Correlation-Robust Analysis of Single Item Auction

Correlation-Robust Analysis of Single Item Auction∗

Xiaohui Bei† Nick Gravin‡ Pinyan Lu‡ Zhihao Gavin Tang§

Abstract

We investigate the problem of revenue maximization insingle-item auction within the new correlation-robustframework proposed by Carroll [2017] and further de-veloped by Gravin and Lu [2018]. In this frameworkthe auctioneer is assumed to have only partial informa-tion about marginal distributions, but does not knowthe dependency structure of the joint distribution. Theauctioneer’s revenue is evaluated in the worst-case overthe uncertainty of possible joint distribution.

For the problem of optimal auction design in thecorrelation robust-framework we observe that in mostcases the optimal auction does not admit a simpleform like the celebrated Myerson’s auction for indepen-dent valuations. We analyze and compare performancesof several DSIC mechanisms used in practice. Ourmain set of results concern the sequential posted-pricemechanism (SPM). We show that SPM achieves a con-stant (4.78) approximation to the optimal correlation-robust mechanism. We also show that in the symmet-ric (anonymous) case when all bidders have the samemarginal distribution, (i) SPM has almost matchingworst-correlation revenue as any second price auctionwith common reserve price, and (ii) when the numberof bidders is large, SPM converges to optimum. In ad-dition, we extend some results on approximation andcomputational tractability for lookahead auctions to thecorrelation-robust framework.

1 Introduction

The monopolist’s theory of Bayesian revenue maximiza-tion for single-item auction is one of the most well stud-ied topics in mechanism design literature. The centralresult in this field is by Myerson [30] that gives the opti-mal auction for the case when the buyers’ private valuesare drawn according to independent prior distributionswhich are known to the auctioneer. As Myerson pointed

∗This work is supported by National NSF of China (No.

61741209) and the Fundamental Research Funds for the Central

Universities.†Nanyang Technological University. [email protected]‡Shanghai University of Finance and Economics. nikolai,

[email protected]§The University of Hong Kong. [email protected]

out, the independence is a strong assumption that is cru-cial for the optimality of his auction and which does nothold in many scenarios. Naturally, the general problemwhere the joint prior distribution of bidders valuationsmay be correlated has attracted significant amount ofattention in economics (for a survey see [27]). A no-table result from this literature is [20] by Cremer andMcLean who showed that the auctioneer can extract fullsocial surplus if the joint prior distribution satisfies cer-tain mild conditions1. This result implies that for thosesufficiently generic joint distributions the auctioneer canalways allocate the item to the highest value bidder andat the same time make the expected utility of each bid-der to be zero by collecting winner’s value for the goodas a payment. This result does not seem to be applica-ble in any practical setting and, therefore, “casts doubton the value of the current mechanism design paradigm”according to [29].

Another practical issue concerning the generalBayesian problem with any (correlated) prior distribu-tion was articulated in [25] for a related but differentcontext of the multi-product monopoly problem with asingle buyer. Namely, the corresponding learning prob-lem of a multi-dimensional prior distribution has expo-nential in the dimension (i.e., in the number of goods)representation and sampling complexity. We note thatthe same issue is pertinent to the Bayesian single-itemauction with general multi-dimensional prior distribu-tion of buyers’ private values. In other words, if anyonedecides to obtain an accurate statistical estimate of thejoint distribution of buyers’ values they would need toobserve unrealistically many examples of the joint valueprofiles. Furthermore, if the seller and the buyers donot have publicly available and statistically accurate es-timate of the joint prior, then they will need to agreeon the common Bayesian prior, each party having muchuncertainty about this distribution. The standard ap-proach in theoretical economics to model such uncer-tainty is via extremely expressive and rich type spacesof [26]. This route is subject to the criticism from [8],who state that “very large type spaces would be needed

1It was later shown in [18] that full surplus extraction is

possible for generic prior in a certain topological space definedon the set of prior distributions.

Copyright c© 2019 by SIAMUnauthorized reproduction of this article is prohibited

and applied work remains highly sensitive to sometimesunexamined modeling choices about types; nowhere isthis more true than in mechanism design”.

The robustness analysis of the single-item auctionis the topic of our paper. To this end we employa new correlation-robust framework of [13] originallyproposed for the monopoly problem with multiple goodsand a single buyer, but which could be easily andnaturally adapted to Bayesian multiple bidder scenariowith possible correlation among the bidders’ types aswas mentioned in [25]. In this framework the valuationprofile v = (v1, . . . , vn) is drawn from a joint general(correlated) distribution J of bidder’s values, which isnot completely known to the auctioneer. It is assumedthat the seller knows the marginal distributions of Jfor the private values vi ∼ Fi of every individualbidder i, but neither the seller nor the bidders have anyknowledge about possible correlation across differentbidders. Any single-item auction is evaluated accordingto the auctioneer’s expected profit derived in the worst-case, over all possible joint distributions consistentwith the given set of individual distributions Fi ofeach separate bidder i ∈ [n]. This evaluation givesthe seller a robust guarantee on the expected profitof his auction which holds for any distribution withpossible dependencies across different bidders. Theseller seeks a truthful auction, i.e., dominant strategyincentive compatible (DSIC) and ex-post individuallyrational (IR)2. On the other hand, because of thevon Neumann minimax theorem, there exists a priordistribution for which the latter guarantee is tighteven in the standard Bayesian framework where theauctioneer besides marginal distributions is given thespecific joint prior.

Interestingly, the correlation-robust framework wasfirst used in [13, 25] as a tractable mathematicalframework to study the unwieldy and difficult multi-dimensional monopoly problem with a single buyer. Onthe other hand, in this setting it is often possible for theseller to acquire more information about dependencies,say, between different pairs or even triplets of items bydoing more extensive market research. However, in thescenarios with multiple buyers these pair-wise depen-dencies might be quite hard to observe and learn. Forexample, a particular pair of buyers could have never

2Truthfulness is a standard assumption usually adopted inthe computer science literature. In economics a more popularassumption for multi-agent environment is a weaker Bayesian

incentive compatibility (BIC) and interim IR. We note that BICand interim IR assumptions are not well compatible with thecorrelation-robust framework. Indeed, the prior joint distribution

J is not explicitly given. It is hard to model and even harder topredict how the bidders would behave under such uncertainty.

participated in the same auction together, or the auc-tioned item is of the type that is usually sold directlyvia one-to-one negotiations or by price posting. An-other common scenario where the auctioneer may onlyknow the marginal distribution for each buyer is the onewhere identities of the participating bidders cannot beobserved. In this case it is reasonable to assume thateach buyer has identical prior distribution, which canbe deduced from an empirical study of a small randomsubgroup of the buyers’ population.

The correlation-robust framework allows one todirectly compare the practical results in Myerson’ssetting 3. One can view both scenarios as a problemwhere the auctioneer only knows prior distribution ofeach individual bidder. The difference is that in ourcase we assume that the buyers’ valuations can becorrelated and take the worst-case approach while inMyerson’s setting the seller simply assumes that buyers’valuations are independent, i.e., speaking in computerscience terms, does the average case analysis.

1.1 Our Results. Our first group of results con-cerns the design of optimal correlation-robust auctions,which, turn out to be not as simple as the Myerson’sauction. In Appendix B we give a few simple numeri-cal examples with an intricate structure of the optimalauction. Next, we turn to the design of approximatelyoptimal, but much simpler and more practical auctions.To this end, we look at a number of single-item DSICauctions used in practice in the correlated-robust frame-work. We show that among them a sequential posted-price mechanism (SPM) can always provide strongcorrelation-robust revenue guarantees. Our main resultis a construction of SPM which is 4.78-approximationto the optimal revenue in this correlation-robust frame-work. In the anonymous case when all buyers share thesame marginal distribution, we show that (1) when thenumber of bidders exceeds the number of types in thesupport of the marginal prior distribution, SPM is theoptimal correlation-robust auction. In other words, asthe number n of bidders grows the SPM’s revenue ap-proaches optimal correlation-robust revenue. We alsoshow for the anonymous case that (2) SPM extracts atleast n−1

n -fraction of the revenue of the best second priceauction with common reserve price.

Our second group of the results is closely related tothe computer science literature on lookahead auctionsinitiated by Ronen [32]. The practical restriction in the

3Here, by saying practical we refer to the comparison betweenMyerson’s setting to the general (correlated) case. Of course, onemay argue that even Myerson’s optimal auction is not practical,

but in certain important special cases its theoretical solution isclose to what is observed in practice.


family of lookahead auctions is that only the highestbidder can get the item. We note that this restrictioncan be quite handy for the designer as together withtruthfulness it guarantees that the outcome is envy-free.On the other hand, it is quite easy to optimize withinthis class of auctions, because the problem reduces toa single-dimensional pricing problem for the highestbidder. It is also known that the revenue of the optimallookahead auction is at least half of the revenue ofthe optimal auction. We note, however, that all theprevious results assume perfect knowledge of the priordistribution of buyer types. In this paper, we study theworst-correlation performance of lookahead auctions. Itis easy to show that the optimal lookahead auctionremains 2-approximation to the optimal revenue in thecorrelation-robust framework. However, it is not thateasy to compute one without the access to the jointprior distribution. We solve this computational questionin the anonymous case. Specifically, we identify thestructure of the corresponding worst-case distribution,which allows us to find a polynomial time algorithmthat computes the optimal correlation-robust lookaheadauction.

1.2 Related work Most of the work on Bayesiansingle-item auction for general (correlated) prior dis-tribution concerns the optimal auction design with re-spect to a given prior. It was shown in [20] that fullsocial surplus can be extracted if the joint prior distri-bution is generic in a certain sense. Their result wasextended in [29] to continuous type spaces, and by [18]to generic prior distributions in a properly defined topo-logical space over the set of all prior distributions. Froma computational perspective, it was shown in [31] thatcomputing the optimal mechanism under ex-post indi-vidual rationality constraint is NP-hard. On the otherhand, for randomized mechanisms [21] showed that op-timum can be computed in time polynomial in the sizeof the distribution’s support.

Another line of work is concerned with the design ofsimple and approximately optimal auctions. In partic-ular, the lookahead auctions introduced by Ronen [32]admits both computational efficiency and good approx-imation to the optimal revenue: it is a 2-approximationto the optimal mechanism. Ronen also introduced thegeneralized k-lookahead auction, which allows allocat-ing the item to one of the k highest bidders. Later, [21]showed that the best k-lookahead auction has 2k−1

3k−1 -approximation and can be computed efficiently in theoracle access model defined in [33]. [17] further im-

proved the ratio to e1−1/k

e1−1/k+1, which was recently proved

to be asymptotically tight in [22]. The family of se-quential posted-price mechanisms is proved to achieve

a e/(e − 1) approximation in the independent multi-dimensional setting by Chawla et al. [15, 16]. Later,Yan [34] generalized the results to matroid environ-ments.

When the correlated prior distribution is unknownand needs to be learned, computational and samplingcomplexity concerns have been raised in several pa-pers [1, 28, 25]. In contrast to the case of independentprior, much less is known about learning problem of acorrelated prior or designing a mechanism from givensamples. Fu et al. [23] proved that the optimal mech-anism can be efficiently learned when there is a finiteset of distributions from which the true distribution isdrawn. However, there is a fundamental distinction be-tween infinite sets of distributions and finite sets, andtheir results do not extend to the general value distri-butions [3].

Our work follows a trend in economics literatureon robust mechanism design [5], where the goal is toprovide performance guarantees even when there is un-certainty in the distribution over bidders’ valuations. Anumber of works in different settings employ a similar toour correlation-robust approach of searching solution toa parametrized optimization problem in the worst-caseover the space of parameters, e.g., [12, 14, 9, 19, 24].In particular, [4] and [11] study the setting of single-item auction. As in the standard Bayesian frameworkthe bidders’ values for the item are drawn from a com-monly known prior, however bidders there may havearbitrary information (certain high-order beliefs) aboutthe prior distribution unknown to the seller. Auction’sperformance is measured according to the lowest ex-pected revenue across a class of incomplete informationcorrelated equilibria termed Bayes correlated equilibria(BCE) in [6, 7]. In our work we completely ignore thebelieves of the buyers by focusing on the DSIC mecha-nisms. The authors in [4] also admit that allowing formodel uncertainty only for the seller but not for thebuyers is a valid concern, and “the possibility that thebuyers face model uncertainty is eminently worthy ofstudy”. When one cannot say much about buyers’ be-liefs, BCE or BIC cannot be applied anymore, and ourapproach with DSIC and ex-post IR requirement seemsto be more appropriate.

In AI community [2] also studied mechanisms thatare robust to uncertainty in the distribution, assumingthat the space of distributions has polynomial numberof dimensions. In contrast, the dimension of the spaceof distributions in our correlation-robust framework isexponential in the number of buyers.


2 Preliminaries

We consider a single-round auction where one item isbeing sold to n bidders. Each bidder i ∈ [n] hasa privately known value vi for the item and submitsa sealed bid bi to the auctioneer. Upon receivingsubmitted bids b = (b1, . . . , bn) from all bidders, theauctioneer decides which bidder i (if any) receives theitem and the amount bidder i pays pi(b). Bidder i utilityis the difference between her value vi and her paymentpi(b) if she gets the item; otherwise, she pays 0 andgets utility of 0. More generally, e.g., for a randomizedmechanism, we write allocation probabilities for thebidders as x(b) = (x1(b), . . . , xn(b)). In general,throughout the paper we use the convention to denotevectors, or any multidimensional objects, with bold facescript. The only exception will be the set of pricefunction pi(b)ni=1 that we simply denote by p.

The type v is drawn from a joint distribution J,which is not completely known to the auctioneer andwhich may admit correlation between different com-ponents vi and vj of v. The auctioneer only knowsmarginal distributions Fi of J for each separate com-ponent i but does not know how these components arecorrelated with each other. We assume that every dis-tribution Fi has finite support4 Vi. We use fi to denotethe probability density function of the distribution Fi.For notational convenience we also use Fi to denote therespective cumulative density function. The joint sup-port of all Fi is V = ×ni=1Vi. We use Π to denote allpossible distributions π supported on V that are consis-tent with the marginal distributions F1, F2, · · · , Fn, i.e.,Π =

π∣∣ ∑

v-iπ(vi,v-i) = fi(vi), ∀i ∈ [n], vi ∈ Vi

.

Since we want to study the worst-case performance ofa mechanism over this uncertainty, we measure perfor-mance of a mechanism in the worst-case over π ∈ Π.

(2.1) minπ∈Π

∑v

π(v) ·n∑i=1

pi(v).

Note that (2.1) is a linear program, since Π is givenby a set of linear inequalities. We also write the

4Similar to [13, 25] all our non computational and some

computational results extend to the distributions with continuoustypes and bounded support.

corresponding dual problem.

min∑v

π(v) ·n∑i=1

pi(v)(2.2)

s. t.∑v-i

π(vi,v-i) = fi(vi) dual var. λi(vi)

π(v) ≥ 0

max

n∑i=1

∑vi

fi(vi) · λi(vi)(2.3)

s. t.

n∑i=1

λi(vi) ≤n∑i=1

pi(v) ∀v

λi(vi) ∈ R

The (2.2) LP can be simplified if we consider identicaldistributions (Fi = F for all i ∈ [n]) and when mech-anisms are symmetric, i.e., mechanisms are invariantunder permutation of bidder identities. Indeed, we cantake the optimal solution π∗ to the primal LP and av-erage it out over all permutations of bidders (note thatevery permutation of bidders again yields an optimal so-lution to the primal LP). The simplified dual LP looksas follows.

max n ·∑v

f(v) · λ(v)

s. t.

n∑i=1

λ(vi) ≤n∑i=1

pi(v) ∀v = (v1, · · · , vn) ∈ V n

λ(v) ∈ R ∀v ∈ V

The seller’s problem. Besides evaluating theperformance of a given mechanism (x, p), the seller’sgoal is to find a mechanism with the maximal worst-correlation revenue for a given set of marginal distri-butions Fini=1. Formally, we want to find a truthfulmechanism (x∗, p∗) such that

(2.4) (x∗, p∗) ∈ argmax(x,p)

minπ(x,p)π∈Π

∑v∈V

π(v)

n∑i=1

pi(v).

We consider only truthful mechanisms (x, p) thatare dominant strategy incentive compatible (IC) andex-post individually rational (IR). Formally, for all i,type v, and vi

′, (x, p) satisfies:

xi(v) · vi − pi(v) ≥ xi(vi′,v-i) · vi − pi(vi′,v-i)(IC)

xi(v) · vi − pi(v) ≥ 0(IR)

For problem (2.4) we can write an LP formulation


based on the dual of LP (2.2).

max

n∑i=1

∑vi

fi(vi) · λi(vi)

(2.5)

s. t.

n∑i=1

λi(vi) ≤n∑i=1

pi(v) ∀v ∈ V

λi(vi) ∈ R(x(v), pi(v)i∈[n]) : (IC), (IR)

Robust Analysis of specific Auctions. Thecorrelation-robust framework can be used to evaluateany specific auction in the worst-correlation metric.This metric serves as another dimension along whichone can compare different auction formats. We notehowever, that it is not a trivial computational taskto evaluate worst-case revenue of a given mechanismover all distributions π ∈ Π. However, for most ofthe truthful mechanisms employed in practice we cancompute in polynomial time worst-correlation revenue(see Appendix A for details): sequential posted pricemechanisms, second price auction with arbitrary set ofreserves, and Myerson’s optimal auction. Our resultfor sequential posted price mechanism continue to holdeven when one can randomize between different set ofprices.

Optimal Auctions: Examples. For any set ofmarginal distributions Fii∈[n], we can calculate (intheory) the optimal mechanism in the correlation-robustframework by solving the LP (2.5). In practice, we cansolve for optimal correlation-robust auctions for the in-stances with a constant number of bidders and priormarginal distributions with reasonably small supportsizes. In Appendix B, we illustrate with a few numericalexamples how optimal correlation-robust auctions maylook like. From these examples, it seems hard to finda reasonable description of such mechanisms other thanexpressing it in the explicit form. These examples donot completely rule out the possibility of a reasonablysimple description for the optimal solution, but theywere enough to refute all our conjectures about opti-mality of already known simple formats for single-itemauction. We believe a better understanding of theseexamples could shed light on the problem of optimalcorrelation-robust mechanism design.

3 Sequential Posted-Price Auctions

In this section, we study the class of sequentialposted price mechanisms (SPM). First, we constructan SPM which is a 4.78-approximation to the opti-mal correlation-robust mechanism over any distributionπ ∈ Π. Next we focus on the anonymous setting where

buyers share an identical marginal distribution. We givean SPM that achieves optimal worst-case revenue whenthe number of bidders n is large enough; we also com-pare SPM with the second price auctions with anony-mous reserve price: for any such second price auctionwe construct an SPM which is n−1

n -approximation tothe worst-case revenue of this auction.

Sequential Posted Price Mechanism. A se-quential posted price mechanism (SPM) M can be pa-rameterized by an ordering over the buyers σ, and avector of prices p. Given a set of marginal distributionsFini=1, the mechanism is implemented by sequentiallyproposing price pσ(i) to buyer σ(i) for i = 1, 2, . . . , n,and selling the item to the first buyer who accepts hisprice.

We will sometimes represent the price offered to abuyer in the quantile space with respect to that buyer’sdistribution. More specifically, given price pi offered tobuyer i, let its distribution quantile be qi = 1− Fi(pi),i.e., qi is the probability that buyer i accepts pricepi. When the function Fi is continuous and strictlymonotone, the mapping 1−Fi : supp(Fi)→ [0, 1] is one-to-one. Hence, we can parametrize each distributionFi in the quantile space as a mapping Vi : [0, 1] →supp(Fi)

5 with Vi(q) = F−1i (1−q). The use of quantiles

is convenient in several places in this section. Weslightly abuse the notations and denote the mechanismas both M = (σ,p) and M = (σ,q) interchangeably,where q = (q1, . . . , qn).

3.1 Worst-Case Revenue Guarantee We first ar-gue that the worst-case distribution for any SPM thatposts non-increasing sequence of prices is always thedistribution π∗ with the maximum positive correla-tion6. To construct π∗ one randomly draws a quantileq ∼ U [0, 1] and takes vi = Vi(q) for all i ∈ [n]. Thus

π∗def= (V1(q), · · · , Vn(q))|q ∼ U [0, 1].

Lemma 3.1. For any sequential posted price mecha-nism M that posts non-increasing sequence of prices,minπ∈Π Rev(M, π) = Rev(M, π∗).

Proof. We first calculate the revenue extracted by Mfor distribution π∗. Observe that buyer i takes the itemand pays pi if and only if vi ≥ pi and vj < pj for

5In general case, when Fi is not a continuous distribution, we

let Vi(q)def= infv | Fi(v) ≥ 1− q.

6From a revenue maximization point of view, the assumptionthat an SPM always posts non-increasing sequence of prices is

without loss of generality. Because otherwise one can always re-

order the prices in non-increasing order and this can only improvethe revenue.


all j < i, which happens with probability maxj≤i(1 −Fj(pj))−maxj≤i−1(1− Fj(pj)). Therefore,

Rev(M, π∗)

=

n∑i=1

(maxj≤i

(1− Fj(pj))− maxj≤i−1

(1− Fj(pj)))· pi.

On the other hand, the probability that the item issold in the first i rounds for any distribution π ∈ Πis Pr[∃j ≤ i, vj ≥ pj ] ≥ maxj≤i(1 − Fj(pj)). Since thesequence of pi is non-increasing, the latter probabilitybound also means that the item is sold at a price equalto, or greater than pi. This gives us a lower bound onthe expected revenue ofM, which matches the revenueRev(M, π∗) of M on π∗. Indeed,

Rev(M, π) =

n∑i=1

pi ·(

Pr [∃j ≤ i, vj ≥ pj ]

−Pr [∃j ≤ i− 1, vj ≥ pj ])

=

n∑i=1

(pi − pi+1) ·Pr [∃j ≤ i, vj ≥ pj ]

≥n∑i=1

(pi − pi+1) ·maxj≤i

(1− Fj(pj))

=

n∑i=1

(maxj≤i

(1− Fj(pj))

− maxj≤i−1

(1− Fj(pj)))· pi

= Rev(M, π∗),(3.6)

where pn+1 = 0.

This characterization of the worst-case distributionprovides a simple and convenient way to analyze theworst-case revenue of an SPM. For example, given aset of marginal distributions Fini=1, we can assumewithout loss of generality that any Pareto-optimal SPMM = (σ,q) has qσ(i) < qσ(i+1) for every 1 ≤ i < n. Thisis because if there exists i < j such that qσ(i) > qσ(j),then when the type v is drawn from distribution π∗,the item will never be sold to buyer σ(j). In this caseone might as well put buyer σ(j) to the very end of thequeue and set qσ(j) = 1 as it will never hurt the revenue.Furthermore, when the offers are in the increasing orderof quantiles, the worst-case revenue of an SPM M issimply

Rev(M, π∗) =

n∑i=1

(qσ(i) − qσ(i−1))pσ(i).

Next we focus on a specific class of sequentialposted price mechanisms, where the quantiles proposedto different buyers are all “spread-out”, in the sense thatany two quantiles are at least factor of 2 apart from eachother.

Definition 3.1. Spread-out SPM is a sequential postedpricing MS = (σ,q) such that for any two i 6= j eitherqi ≥ 2qj, or qj ≥ 2qi.

A good feature of SPM with spread-out quantiles is that,as the following lemma shows, the worst-case revenue ofan SPMM = (σ,p) with spread-out quantiles is withina constant factor of the revenue of the n independentsales in which the seller has n copies of the items andindependently offers one to each buyer i at price pi.This allows us to approximate the revenue of M as thesum of individual revenues from price posting and tosimplify the comparison with the worst-case revenue ofthe optimal mechanism.

Lemma 3.2. Given a set of marginal distributionsFini=1, if an SPM MS = (σ,q) satisfies 2qσ(i) ≤qσ(i+1) for every i ∈ [n − 1], then for any π ∈ Π,

Rev(MS , π) ≥ 12

∑i qipi.

Proof. By Lemma 3.1, it suffices to show thatRev(MS , π∗) ≥ 1

2

∑i qipi. We have

Rev(MS , π∗) =

n∑i=1

(qσ(i) − qσ(i−1))pσ(i)

≥n∑i=1

(qσ(i) − qσ(i)/2)pσ(i)

=1

2

n∑i=1

qσ(i)pσ(i) =1

2

n∑i=1

qipi.

Now we are ready to prove that SPM is a con-stant approximation to the mechanism with the optimalworst-case revenue. Given the marginal distributionsFini=1, we already know the worst-case joint distri-bution for an SPM is π∗, the one with the maximumpositive correlation. Our plan is to construct a specificjoint distribution π ∈ Π, as well as a revenue upperbound U , such that

• U is an upper bound on the revenue of any IC andIR mechanism on distribution π.

• Find a spread-out SPM MS such thatRev(MS , π∗) = Ω(U).

We construct a spread-out SPM MS in the follow-ing iterative algorithm AlgSequence (see below). In


Algorithm 1 AlgSequence

1: Offered← ∅ . Set of buyers who has been offered aprice

2: Qfree[0]← [0, 1] . Initially available quantiles forspread-out pricing

3: for k = 1 to n do4: Pick (i, qi) ∈ argmax(i,qi)qi · Vi(qi)|i /∈ Offered,qi ∈ Qfree[k − 1]..

5: σ(k)← i; Offered← Offered ∪ i . Offerprice Vi(qi) to i

6: Iidef= Qfree[k − 1] ∩ [ qi2 , 2qi] . Reserve quantile

interval around qi

7: Qfree[k]def= Qfree[k − 1] \ Ii . Update available

quantiles for spread-out pricing8: end for9: return (σ,q = (q1, . . . , qn))

the output of AlgSequence, the order σ and quan-tiles q define the specific SPM. We note that in Al-gSequence the sets Qfree[0] ⊃ Qfree[1] ⊃ . . . ⊃ Qfree[n].On the other hand, each Ii is indeed an interval, sincenone of the intervals [

qj2 , 2qj ] is contained in any other

such interval. Also, by construction intervals Ii for alli ∈ [n] are disjoint.

Construction of distribution π. The distribu-tion π is closely connected to the description of thealgorithm AlgSequence. To construct π we define

Rσ(k)def= Qfree[k] for all k ∈ [n − 1]. For nota-

tional convenience, we redefine the last interval Iσ(n)def=

Qfree[n − 1] by including all uncovered quantiles in[0, 1] \

(Iσ(1) ∪ . . . ∪ Iσ(n−1)

)and also define the last

Rσ(n)def= Qfree[n − 1]. Hence, the collection of sets

I = I1, . . . , In becomes a partition of [0, 1]. By con-struction we have Rσ(k) ∩ Iσ(j) = ∅ for all j ≤ k < nand Iσ(1)∪ . . .∪ Iσ(k)∪Rσ(k) = [0, 1] for all k ∈ [n]. Thepartition I of the quantile space [0, 1] and R are usedto generate the distribution π.

Construction of distribution π.

1: Draw a quantile q ∼ U [0, 1]. Find k such thatq ∈ Iσ(k).

2: for i = σ(1) . . . σ(k − 1) do3: Independently draw qi ∼ U [Ri] and let vi =Vi(qi).

4: end for5: for i = σ(k) . . . σ(n) do6: Let vi ← Vi(q).7: end for8: return (v1, . . . , vn)

Figure 1 illustrates the structure of distributionπ. It is easy to check that π is consistent with themarginal distributions Fini=1. Indeed, for each giveni = σ(k), if the quantile q was drawn from the setIσ(j) with j ≤ k then vi = Vi(q). If q was drawnfrom Iσ(j) with j > k, then q was drawn from U [Ri],since Ri = Rσ(k) = ∪j>kIσ(j). The latter means thatvi = Vi(qi) where qi is sampled from U [Ri] has exactlythe same distribution as Vi(q). Hence, the marginaldistribution of vi is Vi(q) for q ∼ U [0, 1], i.e., vi ∼ Fi.

The next lemma sets up the upper bound U for therevenue of any mechanism on distribution π.

Lemma 3.3.

maxM

Rev(M, π) ≤ U def=

n−1∑k=1

∫Iσ(k)

maxj≥kVσ(j)(q)dq +

n∑i=1

maxq∈Riq · Vi(q).

Proof. Let M be the optimal mechanism for distribu-tion π. Fix quantile parameter q ∈ [0, 1] which is usedin π to generate v. Suppose q ∈ Iσ(k) and i = σ(k). Weconsider the following two scenarios based on the buyerswho contribute to the revenue of M.

1. When k 6= n, and the payment comes from the buy-ers in σ(k), . . . , σ(n). In this case, the payment isupper bounded by maxj≥kvσ(j), since it cannotexceed the valuation of the winning agent. Whenwe integrate this over q ranging from 0 to 1, we getthe term

∑n−1k=1

∫Iσ(k)

maxj≥kVσ(j)(q)dq in RHS.7

2. Contribution to the revenue from agentsσ(1), . . . , σ(k − 1), or when k = n paymentof agent j = σ(n). In this case, we allow M toobserve the set Iσ(k) from which quantile q wasdrawn.

If k < n, then we do not need to think aboutagents σ(k), . . . , σ(n), since the payments fromthese buyers are accounted for in the previous case.On the other hand, according to the constructionof π the values of agents σ(1), . . . , σ(k − 1)are drawn independently from each other (eachqσ(j) ∼ U [Rσ(j)] for j ∈ [k − 1]). Similarly, ifk = n, the values of agents σ(1), . . . , σ(n − 1)are drawn independently from each other and alsoindependently from the initial quantile q ∈ Iσ(n)

7Note that this upper bound could be achieved in theory by

M. Because when q ∈ Iσ(k) is selected, the valuations of buyersσ(k), . . . , σ(n) are perfectly correlated. An optimal mechanismwill use other bids to infer the quantile parameter q and hence

the value vi = Vi(q). Therefore, it will extract the full socialsurplus of buyer i.


σ(1)

σ(2)

...

σ(k)

...

σ(n)

0 1

Iσ(1)

Rσ(1)

Iσ(2)

Rσ(2)

... ...

Iσ(k)

Rσ(k)· · ·

Iσ(n) = Rσ(n)· · · · · ·

... ...

q

Figure 1: structure of distribution π: after q ∼ U [0, 1] is drawn, the quantiles of buyers in Iσ(k) region all equal q,i.e. qi = q for i = σ(k)...σ(n); the quantiles of the remaining buyers are drawn independently from their respectiveR region, i.e. qi ∼ U [Ri] for i = σ(1)...σ(k − 1).

(vσ(n) = Vσ(n)(q), where q ∼ U [Iσ(n)]). Eitherway, the values of the agents σ(1), . . . , σ(k − 1)are drawn from independent distributions. In thissituation the Myerson’s auction gives the optimalrevenue. Instead we allow the auctioneer to selln items. Then the best mechanism is to sellitems separately by posting the best individualprices. This gives an upper bound on the revenueof Myerson’s auction. We integrate this bound overall quantiles q ∈ [0, 1] and get an upper bound ofmaxqi∈Riqi ·Vi(qi) on the revenue extracted fromeach agent i ∈ [n] (in fact, the bound is slightlybetter, because the probability that i accepts priceVi(qi), where qi ∈ Ri is smaller than or equal to qi).Summing over all agents i ∈ [n] we get the secondterm

∑ni=1 maxqi∈Riqi · Vi(qi) in RHS.

We conclude the proof by adding these two upperbounds together.

Now we show that the output of algorithm AlgSe-quence the spread-out SPMMS = (σ,q) has expectedrevenue that approximates U for its worst-case distrib-tuion π∗. The upper bound in Lemma 3.3 consists of twoterms. In the following (Lemma 3.4 and Lemma 3.5),we relate Rev(MS , π∗) to each of them separately.

Lemma 3.4.

2 ln(4) · Rev(MS , π∗) ≥n−1∑k=1

∫Iσ(k)

maxj≥kVσ(j)(q)dq.

Proof. We treat Ii separately for each i = σ(k), wherek ∈ [n − 1]. To simplify notations let us define the

function g(q)def= maxj≥kVσ(j)(q). By the selection

rule of qi and i in AlgSequence we have q · g(q) =q · maxj≥kVσ(j)(q) ≤ qi · Vi(qi) for every q ∈ Ii.

It implies that g(q) ≤ qiVi(qi) · 1q for every q ∈ Ii.

Therefore, the terms under summation in the RHS ofLemma 3.4 are not more than∫

Iσ(k)

maxj≥kVσ(j)(q)dq ≤

∫Ii

qiVi(qi) ·1

qdq

≤ qiVi(qi)∫ 2qi

qi2

1

qdq

= qiVi(qi) ln 4,

where the second inequality holds as Iσ(k) ⊆[qi2 , 2qi

].

The revenue of the SPM M = (σ,q) is at least12

∑i qiVi(qi) according to Lemma 3.2.

Lemma 3.5. 2 · Rev(MS , π∗) ≥n∑i=1

maxq∈Riq · Vi(q).

Proof. Let i = σ(k) be the agent chosen at step k ∈ [n]in AlgSequence. By definition Ri ⊂ Qfree[k−1], which


means that qi ·pi = qi ·Vi(qi) ≥ maxq∈Ri q · Vi(q). Wesum these inequalities over all i ∈ [n] and conclude theproof by employing Lemma 3.2.

By putting everything together we obtain the finaltheorem.

Theorem 3.1. The revenue guarantee of MS is a(2 ln 4+2)(≈ 4.78)-approximation to the best worst-caserevenue among all mechanisms.

Proof. We have

(2 ln 4 + 2) ·minπ∈Π

Rev(MS , π)

=(2 ln 4 + 2) · Rev(MS , π∗)

≥n−1∑k=1

∫Iσ(k)

maxj≥kVσ(j)(q)dq +

n∑i=1

maxq∈Riq · Vi(q)

≥maxM

Rev(M, π) ≥ maxM

minπ∈Π

Rev(M, π),

where the first line follows from Lemma 3.1, the secondline is a combination of Lemma 3.4 and Lemma 3.5, andthe third line is by Lemma 3.3.

3.2 Large Market Auctions In this subsection westudy a regime where the number of bidders is largeand the auctioneer cannot observe bidder identities, e.g.,in many auction scenarios on the internet. More con-cretely, we assume that bidders have identical marginaldistributions Fi = F for every i ∈ [n] and consider thecase when support V of F has a small size, not morethan n, where n is the number of bidders. We showthat in this case a sequential posted-price mechanismcan achieve the optimal worst-case revenue. This is be-cause by Lemma 3.1, the worst-case distribution for anySPM is the distribution π∗ with the maximal positivecorrelation. On the other hand, as the number of pos-sible valuations is no more than the number of bidders,the seller running an SPM has enough attempts to queryevery possible value in the support of F , therefore ex-tract the full social surplus.

Descending Sequential Posted Price Mecha-nism DSPM.

1. Let the support of F be V = s1, . . . , sm (m ≤n). W.l.o.g. let s1 > s2 > · · · > sm.

2. For i = 1 through m do:

• Offer take-it-or-leave price si to bidder i

Theorem 3.2. When n bidders have identical marginaldistributions Fi = F with support size |V | ≤ n,sequential posted price mechanism DSPM has optimalworst-case revenue.

Proof. Since DSPM is an SPM with monotonically de-creasing sequence of prices, the worst-case distributionis π∗ by Lemma 3.1. We repeat here the definition ofπ∗ in the case when distributions Fi are identical: ran-domly draw v ∼ F and make vi = v for all i ∈ [n]. Therevenue of DSPM achieved on any profile with vi = v forall i ∈ [n] is exactly v, since the support of F is equalto or smaller than n and DSPM will have a chance tooffer the price of v to a bidder with value v. Therefore,the worst-case revenue achieved by this mechanism isno less than

∑v∈V f(v) · v.

On the other hand, the revenue of any IR mecha-nism with respect to distribution π∗ cannot be largerthan the social welfare, which in this case is exactly∑v∈V f(v) · v.

The above result can be extended to instanceswhere the support of F is larger than the number ofbidders, at a small multiplicative approximation error.To work with the continuous spaces we simply groupthe values into the ranges of [(1 + ε)i, (1 + ε)i+1) andcan get the following corollary.

Corollary 3.1. When n bidders have identicalmarginal distributions Fi = F supported on [1, h],an SPM achieves (1 + ε)-approximation to the worst-correlation revenue of the optimal mechanism, providedthat n ≥ lnh

ln(1+ε) (≈ lnhε ).

3.3 Comparison with Second Price Auctionwith Common Reserve Price In this subsectionwe investigate the relation of the sequential posted-price mechanisms and second price auctions with reserveprices in our correlation-robust framework. These twofamilies of mechanisms are arguably the most importantand well-studied DSIC mechanisms in the mechanismdesign area. In particular, in the classic setting whenbuyers’ valuations are drawn i.i.d, Myerson’s celebratedauction falls into the family of second price auction withcommon reserve price, while SPM is shown to approachoptimal revenue asymptotically [10].

Interestingly, in our correlation-robust framework,under the assumption that buyers share an identicalmarginal distribution, the best SPM has almost match-ing or even better worst-correlation revenue than thebest second price auction with uniform (anonymous)reserve price. We formalize the statement below andprovide the proof in Appendix C.


Lemma 3.6. When buyers’ valuations follow identicalmarginal distributions. The revenue guarantee of thebest sequential posted price auction is always a n

n−1 -approximation to the best worst-case revenue among allsecond price auctions with common reserve price.

4 Lookahead Auctions

In this section, we study mechanisms with an additionalrestriction that the auctioneer must sell the item to thehighest bidder. This family of mechanisms is also knownas the lookahead auctions in the literature. We firstgive the formal definition of lookahead auctions, whichis slightly different from the literature when there aremultiple buyers sharing the same highest bid.

Lookahead Auction Family L. A mechanismM = (x,p) is a lookahead auction if for any v ∈ Vand i ∈ [n] such that vi 6= maxj vj , the allocation to thei-th buyer xi(v) = 0. LetML be the lookahead auctionwith the best robust-correlation revenue.

Theorem 4.1. The revenue guarantee of ML is a 2-approximation to the best worst-case revenue among allmechanisms.

Proof. According to Ronen’s result [32], we know thatfor any joint distribution π, maxM Rev(M, π) ≤ 2 ·maxM∈L Rev(M, π). Therefore,

maxM

minπ(M)∈Π

Rev(M, π) = minπ∈Π

maxM(π)

Rev(M, π)

≤ 2 ·minπ∈Π

maxM(π)∈L

Rev(M, π)

= 2 · maxM∈L

minπ(M)∈Π

Rev(M, π).

Where the first and the last equality follow from the vonNeumann minimax theorem and the convexity of L andthe set of all truthful auctions.

For known correlated distribution and under appro-priate oracle assumption of how to access the distribu-tion (since the distribution itself may have exponentialin n size), the best lookahead auction can be computedin polynomial time.

Is the optimal correlation-robust lookaheadauction computable in polytime?

Note that this question does not require any additionalassumptions on the oracle access model, as the inputto the problem is given by n single dimensional distri-butions Fii∈[n] that can be explicitly described as theinput to an algorithm. It is an interesting open question,which we do not know how to solve. In the remainingpart of this section, we show how to solve this problem

for an important special case of the identical marginaldistributions.

For each permutation σ ∈ Sn, let vσ be the valueprofile that vσi = vσ(i) for all i. As mentioned inSection 2, for each distribution and mechanism, bytaking the average over all permutations of the buyeridentities, we may restrict our study to symmetric joint

distributions and mechanisms. Let Πsymdef= π ∈ Π :

π(v) = π(vσ),∀v, σ) and Lsymdef= M ∈ L : xi(v) =

xσ(i)(vσ), pi(v) = pσ(i)(v

σ),∀v, σ, i. We have that

(4.7) maxM∈L

minπ∈Π

Rev(M, π) = maxM∈Lsym

minπ∈Πsym

Rev(M, π).

Consider the following simple family of value pro-files, which we refer as high-low types:

VHLdef= v : ∃i ∈ [n], v

H≥ v

Lsuch that vi = v

H

and vj = vLfor j 6= i.

In other words, there exists a high-type buyer i withvalue v

Hwhile all other buyers have the same value

vL. Let ΠHL be the family of distributions that are

supported on VHL, i.e. ΠHLdef= π ∈ Πsym : π(v) =

0,∀v /∈ VHL. Our next goal will be to show thatthe worst-correlation distribution with respect to (4.7)belongs to ΠHL. Note that all lookahead mechanismscan be described as posting a randomized price tothe highest bidder based on the other bidders’ values.Intuitively, the worst distribution should reveal the leastpossible information about the highest bid by showingthe remaining bids. We show that following this logicthe most efficient way to hide information is by usingonly high-low types.

Given a distribution π ∈ Πsym, we consider the fol-lowing transformation into a distribution πHL ∈ ΠHL.For each v = (v1, . . . , vn), We distribute the proba-bility mass π(v) uniformly among the following high-low types (vi, vj , . . . , vj)j 6=i, where i is sampled uni-formly among highest bidders and then j is sampleduniformly among all bidders except i. We denotethis partial distribution on VHL transformed from vas vHL. Applying this process to all v, we obtain adistribution πHL supported on VHL. Moreover, by con-struction πHL must be consistent with the marginal f ,i.e., πHL ∈ Π. Indeed, our transformation per anyv ∈ V does not change the average (over the set[n] of bidders) marginal distribution8. We claim thatmaxM Rev(M, π) ≥ maxM Rev(M, πHL).

8Speaking in algebraic terms, our transformation does notchange bidder marginal distributions when applied to all types in

the orbit of any particular v ∈ V under the action of permutationgroup Sn.


Before going to the details of our proof, we intro-duce extra notations to describe a partial lookaheadmechanism defined on high-low types. For v

H> v

L, the

mechanism will look at the value vL

and offer a possiblyrandomized price p ≥ v

Lto the high type bidder. We

denote δ(vL, p), where p ≥ v

L, the probability of posting

price p. IC and IR conditions are satisfied automati-cally in this form, and the feasibility of any symmet-ric single-item lookahead auction can be described bythe two following conditions:

∑p≥v

Lδ(v

L, p) ≤ 1 and

δ(vL, v

L) ≤ 1

n . For any truthful symmetric lookaheadauction M, we define MHL as its restriction on high-low type and we know that it can be represented by theabove δ(·, ·).

Next, we provide a reversal transformation to ex-tend a partial mechanismMHL to a symmetric truthfulmechanism MHL on the full support. For each valueprofile v, let vsecond be the second highest bid, which isequal to the highest bid when there are multiple high-est bidders. Recall that vHL is a partial distribution onVHL with total probability mass π(v). The mechanismMHL on v simply mimics the mechanism MHL on vHL

with the following small modification: when the mech-anismMHL offers a prices which is less than vsecond, themechanism MHL will offer the price vsecond instead. Bythis definition, it is easy to verify that MHL is indeeda symmetric truthful lookahead auction on the wholesupport. Another conclusion from this transformationis that the payment ofMHL on v is at least the expectedpayment of MHL on vHL since MHL can always sell theitem when MHL can sell it and with a same or largerprice. Sum over all v, we get the conclusion that

Rev(MHL, π) ≥ Rev(MHL, πHL).

Now, we are ready to prove maxM Rev(M, π) ≥maxM Rev(M, πHL) by the following chain of inequal-ities:

maxM

Rev(M, πHL) = maxMHL

Rev(MHL, πHL)

≤ maxMHL

Rev(MHL, π) ≤ maxM

Rev(M, π).

Taking the minimum over all π ∈ Πsym, we have

minπ∈Πsym

maxM

Rev(M, π) = minπ∈ΠHL

maxMHL

Rev(MHL, π).

Therefore, we can calculate the optimal partialmechanism by solving the following LP and extend it

into the whole support by the previous procedure.

max n ·∑v

f(v) · λ(v)

s. t. λ(v) ≤ vδ(v, v) ∀vλ(v

H) + (n− 1)λ(v

L) ≤

∑vH≥p≥v

L

pδ(vL, p) ∀v

H> v

L∑p≥v

L

δ(vL, p) ≤ 1 ∀v

L

δ(v, v) ≤ 1

n∀v

Note that this LP involves only quadratic number ofvariables and polynomial number of constraints, hence,is computable in polynomial time.

For notational simplicity, we presented our resultfor the original lookahead auction, which only sells itemto the highest value bidder. Both previous results canbe extend to more general k-lookahead auction, whichallows allocating the item to one of the top k bidders.

Theorem 4.2. The revenue guarantee of k-lookahead

auction is a e1−1/k

e1−1/k+1-approximation to the best worst-

case revenue among all mechanisms. For constant kand symmetric distributions, the optimal k-lookaheadauction can be computed in polynomial time.

5 Conclusion and Open problems

In this work we adapted the correlation-robust frame-work developed for the additive monopoly problem [13]to the well studied setting of single-item auction. Ourapproach can be considered as a worst-case counterpartto the extensively studied Myerson’s setting with in-dependent prior. In this sense the Myerson’s auctioncan be interpreted as the optimal average-case mecha-nism. This framework provides meaningful guaranteesfor the difficult problem with general (correlated) prior,and also is much easier to apply in practice than thestandard Bayesian approach, since it only requires in-formation about single dimensional distributions fromeach agent (which can be efficiently learned from data)and also avoids unrealistically good conclusions of [20].

In this work we showed the power of sequentialposted price mechanisms and proved that SPMs achieveapproximately optimal revenue in the correlation-robustframework. We also studied lookahead auctions andgave some computational results. Our work leaves alot of open questions. Perhaps the most important (asour numerical experiments indicate also quite difficult)is the question of describing/computing the optimalcorrelation-robust auction. On the other hand, wefound that, when the number of bidders is large, theformat of the optimal auction does not matter much and


a simple SPM extracts the optimal worst-correlationrevenue. A simpler but still very interesting questionis to find other regimes where the optimal correlation-robust mechanisms are easy to describe or to compute.

We also showed how to evaluate the worst-caserevenue of most of the existing deterministic DSICmechanisms such as SPM, second price auctions withreserves, and Myerson’s auction. The question towhich extent these computational results extend to therandomized mechanisms remains widely open. We onlyknow the answer for the class of randomized SPM (andthat one only for the prices offered in the decreasingorder).

Correlation-robust framework offers some interest-ing computational questions for lookahead auctions. Inthe classic Bayesian framework the computation of thebest lookahead auction is quite trivial and simply re-quires careful description of the oracle access to theprior distribution. In the correlation-robust frameworkthe input to the problem has a succinct representation,and for special case of anonymous bidders we alreadyknow that optimal lookahead auction can be computedin polynomial time. It is intriguing question to extendthis result to an arbitrary set of distributions. On theother hand, it is unclear if the correlation-robust ap-proximation guarantee of lookahead auctions can be im-proved for the case of identical marginals. Consequently,it is also an interesting question whether the approxi-mation ratio of lookahead auctions can be improved inthe standard Bayesian setting if the prior distributionis symmetric, i.e., invariant under any permutation ofbidders’ identities.

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A Robustness Analysis of Existing Auctions

We emphasize that the correlation-robust frameworkcan be used to evaluate any specific auction in theworst-correlation metric. These evaluations can help tocompare and decide between different auction formats.Motivated by this consideration we study the problemof computing minimal revenue of a given auction overall distributions Π with feasible marginals. Specifically,we show in this section how to calculate the revenue ofsequential posted price mechanisms, second price auc-tion with individual reserves, and Myerson auction inthe correlation-robust framework. Our result for the se-quential posted price mechanisms (SPM) holds even forrandomized mechanisms, provided that randomizationis only used over Pareto-efficient deterministic SPMs,i.e., the sequence of posted prices that are offered to thebuyers is non-increasing.

Theorem A.1. There are polynomial time algorithmsto calculate the worst-correlation revenue for

1. second price auction with individual reserves9,

2. Myerson auction,

3. sequential posted price mechanisms,

4. randomized sequential posted price mechanisms.

For the last result we assume that every deterministicpricing mechanism in the support of the randomizedmechanism offers prices in non-increasing order.

To calculate the revenue in the correlation-robustframework we shall consider the dual LP (2.2). We ob-serve that it has polynomially many variables, namely,λi(vi)i∈[n],vi∈Vi , but exponential in n number of con-straints. Therefore, a naıve approach of solving thisLP directly would take exponential in n number ofsteps. Instead we adopt three different strategies forthe above families of mechanisms. First, for the sec-ond price auction with individual reserves, we show thatλi(vi)vi∈Vi are monotone in the optimal solution ofthe dual LP (2.2) for any i ∈ [n]. To this end, weprove a more general fact that as long as

∑ni=1 pi(v)

is a monotone function of v the set of λi(vi)vi∈Vimust also be monotone for each i ∈ [n]. The mono-tonicity of λi(vi)vi∈Vi allows us to prune redundantconstraints from the dual LP (2.2) and efficiently re-duce it to a manageable size. We note that this ap-proach fails to solve the case of the Myerson auction,since the total payment in the Myerson auction mayexhibit non-monotone behavior. Instead, we introduceextra variables and develop a polynomial size LP that isequivalent to LP (2.2). Our construction builds on theobservation that for the Myerson auction, there are onlypolynomially many possible outcomes, i.e., the winnerof the auction and her payment, and each outcome de-fines a simple-to-describe set of buyers’ types. Finally,we explicitly construct the worst case distribution for allsequential posted price mechanisms with non-increasingprices (SPM) and give closed-form expression for theworst-correlation revenue. Moreover, since all mecha-nisms within this family share the same worst-case dis-tribution, it is easy to compute the worst-correlationrevenue of randomized sequential posted price mecha-nisms.

Proof of Theorem A.1:

9The auction is defined as follows: Each bidder is associated

with a reserve price. Consider the bidders who bid at least theirindividual reserves. If there are at least two such bidders, thehighest bidder wins and pays the larger between his reserve and

the second highest bid. If there is only one such bidder, thatbidder wins and pays the reserve price.


(1) Second price auction with individual reserves.First, we consider second price auction with indi-

vidual reserves. To this end, we show that λivi∈Vimust be monotone for all i ∈ [n] for a large family ofmechanisms.

Lemma A.1. For any M with monotone payments(i.e., ∀v v,

∑i pi(v) ≤ ∑

i pi(v)), there exists anoptimal λ = λi(vi)i∈[n],vi∈Vi such that λi(vi)vi∈Viare non decreasing for all i.

Proof. Let λ be the optimal solution to the LP. Weargue that λi(vi) = maxv≤vi λi(v) is also an optimalsolution. Note that λi is monotonically increasing forall i and λi(vi) ≥ λi(vi) for all i and vi. Hence, theobjective function

∑ni=1

∑vifi(vi) · λi(vi) is equal to or

greater than the optimal. It suffices to show that λ =λii∈[n],vi∈Vi is feasible, i.e.,

∑ni=1 λi(vi) ≤

∑ni=1 pi(v)

for all v.Fix a value profile v. Suppose λi(vi) = λi(vi) for

all i, where vi ≤ vi. Let v = (v1, · · · , vn). Note thatv v. By the original constraint corresponding to v,we have

n∑i=1

λi(vi) =

n∑i=1

λi(vi) ≤n∑i=1

pi(v) ≤n∑i=1

pi(v).

We note that any second price auction with individ-ual reserves has monotone payments. By Lemma A.1the solution to the dual LP (2.2) does not change ifwe add more linear constraints saying that λ is mono-tone. Now we are going to point out only a few (poly-nomially many) important constraints in the latter LP,so that all the remaining constraints do not affect theoptimal solution. Let ri be the reserve price for eachbuyer i ∈ [n]. We partition the space of all value pro-files V into Vi,zii∈[n],zi∈Vi according to the winner ofthe auction and her payment zi and V0 when no buyergets the item. Specifically, in Vi,zi we consider all valueprofiles v such that buyer i wins the auction and paysexactly zi. In this case, we know that buyer i has thehighest bid and every other buyer j 6= i has a valueeither smaller than her individual reserve price rj , orsmaller than the price zi. That is,

Vi,zi ⊆v∣∣ vi ≥ zi, ∀j 6= i, vj ≤ maxzi, r−j

,

where r−jdef= maxvj<rj vj .

Then any constraint in the dual LP (2.2) for v ∈Vi,zi is implied by the monotonicity of λ and thefollowing inequality∑

j 6=i

λj(maxzi, r−j ) + λi(vmaxi ) ≤ zi,

where vimax is the maximum value in the support of

Fi. Moreover, the latter inequality is weaker than theconstraint of dual LP (2.2) corresponding to the valueprofile v, where vi = vi

max and vj = maxzi, r−j forall j 6= i. For V0, we know that all buyers bid smallerthan their individual reserves, i.e. V0 = v | ∀i, vi <ri. Then, due to monotonicity of λ it suffices touse only

∑i λi(r

−i ) ≤ 0 constraint. We observe that

there are only polynomially many different categoriesVi,zii∈[n],zi∈Vi and the monotonicity of λ can bedescribed with only

∑i∈[n] |Vi| linear constraints. Thus

we can prune the remaining constraints and solve theLP efficiently.

(2) Myerson auction.Next, we study the worst-correlation revenue of the

Myerson’s auction. Let ri be the reserve price and φi bethe ironed virtual valuation for each buyer i. Followingthe proof for second price auction with individual re-serves, we partition V into Vi,zii∈[n],zi∈Vi accordingto the winner of the auction and her payment zi ≥ riand V0 when no buyer wins the item. Specifically, weconsider all value profiles v so that buyer i wins theauction and pays exactly zi. We know that the buyer ihas the highest virtual value φi(zi) ≥ 0 and every otherbuyer j 6= i has a virtual value smaller than φi(zi)

10.That is

Vi,zi ⊆ v | vi ≥ zi, ∀j 6= i, φj(vj) < φi(zi) .Then any constraint in dual LP (2.2) for v ∈ Vi,zi isimplied by the following inequality

(A.1)∑j 6=i

maxvj :φj(vj)<φi(zi)

λj(vj) + maxvi≥zi

λi(vi) ≤ zi.

Moreover, as in the previous case this inequality is im-plied by the constraint of the dual LP (2.2) correspond-ing to the value profile v, where vi = argmaxvi≥zi λi(vi)and vj = argmaxvj :φj(vj)<φi(zi) λj(vj) for all j 6= i. AlsoV0 = v | ∀i, vi < ri . Hence, the constraints for allv ∈ V0 are implied by

(A.2)∑i

maxvi≤r−i

λi(vi) ≤ 0.

We observe that there are only polynomially manydifferent categories Vi,zii∈[n],zi∈Vi . To succinctlydescribe the max operators in (A.1) and (A.2), weintroduce extra variables λi(vi)i∈[n],zi∈Vi with a fewextra constraints:

λi(·) is monotone ∀i ∈ [n]

and λi(vi) ≥ λi(vi), ∀i, vi ∈ Vi.

10For simplicity of the presentation we assume that there areno ties for virtual values.


Then constraints (A.1) and (A.2) can be expressed as∑j 6=i

λj(φ−1j (φ(zi)))+λi(v

maxi ) ≤ zi and

∑i

λi(r−i ) ≤ 0,

where φ−1j (φi(zi))

def= maxvj | φj(vj) < φi(zi). Thus,

we obtain a polynomial size LP that is equivalent to thedual LP (2.2) and can solve this new LP efficiently.

(3) & (4) Deterministic and randomized sequentialposted price mechanisms.

For this case, we first consider SPM that postsnon-increasing sequence of prices. It has already beenshown in Lemma 3.1 that the worst-case distribution forSPM is the distribution π∗ with the maximal positivecorrelation. Therefore, to calculate the worst revenueguarantee, it suffices to calculate the sum in (3.6)in Section 3. Furthermore, observe that the worstdistribution π∗ does not dependent on the mechanism.Hence, for any randomized mechanism with only nonincreasing sequences of posted prices the worst-casedistribution is again π∗. Then it is easy to compute theworst-correlation revenue of the mechanism by summingthe expression (3.6) over all random price sequences11.

Finally, the approach used in Myerson auction ofpartitioning the value space according to the winnerof the auction and her payment, can also be used toget a polynomial time algorithm for sequential postedprice mechanisms with arbitrary sequence of prices (e.g.,which is not necessarily monotone).

B Optimal Auctions: Examples

This section provides numerical examples of severaloptimal correlation-robust auctions. In order to betterunderstand the framework, we run experiments fordifferent marginal distributions to explore the structureof the optimal mechanisms.

In our first example there are two buyers withidentical marginal distributions U [0, 1]. In this case, wecould analytically prove that the optimal mechanismhas a reasonably simple description: it is the secondprice auction with a random anonymous reserve, wherethe reserve price is drawn from U [0, 3

4 ]. The optimal

λ(v) = 2v2

3 for v ∈ [0, 34 ] and λ(v) = 3

8 for v ∈ ( 34 , 1]

and the worst-correlation revenue equals to 38 . It is

worthwhile to mention that second price auction andMyerson auction achieve worst-correlation revenue 1

4and 5

16 respectively. Given the simple format of theoptimal mechanism in the latter setting, it is natural toask:

11If there are exponentially many price sequences in the support

of the randomized mechanism, one can simply do Monte-carlosimulation by sampling valuations from π∗ and running M.

Does the result generalize to (1) more buyersand (2) other distributions?

Our further numerical experiments give a strong evi-dence that the answer to both of these questions isnegative. Consider a minimal extension of the previ-ous example to the case of n = 3 buyers with identicalmarginal distribution U [0, 1]. We observe complicatedstructure of the optimal mechanism12: the mechanismnot only allocates the item to the highest bidder, butalso allocates to the second highest bidder with signifi-cant probability. The latter fact is quite disconcerting, asthe optimal auction exhibits such unruly behavior evenin this simple symmetric setting. Furthermore, this factalso rules out the possibility that any lookahead auc-tion can be optimal in general. We remark that we onlydo not know how to describe the optimal mechanismin a simple language, but still admit that there couldbe interesting special cases where the optimal auctionsare well behaved. We warn our readers about possi-ble difficulties and leave the two latter questions as aninteresting open research direction.

Next, we consider the case of two symmetric buyerswith marginal uniform distribution U(1, 2, 3), i.e., theuniform distribution over the discrete values 1, 2, and3. It might seem that at least in this extremelysimple setting the optimal mechanism must have aclean description. Surprisingly, it is not the case. Wepresent the optimal mechanism in the following matricesdescribed in a symmetric way, i.e., independent of thebuyer identities, where x(v1, v2) and p(v1, v2) are theallocation and payment to the buyer who bids v1 whilethe other one bids v2:

x :

1 2 3

1 13 0 0

2 56

12 0

3 1 1 12

and p :

1 2 3

1 13 0 0

2 43 1 0

3 116

52

32

The optimal λ(1) = 1

3 , λ(2) = 1, λ(3) = 32

and the worst-correlation revenue is 179 . One pos-

sible description of this mechanism is as a secondprice auction with random individual reserves. How-ever, the reserve prices are much trickier as we il-lustrate below. First, one chooses a pair of re-serve prices uniformly at random from the multiset(1, 2), (1, 2), (2, 1), (2, 1), (2, 3), (3, 2). Then we runsecond price auction with tie broken in favor of thebidder with the larger reserve. For instance, when thebuyers have values (3, 2) and reserve prices are (1, 2),

12We do not have an analytical expression for the optimal

mechanism in this case. Instead, we discretize the uniformdistribution and observe the numerical solution to the LP (2.5).


the first buyer wins and pays 3, since if she bids 2, thesecond buyer would win the auction by the higher re-serve tie-breaking rule. Some part of the auction’s com-plexity could be explained as a result of the tricky tie-breaking issues for the discrete types. Nonetheless, thecomplicated structure we already observe in such smallinstances precludes us from making any good conjectureabout the optimal mechanism.

Finally, we consider an asymmetric setting to avoidany tie-breaking complications from the previous exam-ple. In this setting there are n = 2 buyers with marginaldistributions U(1, 3, 5) and U(2, 4, 6) respectively. Wegive the optimal mechanism by the following matrices.

x1 :

2 4 6

1 0 0 0

3 23 0 0

5 1 1 0

, x2 :

2 4 6

1 12

56 1

3 0 56 1

5 0 0 1

and p1 :

2 4 6

1 0 0 0

3 2 0 0

5 113 5 0

, p2 :

2 4 6

1 1 73

103

3 0 103

133

5 0 0 6

C Proof of Lemma 3.6

Lemma C.1. When buyers’ valuations follow identicalmarginal distributions. The revenue guarantee of thebest sequential posted price auction is always a n

n−1 -approximation to the best worst-case revenue among allsecond price auctions with common reserve price.

Proof. Let SPAr be the second price auction with re-serve price r. By Lemma 3.1, it suffices to show that forall r, there exists an SPM M∈ S such that

n

n− 1Rev(M, π∗) ≥ min

π∈ΠRev(SPAr, π).

Again, it would be easy for us to work in the quantilespace. We first construct distribution π to establish anupper bound on minπ∈Π Rev(SPAr, π).

Construction of distribution π.

1: Draw a quantile q ∼ U [0, 1]

2: if q ≤ q∗ def= 1− F (r) then . All bids are at least r

3: Draw a buyer h ∼ U [n]. . Pick the winningbuyer

4: Let vh ← V ( qn )

5: Let v` ← V ( (n−1)qn + q∗

n ) for all ` 6= h.6: end if7: if q > q∗ then . All bids are less then r and the

item remains unsold8: Let vi ← V (q) for all i.9: end if

We first explain why π is consistent with themarginal distribution F . Fix any buyer i.

• When q ≤ q∗ and i was selected as the winningbuyer h, vi ← V ( qn ) where q

n was drawn from

U [0, q∗

n ]. This event happens with probability q∗

n .

• When q ≤ q∗ and i was not selected as the winning

buyer, vi ← V (q′) where q′ = (n−1)qn + q∗

n was

drawn from U [ q∗

n , q∗]. This event happens with

probability q∗ − q∗

n .

• When q > q∗, vi = V (q) where q was drawn fromU [q∗, 1]. This event happens with probability 1−q∗.

Combining these three cases together, vi = V (q) whereq ∼ U [0, 1], i.e., vi ∼ F .

When SPAr is run on π, if q > q∗, all bids areless than r and the item remains unsold. On the otherhand, if q ≤ q∗, we always have q

n ≤(n−1)qn + q∗

n ,which implies vh ≥ v`. In this case bidder h wins theauction and pays the price of v`. Therefore we have

Rev(SPAr, π) =∫ q∗

0v`dq =

∫ q∗0V(

(n−1)qn + q∗

n

)dq. By

letting q′ = (n−1)qn + q∗

n , we have

Rev(SPAr, π) =

∫ q∗

q∗n

V (q′)d

(nq′ − q∗n− 1

)=

n

n− 1

∫ q∗

q∗n

V (q)dq

=n

n− 1

n−1∑i=1

∫ (i+1)q∗n

iq∗n

V (q)dq

≤ n

n− 1

n−1∑i=1

V

(iq∗

n

)· q∗

n

=q∗

n− 1

n−1∑i=1

V

(iq∗

n

).

Now consider an SPM M that posts price V ( iq∗

n )to the i-th buyer in sequence for 1 ≤ i ≤ n − 113. ByLemma 3.1, we know the worst-correlation revenue ofM is

Rev(M, π∗) =q∗

n

n−1∑i=1

V

(iq∗

n

).

We conclude the lemma by combining the above twobounds.

13We only post prices to the first n− 1 buyers in this SPM.


Correlation-Robust Analysis of Single Item Auction

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