a dominant strategy in that auction, theauthors aim to identify and analyze bidding strategiesthat lead to a Bayes–Nash equilibrium. To render theiranalysis tractable, they restrict themselves to a highlystylized setting, including only three agents: two havingno exclusivitymargin and onewhose exclusivitymarginis a fixed portion of the shared valuation. They find thatallowing advertisers to bid for exclusivity usually in-creases the search engine’s revenue (because of an

increased competition effect) but that revenue mayfall under certain conditions.Cai et al. (2011, 2013) investigate multidimensional

mechanisms that are Bayes–Nash incentive compat-ible. They also admit randomizedmechanisms,whichallows them to use linear programming to acquire apolynomial-time approximation scheme. Their methodis not applicable for our problem because it does nothandle allocational externalities; that is, the value fromwinning depends on allocations of other bidders. More-over, if their approach is employed to construct de-terministic and dominant strategy incentive-compatiblemechanisms, it yields an integer linear programwith anexponential number of variables and constraints, whichrenders this approach impractical.Another line of research focuses on heuristic mecha-

nisms, which possess a succinct description and mightexhibit a guaranty on expected revenue. In relation toour problem, the work of Devanur et al. (2011) andDhangwatnotai et al. (2015) bears relevance. Theydeal with multiparameter mechanism design involvingunit-demand bidders, regular-type distributions, andmatroid or downward closed feasibility constraints.They derive simple mechanisms that achieve constantapproximations of the optimal revenue. The key pointof their proof is to employ the solution from a single-dimensional analogue as anupper bound for the optimalrevenue. Then the revenue of their simple mechanismis compared with that upper bound. Despite the simi-larities, these results cannot be used directly for ourproblem because their setting cannot accommodateallocational externalities. Even if one treats the differentallocations as items, the corresponding feasibility con-straints are not downward closed, which is a necessaryassumption in theirmodel. Thewaywederive the secondupper bound is reminiscent of the technique developedin Devanur et al. (2011) in the sense that we also splitthe multidimensional agents into single-dimensionalrepresentatives. The twist in our method is that tohandle allocational externalities, we endow the rep-resentatives with interdependent valuations.Allocational externalities are similar in nature to

interdependent valuations and identity-dependentexternalities in the sense that they all try to catch theimpact of the agents on each other. The implications ofexternalities have been studied in various settings;see, for example, Jehiel et al. (1996), Segal (1999), andFigueroa and Skreta (2011). InAseff andChade (2008),the optimal auction for selling two identical goods isconsidered; bidders impose externalities on each other.Their model can be parameterized such that it coincideswith a special case of Deng and Pekec (2013), but it islimited to selling two items, multiplicative externality,and single-dimensional valuations.The idea of using generalizations of VCG mecha-

nisms, called affine maximizers, to improve on the


revenue of the VCG mechanism has appeared mostlyin connection with combinatorial auctions (Nisanet al. 2007). Likhodedov and Sandholm (2004) spec-ify a special class of affine maximizers and try to fine-tune its parameters. Tang and Sandholm (2012) con-sider the case of two bidders and two items and searchfor the best parameters for a given affine maximizer.RM is also an affine maximizer, but its parameteriza-tion idea differs from that of the two previous papers.Moreover, by deriving upper-bound mechanisms, wecan demonstrate in a novel way how small the gap isbetween the affine maximizer and the optimal revenue.

3. ModelSuppose that an item can be sold either exclusively toone bidder or shared simultaneously with k out of nbidders (in the set N ! {1, . . . , n}), who each have unitdemand for the item. The maximum level of sharing kis predecided based on regulation, seller policy, buyerpreferences, or other such constraint. For example, thedisplay of video ads on a smartphone or a YouTubevideo would permit at most a handful of simultaneousads. The value of bidder (or agent) i for obtaining theitem is characterized by the bidder’s type ti ! (si,mi),where si is the valuation for getting the item sharedwith other agents, and mi is the margin for exclusivepossession; that is, si+mi is the valuation for receivingthe item exclusively. Each bidder’s type ti ! (si,mi) isprivate information to the bidder. This two-dimensionaltype explicitly captures valuation for exclusive owner-ship and for sharing with k−1 others. What about theshared valuation when sharing with fewer than k−1others? Shared allocation with fewer than k biddersmay be optimal under certain circumstances (e.g., whenthere are fewer than k shared bids that meet the re-serve price). To allow such allocations, we assume thesimplification that si also represents i’s value whensharing with fewer than k−1 others rather than solic-iting all values, which would lead to a k-dimensionaltype and create more complexity for bidders.

To the platform (and other bidders), (si,mi) are ran-domvariables independently drawn from cumulativedistributions F and G, respectively (both strictly in-creasing with corresponding densities f and g), oversets S andM. The set of possible types ti for each agentis Ti ! S ×M. Let t ! (t1, t2, . . . , tn) be the vector ofagent types drawn from the cross-product T ! ∏i T

i !(S ×M)n. Let A be the set of all feasible allocations sothat each a(t) represents the set of agents who receivethe item under bid vector t (which might, in principle,differ from the true types of the agents). In our setting,A is contained in the power set of N, covering subsetsof size 0, 1, . . ., k. The distribution of type ti is F × G,and the distribution of type tuples is (F × G)n. Withrespect to optimal reserve price computation (seeEquation (1)) in this two-dimensional context, the

valuation v could correspond to one of s, m, or s +m sothat D would be F, G, or C (convolution of F and G).The basic notation used in this paper is summarizedin Table A.1, which also has a list of acronyms(Table A.2).Following the revelation principle (Myerson 1981),

we restrict our attention to direct auctions. Each directauction (a, p) can be characterized by its allocationrule a : T → A and its payment scheme p : T → Rn.Note that a(t) would be a singleton under exclusiveallocation, and |a(t)| ≥ 2 represents shared allocation.Given an auction (a, p) and report profile t ! (s, m), therealized valuation v of agent i is

vi(a(t), ti)

!

si if 1< a(t)

≤k and i ∈ a(t),agent i receives a shared allocation( )

si +mi if a(t) ! {i},agent i receives exclusive allocation( )

0 otherwise.

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

The net utility (after subtracting payment) is ui a(t),(

p(t), ti) ! vi(a(t), ti) − pi(t).

3.1. Design Goals and ChoicesConsider a particular category of the auction-designproblem faced by a mixed-format matching platform,that is, with specific distributions F and G that de-scribe shared valuations and exclusivity margins. Ide-ally, the platform chooses a revenue-maximizing designwith respect to these distributions, meaning that itchooses, among all imaginable allocation and pricingrules, a combination that maximizes expected revenue,assuming that agents follow a utility-maximizing bid-ding strategy. As outlined in Section 2, finding suchrules in our setting is not only a notoriously difficultproblem given the multidimensional nature of valu-ations but also may give complex designs that aredifficult to execute and communicate to bidders and,last but not least, to play. Therefore, our design objectivecombines both profitability and practicality, that is,maximizing auction-revenue subject to designs that aresimple enough and impose low costs on all participants.From the platform’s point of view, the design di-

lemma implies choosing between mechanisms thatpromote truthful bidding versus those under whichagentsmight schemeandshade their bids in equilibrium.In general, the best among the latter kinds of auctionsmay produce higher revenue but at higher partici-pation costs for agents because agents must computehow to shade their bids while making assumptionsabout and anticipating the behavior of other agents.These costs may be substantial (relative to the simpler“bid your true value” rule) and can impose high costs


on the auctioneer as well. If agents cannot reliablyidentify their optimal bidding strategies, then realizedrevenue may fall short of predicted revenue. Hence, ourjoint objective of profitability and simplicity steers us inthe direction of incentive-compatible mechanisms, thatis, rules that maximize profits while strongly encour-aging truthful bidding and thereby being easy for agentsto participate in. Formally, this defines a feasible spaceof auction designs in which truth telling is a dominantstrategy for agents.

Definition 1 (Dominant Strategy Incentive Compatible). Adirect auction is dominant strategy incentive compatible(DSIC) if truth telling is a dominant strategy for eachagent: given the other agents’ bids, every agent’s utilityis maximized by bidding truthfully. Formally, a directauction (a, p) is DSIC if, for every i, t−i, ti, and ti, it holdsthat ui a(t), p(t), ti( ) ≥ ui a(ti, t−i), p(ti, t−i), ti( )

.Next, we specify another natural requirement that

we impose on the auction design, which is that truthtelling leads to nonnegative utility for every agent.

Definition 2 (Ex Post Individual Rational). A direct auc-tion is ex post individual rational (EPIR) if a truth-tellingagent has nonnegative utility for every report of otheragents. Formally, a direct mechanism (a, p) is EPIRif, for every i and t ! (ti, t−i), it holds that ui a(t),(p(t), ti) ≥ 0.

3.2. Mathematical Programming FormulationThe design goals stated can be summarized as follows:given distributions F and G, find a direct auction (al-location and pricing rules) that maximizes expectedrevenue while preserving individual rationality (EPIR)and truthful bidding (DSIC). We encode these goalsinto a mathematical programming formulation thatwe call the dominant-strategy auction (DSA). The pa-rameters in the objective function are determined bythe priors of the distributions, and the constraintsrepresent incentives and allocation feasibility. Oncesolved, the decision variables tell, for each report oftypes, who receives the item and prices to be paid.Because it is DSIC, agents’ reports are simply theirtrue values. Then the mathematical program is

maxa(t),p(t)

Et∑

ipi(t)

[ ](DSA)

subject to

ui a(t), p(t), ti( ) ≥ ui a(ti, t−i), p(ti, t−i), ti( )

∀i,∀t−i,∀ti,∀ti. (DSIC)

ui a(t), p(t), ti( ) ≥ 0 ∀i,∀t, (EPIR)a(t)| | ≤k ∀t. (SUPPLY)

The formulation can easily be transformed into theusual integer-programming format by defining binary

variables xi(t) ! 1 if i ∈ a(t) and zero otherwise. Theoptimal solution (a, p) for (DSA) is denoted as DSA∗.We note that the DSA formulation is, for us, pri-

marily a theoretical construct. It has a huge number ofvariables and constraints, exponential in input size.Even if we could solve it, that solution would not bepractical because it would only provide a full list ofa(t) and p(t)vectors, one for each t. For us, the programserves a different purpose. On the one hand, it enablesus to generate alternative practical mechanisms, eitherby adding certain constraints to this program (seeSection 4.1) or via a special class ofDSIC two-dimensionalauctions (presented in Section 4.2). Optimal solutionsto these refined programs are, therefore, feasible so-lutions to the original DSA program and providelower bounds to the optimal DSA revenue. On theother hand, we develop relaxations of this program thatreadily can be optimized, thereby generating upperbounds for the optimal solution of (DSA) and en-abling performance evaluation of the heuristic solu-tions (Section 5). Combining these two features, weshow how to achieve an implementable mechanismwith provable performance, and indeed, we dem-onstrate in Section 6 that this mechanism has excellentrevenue performance.

3.3. MonotonicityBefore proceeding to auction design, we demonstratehow the search for feasible solutions of (DSA) can beguided by adding some superfluous constraints. Con-sider a pair of DSIC constraints on two potential bidvectors that differ in the ith agent only while keepingt−i fixed. Bid vector 1 has an ith type ti, and bid vector2 has an ith type ti. Apply type ti’s DSIC constraint tothe report ti and apply type ti’s DSIC constraint to thereport ti to get

vi a(ti, t−i), ti( ) − pi ti, t−i( ) ≥ vi a(ti, t−i), ti( ) − pi ti, t−i

( ),

vi a(ti, t−i), ti( ) − pi ti, t−i( ) ≥ vi a(ti, t−i), ti( ) − pi ti, t−i

( ).

Adding these two constraints together and reorder-ing their terms yields the weaker but more intuitiveconstraint

vi a(ti, t−i), ti( ) − vi a(ti, t−i), ti( ) ≥ vi a(t), ti( ) − vi a(t), ti( ).

(MON)

Allocation rules satisfying (MON) will be calledmonotone. By construction, designs that fail this mono-tonicity property will not permit an incentive-compatiblepayment rule.

4. Candidate Auctions (RevenueLower Bounds)

Considering the difficulty of identifying the optimalDSA auction, a one-to-many platform provider could


start by accepting heuristic solutions of the designproblem.We start by describing a few auction designsthat can be derived from the literature and that canserve as benchmarks (lower bounds) for any newdesigns and for the optimal auction. The first two ofthese involve a precommitment to OE or OS alloca-tions across all (F, G) variations that might occur for amatching platform. The third one, MaxSimple, pre-commits the platform to either the exclusive or sharedformat with the choice based on information (distri-butions F and G) particular to each product category.These three designs are single-dimensional auctions;that is, they involve only one piece of private in-formation and require only a single-dimensional bid.These three benchmarks are presented in Section 4.1.

Our fourth benchmark is a two-dimensional auc-tion whose design is based on VCG principles, whichwe present in Section 4.2. Next, we develop a newtwo-dimensional format inspired by fundamentalprinciples in optimal auction design. Specifically, itapplies theoretical principles of affine maximizerdesigns (because these guarantee truthful bids) andextends the optimal reserve price approach to two-dimensional bids. Accordingly, we call this mecha-nism RM (for reserve-price mechanism). The readerinterested in a working example of any or all of thesemechanisms may jump ahead to Section 4.3. Section 6demonstrates that RM outperforms VCG and the single-dimensional optimal designs, and we also show that itsrevenue is a close approximation to the optimal revenuefromDSA∗. For the latter, because DSA∗ is not known,we construct a new, tight upper bound on DSA∗ (inSection 5) and show (in Section 6) that RM revenue isrobustly close to this upper bound across a variety ofparameter settings.

4.1. One-Dimensional Benchmark AuctionsWe start with three simple auction designs that, byconstruction, provide lower bounds on the revenuefrom the optimal auction. These designs (1) alwaysallocate exclusively to a maximum of one bidder,sacrificing the possibility that shared allocation isbetter, (2) always allocate shared and never allocate toonly one bidder, sacrificing the possibility that ex-clusive allocation is better, and (3) pick the optimalshared versus exclusive allocation auction based onpriors about distribution of valuations. The formu-lation as a mathematical program is (DSA) with anadditional constraint, either a(t)| | ≤1 (exclusive only)or 1< a(t)| |< k for all j ∈ N (shared only). All thesedesigns require only one-dimensional bids (eithersi+mi or si), are solvable with the Myerson (1981)apparatus, and are simple to implement (becauseagents can bid truthfully). If any of these designsproduced sufficiently good revenue performanceacross the problem parameters, then we would have an

acceptable design. Unfortunately, as we show in Sec-tion 6, none does. However, they do provide lowerbounds for the original optimal auction problem (DSA).We capture the details of the two Myerson (1981) de-signs and their properties in the following two the-orems. The proofs are presented in the appendix.

Theorem 1 (Optimal Only Exclusive Auction). If F and Gare MHR (and, consequently, C is the convolution of thetwo), then the optimal expected revenue from always allo-cating exclusively to one agent is

Rev(OE) ! Et∑

i∈aOE(t)φC(si +mi)

[ ]. (2)

The optimal auction that achieves this revenue has allo-cation rule “award to bidder with highest exclusive virtualvaluation if nonnegative,” that is,

aOE(t) ! argmaxα⊆ Aα| |≤1

∑

i∈αφC(si +mi)

( )(3)

(with ties broken arbitrarily in case of multiple optimalarguments), and payment rule “lowest bid necessary tobecome a winner,” that is,

piOE(t) !inf{si + mi | i ∈ aOE((si, mi), t−i)} i ∈ aOE(t),0 otherwise.

{

(4)

ForOE, the design ensures that the item is never allocatedif the highest virtual valuation is negative, that is, ifthe highest bid is below the reserve price. With sharedallocations, we see that this intuitive requirement maybe defeated.

Theorem 2 (Optimal Only Shared Auction). If F is MHR,then the optimal revenue from sharing among at most kagents (but never exclusively to one) is

Rev(OS) ! Et∑

i∈aOS(t)φF(si)

[ ]. (5)

The optimal auction that achieves this revenue has allo-cation rule “award based on up to k highest shared virtualvaluations,” that is,

aOS(t) ! argmaxα⊆ N

α| |≤k, α| |!1

∑

i∈αφF(si)

{ }(6)

(with ties broken arbitrarily in case of multiple optimalarguments), and payment rule “lowest bid necessary tobecome a winner” (aOS, pOS) because

piOS(t) !inf{si | i ∈ aOS((si,mi), t−i)} if i ∈ aOS(t),0 otherwise.

{(7)

The specification and allocation rule of OS deservessome explanation. The requirement that we always


allocate to at least two bidders exists to prevent ex-clusive allocations (i.e., to obtain incentive compati-bility in shared bids) and has the consequence thatsometimes a bidder might share an item even if thevirtual value is below zero, or in other words, thevalue is below the reserve price, and the bidders payprices that are below the reserve prices. Although thissounds counterintuitive, never allocating to a singlebidder is necessary to preserve incentive constraints.If not, a bidder might get an exclusive allocation andwould enjoy extra value from this allocation. In asetting in which the bidder and all other bidders haveshared value below the reserve price, the bidder wouldhave incentives to report a higher shared value than thebidder’s true one, securing such an exclusive alloca-tion. This demonstrates that platforms that limit theexpressiveness of bids for the sake of simplicity needto design allocations carefully so as not to invite strate-gic bidding related to information that is not revealedin the auction.

Designs that commit the platform to either exclu-sive- or shared-only allocations for all instances orproblem categories require only a one-dimensionalbid. Between these two, intuitively, the choice de-pends on the magnitude and distribution of s and msets across multiple instances. Alternately, a platformmight wish to exploit category-specific prior infor-mation (e.g., leads for BMW buyers in zip code 93940during June) and then pick the better of exclusive andshared based on expected revenue performance giventhe prior and then commit to it for that category inadvance. This idea is implemented in the MaxSimplemechanism in our framework. For each category, partici-pants need to provide only one bid (for exclusive orshared allotment as specified by the platform), but whichkind it is can vary across categories.

Definition 3 (MaxSimple Mechanism). For any instanceof the exclusivity auction problem, MaxSimple calcu-lates the expected revenue of OS and OE AND thencommits the mechanism with the highest expectedrevenue.

Note that the choice in MaxSimple is made up frontfor given priors F and G and not for every type reali-zation; therefore, the actions of the agents do not influ-ence which of the two single-item auctions is executed.Despite the fact that MaxSimple capitalizes only on onetype of bid and allocation, a later result (Theorem 5,which can be established only after setting up addi-tional results) reveals that the seller can still capture aconstant fraction of the optimal revenue by imple-menting it. Constant factor results are useful in thedesign of approximate auctions because they providea performance assurance (Alaei et al. 2012). Note thatalthough OE and OS are the revenue-maximizingauctions within their classes of allocations (always assign

to just one or always adjust to multiple merchants), thebenefit of MaxSimple is that it enables the platform tomake a category-specific choice between the two. More-over, MaxSimple can potentially be a better lower boundfor (DSA) than either OE and OS. Although both pro-vide useful lower bounds, the quality of these boundsvaries considerably across the parameter regions, andthe quality variation generally moves in opposite di-rections, making it worthwhile to examine the use ofMaxSimple as a lower bound. The working of OE, OS,andMaxSimple is illustrated inExample 3 in Section 4.3.Although MaxSimple is intuitive and simple, it is a

combination of two different mechanisms. Therefore,if a platformprovider hasmultiple similar products tooffer, then changing the rules and the type of allocationproduct by product to derive a revenue advantage fromthe alternate rule5 can create confusion among the buyers.In addition,MaxSimple is only singledimensional; thus, itdoes not take advantage of the extra information andcreate awider offermenu thatmight be possible by takinginto account both types of valuations and allowing forboth types of allocations ex interim. The next naturalstep, accordingly, is to devise mechanisms that con-sider two types of bids: one for shared and one forexclusive allocations.

4.2. Two-Dimensional Auctions: VCG and RMTurning toward using both the dimensions of privateinformation, one appealing candidate design is tocombine a VCG pricing rule with allocations thatmaximize agent value (i.e., to allot exclusive or sharedbased on whether the highest exclusive bid exceedsthe sum of the k highest shared bids). Formally, theallocation rule assigns to agent i exclusively if (si + mi)exceeds (or equals) the sum of the highest k s valuesor, otherwise, to the first k agents in descending orderof sj values (using some tie-breaking rule when nec-essary). The VCG pricing rule charges every bidderthe externality the bidder poses on other bidders andthereby eliminates bid-shading incentives that ariseunder a pay-your-bid pricing rule.More precisely, theprice bidder i has to pay equals the total value of thewelfare-maximizing allocation for other bidders hadthebiddernotparticipated in the auctionminus thevaluethat all other bidders get when the bidder is partici-pating. Note that for bidders who do not win the item,this results in zero payments. The working of VCG isillustrated in Example 3 in Section 4.3.Although VCG maximizes total welfare (sum of

valuations) in each allocation and is incentive com-patible, it can have poor revenue performance, as weshow in Section 6. Further, because of revenue equiva-lence, there is noway to increase paymentswithout losingDSIC and EPIRwhilemaintainingwelfaremaximization.We therefore need to give up on the welfare-maximizingallocation and include the allocation rule in our search


space to improveon the revenueperformanceofVCG. Tothis end, we propose a new mechanism (RM) alongthe lines developed in the OE and OS methods whileemploying both sets of valuations and preservingtruthful bidding.

To define the RM auction, we recall the definition ofan affine maximizer, also called a generalized VCGauction. Recall thatA is the set of all feasible allocationsfor agents inNwith index α ranging over this set, andA−i is defined analogous to A for N\i.Definition 4 (Affine Maximizer Auctions). A mechanismwhose allocation is described by a(t) for each t is anaffine maximizer if it maximizes a weighted sum of val-uations across a setA of feasible allocations. That is, if thereexists a vector of γα values (each γα ∈ R) and a vectorof λi values (λi ∈ R+ across all agents i ∈ N) such that

∀ t ∈ T, a(t) ! argmaxα∈A

γα(t) +∑

i∈Nλivi(α(t), ti)

( ). (8)

The payment rule of the affine maximizer mechanismis that the winner pays the winner’s impact on theaffine welfare of others equal to

pi(t) ! 1λi γa(t−i) +

∑

j∈N\iλjvj(a(t−i), tj)

( )[

− γa(t) +∑

j∈N\iλjvj(a(t), tj)

( )], (9)

where

∀ t ∈ T−i, a(t) ! argmaxα∈A−i

γα(t) +∑

j∈N\iλivi(α(t), tj)

( ).

Because affine maximizer mechanisms are DSICand EPIR (Roberts 1979), every affine maximizer is afeasible solution of (DSA). Choosing λi ! 1 for all i ∈N and γb ! 0 for all b ∈ AN yields VCG. Using theseobservations, we define the RM as a mechanism that,for each bid combination, decides on a shared or anexclusive allocation depending onwhichmaximizes aparticular affine function.

Definition 5 (RM). Let rF and rC denote the shared andexclusivity reserve values, respectively (see Equa-tion (1)). Then RM is defined as an affine maximizer,where λi ! 1 for all i, A ! {a ⊆ N | a| | ≤k}, γ{i} ! −rCfor all i, γ∅ ! 0, and γa ! −rF a| | for all a ⊆ N such that2 ≤a| | ≤k.

To explain how RM works, let us define the affinevalue under RM. For any subset of bidders α such thatα| | ≤k, the affine value under RM is

−rC + (si+mi) if α ! {i},∑

i∈α(si−rF) if α| | ≥ 2. (10)

Then RM makes an exclusive allocation a ! {i} forsome bidder i only if (si+mi) ≥ rC, and the affine value(si+mi−rC) dominates the affine value of any othersubset α. The basic scenario for shared allocation with|a| ≥ 2 is that all bidders in a have a shared value abovethe reserve price rF, and their joint value dominatesany affine value of any other bid. In addition, sharedallocation with two bidders, say a ! {i, j}, might bechosen if one bidder, say i, has a shared value belowrF. This could happen when (si + sj − 2rF) ≥ 0 is thehighest among affine values of shared allocations, andit dominates the affine value for a ! {j} (i.e., sj +mj − rC).This illustrates that, similar to OE, we cannot simply

set reserve prices for exclusive and shared bids, eliminatebids below the reserves, and then allocate in a welfare-maximizing waywith respect to the remaining bids. Thisrule would violate the monotonicity condition (MON)and not be incentive compatible in general, as dem-onstrated by Example 2.

Example 2. Let N ! {1, 2}, and consider a type distri-bution that has rF ! 0, rC ! 5.5 and yields t1 ! (3, 3),t2 ! (2, 1) in a particular realization. With k ! 2, thewelfare-maximizing allocation that meets the reserve isthat agent 1 receives the item exclusively. Now set t1 !(1, 4) and note that the welfare-maximizing allocationthat satisfies the reserve is to share the item betweenagents 1 and 2. This allocation violates (MON).

RM is an appealing choice for platform providersbecause it is simple and fast enough to be imple-mented but complex enough to be able to capitalize onthe particularity of the multidimensional valuations.The reserve prices rF and rC are determined uniquelyfor each “product” based on knowledge about thedistribution of reservation prices for that product. Forinstance, for BuyerLink.com’s lead-marketing auctionsfor automotive sales, this would mean computingthesevalues for each“make” and “model” combinationfor groups of geographic locations that are similar intheir distribution of reservation prices. Notably, thisrequires no more information than that needed fordesigning a single-dimensional mechanism, which isin contrast to MaxSimple, for which one would haveto also compute the expected revenue for each productto see whether OE or OS has to be applied. Moreover,as the experimental evaluations in Section 6 highlight,the expected revenue of RM is very close to the op-timal one regardless of the number of agents.

4.3. Illustrative ExampleWe provide a single comprehensive example thatillustrates the five discussed mechanisms.

Example 3. Let k ! 3, n ! 4, si ≈ U(0, 10), and mi ≈U(0, 10), for all i ∈ {1, 2, 3, 4} with rF ! 5 and rC ! 9.


Suppose that the following type profile t isrealized:

Given these profiles, the following configurationsand payments emerge for each mechanism:

The reserve prices rF and rC are computed from thedistributions for shared and exclusive values, respectively.RM beats both the optimal exclusive auction and theoptimal shared auction because the platform’s opennessto allocate either exclusive or shared in combinationessentially creates more competition in the auctionprocess. RM is able to leverage this added competitionby employing full information about the distributions ofvaluations (i.e., on both dimensions) in configuring thetwo reserve prices (agent 4 has low shared value; hence,VCG cannot exact more payments). To see this, let uscompute the payments for agents 1 and 2 with thehelp of (9) and (10). In the case of agent 1, a(t−1) ! {2}and γa(t−1) ! −rC ! −9, and a(t) ! {1, 2, 3} and γa(t) !−3rF ! −15 because RM chooses exclusive allocation foragent 2 when agent 1 is not present and a shared allo-cation foragents1–3 otherwise.Then∑j∈N\1 vj(a(t−1), tj) !8 + 9 and ∑

j∈N\1 vj(a(t), tj) ! 8 + 6; therefore,

p1 ! (−9 + 8 + 9) − (−15 + 8 + 6) ! 9.

In a similar manner as agent 2, a(t−2) ! {1, 3} andγa(t−2) ! −2rF !−10, and a(t)! {1,2,3} and γa(t) !−3rF !−15 because RM chooses shared allocation for agents1 and 3 when agent 2 is not present and a sharedallocation for agents 1–3 otherwise. Then

p2 ! (−10 + 10 + 6) − (−15 + 10 + 6) ! 5.

It can be seen that the reserve price rF increases theprice for agent 2 compared with the case under VCGfrom 4 to 5, that is, from the shared value of agent 3 tothe reserve price. Moreover, the competition fromexclusive allocation further drives up the price foragent 1 from 5 to 9.

Further, MaxSimple mimics OE because, for the prob-lemparameters, the revenue ofOE is higher than that ofOS in expectation: approximately 11.3 versus 10.1, respec-tively, which can be verified, for example, via simulation.

Note that these examples are constructed only toillustrate the inner workings of the mechanisms, notto derive a general conclusion about their revenue-generating prowess. In general, the revenue orderingwill vary by type profile, and there is no mechanismthat dominates the others for all type realizations. Forexample, because of the reserve prices for some lowtype profiles, only VCG would give away the itemand, hence, would collect payments. Revenue has to becompared in expectation, which is done in Section 6.

5. Upper Bounds on RevenueThe preceding section has developed, for the one-to-many matching problem, several potential auctionformats that are easy to implement and computa-tionally tractable. As we show later, the relative rev-enue performance of these auctions can be readilycompared, providing a basic understanding underwhich each design outperforms the others. This,however, does not inform whether the best designperforms well enough relative to the optimal solutionDSA∗ of the auction design problem (DSA). Becausethere are no known techniques for identifying theoptimal design for this two-dimensional problem, ourapproach to estimate the distance from the optimalsolution is to establish and compare against upperbounds to (DSA). One straightforward upper boundis the optimal expected welfare because it is alwayslarger than or equal to the optimal revenue as a resultof the EPIR assumption. However, this is a weakbound, and the gap between the two can be signifi-cant. For example, we show in Section 6 that theoptimal welfare can be twice as high as the revenue of aclose-to-optimal mechanism. To find tighter bounds, wepropose a different, novel approach. We show in thissection two ways of relaxing the conditions of (DSA)such that the resulting optimization problem admits aclosed-form optimal solution. The idea is to relax theproblem such that the resulting setting is single di-mensional and then use the framework of Myerson(1981) to solve the relaxed problem exactly. For no-tational convenience, the expected revenue achievedby anymechanismMech is symbolized by Rev(Mech).5.1. Relaxation 1: Public Exclusivity MarginWe establish the first upper bound by solving (DSA)under the assumption that the exclusivity margin ispublic knowledge. That is, si and mi are still distributedaccording to F and G but pretend that the realizationof mi is observable. Therefore, any incentive constraintinvolving two differentmi can be omitted from (DSA).This yields the following relaxed mathematicalprogram:

maxEt∑

ipi(t)

[ ](UBM)

Valuation Agent 1 Agent 2 Agent 3 Agent 4

s 10 8 6 4m 5 9 5 7e 15 17 11 11

Auction type Reserve Individual payments Total revenue

OE rC ! 9 [0, 15, 0, 0] 15OS rF ! 5 [5, 5, 5, 0] 15MaxSimple 9 [0, 15, 0, 0] 15VCG [4, 4, 4, 0] 12RM rF ! 5, rC ! 9 [9, 5, 5, 0] 19


subject to

ui(a(si,mi, t−i), p(si,mi, t−i), ti)≥ ui(x(si,mi, t−i), p(si,mi, t−i), ti) ∀i,∀t−i,∀mi,∀si,∀si.

ui a(t), p(t), ti( ) ≥ 0 ∀i,∀t,1 ≤a(t)| | ≤k ∀t.

(DSIC2)

We refer to the optimal mechanism for (UBM) asUBM∗. Clearly, (UBM) is a relaxation of (DSA), whichmeans that each feasible solution for (DSA) is feasiblefor (UBM); moreover, Rev(UBM∗) ≥ Rev(DSA∗). Toidentify UBM∗ succinctly, some additional notation isintroduced. For given type profile t ! (s,m), defineas(t) and ae(t) to comprise the set of agents with thehighest virtual values with respect to F. That is,

as(t) ! argmaxa⊆ Aa| | !1a| |≤k

∑

i∈aφF(si)

{ }

(set of agents with k-highest shared virtual values),ae(t) ! argmax

iφF(si) +mi{ }

(agent with highest sum of shared virtual valueand exclusivity margin).

Ties are broken arbitrarily in case of multiple optimalarguments. Note that ∑

i∈as(t) φF(si) ≥ 0 as the emptyset is also a solution. The next theorem shows thatRev(UBM∗) can be expressed in terms of virtual valua-tions with respect to F. Moreover, the set of winners forreport t in the optimal auction is equal to either as(t) orae(t), dependingonwhichoneprovideshighervirtualvalue.

Theorem 3. If F has MHR and the exclusivity margins arepublic information, then the allocation rule of UBM∗ giventype profile t ! (s,m) can be characterized as

a(t) !as(t) when

∑

j∈as(t)φF(sj) ≥ φF(sae(t)) +mae(t),

ae(t) when∑

j∈as(t)φF(sj)<φF(sae(t)) +mae(t).

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

The expected revenue under this allocation rule is equal to

Et∑

ipi(t)

[ ]! Et

∑

i∈ae(t)φF(ti) +mi( ) +

∑

i∈as(t)φF(ti)

[ ]. (12)

The proof can be found in the appendix. It builds onresults in Deng and Pekec (2013) for a similar model.Using the framework of Myerson (1981), Deng andPekec provide the optimal auction when shared val-uations are private information and exclusivity marginsare simple functions of other bidder’s shared value.

In general, the optimal solution of (UBM) mightyield strictly higher revenue than (DSA). This is becausethe optimal allocation rule for (UBM) is not alwaysfeasible for (DSA), as the following example shows.

Example 4. LetN ! {1, 2}, si ∼ U(0, 1), andmi ∼ U(0, 1),for all i ∈ N. Assume that t1 ! (0.3, 0.3), t2 ! (0.4, 0.1).Then we have that φU(0,1)(s1) +m1 ! −0.1, φU(0,1)(s2) +m2 ! −0.1, φU(0,1)(s1)!−0.4, and φU(0,1)(s2)!−0.2; there-fore, according to UBM∗, nobody gets anything. Nowchange the valuations of agent 1 such that t1 ! (0.5,0.05).Because φU(0,1)(s1)+ m1 ! 0.05 and φU(0,1)(s1)! 0, agent 1receives the item exclusively. This allocation rule vio-lates monotonicity condition (MON), whichmeans thatthere is no payment scheme such that the incentivecompatibility constraints are satisfied. Therefore, itcannot be part of a solution for (DSA).

5.2. Relaxation 2: Bound with RepresentativesThe derivation of the second upper boundmechanismis more convoluted. For every instance of (DSA), wedefine an instance of an alternative setting by in-troducing two representatives for each agent: one foreach dimension of the agent’s type. To distinguishthis setting in the notation, we add a bar to each item,as in N.

Definition 6 (Representative Environment). For any in-stance of (DSA) given by the set of agents N, the set oftypes T, and distributions of types F,G, the represen-tative environment is defined as follows:• For each i ∈ N, introduce is and ie such that the set

of agents is N ! Ns ∪Ne, where Ns ! {1s, . . . , is, . . . , ns}and Ne ! {1e, . . . , ie, . . . ,ne}.• Let As ! {α ⊆ Ns | α| | ≤k} and Ae ! {α ⊆ Ne | α| | !

1} so that A ! As ∪ Ae is the set of feasible allocations.Let αs and αe denote elementsofAs andAe, respectively.• Let Tis ! Si for all is ∈ Ns, T

ie ! Mi for all ie ∈ Ne,and T ! ×i∈NT

i, the set of type profiles.• Let the valuations for all i ∈ N, for all t ∈ T, for all

ti ∈ Ti, and for allocation rule a be

vi(a( ti, t−i), t) !sis ∀i ∈ Ns ∩ a(ti, t−i),sis +mie ∀i ∈ Ne ∩ a(ti, t−i),0 ∀i ∈ N otherwise.

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

For payment function p,

ui(a(ti, t−i), ti), p ti, t−i)t( ) ! vi(a(ti, t−i), t) − pi(ti, t−i)

is the utility function.• Let ti ∼ F for all is ∈ Ns and ti ∼ G for all ie ∈ Ne.Note that agent ie exhibits informational externality

in the agent’s valuation, meaning that the agent’svalue for a particular allocation may depend on thetrue type of other agents. More specifically, the valuethat an agent ie observes if assigned an item depends


on the value of agent is. In such settings, dominant strat-egy incentive compatibility is too demanding becauseit requires truthfulness for every possible report of theother agents, leaving little freedom for nontrivial mech-anisms (see, e.g., Roughgarden and Talgam-Cohen2013). Therefore, we relax the incentive compatibilitycondition to EPIC, which requires truthfulness onlyfor every truthful report of the other agents.

Definition 7 (EPIC). A direct mechanism is EPIC if, foreach agent, truth telling is a dominant strategy given thatthe others report truthfully. Formally, a directmechanism(x, p) is EPIC if, for every i, t−i, ti, and ti, it holds that

ui a(t), p(t), t)( ) ≥ ui a(ti, t−i), p(ti, t−i), (ti, t−i)( ).

Another deviation from (DSA) is in permitting al-locations in which only one shared agent receives theitem. This increases the set of feasible allocations and,hence, bears the possibility to generate more revenue.The revenue-optimization problem for the repre-sentative environment can be stated as

maxEt∑

ipi(t)

[ ](UBR)

subject to

ui a(t), p(t), t( ) ≥ ui a(ti, t−i), p(ti, t−i), (ti, t−i)( )

∀i,∀ti,∀ti,∀t−i, (EPIC)

ui a(t), p(t), t( ) ≥ 0 ∀i,∀t, (EPIR2)

a(t) ∈ A ∀t. (Feas)

Thus, (EPIC) is responsible for ex post incentive com-patibility and (EPIR2) for ex post individual rationality,and (Feas) ensures that only feasible allocations can bemade; that is, there cannot be allocations in whichagents both fromNs andNe receive an item, and that atmost one agent from Ne wins or at least two agentsfrom Ne win or no one wins. We refer to the optimalmechanism for (UBR) as UBR∗.

Strictly speaking, (UBR) is not a relaxation of (DSA)because it has more variables than (DSA). Still, the fol-lowing proposition shows that the optimal solution of(UBR) can be used as an upper bound for (DSA).

Proposition 1. For every feasible mechanism (a, p) for(DSA), there is a feasible mechanism (a, p) for (UBR) suchthat Rev(a, p) ≥ Rev(a, p).

Here, only the intuition behind the proof is given;for technical details, refer to the appendix. Because eachrepresentative corresponds to a particular dimension ofan original agent’s valuation, any allocation rule in afeasible solution (a, p) for (DSA) induces an allocationrule a for (UBR). Because each agent in N has a one-dimensional type, x is part of a feasible solution of(UBR) if and only if it satisfies (MON) (this is a well-

known fact of one-dimensional mechanisms). It canbe shown that feasibility of a implies (MON), exceptfor a null set of types in the original problem and that,except for that null set, the prices used to make atruthful are at least asmuch as in p. On such null sets, acan be adjusted to be a feasible solution of (UBR).It is immediate from Proposition 1 that the optimal

expected revenue of (UBR) serves as another non-trivial upper bound for Rev(DSA∗). Because (UBR) issingle dimensional, it is possible to provide a closed-form solution for UBR∗ in a similar way as before. Forgiven type profile t, let

as(t) ! argmaxa⊆ Nsa| |≤k

∑

is∈aφF(tis )

{ },

ae(t) ! argmaxie∈Ne

tis + φG(tie ){ }

.

Ties are broken arbitrarily in the case of multipleoptimal arguments. Note that ∑i∈as(t) φF(tis) ≥ 0 becausethe empty set is also a solution.

Theorem 4. When F and G have MHR, the allocation rule ofUBR∗ represented by a for type profile t is computed as follows:

a(t)

!

as(t) when∑

js∈as(t) φF(tjs )≥ maxje∈Ne

tjs{ + φG(tje )

},

ae(t) when∑

js∈as(t) φF(tjs )< maxje∈Ne

tjs{ + φG(tje)}.

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

The optimal revenue under this allocation is

Rev(UBR∗) ! Et∑

is∈Ns∩a(t)φF(tis )

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

+∑

ie∈Ne∩a(t)tis + φG(tie)( )⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

and it holds that Rev(UBR∗) ≥ Rev(DSA∗).

5.3. Significance of RelaxationsThe two relaxations provide us with an alternative tousing the expected maximum welfare as an upperbound. To get a sense of the quality of these bounds,we calculated expected revenue of UBR∗ and UBM∗

along with expected maximal welfare for differentinstances by means of simulation (more details on thetechnique are given in Section 6). The results aredepicted in Figure 1. Because the displayed revenueratios are taken over the revenue of RM, it is apparentthat both relaxation bounds are much tighter than thewelfare upper bound. In particular, for the exponentialdistribution case, the welfare can be twice as muchas UBR∗ or UBM∗. Furthermore, depending on the


support of the two valuations, the difference betweenUBR∗ and UBM∗ might be significant. In general, whenexclusivity margins are small relative to shared valua-tions, UBM∗ is tighter, and UBR∗ is closer to the optimalrevenue when exclusivity margins are relatively large.

Now, (UBM) can as well be used to prove thatMaxSimple, introduced previously as a lower bound,achieves a constant factor approximation of DSA∗ (seethe appendix for proof).

Theorem 5. Take an instance of the exclusivity auctionproblem in which F and G are MHR distributions. Thenexpected revenue of MaxSimple is at least (1 − 1/Hn)/(2 − 1/Hn), the expected revenue of the optimal mechanism.The minimum of this approximation ratio is 1/4, attained inthe case of two bidders. Moreover, this approximation ratioholds for the optimal Bayes–Nash implementable mechanismas well.

6. Performance AnalysisThis section describes and compares the performanceof the discussed mechanisms across multiple di-mensions: the number of agents n, the number oftimes k an item can be allocated in a one-to-manymatch, the type of distribution for shared and ex-clusive valuations, and the distribution parameter.Specifically, we consider two types of distributionsfor both s and m, uniform and exponential. To makecomparisons easier and more intuitive, we parame-terize each distribution with its expected value, thatis, means and meanm. This choice is trivial for the ex-ponential distribution, andwemake it possible for theuniform distribution by setting the lowest possiblevalue to zero. In case of the exponential distribution,we let both means and meanm be taken from the set{1, 2.5, 5}, and for the uniform distribution, we fixthem to 5. The reason for these choices is that the

convolution (which we need for the exclusive valu-ation) of two exponential distributions with differentparameters is easy to derive, and for the uniformdistribution, only the identical case is trivial. Thenumber of agents and number of items each variesbetween 2 and 10 (step size 1) with k ≤n. For eachdistribution type (uniform and exponential), an ob-servation in the computational data set is a quadrupleof the four factors (n, k,means,meanm), resulting in 450unique observations. Each observation is the expec-tation across multiple simulations under each auctionmethod for each quadruple. We sample from the typedistributions 100,000 times and calculate the paymentfor each type report. The expected revenue is then es-timated by the average of the simulated payments. Tomake comparisonsmore compact,wedefineUB∗ asthe minimum of UBR∗ and UBM∗ for each instance. Aswith the model development and analysis, whichemployed a general (but predecided) k, we conductthe analysis for a range of k values to confirm that ourresults hold across the board. In reporting key results,we limit the visualizations to scenarios that are morelikely and relevant to practice (we use k ! 3) to avoidclutter and obscure the main points.The direct and most compelling evidence for the

superiority of the RM auction format is also the sim-plest to present. Figure 2 displays the revenue cap-tured by each design relative to the revenue possibleunder UB∗, which is an upper bound on DSA∗,across the computational scenarios described in theexponential distribution case. It is evident that therevenue ratio for RM lies in a tight range (0.8–1),whereas the other designs have poorer performanceon average, best case, and worst case. Table 1 sum-marizes the computational statistics. Across all compu-tations, RM achieves 96% of the revenue received

Figure 1. (Color online) Revenue Over Optimal Welfare of the Two Upper Bounds for Two Different Product Instances

Notes. (a) Case 1 (si ~ Exp(1), mi ~ Exp(2)). (b) Case 2 (si ~ Exp(1), mi ~ Exp(0.5)).


under UB∗. Given how close to 100% this is, it notonly implies that RM has high performance for DSA,but it also implies that RM performance is quiteuniformly high across all scenarios. This is confirmed byobserving that RM captures more than 94% of possiblerevenue in 75% of the observations. In contrast, theOE, OS, and VCG auctions produce averages of 84%,67%, and69%, respectively, and the75%revenue thresholdis achieved only in 77%, 50%, and 52% of cases, respec-tively. MaxSimple performs better than these but stillpoorer than RM (and recall the practical limitations ofMaxSimple discussed at the end of Section 4.1). Theseformats are not only inferior to RM on average or best-case performance but also suffer with respect to worst- orunfavorable-case settings. Hence, the evidence is that RMis an excellent mechanism, and none of the simpler or“obvious” designs come close.

What value does RM, as a two-dimensional design,add relative to asking the bidder for just one piece ofprivate information? Then the choices are OE and OSif the platform wishes to maintain the same formatacross all its product categories or MaxSimple if it canpick and announce the format (exclusive or shared)based on priors for each category. In contrast, RMrequires two pieces of private information and thenmakes the choice of exclusive or shared based on thebids. Figure 3 illustrates the cost of precommitment toexclusive or shared: that a substantial amount ofrevenue is lost under the single-dimensional mech-anisms relative to RM. Moreover, the advantage ofRM is amplified on account of one additional factor.With MaxSimple, the choice of exclusive or sharedrequires knowledge of the number of agents n and theexpected revenue for each instance, whereas config-uration of RM (in terms of the reserve prices) does not.Hence, when n is not precisely known or the com-putational capacity is limited, the performance ad-vantage of RM is even higher.

To further understand how the four simulationfactors influence performance of different formats,we

dig deeper into the computational results. Figure 4explores how the number of agents impacts relativeperformance for the case in which shared valuationsandmargins are distributed exponentially (and fixingk = #items to 3). Note that MaxSimple is not displayedbut is simply the upper frontier of OS and OE. Thepanels cover nine pairs of parameters for these dis-tributions, and comparison across these nine panelsreveals a few common threads. When exclusivitymargins are very high relative to shared valuations(e.g., when n ormeans is small), then, not surprisingly,OE is nearly as good as RM. Because OE is optimal forsingle-dimensional exclusive allocations, this in-dicates that when RM has a maximum gap from theupper bound (which is coincident with when OE isclose to RM), it is the upper bound that is weak. In theconverse case (high shared valuations), OS gets rel-atively stronger but never quite close enough to RM.OE gets weaker as there are more agents (because itbecomes more likely for RM to earn higher revenuethrough a sum of shared valuations). Compared withOE, OS and VCG perform more poorly, especiallywhen n (number of agents) is small and close to k. ForRM, which is overall excellent relative to the boundUB∗, the least excellent performance occurs whenn is small and the parameters for s andm are balanced.Figure 5 confirms that these observations hold evenwhen valuations are distributed uniformly.

Figure 2. (Color online) Revenue Performance of Auction Formats: Density of Exponentially Distributed Simulation InstancesThat Capture a Certain Fraction of Revenue Relative to UB∗, an Upper Bound on DSA∗

Table 1. Revenue Performance of Various Auction FormatsRelative to UB∗, an Upper Bound on Optimal DSARevenue DSA∗

RM OE OS VCG MaxSimple

Mean, % 96 84 67 69 90Median, % 97 84 71 78 91First quartile, % 94 77 50 52 84Minimum, % 80 58 13 0 79

Note. The statistics represent 405 observations ranging over fourdimensions (n, k, and parameters means and meanm of exponentialdistributions for shared valuations and exclusivity margin).


Overall, from the previous analyses and visuali-zations, we find that RM exceeds single-format designsand has high revenue performance across a spectrum ofmarket scenarios. Its advantage is lower only when

conditions are homogeneously ripe either for alwaysexclusive or always shared allocation (e.g., extremelylow values under sharing and extremely low exclusivitymargin, respectively; see Figure 6). Most platforms

Figure 3. (Color online) The Cost of Precommitment for Exponentially Distributed Cases

Notes. Percentage of simulation instances losing at least the given fraction of revenue, relative to RM, when it is predecided to match alwaysexclusively (OE) or shared (OS) or to prepick between the two based on product category priors (MaxSimple). For instance,MaxSimple (which isthe best among the one-dimensional designs) loses more than 10% of revenue in more than a quarter of instances.

Figure 4. (Color online) Comparing the Expected Revenues of Different Mechanisms to the Expected Revenue of RM

Notes. Comparisons aremade for different numbers of agents, different expected value of shared valuations, and exclusivitymargins (means andmeanm, respectively). The number of items is assumed to be three, and both valuations are exponentially distributed.


indeed feature multiple product categories because,inmanyof them, each participant or group of participants(e.g., a consumer looking for financial products, a Face-book user, or a make–model–zip combination in leadmarketing) is a unique “category” in the sense that itwould generate different priors, attract a different num-ber of bidders, or produce varying levels of differences inshared versus exclusive allocation.

Finally, Figure 7 demonstrates the value of the upperbound that we developed relative to using an obviousupper bound, such as optimal welfare, by showing thebound to the revenue of RM ratios across all simulationinstances. The gap between the revenue of RM and therespective upper bounds shows that using optimalwelfare as a bound would not yield information re-garding quality of the heuristic designs (specificallyRM),whereas using the UB∗ helps establish the excel-lent performance of RM.

7. ConclusionThis paper has developed results for a new kind ofone-to-manymatching auction format that is relevantfor many of today’s platforms. The complexity arisesfrom the fact that some auction participants couldpotentially have very high value for exclusive purchase,and this raises the need for a method that choosessmartly between exclusive and shared allocation. Al-though the analysis has roots in optimization and auc-tion theory, the fundamental contribution of the paperis with respect to pricing and revenue management inplatforms: an auction format that delivers high revenueperformance, predictability (because truth telling is thedominant strategy for bidders), and ease of bidder par-ticipation (with simple transparent rules for winnerdetermination and pricing). The platform owner re-quires only a reasonable prior about the distributions

Figure 5. (Color online) Comparing the Expected Revenues of Different Mechanisms with the Expected Revenue of RM

Notes. MaxSimple, not displayed, is the upper frontier of OS and OE. Comparisons are made for different numbers of agents. The number ofitems is assumed to be three, and both valuations are uniformly distributed between 0 and 10.


of valuations to configure the auction (specifically, bysetting reserve prices). Once configured, the algo-rithm is very efficient at computing the actual alloca-tions and prices for any particular configuration of bids.Moreover, as the platform owner obtains improvedinformation about the distribution, the algorithm canefficiently be reconfigured with new reserve prices.

Finally, the auction is straightforward for bidders be-cause their optimal equilibrium strategy is simply tobid their true valuations.Our analysis has focused on auction designs in

which truth telling is a dominant strategy and thatensure ex post individual rationality. Alternatives doexist; for instance, it is folklore knowledge that, in

Figure 6. Influence of Parameters (n,means,meanm) on RM’s Advantage Measured as Percentage of Revenue Gap Over Single-Dimensional Optimal Auctions for Exponentially Distributed Cases, OE and OS (with k ! 3)

Note. Bubble area is proportionate to revenue gap, relative within the panel, and not comparable across panels.

Figure 7. Comparing Different Upper Bounds with the Expected Revenue of RM Across All Simulation Instances for BothUniform and Exponential Distributions


multidimensional settings, randomization improvesrevenueandthat theclassofBayes–Nash-implementablemechanisms allows more flexibility for the designer(e.g., Cai et al. 2011). However, we maintain faith inthe incentive-compatible design because, in practice,both the auctioneer and bidders desire fast, predict-able, and simple mechanisms that require minimal userinteraction and that make the available strategies clearfor the users. Moreover, the dominant strategy imple-mentation is prior-free for bidders; hence, it is morerobust from a practical perspective. In contrast, Bayes–Nash implementation requires a lot more of the partic-ipants: all of them should have the same prior aboutthe type distributions, and they have to be able tocompute expected utilities based on their actions. Theuse of lotteriesmight be acceptable in certain applications,but it is not yet desired in everyday business. Payingmore than your bid is likely to alienate users from asystem because they may regret their participation inthe auction; therefore, ex post individual rationality canbe seen as a natural property of an applied mechanism.

In addition to platforms that already employ one-to-many matching, our research can also help one-to-one matching platforms. For instance, CreditKarmapasses each customer lead exclusively to a single“best” merchant, but it would be feasible to imple-ment a business practice that allows for both exclusiveand shared matches. Another application category istelevision advertising,which has primarily been a one-to-one matching process (each ad slot on a programsold to one advertiser), but modern technology allowsa one-to-many match with which video-serving plat-forms and publishers could display a static of foursimultaneous video ads and ask the viewer to click onone. Video advertising is ripe for such transformationbecause of recent changes in consumption habits andthe consequent decline in demand for traditional TVadvertising.6 Publishers therefore could offer adver-tisers a choice between an exclusive sale and sharingan impression with a handful of other advertisers whilegiving the viewer a choice on which ad to click. Moregenerally, the mixed-format auction works well whenpayment is for an impression (because it easily accom-modates multiple winners) rather than a click or a sale.Mixing 1:1 and 1:k auctions is still applicable undercost-per-click payment (the auctioneer would taketwo bids, again, for shared and exclusive display) whenbidders anticipate higher click-through rates under ex-clusive display. A more comprehensive analysis of thissetting is a useful topic for future research.

The underlying construct of our auction format(mixing exclusive and shared sales when buyers havea margin for exclusive purchase) is applicable to manyinformation goods because of their nonrivalrous prop-erty. Higher value for exclusive purchase can be fueledby the threat of competition, by a sense of privilege, or

by special customer preferences (e.g., luxury goods).For example, a prospector would derive higher valuefrom exclusive possession of information regarding anatural resource repository. A retailer may perceivegreater value from the exclusive right to sell a goodbecause it avoids competition with other shops. Anewspaper advertiser could buy exclusive access on apage or split the audience by purchasing a fraction ofthe advertising space. Mobile search advertisers may,because of the device’s limitations in display size andnavigation, be willing to pay substantially more forexclusive promotion; thus, major search engines havealready examined designs by which advertisers canplace two-dimensional bids for receiving the click exclu-sively or shared (Sayedi 2012, Sayedi et al. 2018). Givensuchwide relevance of one-to-manymatching, this papercontributes by creating insights and an implementableapparatus that can improve profits and welfare.

AcknowledgmentsThe authors acknowledge feedback and comments fromseminar participants at the University of Maryland; the Plat-form Symposium in Boston; the University of California, SanDiego; and the University of California, Davis (MathematicsDepartment). The authors are particularly grateful to the entirereview team for thorough and constructive feedback, whichhas inspired substantial improvements throughout the re-view process.

Appendix.Technical Details and Proofs

Table A.1. Summary of Notation

Notation Meaning

k Maximum number of sold itemsN ! {1, . . . ,n} Set of agentsA Set of possible allocationsα Index for set ASi,Mi Set of shared valuations and exclusivity margins,

respectively, of agent iTi ! Si ×Mi Set of types of agent iT ! T1 × · · · × Tn Set of type profilesti ! (si,mi) Type of agent it−i ! ×j!itj Type profile excluding the type of agent iF,G Cumulative distribution function (CDF) of shared

valuations and exclusivity margins, respectivelyC CDF of the exclusive value, that is, the convolution

of F and GφD(ti) Virtual valuation of agent i for random variable

ti distributed as Da : T → A Allocation rulep : T → RN Payment schemevi(a(t), ti) Valuation of agent i for allocation x(t) having

actual type ti

ui a(t), p(t), ti( )Utility of agent i for allocation x(t), payment p(t)

having actual type ti

SWN(a | t) Total welfare of agents in N for allocation a andtype profile t

xi(t) Whether i wins, xi(t) ! 1 ⇐⇒ i ∈ a(t); otherwise,xi(t) ! 0


Note About NotationNote the equivalence between binary vector x(t) and the setof agents that receives the item a(t). Specifically, xi(t) !1 ⇐⇒ i ∈ a(t); otherwise, xi(t) ! 0. Although this paper em-ploys just the set notation a(t), presentation of technical detailsandproofs requiresus toalsousex(t) to facilitate the exposition.Proof of Theorems 1 and 2. The proof of both results de-pends on the Myerson (1981) framework, for which we usethe generalization of Devanur et al. (2011) of the original result.Myerson’s (1981) result requires that agent i’s type ti is singledimensional and drawn from set Ti ⊂R with regular distri-bution Di that satisfies MHR. Let T ! ×iTi, and let D ! D1

× · · · ×Dn be the distribution overT. Then the single-parameterenvironment (NU,A,T,D) has the following properties:(1) For every DSIC and EPIR mechanism (a, p), where a is

the allocation rule and p is the payment rule, the expectedvalue of total payments can be written as

Et∑

ipi(t)

[ ]! Et

∑

i∈a(t)φDi (ti)

[ ],

the expected total virtual valuation.(2) A revenue-maximizing DSIC and EPIR auction design is

given by the allocation rule “pick the agent with highest virtualvalue,” that is,

a(t) ! argmaxα∈A

∑

i∈αφDi (ti)

{ },

and payment rule “charge the social cost on others (=minimum needed for this agent to win),” that is,

pi(t) ! inf{ti | i ∈ a(ti, t−i)} i ∈ a(t),0 otherwise.

{(A.1)

The main message is that awarding item(s) to agents whohave the highest virtual valuations and charging them thesocial cost they impose on others produce the optimal auctionthat maximizes total expected revenue. The application toTheorem 1 (OE) is straightforward: D is interpreted as theconvolution of F and G (i.e., C) and follows because C sat-isfies MHR because the set of MHR distributions is closed

under convolution (Barlow et al. 1963). Note that OE makesno allocation if all virtual values happen to be strictly lessthan zero.

Theorem 2 also follows directly. Note that Myerson (1981)requires only that for each agent, the allocation is monotonenondecreasing in its own type given the other types; there-fore, the fact that OS never allocates for singletons is not anissue for invoking the Myerson theorem. For both results, theexpected revenue equation should not be interpreted as spec-ifying a point-wise mapping between prices paid and virtualvaluations. Instead, prices paid are the incentive-compatiblepayments given in the second part of the result (A.1): eachwinning merchant pays the minimum amount the merchantwould have to bid to secure that allocation. □

Proof of Theorem 3. (UBM) can be decomposed by treatingall possible exclusivity margin profiles separately becausenone of the constraints involve variables that are related todifferent exclusivity margins; furthermore, the probabilitiesare independent among agents and among the dimensions oftheir type. Therefore, it is sufficient to solve the subproblemsseparately for each fixed m and acquire the expected revenueas the expectation over the optimal objective values of thesubproblems. From proposition 6 in Deng and Pekec (2013), itfollows that for fixed m, the allocation rule x of the optimalDSIC and EPIR mechanisms given type profile t ! (s,m) is

xi(t) ! 1 for

i ∈ as(t) when∑

j∈as(t)φF(sj) ≥ φF(sae(t)) +mae(t),

i ! ae(t) when∑

j∈as(t)φF(sj)<φF(sae(t)) +mae(t),

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A.2)

and xi(t) ! 0 otherwise. The expected revenue under thisallocation rule is equal to

Es∑

ipi(s,m)

[ ]! Es

∑

iφF(si) +mi ∏

j !i(1 − xj(s,m))

( )xi(s,m)

].

[

(A.3)

Because (A.2) holds for any fixed m, it concludes (11). Fi-nally, (12) follows from taking expectation over (A.3) withrespect to the exclusivity margins.

Note that Deng and Pekec (2013) originally studied EPICmechanisms because their exclusivity margin is assumed tobe a linear combination of the shared valuations of the otheragents. In our case, exclusive margins do not depend on thevaluation of the other agents; hence, the notions DSIC and EPICcoincide (for formal definition of EPIC, see Definition 7). □

Proof of Proposition 1. We start with a lemma that uses(MON) to give insight into how an incentive-compatible expost individual rational mechanism (x, p) for (DSA) assignsallocations and prices to types.

Lemma A.1. Let (x, p) be a feasible mechanism for (DSA). Leti ∈ N, t−i ∈ T−i, and assume that pi(0, 0, t−i) ! 0. Then there exists∗ ∈ R ∪ {∞} and m∗ ∈ R ∪ {∞}, depending on i and t−i, such that(1) For all (si,mi) such that si < s∗ and si +mi < s∗ +m∗, x does

not allocate the item to i; that is, xi(si,mi, t−i) ! 0.

Table A.2. List of Acronyms

Acronym Expansion

DSA Dominant strategy auctionDSIC Dominant strategy incentive compatible, property of auctionEPIC Ex post incentive compatible, property of auctionEPIR Ex post individually rational, property of auctionMHR Monotone hazard rate h(x)

1−H(x) when x has cumulativedistribution H

MON Monotonicity rule among allocationsOE Optimal exclusive auction (precommitment to exclusive

allocations)OS Optimal shared auction (precommitment to shared

allocations)RM Reserve-price based mechanism (our design)UBM Upper bound with (public) marginsUBMin Minimum of (UBM, UBR)UBR Upper bound with representative agentsVCG Vickrey–Clarke–Grove auctions


(2) For all (si,mi) such that si > s∗ and mi <m∗, x allocates theitem shared to i; that is, xi(si,mi, t−i) ! 1 and |a(si,mi, t−i)|> 1.(3) For all (si,mi) such that si +mi > s∗ +m∗ andmi >m∗, x allocates

the item exclusively to i; that is, xi(si,mi, t−i) ! 1 and |a(si,mi, t−i)| ! 1.(4) If x allocates no item to i for some type (si,mi), then

p(si,mi, t−i) ! 0.(5) If x allocates the item shared to i for some type (si,mi), then

p(si,mi, t−i) ! s∗.(6) If x allocates the item exclusively to i for some type (si,mi),

then p(si,mi, t−i) ! s∗ +m∗.

Proof. Fix agent i and type profile t−i. We begin by defining(s∗,m∗) for four possible cases:

a. If there exists no (si,mi) for which x allocates eithershared or exclusively to i, then let m∗ ! s∗ ! ∞.

b. If there exists a (si,mi) for which x allocates exclusively,but there is no type such that x allocates shared, including i, then

m∗ ! inf{mi| ∃ si such that x allocates exclusive at (si,mi)}ands∗ ! inf{si +mi| x allocates exclusive at (si,mi)}.

c. If there is no (si,mi) for which x allocates exclusively,but there is a type for which x allocates shared, including i,then m∗ ! ∞ and

s∗ ! inf{si| ∃ mi such that x allocates shared at (si,mi)}.

d. If there are types for both exclusive and shared allo-cations for i, thenm∗ ! inf{mi| ∃ si such that x allocates exclusive at (si,mi)}

and

s∗ ! inf{si| ∃ mi such that x allocates shared at (si,mi)}.

As a visual aid, Figure A.1 depicts the different allocationsituations and the corresponding (s∗,m∗) tuples.

Recall that (MON) says that for any two types (si,mi),(si, mi), we have

vi(x(si,mi, t−i), (si,mi)) − vi(x(si, mi, t−i), (si,mi)) +vi(x(si, mi, t−i), (si, mi)) − vi(x(si,mi, t−i), (si, mi)) ≥ 0.

We prove first parts (1)–(3) of the lemma for different cases.We start with the case in which 0<m∗ <∞ and 0< s∗.

To see (1), observe that x cannot allocate shared at (si,mi)because of the definition of s∗. If it allocates exclusively, wemust have mi ≥ m∗ by the definition of m∗ and that s∗ <∞. Let(si, mi) be a type such that

si + mi > si +mi

mi <m∗

si < s∗.

By the definition of s∗ and m∗, x does not allocate an item to iat (si, mi). By (MON) for the two types (si,mi) and (si, mi), weget

si +mi − (si + mi) ≥ 0,

which yields a contradiction. Therefore, xi(si,mi, t−i) ! 0.To see (2), observe that x cannot allocate exclusively an

item to i at (si,mi) by the definition of m∗. Furthermore, theexistence of a type as in (2) implies s∗ <∞. Suppose that x does

not allocate any item at (si,mi). By the definition of s∗, thereexists (si, mi) such that si < si, and x allocates shared at (si, mi).Using (MON) again, we get

−si + si ≥ 0,

acontradiction.Therefore,xmust allocate shared to i at (si,mi).To see (3), observe first that the existence of such a type

implies s∗ <∞, and the assumption that m∗ <∞ implies thatthere is a type for which x allocates exclusively to i. As a nextstep, we show that for any (si,mi) with mi >m∗, x cannotallocate shared at (si,mi). Suppose that it does; then let (si, mi)be a type such that mi <mi, and x allocates exclusively at(si, mi). Using (MON), we get

si − (si +mi) + si + mi − si ≥ 0,

yielding mi ≥ mi, a contradiction.Now let (si,mi) as in (3). It remains to show that x cannot

allocate nothing to i at this type. Suppose that it does. Let(si, mi) be a type such that

si + mi < si +mi

mi >m∗

si > s∗.

By what we yet showed, x cannot allocate shared at thistype. By using the same construction as in the proof of part(2), x can also not allocate nothing at this type. Therefore,x must allocate exclusively at (si, mi). Invoking (MON) oncemore, we get

−(si +mi) + si + mi ≥ 0,

yielding a contradiction. Therefore, x must allocate exclu-sively to i at (si,mi).

Figure A.1. AllocationMaps and the Defined (s∗,m∗) Tuplesas a Function of (si,mi) for Fixed i and t−i


Next, we consider the case in which 0<m∗ <∞ and s∗ ! 0.For this case, there is no type that satisfies (1), and we canfocus on (2) and (3). We start with (2). Suppose that mi <m∗.By definition ofm∗, x cannot allocate exclusively at (si,mi). If xallocates nothing at (si,mi), let (si, mi) be a type with si < si, atwhich x allocates shared to i. As in the proof of (2) before, wecan show that this contradicts (MON). Next, consider (3). If xallocates shared at (si,mi), we can construct a contradiction asin the proof of (3) before. If x allocates nothing to i at (si,mi),we can use the same construction as in the second part of theproof of (3) before to yield a contradiction.

Next, consider the case m∗ ! 0. For this case, there is notype that satisfies (2), and we can focus on (1) and (3). First,observe that x cannot allocate shared at any type (si,mi) withmi > 0. This follows from a similar construction as in the proofof (3) before.

To see (1), note that x cannot allocate exclusively by thedefinition of s∗. Next, suppose that it allocates shared; thenmi ! 0. Let (si, mi) be a type such that

si + mi < s∗ +m∗

mi > 0

si > si.

Note that x cannot allocate anything at (si, mi). Thus, (MON)yields

si − si ≥ 0,

a contradiction.To see (3), note that, if x allocates nothing at (si,mi), we can

find a type (si, mi) for which si + mi < si +mi, at which x al-locates exclusively, yielding a contradiction to (MON). Be-cause mi > 0, a shared allocation is not an option either.

It remains the case that m∗ ! ∞, meaning that x does notallocate exclusively to i for any type (si,mi) given t−i. If, inaddition, s∗ ! ∞, no type exists satisfying (2) or (3), and noitem is allocated for any type, showing that (1) holds trivially.Thus, assume that s∗ <∞. We can focus on (1) and (2). Bothfollow from very similar arguments as in the previous cases,utilizing (MON). This finishes the proof of (1)–(3).

It remains to show parts (4)–(6). First, recall that for anytwo types that yield the same allocation, the prices must beequal; otherwise, the type with the higher price would have anincentive to report the type that yields a lower price. Therefore,we denote p0, ps, and pe as the prices that are charged fornonshared, shared, and exclusive allocation, respectively.

As for (4), if the price i has to paywas strictly positive whengetting no item, (EPIR) would be violated; if the price wasstrictly negative at such a type, a bidder of type (0, 0) wouldhave an incentive to misreport this type.

As for (5), note that s∗ <∞ because otherwise there is notype for which x allocates shared. Observe that by (EPIR), theprice ps cannot be strictly larger than s∗. If s∗ ! 0, we are donebecause the price cannot be negative. If s∗ > 0 and ps < s∗,consider a type (si, 0)with ps < si < s∗. By (2), x does not allocatean item at this type, and therefore, this type would have anincentive to pretend to be of a type at which x allocates shared.

As for (6), note that s∗ <∞ and m∗ <∞ because otherwisethere is no type at which x allocates exclusively. By (3), itfollows that s∗ +m∗ ! inf{si +mi| x allocates exclusive at (si,mi)}. Therefore, pe ≤s∗ +m∗. If s∗ +m∗ ! 0, we are done be-cause the price cannot be negative. If s∗ +m∗ > 0, consider firstthe case s∗ > 0. If pe < s∗ +m∗, by (3), there is a type (si,mi) suchthat si +mi > pe, at which x allocates nothing. This type wouldhave an incentive to pretend to be of a type that allocatesexclusively. Finally, consider the case s∗ ! 0. If pe <m∗, there is atype (si,mi) such that mi > pe, at which x allocates shared. Thistype would have an incentive to pretend to be of a type thatallocates exclusively because it gets utility si if being truthfuland si +mi − pe > si if not being truthful. □

Now we are ready to prove Proposition 1. Let (x, p) befeasible for (DSA). First, we may assume that (x, p) satisfiesthe conditions of Lemma A.1 because the payment of anagent i with type (0,0), given any type of other agents,cannot be larger than zero by (EPIR), and if it was smallerthan zero, we can change the payment of this agent by apositive amount, thereby not decreasing the revenue of themechanism. Given this assumption, we construct a truthful(x, p). Take any type t and any agent i.

Suppose that there exists no type (si,mi) such thata(si,mi, t−i) ! {i}; that is, m∗ ! ∞. Then assign an item to is ifand only if x assigns an item to i. Assign no item to ie. Set thepayment for is equal to s∗ if is gets an item and zero oth-erwise. Set the payment for ie equal to zero.

Suppose that there exists (si,mi) such that a(si,mi, t−i) ! {i}.Letm∗ be as defined in LemmaA.1. Ifmi ! m∗, assign an itemto is if and only if x assigns shared to i, and assign an item toie if and only if x assigns exclusive to i. If x assigns nothing toi, then neither assign an item to is nor to ie. Set the price for isequal to s∗ if is is assigned an item and zero otherwise. For ie,let the price be si + inf{mi | x assigns exclusive at (si,mi, t−i)}if ie gets an item and zero otherwise. Ifmi ! m∗, assign neitheran itemto ie nor to is, and set the price for both agents equal tozero.Therebynote that,byLemmaA.2, for all si, the price that iepays when receiving the item is larger than or equal to s∗ +m∗.

Note that pi(t) ≤pis (t) + pie (t), except for ti such that mi !m∗. Therefore, the expected revenue from agent i satisfies

Et−i Eti p ti, t−i([ ][ ] ≤Et−i E(si ,mi) pis (si,mi, t−i) + pie (si,mi, t−i)[ ][ ]

.

Indeed, the argument of the inner expected value on the left-hand side is, except for a null set, smaller than or equal to thecorresponding argument on the right-hand side.

By Lemma A.1, it is immediate that given any type of theother agents, the payment for is and ie, respectively, if winningthe item equals the infimum over all valuations of that agentsuch that the agent wins the item. It is well known but can alsoeasilybe verifiedalong the lines of theproof ofLemmaA.1 thatsuch a payment rule satisfies (EPIR) and (EPIC). □

Proof of Theorem 4. Roughgarden and Talgam-Cohen (2013)extend the results of Myerson (1981) to settings with informa-tional externalities and correlated type distributions for mech-anisms that are ex post incentive compatible and ex postindividual rational. In particular, it is shown that the expectedrevenue of any EPIC and EPIR mechanism equals the expectedsum of virtual valuations, provided that the payments are


maximal; that is, the utility of an agent with zero type iszero. Moreover, the revenue-maximizing mechanism allocatessuch that the sum of virtual valuations is maximal for eachtype profile given that the resulting allocation is monotonenondecreasing for each agent in the agent’s own type. Notethat the virtual valuation of agent ie having type tie is tis +φG(tie ) because of the informational externality. Having allthese in mind, the only thing left to show is that all agents’probability of receiving the item is monotone nondecreasingin their reported types.

Because F has a monotone hazard rate, we have that φF(tis )is monotone nondecreasing in tis for all is ∈ Ns. Therefore, ifis ∈ as(tis , t−is ), then is ∈ as(tis , t−is ) for any tis > tis . Similarly,because G has a monotone hazard rate, we have that φG(tie ) isalso nondecreasing in tie for all ie ∈ Ne. This ensures that onceie ∈ ae(tie , t−ie ), then ie ∈ ae(tie , t−ie ) also for any tie > tie . The lastcritical point is when any agent is ∈ Ns increases tis becausethen both φF(tis ) and tis + φG(tie ) increase at the same time. Tosee why this is not a problem, observe that the MHR as-sumption ensures that φF(tis ) increases at least as fast as tis +φG(tie ) in tis . □

Proof of Theorem 5. To prove this theorem, we need toelaborate first in more detail on the VCG auction and prove acouple of results. Let SWN(a | t) ! ∑

i∈N vi(a, ti) denote thewelfare of agents in N for allocation a and type profile t.

Definition A.1 (VCG Mechanism). Let AN be the set of fea-sible allocations for agents in N, and let a(t) ∈ AN be an al-location for each type profile t such that

a(t) ∈ argmaxa∈AN

SWN(a | t).

Similarly, let AN\{i} be the set of feasible allocations foragents in N \ {i}, and let a(t−i) ∈ AN\{i} be an allocation foreach type profile t−i such that

a(t−i) ∈ argmaxa∈AN\{i}

SWN\{i}(a | t−i).

Then, for each t, VCG chooses a(t) as allocation and elicitspayment

pi(t) !∑

j !ivj(a(t−i), tj) −

∑

j !ivj(a(t), tj). (A.4)

VCG maximizes the total welfare point-wise; moreover, itis EPIR. The following lemma helps compute its expectedrevenue.

Lemma A.2. Let N ! {1, . . . , n} be the set of agents with in-dependent and identically distributed (i.i.d.) types, and let Q !{1, . . . , n − 1} denote the same set of agents with one less member.Let TN ! ×i∈NTi denote the set of possible type profiles of agents inN, and similarly, let TQ ! ×i∈QTi be the set of possible type profilesof n − 1 agents. Then

Rev(VCG) ! nEt∈TQ SWQ(a(t) | t)[ ] − (n − 1)

· Et∈TN SWN(a(t) | t)[ ]. (A.5)

Proof. According to (A.4), the expected value of the VCGpayments can be written as

Rev(VCG) ! Et∈TN

∑

i∈N

∑

j∈N\{i}vj(a(t−i), tj)

([−

∑

j∈N\{i}vj(a(t), tj)

)]

! Et∈TN

∑

i∈N

∑


[ ]

− Et∈TN

∑

i∈N

∑

j∈N\{i}vj(a(t), tj)

[ ]

!∑

i∈NEti∈Ti Et−i∈T−i

N

∑


[ ][ ]

− (n − 1)Et∈TN SWN(a(t) | t)[ ]

!∑

i∈NEti∈Ti Et∈TQ

∑

j∈Qvj(a(t−i), tj)

[ ][ ]

− (n − 1)Et∈TN SWN(a(t) | t)[ ]

! nEt∈TQ

∑

j∈Qvj(a(t), tj)

[ ]− (n − 1)Et∈TN SWN(a(t) | t)[ ]

! nEt∈TQ SWQ(a(t) | t)[ ] − (n − 1)Et∈TN SWN(a(t) | t)[ ].

The equalities are direct consequences of the assumptionthat types are i.i.d. □

Corollary A.1. If

Et∈TQ SWQ(a(t) | t)[ ]

Et∈TN SWN(a(t) | t)[ ] ≥ 1 − ϱ,

then the expected revenue of VCG is at least 1 − nϱ times theexpected optimal welfare.

The message of Corollary A.1 is that if one wants tocompare Rev(VCG) to the optimal expected welfare, then itis sufficient to know the added value of an extra agent to thewelfare. Let Hi ! ∑i

j!1 1/j represent the ith harmonic num-ber, and set H0 ! 0. The next lemma is useful for providinglower bounds on the revenue–welfare ratio.

Lemma A.3 (Lemma 3 from Roughgarden and Sundararajan(2007)). Draw independently n times from an MHR distribution.Then the expected value of the lth largest value of n samples is atleast (Hn −Hl−1)/(Hn+j −Hl−1) times that of the lth largest value ofn + j samples.

Theorem A.1. Consider the single-item auction problem withn ≥ 2 agents who have unit demand and single-dimensional valu-ations i.i.d. according to an MHR distribution. Then VCG ex-tracts at least 1 − 1/Hn fraction of the optimal welfare in terms ofexpected revenue.

Proof. LetN ! {1, . . . , n} be the set of agents with i.i.d. types,and let Q ! {1, . . . , n − 1} denote the same set of agents withone less member. Furthermore, denote the lth largest valuefrom n samples by v[l:n]. Lemma A.3 implies that

E v[l:n−1][ ] ≥ (Hn−1 −Hl−1)/(Hn −Hl−1)E v[l:n]

[ ].


Therefore, we have that

Et∈TQ SWQ(x(t) | t)[ ]

Et∈TN SWN(x(t) | t)[ ] !E v[1:n−1][ ]

E v[1:n][ ]

≥ (Hn−1/Hn)E v[l:n][ ]

E v[l:n][ ]

! 1 − 1nHn

.

The proof is concluded by invoking Corollary A.1 and

letting ϱ ! 1nHn

. □

We note that according to theorem 4 of Roughgarden andSundararajan (2007), the ratio of the VCG revenue to theoptimal welfare is at least 1 − 1/n for monotone hazard ratedistributions. Their result is apparently not precise becauseour bound is tight for exponential distributions, and 1 − 1/Hnis generally lower than 1 − 1/n.

Now we are ready to prove Theorem 5. According to The-orem 3, Rev(UBM∗) is an upper bound on Rev(DSA∗); there-fore, it is sufficient to show that Rev(MS) approximatesRev(UBM∗). Let xUBM∗ represent the allocation rule of UBM∗.Define Te

UBM∗ ! {t ∈ T | ∃i : xiUBM∗ (t) ! 1,∀j ! i : xjUBM∗ (t) ! 0}and Ts

UBM∗ ! {t ∈ T | ∃i, j ! i : xiUBM∗ (t) ! xjUBM∗ (t) ! 1}. Then,according to Theorem 3, the revenue of UBM∗ can be splitsuch that Rev(UBM∗) ! Rev(UBM∗)s + Rev(UBM∗)e, where

Rev(UBM∗)s ! Et∈TsUBM∗

∑

iφF(si)xiUBM∗

[ ](A.6)

and

Rev(UBM∗)e ! Et∈TeUBM∗

∑

iφF(si) +mi( )

xiUBM∗

[ ]. (A.7)

The idea of the proof is to bound the two terms separately.We start with Rev(UBM∗)e.

BecauseφF(si) +mi ≤si +mi for all i,wehave thatRev(UBM∗)eis less than or equal to the optimal welfare of a single-itemauction. Note that OE achieves at least as much expectedrevenue as VCG does for the single-item auction. Therefore,because of Theorem A.1, Rev(OE) is at least 1 − 1/Hn timesthe optimal welfare of a single-item auction. This leads tothe conclusion that Rev(OE) ≥ 1 − 1/Hn( )Rev(UBM∗)e.

To bound Rev(UBM∗)s, observe that whenever UBM∗ al-locates shared for type report t, then xUBM∗ (t) ! xOS(t). This isbecause as(t) is defined the same way for both mechanisms.Using this observation together with LemmaA.2, we can write

Rev(OS) ! Et∑

iφF(si)xiOS(t)

[ ]

! Et∈TsUBM

∑

iφF(si)xiOS(t)

[ ]+ Et/∈Ts

UBM

∑

iφF(si)xiOS(t)

[ ]

! Rev(UBM∗)s + Et /∈TsUBM

∑

iφF(si)xiOS(t)

[ ]

≥ Rev(UBM∗)s.

The last inequality holds because, under OS, allocation onlyoccurs when the sum of virtual valuations is nonnegative.Putting together the two bounds results in

11 − 1/Hn

Rev(OE) + Rev(OS) ≥ Rev(UBM∗)s + Rev(UBM∗)e! Rev(UBM∗) ≥ Rev(DSA∗).

To finish the proof, note that Rev(MS) !max{Rev(OS),Rev(OE)}; hence,

11 − 1/Hn

+ 1( )

Rev(MS) ≥ Rev(DSA∗).

Finally, note thatwe can replaceDSIC to Bayes–Nash incentivecompatibility in (DSA) and relax the resulting mathematicalprogramby letting the exclusivitymargin be public information,as in (UBM). As we arrive at a single-dimensional setting, itis folklore knowledge that the optimal Bayes–Nash mecha-nism is DSIC. This means that UBM∗ is optimal even amongBayes–Nash-implementable mechanisms; therefore, it is alsoan upper bound for the Bayesian relaxation of (DSA). □

Endnotes1The terms shopper and merchant are used as a placeholder for tworoles, which could be patient–provider, content consumer–contentproducer, app developer–smartphone operating system, etc. Spe-cifically, the shopper could be a consumer or a buyer firm, and themerchant could be a firm or an individual. Later in this paper, whenwe discuss the auction, the word agent is used to represent a mer-chant, and a shopper is the item being auctioned or matched.2The limit k is usually invariant and set by policy. In practice, a valueof 3 or 4 provides sufficient competition and variety without over-loading customers with unwanted sales calls. For instance, lead-generation platforms for automobile trades typically match a customerwith three car dealers.3The optimal reserve price and the expression correspond to the op-timality rule (“inverse elasticity = optimalmarkup”) for pricing a singleitem to a buyer with unknown value v drawn from a distribution D.4Other assumptions are similar to ours: private values are distributedaccording to a monotone hazard rate distribution, and they search fordeterministic mechanisms.5Note: Even for the same product, the demand profiles or the numberof bidders can vary when the product sale is repeated at a differentdate, day of the week, time, location, etc., potentially motivatingswitching the auction format within the same product.6 SeeNetworks Offer Taste of TV’s Ad Future.Marketers Are Hungry forMore,https://www.nytimes.com/2017/05/21/business/media/networks-offer-taste-of-tvs-ad-future-marketers-are-hungry-for-more.html.

ReferencesAlaei S, FuH,HaghpanahN,Hartline JD,MalekianA (2012) Bayesian

optimal auctions via multi- to single-agent reduction. Proc. 13thACM Conf. Electronic Commerce (ACM, New York), 17.

Aseff J, Chade H (2008) An optimal auction with identity-dependentexternalities. RAND J. Econom. 39(3):731–746.

Barlow RE, Marshall AW, Proschan F (1963) Properties of probabilitydistributions with monotone hazard rate. Ann. Math. Statist.34(2):375–389.

Cai Y, Daskalakis C,Weinberg SM (2011) On optimal multi-dimensionalmechanism design. SIGecom Exchange 10(2):29–33.


https://www.nytimes.com/2017/05/21/business/media/networks-offer-taste-of-tvs-ad-future-marketers-are-hungry-for-more.html

https://www.nytimes.com/2017/05/21/business/media/networks-offer-taste-of-tvs-ad-future-marketers-are-hungry-for-more.html

Cai Y, Daskalakis C, Weinberg S (2013) Understanding incentives:Mechanism design becomes algorithm design. FoundationsComput. Sci. (FOCS), 2013 IEEE 54th Annual Sympos. (IEEEComputer Society, Washington, DC), 618–627.

Choudary SP, Van Alstyne MW, Parker GG (2016) Platform Revolution:How Networked Markets Are Transforming the Economy–And How toMake Them Work for You (WW Norton & Company, New York).

Clarke EH (1971) Multipart pricing of public goods. Public Choice11(1):17–33.

Deng C, Pekec S (2013) Optimal allocation of exclusivity contracts.Accessed January 15, 2014, https://www.semanticscholar.org/paper/Optimal-Allocation-of-Exclusivity-Contracts-Deng/0172084825d54099e8a52136686caefb1eda1b19.

Devanur NR, Hartline JD, Karlin AR, Nguyen CT (2011) Prior-independent multi-parameter mechanism design. ChenN, Elkind E,Koutsoupias E, eds.WINE’11 Proc. 7th Internat. Conf. Internet NetworkEconom. (Springer-Verlag, Berlin, Heidelberg), 122–133.

Dhangwatnotai P, Roughgarden T, Yan Q (2015) Revenue maximi-zation with a single sample. Games Econom. Behav. 91:318–333.

Evans DS, Schmalensee R (2016)Matchmakers: The New Economics of Mul-tisided Platforms (Harvard Business Review Press, Brighton, MA).

Figueroa N, Skreta V (2011) Optimal allocation mechanisms withsingle-dimensional private information. Rev. Econom. Design15(3):213–243.

Groves T (1973) Incentives in teams. Econometrica 41(4):617–631.Hartline JD (2012) Approximation in mechanism design. Amer. Econom.

Rev. 102(3):330–336.Jehiel P, Moldovanu B, Stacchetti E (1996) How (not) to sell nuclear

weapons. Amer. Econom. Rev. 86(4):814–829.Likhodedov A, Sandholm T (2004) Methods for boosting revenue in

combinatorial auctions. Proc. 19th Natl. Conf. Artificial Intelligence,AAAI’04 (AAAI Press, San Jose, CA), 232–237.

Milgrom PR (2004) Putting Auction Theory to Work, Churchill Lecturesin Economics (Cambridge University Press, Cambridge, UK).

Myerson R (1981) Optimal auction design.Math. Oper. Res. 6(1):58–73.Nisan N, Roughgarden T, Tardos E, Vazirani VV (2007) Algorithmic

Game Theory (Cambridge University Press, New York).Pei J, Klabjan D, Xie W (2014) Approximations to auctions of digital

goods with share-averse bidders. Electronic Commerce Res. Appl.13(2):128–138.

Roberts K (1979) The characterization of implementable choice rules.Laffont JJ, ed. Aggregation and Revelation of Preferences (North-Holland, New York), 321–349.

Roughgarden T, SundararajanM (2007) Is efficiency expensive? ThirdWorkshop Sponsored Search Auctions, Banff, Alberta, Canada.

Roughgarden T, Talgam-Cohen I (2013) Optimal and near-optimalmechanism design with interdependent values. Proc. 14thACM Conf. Electronic Commerce, EC ’13 (ACM, New York),767–784.

Salek M, Kempe D (2008) Auctions for share-averse bidders.Papadimitriou CH, Zhang S, eds. Internet and Network Economics—WINE 2008, Lecture Notes in Computer Science, vol. 5385(Springer, Berlin, Heidelberg), 609–620.

Sayedi A (2012) Essays on sponsored search advertising. Unpub-lished doctoral dissertation, Carnegie Mellon University,Pittsburgh.

Sayedi A, Jerath K, Baghaie M (2018) Exclusive placement in onlineadvertising. Marketing Sci. 37(6):970–986.

Segal I (1999) Contracting with externalities. Quart. J. Econom. 114(2):337–388.

Tang P, Sandholm T (2012) Mixed-bundling auctions with reserveprices. Proc. 11th Internat. Conf. Autonomous Agents MultiagentSystems, vol. 2, AAMAS ’12 (International Foundation for Auton-omous Agents and Multiagent Systems, Richland, SC), 729–736.

Varian HR (2009) Online ad auctions. Amer. Econom. Rev. 99(2):430–434.

Vickrey W (1961) Counterspeculation, auctions, and competitive sealedtenders. J. Finance 16(1):8–37.


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