The Limits of an Information Intermediary in Auction Design
Reza Alijani, Siddhartha Banerjee, Kamesh Munagala, Kangning Wang
TThe Limits of an Information Intermediary in Auction Design
Reza Alijani ∗ Siddhartha Banerjee † Kamesh Munagala ∗ Kangning Wang ∗ Abstract
We study the limits of an information intermediary in Bayesian auctions. Formally, weconsider the standard single-item auction, with a revenue-maximizing seller and n buyers withindependent private values; in addition, we now have an intermediary who knows the buyers’true values, and can map these to a public signal so as to try to increase buyer surplus. Thismodel was proposed by Bergemann et al., who present a signaling scheme for the single-buyersetting that raises the optimal consumer surplus, by guaranteeing the item is always sold whileensuring the seller gets the same revenue as without signaling. Our work aims to understandhow this result ports to the setting with multiple buyers.Our first result is an impossibility: We show that such a signaling scheme need not existeven for n = 2 buyers with 2-point valuation distributions. Indeed, no signaling scheme canalways allocate the item to the highest-valued buyer while preserving any non-trivial fraction ofthe original consumer surplus; further, no signaling scheme can achieve consumer surplus betterthan a factor of compared to the maximum achievable. These results are existential (andnot computational) impossibilities, and thus provide a sharp separation between the single andmulti-buyer settings.On the positive side, for discrete valuation distributions, we develop signaling schemes withgood approximation guarantees for the consumer surplus compared to the maximum achievable,in settings where either the number of agents, or the support size of valuations, is small. For-mally, for i.i.d. buyers, we present an O (cid:0) min(log n, K ) (cid:1) -approximation where K is the supportsize of the valuations. Moreover, for general distributions, we present an O (cid:0) min (cid:0) n log n, K (cid:1)(cid:1) -approximation. Our signaling schemes are conceptually simple and computable in polynomial(in n and K ) time. ∗ Computer Science Department, Duke University, Durham NC 27708-0129. Supported by NSF grantCCF-1637397, ONR grant N00014-19-1-2268, and research awards from Adobe and Facebook. Email: { alijani,kamesh,knwang } @cs.duke.edu . † School of Operations Research and Information Engineering, Cornell University, Ithaca, NY. Supported by NSFgrants ECCS-1847393, DMS-1839346 and CNS-1955997. Email: [email protected] . a r X i v : . [ c s . G T ] S e p Introduction
Consider a seller selling an item to a buyer, whose private value V is drawn from some knowndistribution D . The overall social welfare is maximized when the seller sells the item for $0. Incontrast, to maximize the (average) revenue, the seller’s optimal strategy is to sell at a revenue-maximizing price, which may lead to welfare loss due to the item going unsold.More generally, in a single-item Bayesian auction with n buyers with independent private valu-ations, a welfare-optimal mechanism is the second-price (or VCG) auction, which always gives theitem to the highest-valued buyer. In contrast, even when the buyers have i.i.d. regular (continuous)valuations, the revenue-optimal mechanism was shown by Myerson [15] to be a second-price auctionwith a reserve price ; this may lead to the item going unsold. The situation is more complex fornon-i.i.d. buyers, where the revenue-optimal mechanism may in addition sell the item to a buyerwith lower value than the highest, leading to more welfare loss. We visualize this via a revenue-CStrade-off diagram (Fig. 1b), where, for different mechanisms and value distributions, we plot ex-pected consumer-surplus (i.e., value minus payment), denoted CS , versus expected seller-revenue,denoted by R . Any welfare-maximizing mechanism including VCG (point V ) lies on the 135 ◦ linewith intercept W ∗ , the maximum welfare. In contrast, Myerson’s mechanism (point M ) has revenue R M greater than that under VCG, but also lies strictly below the maximum-welfare line. Information Intermediary.
Now consider the same setting, but with an additional informationintermediary : a third-party who knows the true buyer values (cid:126)V = ( V , V , . . . , V n ) and can providea “signal” or side-information to the seller and the buyers. Both the signal and signaling schemeare common knowledge to all agents (buyers and seller), who can thus use Bayes’ rule to updatethe prior over valuations given the signal. The signal “re-shapes” the joint prior over the buyervaluations in a Bayes-plausible manner (i.e., such that the posterior averaged over signals equalsthe prior). Given the signal, a seller can then propose the revenue-maximizing mechanism, andbuyers bid optimally, under the posterior distribution. We illustrate this in Fig. 1a.Formally, consider a setting where n buyers have independent private valuations (cid:126)V drawn froma distribution D = D × D × · · · × D n . The valuations (cid:126)V are known to the intermediary, who mapsthem to a signal σ via a public signaling scheme Z . Given σ , all agents compute the posterior S over buyer values; note these can now be correlated. The seller then proposes a mechanism M S (comprising allocation and payment rules) which maximizes the expected revenue assuming buyersact in a manner which is ex-interim incentive-compatible (IC) and individually-rational (IR) given S . If σ is such that S = D , then M S is Myerson’s auction (point M in Fig. 1b); on the otherhand, if the signal fully reveals (cid:126)V , then the seller can extract full surplus (i.e., get revenue W ∗ ,point A in Fig. 1b). Moreover, the seller gets revenue at least R M under any signaling scheme, asshe can always ignore the signal (see Section 2). Thus any signaling scheme Z must give a point inthe shaded triangle with consumer surplus CS ( Z ) and revenue R ( Z ), and the maximum possiblesurplus Opt is achieved at point O in Fig. 1b. Now we can ask:What revenue-CS trade-offs can an information intermediary achieve via signaling?More specifically, what is the maximum possible consumer surplus that is achievable?In the single-buyer case, Bergemann, Brooks and Morris [1] completely answer these questionsby showing that the entire shaded region is always achievable. In particular, the point O is met bya simple signaling scheme where the revenue is exactly R M , and the item is always sold thus themechanism is efficient.In this work we study the effectiveness of an information intermediary in a multi-buyer (i.e., n ≥ B B n ... S ( V , V , . . . , V n ) σ M S V |S V |S V n |S ... InformationIntermediary (a) Auction with Information Intermediary R R + C S = W ∗ CS R = R M M O VA B
Opt (b) Revenue-CS Trade-off Diagram
Figure 1: (a) The auction with information intermediary setting, where the intermediary has full knowledgeof valuations (cid:126)V , and can use this to provide a signal σ to the seller and the buyers. The seller then uses therevenue-optimal mechanism M S for the posterior distribution over valuations S given σ .(b) Two-dimensional space of seller revenue, R , and consumer surplus, CS , of different signaling schemes.The points M and V correspond to the Myerson and VCG mechanisms, and the point A corresponds toselling to the highest-valued buyer at her value when the seller has full information. multi-buyer settings, as in the latter, no signaling scheme can guarantee more than a constantfraction of the optimal consumer surplus ( Opt in Fig. 1b). On the positive side, we obtain a novelyet simple signaling scheme with strong approximation guarantees for a wide range of settings.While our main focus is on theoretical results, our work has broader practical relevance. Con-sider for example an agency like the FCC with privileged information about bidders in a spectrumauction, or a bid optimizer working for multiple competing clients in an ad-exchange. These inter-mediaries have private information about the buyers, and can selectively release it to influence theauction. Our work also fits in a broader space of multi-criteria optimization where a third-partyplatform or government agency can release information about agents to a principal in charge of anactivity such as admissions or hiring, so as to trade-off the principal’s objective such as maximizingquality of hire, with a societal objective such as fairness or diversity.
We consider a single-item auction with n buyers with discrete valuations. We assume the buyervaluations are independent, so D = D × D × · · · × D n , where D i has support size K i , and the sizeof the union of the supports is K . We parametrize our results in terms of n, K i , and K .Our first set of results (Section 3) shows a sharp demarcation between the cases of n = 1 and n ≥ BO is not achievable; indeed, the onlyachievable points on segment AO are arbitrarily close to A . Therefore, achieving full welfare needssacrificing an arbitrarily large fraction of consumer surplus compared to the no-signaling baseline. Theorem 1.
For n = 2 buyers each with K i = 2 , for any given constant ε > , there are instanceswhere any signaling scheme Z under which the revenue-optimal auction obtains full welfare (i.e.,allocates to highest-value buyer), has CS ( Z ) ≤ ε · CS ( D ) , where CS ( D ) is the consumer surplus ofMyerson’s auction without signaling. We next ask if we can sacrifice on welfare, but raise a consumer surplus arbitrarily close to
Opt ? We again answer in the negative, and show a lower bound of 2 on the approximation ratio.2 heorem 2.
For n = 2 buyers each with K i = 2 , for any constant ε > , there are probleminstances where any signaling scheme Z has CS ( Z ) ≤ (cid:0) + ε (cid:1) · Opt . We note that the above results are existential impossibility results , and do not depend on thecomplexity of the signaling scheme. Given these negative results, we focus on approximating theconsumer surplus. To this end, we propose novel signaling schemes, which, informally, achieve aconstant-approximation to
Opt when either the number of buyers n is a constant (with arbitrarysupport size K ), or when K is a constant (for arbitrary n ). Formally, our main result, presented inSections 4 and 5, is the following: Theorem 3.
There are signaling schemes that achieve the following approximations to
Opt :1. O (cid:0) min(log n, K ) (cid:1) when D i ’s are identical; and2. O (cid:0) min (cid:0) n log n, K (cid:1)(cid:1) when D i ’s are arbitrary.Further, this signaling scheme has computation time polynomial in n and K . For any n , the optimal signaling scheme for maximizing surplus can be obtained via an infinite-sized linear program (see Eq. (2) in Section 3) with variables for every possible signal, i.e., everypossible joint distribution over buyer valuations. Further, for each such signal, the quantity ofinterest is the consumer surplus of the revenue-optimal auction given the signal. For n = 1 case,Bergemann et al. show this LP has a special structure in that it admits a basis comprising of“equal-revenue distributions” containing the revenue-maximizing price (see Section 2.2). Our workshows that this breaks down for optimal auctions with signaling involving n ≥ n = 1 to n ≥ D . However, with multiple buyers, the optimal auction does not have simple structureeven for independent buyers (see Algorithm 1), and we need to analyze the consumer surplus of thisauction, which can be a discontinuous function of the prior. (See Section 3 for examples.) Further,for correlated buyers, the revenue of the auction itself may not be continuous in the prior! Indeed,a celebrated result of Cr´emer and McLean [7] shows that slightly perturbing an independent priorto a correlated one can discontinuously increase the revenue to W ∗ , hence decreasing consumersurplus to 0. (See Theorem 4 in Section 2.) This makes it tricky to reason about the optimalsignaling scheme, leading to the gap between our upper and lower bounds.In more detail: Our proofs of Theorems 1 and 2 use a special case of the Cr´emer-McLeancharacterization [7]: for n = 2 buyers each with K i = 2, under any non-independent prior theseller can extract full social surplus as revenue. This lets us focus on signaling schemes wherebuyers remain conditionally independent given each signal. Using Myerson’s characterization ofthe optimal auction for discrete valuations [12], we show a structural characterization that reducesthe space of optimal signals to a sufficiently simple form, yielding the desired counterexamples.Theorem 3 is technically the most interesting result in the paper. At a high level, our schemebalances the trade-off between revealing enough information about valuations so that the item issold to a high-value buyer, and revealing too much information such that the seller extracts allthe surplus. Balancing these is delicate; nevertheless, we show simple schemes, with polynomialcomputation time and signal complexity.Our schemes involve choosing a threshold value corresponding to that of the ( t + 1) st buyer indescending order (for carefully chosen t ), and then applying the single-buyer signaling scheme in [1]3o the excess value (i.e., value minus threshold) of a randomly chosen buyer in the top t (whileleaving the rest unchanged). Note that though the posterior conditioned on signal is a productdistribution, it requires the intermediary to observe all the buyer valuations. We present this basicscheme in Section 4. To show our bounds that depend on n , we use an appropriate randomizationover such schemes, while the bounds depending on K follow from a concentration lemma overindependent Bernoulli trials (Lemma 6 in Section 5.1), which may be of independent interest. The general problem of information structure design considers how sharing additional infor-mation can influence the outcome of a mechanism. Different variants of this problem have beenformulated and studied; we refer the reader to [3, 9] for surveys. Of particular importance to usis the
Bayesian persuasion problem formulated by Kamenica and Gentzkow [14], where a receiverselects a utility-maximizing action based on incomplete information about the state of nature. Asender who knows the state of nature can signal side-information to the receiver so that the actiontaken by the receiver is utility-maximizing for the sender. This general problem has been widelystudied in different domains such as bilateral trade and advertising [1, 6, 18]. For this problem,there is a distinction between existence and computational results, and the work of [11] studies thecomputational complexity of finding the optimal signaling scheme under different input models.The restriction of our problem to one buyer is termed “bilateral trade”. Here, the intermediaryis the sender whose utility is consumer surplus, and the seller is the receiver whose action spaceis take-it-or-leave-it prices and whose utility is revenue. Beginning with the work of Bergemannet al. [1], several works [10, 16, 8, 5] have considered various extensions and modifications to thisbasic problem. Unlike bilateral trade where the buyer is perfectly informed, in our setting, not onlythe seller, but also all the buyers are receivers, in the sense that they have imperfect knowledgeof the true valuations of other buyers, and modify their respective bidding strategies in responseto the intermediary’s signal to maximize their own utilities. Our setting is therefore a Bayesianpersuasion problem with multiple receivers, and this aspect makes it significantly more complex.There has been work on signaling in auctions that cannot be modeled as Bayesian persuasion,i.e., in which the common signal is not generated by an intermediary who knows all the true valuesof the buyers. For instance, in [4], the auctioneer has perfect information about buyer valuationsand controls the precision to which buyers can learn it, and in [13], the seller’s signal is drawnfrom a distribution that is correlated with the buyer’s value, In both these works, the goal isto maximize seller revenue. Finally [17] studies equilibria of optimal auctions when each buyercommits to a signaling scheme with imperfect knowledge of other buyers’ valuations, while [2]studies equilibria in first price auctions when buyers are provided correlated signals about otherbuyers’ valuations. In contrast with the former, our work considers a richer space of signals via aninformation intermediary, while compared to the latter, in our setting the seller’s mechanism is notfixed, but is instead also a function of the information structure.
We consider Bayesian single-item auctions with n buyers, with independent private valuations (cid:126)V = ( V , V , . . . , V n ) drawn from a known product distribution D = D × · · · × D n . Unless otherwisestated, we present our results for the setting in which each D i is discrete. We denote by K i the sizeof the support of D i , and by K the size of the union of these supports.For distribution D i , we use f D i to denote its probability mass function, and define S D i ( x ) = Pr V ∼ D i [ V ≥ x ] and F D i ( x ) = Pr V ∼ D i [ V ≤ x ]. For a joint distribution D and vector (cid:126)v , we use4 r [ D = (cid:126)v ] = f D ( (cid:126)v ) as shorthand for denoting the probability of (cid:126)v drawn from D . Given any shared prior D (cid:48) on the valuations of the buyers, which in the case of signaling, can bedifferent from D and arbitrarily correlated, the seller runs an optimal (revenue maximizing) auctionthat satisfies ex-interim incentive compatibility and individual rationality. Using the revelationprinciple [15], such an auction is specified by an allocation rule x ∗ ( (cid:126)v ) ≥ p ∗ ( (cid:126)v )(that can be positive or negative) given any realized valuation profile (cid:126)v . The quantity x ∗ i ( (cid:126)v ) is theprobability buyer i gets the item given the valuation profile (cid:126)v .The incentive compatibility (IC) constraint states that the expected utility of any buyer doesnot increase by misreporting its valuation. Formally, for every agent i , for every value q , and forevery valuation vector (cid:126)v = ( q, (cid:126)v − i ) (where (cid:126)v − i denotes the valuations of the other buyers), andevery other possible report r , we have (cid:88) (cid:126)v − i Pr [ D (cid:48) = ( q, (cid:126)v − i )] Pr [ D (cid:48) i = q ] · ( q · x i ( q, (cid:126)v − i ) − p i ( q, (cid:126)v − i )) ≥ (cid:88) (cid:126)v − i Pr [ D (cid:48) = ( q, (cid:126)v − i )] Pr [ D (cid:48) i = q ] · ( q · x i ( r, (cid:126)v − i ) − p i ( r, (cid:126)v − i )) , ∀ i, q. The individual rationality (IR) constraint says that for any buyer and any valuation, the expectedutility under the mechanism is non-negative: (cid:88) (cid:126)v − i Pr [ D (cid:48) = ( q, (cid:126)v − i )] Pr [ D (cid:48) i = q ] · ( q · x i ( q, (cid:126)v − i ) − p i ( q, (cid:126)v − i )) ≥ , ∀ i, q. Finally, for any prior D (cid:48) , let ( R ( D (cid:48) ) , W ( D (cid:48) ) , CS ( D (cid:48) )) denote the expected revenue, welfare(or total surplus ) and consumer surplus under the revenue-maximizing auction. Then we have CS ( D (cid:48) ) = W ( D (cid:48) ) − R ( D (cid:48) ), and: R ( D (cid:48) ) = max (cid:88) (cid:126)v Pr [ D (cid:48) = (cid:126)v ] · (cid:88) i p ∗ i ( (cid:126)v ) and W ( D (cid:48) ) = (cid:88) (cid:126)v Pr [ D (cid:48) = (cid:126)v ] · (cid:88) i v i x ∗ i ( (cid:126)v ) . Our work builds on two special cases – independent valuations, and full surplus extraction.
Optimal auction for independent valuations.
When D = D × . . . D n is a product distri-bution, the optimal auction has a simple form given by Myerson [15]. For distribution D i withsupport z < z < · · · < z k , its virtual value function ϕ D i is defined as: ϕ D i ( z k ) = z k and ϕ D i ( z (cid:96) ) = z (cid:96) − ( z (cid:96) +1 − z (cid:96) ) S D i ( z (cid:96) +1 ) f D i ( z (cid:96) ) , ∀ (cid:96) < k. (1)If buyer i is the only buyer in the system, the optimal auction sets a fixed price, and the buyerbuys the item when her valuation is at least this price. The reserve price of D i , denoted r D i is thesmallest value r in the support of D i that maximizes the corresponding revenue rS D i ( r ). It is easyto check that ϕ D i ( r D i ) ≥ D i are regular , sothat ϕ D i ( z ) is a non-decreasing function of z . Therefore, for all v < r D i , we have ϕ D i ( v ) < For discrete regular distributions, Myerson’sauction takes the form [12] in Algorithm 1. Note that this auction is also ex-post
IC and IR. When the valuations are non-regular, we use the non-decreasing ironed virtual value function [15, 12] instead. lgorithm 1: Myerson’s Auction with prior D and valuations (cid:126)v . Sort the buyers in decreasing order of q i = ϕ D i ( v i ). Assume no two values are identical (canbe ensured by using a fixed tie-breaking rule). Allocate to the bidder j with highest virtual value q j , provided q j ≥ Let m be the bidder with second highest virtual value, and let w = max(0 , q m ). Charge j the smallest value z in the support of D j such that ϕ D j ( z ) > w . Extracting full surplus as revenue.
At the other extreme, a celebrated result of Cr´emer andMcLean [7] shows that for distributions D (cid:48) which are “sufficiently correlated”, the optimal auctionextracts full surplus (i.e., the revenue equals the maximum valuation in each valuation profile).Formally, the result requires that for each agent, their conditional distribution over others’ valuesgiven their own value is full rank; for our purposes, we require a restriction of their result to n = 2buyers, each with two possible valuations. Theorem 4 (Cr´emer-McLean [7]) . For n = 2 buyers, where each buyer i has K i = 2 and the jointdistribution over the valuations is D (cid:48) , the seller can extract the entire social welfare (expected valueof the maximum of the buyer’s valuations) as her revenue when D (cid:48) is a correlated distribution. We next formalize the model of an information intermediary illustrated in Fig. 1a. Since theeffect of the intermediary’s signal is captured by the resulting posterior distribution over valuations,for ease of notation, we henceforth use “signal” to refer to a distribution S over valuations.A signaling scheme Z = { γ q , S q } q ∈ [ m ] comprises a collection of signals (i.e., joint distributionsover valuations) S , S , . . . , S m and corresponding non-negative weights γ , γ , . . . , γ m . The scheme Z is feasible (or “Bayes plausible” [14]) if it satisfies (cid:80) q γ q = 1 and (cid:80) q γ q S q = D . The intermediarycommits to scheme Z before the auction, and it is known to the seller and all buyers.The intermediary maps observed valuation profile (cid:126)v ∼ D to signal S q with probability γ q Pr [ S q = (cid:126)v ] Pr [ D = (cid:126)v ] .The seller uses S q as the shared prior and runs an optimal auction on the buyers. Note that though D is a product distribution, the {S q } can be correlated. Abusing the notation introduced in Eq. (1),we denote the revenue generated by signaling scheme Z as R ( Z ) = (cid:80) q γ q R ( S q ), its consumersurplus by CS ( Z ) = (cid:80) q γ q CS ( S q ), and its welfare by W ( Z ) = (cid:80) q γ q W ( S q ).When D is a product distribution, the revenue from any signaling scheme must be at least theoptimal revenue of Myerson’s auction without signaling, R ( D ). To see this, we note that Myerson’sauction on D is ex-post IC and IR. This means that this allocation and payment rule is still afeasible (ex-interim IC and IR) mechanism conditioned on receiving any signal, completing theargument. Therefore, the consumer surplus CS ( Z ) under any signaling scheme Z is bounded bythe difference of the maximum possible welfare W ∗ = E (cid:126)V ∼D [max i V i ] and the maximum revenuewithout signaling R ( D ). We henceforth denote this bound as Opt , which is defined as follows:
Opt = W ∗ − R ( D ) . We say that Z is a θ -approximation signaling scheme if CS ( Z ) ≥ Opt θ . Our goal is to find the bestapproximation factor θ via a signaling scheme whose computation time is polynomial in n and K .In the rest of the paper, we omit the dependence on D when clear from context. Optimal signaling for a single buyer.
For n = 1 buyer, Bergemann et al. [1] present asignaling scheme with consumer surplus exactly equal to Opt (i.e., implementing the point O
6n Fig. 1b. Their signaling scheme constructs distributions (signals) S , S , . . . S m and assignsweights γ , γ , . . . γ m to them such that (cid:80) q γ q S q = D .Let prior D takes value v i with probability p i , where 0 < v < . . . < v k . Let (cid:126)p = ( p , p , . . . , p k ).In each iteration (cid:96) , the algorithm constructs an equal revenue distribution S (cid:96) and subtracts it fromthe prior D . This equal revenue distribution assigns positive probability p i(cid:96) to v i if p i > p i(cid:96) = 0 if p i = 0. In S (cid:96) , the seller raises equal revenue by setting the price to be any ofthe values v i with p i >
0. It is easy to see that the equal revenue condition specifies a uniquedistribution S (cid:96) . Note that since this signal is equal revenue, the seller sets the lowest value as price,so that the item always sells and the consumer surplus is maximum possible.Let (cid:126)p (cid:96) be the probability vector of S (cid:96) . We set the largest weight γ (cid:96) such that (cid:126)p − γ (cid:96) (cid:126)p (cid:96) ≥
0. Weupdate D by setting (cid:126)p to (cid:126)p − γ (cid:96) (cid:126)p (cid:96) , normalize it so that (cid:80) i p i = 1, and increase (cid:96) by one. We repeatthis till the support of D becomes empty. The { γ (cid:96) , S (cid:96) } specifies the signaling scheme.We illustrate this procedure by an example below. Example 1.
Suppose the type space is { , , } and D = (cid:104) , , (cid:105) are the probabilities of thesetypes. The monopoly price is p = 2 with revenue R ( D ) = , while the point A in Fig. 1b hassocial welfare R ( A ) = W ∗ = E [ D ] = 2. Suppose S = (cid:104) , , (cid:105) with γ = ; S = (cid:104) , , (cid:105) with γ = ; and S = (cid:104) , , (cid:105) with γ = . It is easy to check that the monopoly price for each signalis the lowest price in its support so that the item always sells, and (cid:80) γ i R ( S i ) = . Therefore, (cid:80) γ i CS ( S i ) = 2 − = = Opt , which corresponds to point O in Fig. 1b.We henceforth use BBM ( v, D ) to refer to this scheme when the buyer has valuation distribution D and the realized value is v ∼ D . Below we state some critical properties of the BBM schemewhich we use in our results.
Lemma 1 (Implicit in Bergemann et al. [1]) . For a single buyer with value distribution D (withreserve price r D ), the BBM mechanism satisfies the following properties:1. For any signal S q , ϕ S q ( v ) ≥ for all v in the support of S q .2. CS ( BBM ) =
Opt ≥ Pr V ∼D [ V < r D ] · E V ∼D [ V | V < r D ] = (cid:80) v
By Theorem 4, we know any signal that correlates thebuyers raises zero consumer surplus. Therefore, the only signals S of interest are those under whichbuyer values are independent. Abusing notation we denote such a signal as s = ( α (cid:48) , β (cid:48) ), where Pr [ v = a ] = α (cid:48) and Pr [ v = c ] = β (cid:48) . Note that in this instance, for a signal to get maximumwelfare the resulting optimal mechanism must always award buyer 1, and for non-zero consumersurplus it must award the item to buyer 1 at price b , or buyer 2 at price d .Let CS ( s ) denote the consumer surplus under any such a signal s , and ϕ ( b | s ) and ϕ ( d | s )denote the new virtual values (note that by definition, ϕ ( a | s ) = a and ϕ ( c | s ) = c under anysignal s with α (cid:48) , β (cid:48) > ϕ ( b | s ) , ϕ ( d | s )):7 ϕ ( v ) a ϕ ( b ) b ϕ ( d ) cd ϕ ( a ) ϕ ( c ) v ϕ ( v ) ϕ ( b | s ) ϕ ( d | s ) ϕ ( a | s ) ϕ ( c | s ) ϕ ( v ) ϕ ( b | s ) ϕ ( d | s ) ϕ ( a | s ) ϕ ( c | s ) s Figure 2:
Illustrating the setting for Theorems 1 and 2: On the left (below the axis) we show the settingwithout signaling, where buyer 1 (blue) has values ( a, b ) and buyer 2 (red) has values ( c, d ) ; we also showthe corresponding virtual values (above the axis). On the right, we show the two settings characterized byTheorem 5 under which a signal s has non-zero consumer surplus (the changed virtual values are highlighted). Proposition 1.
Conditioned on receiving a signal s , we have the following cases:1. If ϕ ( b ) ≥ c , then the optimal mechanism is to sell to Buyer at price b . CS ( s ) = ( a − b ) α (cid:48) .2. If ϕ ( d ) ≥ max(0 , ϕ ( b )) , then the optimal mechanism is to try selling to Buyer at price a then to Buyer at price d . CS ( s ) = (1 − α (cid:48) ) β (cid:48) ( c − d ) .3. If ϕ ( b ) ≤ and ϕ ( d ) ≤ , then the optimal mechanism is to try selling to Buyer at price a then to Buyer at price c . CS ( s ) = 0 .4. If ≤ ϕ ( b ) ≤ c and ϕ ( d ) ≤ ϕ ( b ) , then the optimal mechanism is to sell to Buyer at price b if Buyer has valuation d ; otherwise, it tries selling to Buyer at price a then to Buyer at price c . CS ( s ) = α (cid:48) (1 − β (cid:48) )( a − b ) . Our main insight, however, is that the setting can be further simplified to get the followingstructural property for the optimal signaling scheme.
Theorem 5 (Structural Theorem) . In an optimal signaling scheme, the only signals s = ( α (cid:48) , β (cid:48) ) that raise non-zero consumer surplus have the following form:(1’) Under signal s , ϕ ( b | s ) = ϕ ( c | s ) = c and CS ( s ) = α (cid:48) ( a − b ) .(2’) Under signal s , ϕ ( d | s ) = ϕ ( b | s ) ≥ and CS ( s ) = α (cid:48) (1 − β (cid:48) )( a − b ) .Proof. Recall that we restrict ourselves to signals S under which the buyer valuations remainindependent. Any such signal can be alternately written as s = ( α (cid:48) , β (cid:48) ) where α (cid:48) = Pr [ v = a ] and β (cid:48) = Pr [ v = c ]. For ease of notation, we henceforth drop the conditioning of virtual valuations onsignal s (i.e., write ϕ ( · ) for ϕ ( ·| s )) when clear from context.Next, let γ s denote the weight of any signal s = ( α (cid:48) , β (cid:48) ). The signaling scheme that maximizesconsumer surplus is the solution to the following linear program written over signals s = ( α (cid:48) , β (cid:48) ):Maximize (cid:88) s γ s CS ( s )Subject to (cid:80) s =( α (cid:48) ,β (cid:48) ) γ s α (cid:48) β (cid:48) ≤ αβ (cid:80) s =( α (cid:48) ,β (cid:48) ) γ s α (cid:48) (1 − β (cid:48) ) ≤ α (1 − β ) (cid:80) s =( α (cid:48) ,β (cid:48) ) γ s (1 − α (cid:48) ) β (cid:48) ≤ (1 − α ) β (cid:80) s =( α (cid:48) ,β (cid:48) ) γ s (1 − α (cid:48) )(1 − β (cid:48) ) ≤ (1 − α )(1 − β ) γ s ≥ ∀ s (2)We examine the cases in Proposition 1 with positive consumer surplus, and characterize theoptimal solution: 8 In Case (1), we have ϕ ( b ) = c . To see this, consider any signal s with ϕ ( b ) > c . Suppose weincrease α (cid:48) and decrease γ s while preserving the product α (cid:48) γ s . Since γ s CS ( s ) = γ s α (cid:48) ( a − b ),this is preserved by the change. Therefore, the objective of LP (2) is preserved, and so are thefirst two constraints. Further, since (1 − α (cid:48) ) decreases, this only makes the third and fourthconstraints more feasible. This transformation decreases ϕ ( b ). • In Case (2) and (4), we have ϕ ( d ) = ϕ ( b ). It does not help to make them unequal by a similarargument as above: In case (2), if ϕ ( d ) > ϕ ( b ), we can increase β (cid:48) while preserving γ s β (cid:48) .Since γ s CS ( s ) = γ s (1 − α (cid:48) ) β (cid:48) ( c − d ), this does not change the contribution to the objectiveof LP (2), and preserves all constraints. This transformation decreases ϕ ( d ) In case (4), if ϕ ( d ) < ϕ ( b ), we can increase α (cid:48) while preserving γ s α (cid:48) . Since γ s CS ( s ) = γ s α (cid:48) (1 − β (cid:48) )( a − b ),this does not change the contribution to the objective of LP (2), and preserves all constraints.This transformation decreases ϕ ( b )Therefore, the only two types of signals s that give positive CS are(1’) If ϕ ( b ) = c , then CS ( s ) = α (cid:48) ( a − b ).(2’) If ϕ ( d ) = ϕ ( b ) ≥
0, then CS ( s ) = max((1 − α (cid:48) ) β (cid:48) ( c − d ) , α (cid:48) (1 − β (cid:48) )( a − b )).As (1 − β (cid:48) )( b − d ) ≥
0, we have (cid:18) b − d + β (cid:48) − β (cid:48) ( c − d ) (cid:19) (1 − β (cid:48) ) ≥ β (cid:48) ( c − d ) . Notice that in Case (2’), we have ϕ ( b ) = b − α (cid:48) − α (cid:48) ( a − b ) = d − β (cid:48) − β (cid:48) ( c − d ) = ϕ ( d ). This gives α (cid:48) (1 − β (cid:48) )( a − b ) ≥ (1 − α (cid:48) ) β (cid:48) ( c − d ) . Thus, the two types of signals s that give positive CS become(1’) If ϕ ( b ) = c , then CS ( s ) = α (cid:48) ( a − b ).(2’) If ϕ ( d ) = ϕ ( b ) ≥
0, then CS ( s ) = α (cid:48) (1 − β (cid:48) )( a − b ).Using the above structural theorem, the proofs of Theorems 1 and 2 follow by different choicesof the parameters ( a, b, c, d ). For Theorem 1, we set d = (cid:0) − ε (cid:1) b , with ϕ ( b ) and ϕ ( d ) slightlyabove zero; for Theorem 2, we set α = β = 1 − δ ; a = − δ ) , b = − δ , c = 1, d = 1 − δ with δ → + . Proof of Theorem 1.
In the instance presented in the beginning of Section 3, suppose the virtualvalues of b and d are slightly above zero with ϕ ( b ) > ϕ ( d ) so that Case (4) in Proposition 1is uniquely optimal for the seller. The optimal auction generates consumer surplus CS ( D ) = α (1 − β )( a − b ) = bc · c − da · ( a − b ) according to Proposition 1.To prove Theorem 1, we set b → c + . Now in Proposition 1, in Case (1), we must have α (cid:48) → + since b → c + , so that CS →
0. Also if α (cid:48) = 1 in a signal then CS = 0 here. The only other signalwhere the item is allocated to the higher bidder is in Case (4) when β (cid:48) = 0. Let γ denote theprobability of the signal of this type s = ( α (cid:48) , ϕ ( b ) ≥ ϕ ( d ), we have α (cid:48) ≤ b − da − d . ϕ ( b ) − ϕ ( d ) and ϕ ( d ) can be arbitrarily small as long as positive, so we take the limits for them first, i.e., weare calculating lim b → c + lim ϕ ( d ) → + ,ϕ ( b ) → ϕ ( d ) + CS in the following part of the proof. This allows us to treat α = ba and β = dc in calculating (1 − α )(1 − β ), as α and β are not infinitesimally close to 1 for any fixed ε .
9y the constraints of LP (2), we have: Pr [ v = b ∧ v = d ] = γ (1 − α (cid:48) ) ≤ (1 − α )(1 − β ) , which simplifies to γ ≤ ( a − d ) · ( c − d ) ac . The consumer surplus in this case is therefore: CS = γ CS ( s ) = γα (cid:48) ( a − b ) ≤ ( b − d ) · ( c − d ) ac · ( a − b ) ≤ b − db · CS ( D ) . Setting d = (cid:0) − ε (cid:1) b and combining with the fact that CS → CS → ε · CS ( D ) so CS < ε · CS ( D ). Proof of Theorem 2.
Without signaling, E [max v i ] = αa + (1 − α ) b and R ( D ) = αa + (1 − α ) βc .(Case (2), (3) and (4) in Proposition 1 give the same revenue R ( D ).) Therefore Opt = E [max v i ] − R ( D ) = (1 − α )( b − βc ) . Now we assign the values as: α = β = 1 − δ ; a = − δ ) , b = − δ , c = 1, d = 1 − δ with δ → + .Then we plug in the values and the two possible types of signals s in Theorem 5 become(1’) If α (cid:48) = − δ − δ < , then CS ( s ) ≤ δ (1 + o (1)).(2’) If α (cid:48) ≤ − δ ; β (cid:48) = − − α (cid:48) )+3 δ (1 − α (cid:48) ) − δ (1 − α (cid:48) )1 − − α (cid:48) )+ δ (1 − α (cid:48) ) > − − α (cid:48) )1 − − α (cid:48) ) , then CS ( s ) ≤ α (cid:48) (1 − β (cid:48) ) δ (1 + o (1)).The consumer surplus maximizing signaling scheme should use t signals S ,i of type (2’), with α (cid:48) ,i , β (cid:48) ,i and weight w ( S ,i ). There is an additional signal S (with weight w ( S )) of type (1’) with α (cid:48) and β (cid:48) . (Having multiple signals of type (1’) gives the same CS as having a single signal astheir average.) Denoting the valuation of the first buyer by v and the second buyer by v , theconstraints in LP (2) imply the two constraints: Pr [ v = b ] = (1 − α (cid:48) ) w ( S ) + t (cid:88) i =1 (1 − α (cid:48) ,i ) · w ( S ,i ) ≤ − α = δ, (Constraint (I)) Pr [ v = b ∧ v = d ] = t (cid:88) i =1 (1 − α (cid:48) ,i )(1 − β (cid:48) ,i ) · w ( S ,i ) ≤ (1 − α )(1 − β ) = δ . (Constraint (II))Note that Opt = (1 − α )( b − βc ) = 2 δ (1 + o (1)).The total consumer surplus therefore is: CS ≤ δ (1 + o (1)) · w ( S ) + t (cid:88) i =1 α (cid:48) ,i (1 − β (cid:48) ,i ) δ (1 + o (1)) · w ( S ,i ) ≤ δ (1 + o (1)) · (cid:32) δ − t (cid:88) i =1 (1 − α (cid:48) ,i ) · w ( S ,i ) (cid:33) + δ (1 + o (1)) t (cid:88) i =1 α (cid:48) ,i (1 − β (cid:48) ,i ) · w ( S ,i )= δ (1 + o (1)) + δ (1 + o (1)) t (cid:88) i =1 ( α (cid:48) ,i (1 − β (cid:48) ,i ) − (1 − α (cid:48) ,i )) · w ( S ,i ) ≤ δ (1 + o (1)) + δ (1 + o (1)) t (cid:88) i =1 (1 − α (cid:48) ,i )(1 − β (cid:48) ,i ) · w ( S ,i ) ≤ δ (1 + o (1)) + δ (1 + o (1)) · δ = δ (1 + o (1)) . α (cid:48) < . The third inequality uses theimplication of ϕ ( d ) = ϕ ( b ) that β (cid:48) ,i > − − α (cid:48) ,i )1 − − α (cid:48) ,i ) . The fourth inequality uses Constraint (II).This establishes a lower bound of 2, since Opt = 2 δ (1 + o (1)). In this section, we introduce a family of signaling schemes, which forms the basic subroutinefor obtaining our upper bounds. We first need one additional definition. For agent i with V i ∼ D i ,we use D i | >a to denote the conditional distribution of V i given V i > a , and D i | a − b to denote the distributionof V i − b given V i > a ; we refer to it as the distribution of V i truncated at a and reduced by b . Wenow state a simple result relating the reserve price of the truncated and the original distributions. Lemma 2.
Let D (cid:48) i = D i | >v ◦ − v ◦ . Then we have r D (cid:48) i ≥ r D i − v ◦ , and moreover, for any v > v ◦ , wehave ϕ D (cid:48) i ( v − v ◦ ) = ϕ D i ( v ) − v ◦ .Proof. We first prove the result about reserve prices. Let r = r D i and r (cid:48) = r D (cid:48) i . If r ≤ v ◦ theinequality is trivial. Otherwise, suppose for contradiction that for some r (cid:48) < r − v ◦ , r (cid:48) · S D (cid:48) i ( r (cid:48) ) ≥ ( r − v ◦ ) · S D (cid:48) i ( r − v ◦ ) . Then we would have: ( r (cid:48) + v ◦ ) · S D (cid:48) i ( r (cid:48) ) ≥ r · S D (cid:48) i ( r − v ◦ ) , since S D (cid:48) i ( r (cid:48) ) ≥ S D (cid:48) i ( r − v ◦ ).Thus, ( r (cid:48) + v ◦ ) · S D i ( r (cid:48) + v ◦ ) ≥ r · S D i ( r ) , a contradiction to the assumption that r is the smallestoptimal reserve price of D i .To see the second part, if we condition the distribution on v > v ◦ , this does not change thevirtual value function for values v j > v ◦ , since both the numerator and denominator in Eq. (1)scale by the same amount. If we now subtract v ◦ from the support of the distribution, it reducesthe virtual value by the same amount. This completes the proof.Note that D (cid:48) i = D i | >v ◦ − v ◦ represents buyer i ’s excess value compared to v ◦ . Lemma 2 showsthat for any threshold v ◦ and any buyer i , given the side-information that V i > v ◦ , her new reserveprice is greater than her original reserve price. The
Rank t signaling scheme. We now introduce a family of signaling schemes
Rank t param-eterized by t ∈ { , . . . n } , For any realized joint valuation profile (cid:126)v = ( v , v , . . . , v n ), the signal sentby Rank t consists of two parts. First, Rank t observes (cid:126)v and outputs ( v ◦ , T ), where v ◦ is the valueof ( t + 1) st largest realized value (or 0 when t = n ), and T is the subset of buyers with realized valuestrictly greater than v ◦ . For the second part of the signal, Rank t chooses a buyer j uniformly atrandom from T , and computes her excess distribution D j | >v ◦ − v ◦ . It then reveals both the identityof j , as well as the signal BBM ( v j − v ◦ , D j | >v ◦ − v ◦ ) generated by the single-buyer BBM schemeon a buyer with value distribution D j | >v ◦ − v ◦ . The scheme is formalized in Algorithm 2. Algorithm 2:
Rank t ( (cid:126)v, D ) v ◦ ← ( t + 1) st largest value in (cid:126)v T ← { i : v i > v ◦ } if T (cid:54) = ∅ then j ← Buyer chosen uniformlyat random from T s ← BBM ( v j − v ◦ , D j | >v ◦ − v ◦ ) return v ◦ , T , j , and s else return v ◦ , and T = ∅ Optimal mechanism under
Rank t . Conditioned onreceiving the signal generated by
Rank t , the seller isguaranteed a revenue of v ◦ from the ( t +1) st largest buyer,and knows that only buyers in T can pay more than v ◦ .The seller can now charge at least v ◦ to any buyer in T ,and can further run an auction over the excess value ofbuyers in T , where for buyer i ∈ T , her excess value hasdistribution D (cid:48) i = D i | >v ◦ − v ◦ . Note that for any buyer i ∈ T except the randomly chosen buyer j , a value drawnfrom D (cid:48) i represents how much more than v ◦ she is willing11o pay. Moreover, distributions D (cid:48) i are independent, andalso, since the identity of j is chosen uniformly at ran-dom, the BBM scheme modifies the distribution of buyer j in a fashion that is independent of D (cid:48) i .By Lemma 1, we know that the BBM scheme ensuresthe virtual value of buyer j is always non-negative. From the characterization of the optimalauction [15, 12], since the item is always allocated to the highest virtual value bidder as long as thisvalue is non-negative, the item will always be allocated to buyer j if all other buyers i ∈ T, i (cid:54) = j have excess values v i − v ◦ < r D (cid:48) i (and hence, negative virtual values). Consumer surplus under
Rank t . Let p i := 1 − S D i ( r D i ) = Pr v i ∼D i [ v i < r D i ] for any buyer i ,where r D i is the reserve price of D i . Let Y i := D i | For ≤ t ≤ n , the consumer surplus of Rank t satisfies: CS ( Rank t ) ≥ (cid:32) n (cid:89) i =1 p i (cid:33) · (cid:32)(cid:32) t · t (cid:88) (cid:96) =1 E [ Z (cid:96) ] (cid:33) − E [ Z t +1 ] (cid:33) . Proof. Fix a buyer b , and any valuation profile (cid:126)v − b = { v i , i (cid:54) = b } such that v i < r D i ∀ i (cid:54) = b . Define v tb to denote the t th largest value in { v i , i (cid:54) = b } . Now consider the event Q ( (cid:126)v − b , b, t ) = { V b > v tb AND b selected for BBM signaling } . Conditioned on Q ( (cid:126)v − b , b, t ), we have that the Rank t scheme (Algorithm 2) with parameter t setsthreshold value as v ◦ = v tb . By Lemma 2, we have that for every i ∈ T, i (cid:54) = b , their value v i issmaller than their new reserve price v ◦ + r D (cid:48) i , since v i < r D i , and modifying D i to D (cid:48) i = D i | >v ◦ − v ◦ does not decrease the reserve price. Therefore, conditioned on Q ( (cid:126)v − b , b, t ), the auction behaves likethe single item mechanism BBM ( v (cid:48) b , D (cid:48) b ), where v (cid:48) b = v b − v tb , D (cid:48) b = D b | >v tb − v tb . Let r (cid:48) b denote thereserve price of D (cid:48) b ; again using Lemma 2 we have r (cid:48) b ≥ r b − v tb . Now, using Lemma 1, we get thatthe expected consumer surplus generated by Rank t under Q ( (cid:126)v − b , b, t ) is at least: E [ CS ( Rank t ) | Q ( (cid:126)v − b , b, t )] ≥ (cid:88) v (cid:48) b We now prove our main theorem. For this, we first decompose our benchmark consumer surplus Opt , and then use Lemma 3 to quantify how we recover each term via signaling. Recall we start witha product distribution D = D ×D ×· · ·×D n , where D i has reserve price r i . Let x < x < . . . < x K be the union of the supports of value distributions. R ( D ) denotes the revenue of the optimal auctionon D , and Opt = E (cid:126)v ∼D [max i v i ] − R ( D ). We can now decompose Opt into three components: Myerson’s surplus. The consumer surplus generated by Myerson’s auction, denoted by CS ( D ). Non-allocation surplus. The loss in consumer surplus due to Myerson’s auction not allocatingthe item, denoted by CS . In Lemma 4 (Section 5.1), we show that CS = (cid:89) i Pr v i ∼D i [ v i < r i ] · E (cid:126)v ∼D (cid:20) max i v i (cid:12)(cid:12)(cid:12)(cid:12) v i < r i ∀ i (cid:21) . is-allocation surplus. The loss in surplus because the highest value and highest virtual-valuebuyers are different. We denote the expected loss due to mis-allocating the item to buyer i as CS i ,and the expected loss due to mis-allocating to a buyer with value x k as (cid:99) CS k .For i.i.d. values, we can write Opt = CS ( D ) + CS , since the highest virtual-value and highestvalue buyers coincide. For non-i.i.d. settings, we have two ways of decomposing Opt : Opt = CS ( D ) + CS + n (cid:88) i =1 CS i and Opt = CS ( D ) + CS + K (cid:88) k =1 (cid:99) CS k . (4)In the remainder, when seeking a θ -approximation to Opt , we assume CS ( D ) is not already a θ -approximation. We first present an O (min(log n, K ))-approximation for the non-allocation surplus CS , and then use it to approximate the remaining terms CS i and (cid:99) CS k in the above expressions. Algorithm 3: S ( (cid:126)v, D ) Choose t = argmax nt (cid:48) =1 CS ( Rank t (cid:48) ). return Rank t ( (cid:126)v, D )The scheme we use to approximate CS is simple: Wechoose the parameter t ∈ { , , . . . , n } that maximizes CS ( Rank t ) and run Rank t . This choice of t only de-pends on the distribution D . Formally, we denote thisscheme as S ( (cid:126)v, D ), and present it in Algorithm 3. Thefollowing theorem is our main technical contribution: Theorem 6. The consumer surplus of the signalingscheme S ( (cid:126)v, D ) is an O (cid:0) min(log n, K ) (cid:1) -approximation to the non-allocation surplus, CS . The proof of the O (log n )-approximation involves analyzing an appropriate randomization of Rank t over different t (Algorithm 4), while the proof of the O ( K )-approximation involves showinga stronger version of Markov’s inequality for independent Bernoulli random variables (Lemma 6),which may be of independent interest. Identically distributed buyers. For i.i.d. values, we use the better of two schemes: Using S ( (cid:126)v, D ) or sending no signal. This immediately implies an O (cid:0) min(log n, K ) (cid:1) -approximation to Opt , as in this setting we have: Opt = CS ( D ) + CS . This completes the first part of Theorem 3. O (log n ) -Approximation of CS To prove Theorem 6, we first derive an expression for CS . We denote a realization from D by (cid:126)v = { v i } . Let p i , r i , Y i , Z t be as defined in Section 4. We have the following lemma: Lemma 4. Let P = (cid:81) ni =1 p i . Then, CS = P · E [ Z ] .Proof. Note that CS is the expected surplus lost due to not allocating the item in Myerson’smechanism. This happens only when all realized values are below their corresponding reserveprice. In this case, the value lost is the maximum valuation, since this value contributes to thewelfare, and the revenue raised is zero. Therefore, we have: CS = (cid:32) n (cid:89) i =1 p i (cid:33) · E (cid:20) max i =1 , ,...,n v i (cid:12)(cid:12)(cid:12)(cid:12) ∀ i, v i < r i (cid:21) where the expectation is over (cid:126)v ∼ D . This is equal to P · E [ Z ].14e first show CS ( S ) is an O (log n )-approximation to CS . We construct a signaling schemedenoted by S such that CS ( S ) ≤ CS ( S ). We will then prove that consumer surplus of S is an O (log n )-approximation to CS .This scheme can be constructed by randomizing over different values for parameter t in Rank t .We assign a weight w j to each rank j ∈ { , , . . . , n } and pick a rank t with probability proportionalto these weights. Note that this choice of the rank t does not depend on (cid:126)v . Subsequently, we run Rank t . Formally, the signaling scheme is as follows. Algorithm 4: S ( (cid:126)v, D ) w j ← j +1 for j ∈ { , , . . . , n − } w n − ← w n ← Choose rank t ∈ { , , . . . , n } where rank j is chosen with probability proportional to w j . return Rank t ( (cid:126)v, D ) Lemma 5. The consumer surplus of S is an O (log n ) -approximation to CS .Proof. We have: CS ( S ) = (cid:80) nt =1 w t · CS ( Rank t ) (cid:80) nt =1 w t . Recall the definition of P from Lemma 4. We now use Lemma 3 to lower bound the numerator ofthe above formula: n − (cid:88) t =1 w t CS ( Rank t ) + w n CS ( Rank n ) ≥ P n − (cid:88) t =1 E [ Z t ] · n − (cid:88) j = t j ( j + 1) − (cid:32) n − (cid:88) t =2 E [ Z t ] · t (cid:33) + (cid:32) n − (cid:88) t =1 E [ Z t ] · n − (cid:33) − E [ Z n ] + CS ( Rank n )= P · ( E [ Z ] − E [ Z n ]) + CS ( Rank n ) . It is easy to see that CS ( Rank n ) ≥ P · E [ Z n ]. Using Lemma 4, we have: n (cid:88) i =1 w i CS ( Rank i ) ≥ P · E [ Z ] = CS . Now, we compute the denominator: n (cid:88) t =1 w t = 1 + n − (cid:88) t =1 t ≤ n. Therefore, we have: CS ( S ) = (cid:80) nt =1 w t · CS ( Rank t ) (cid:80) nt =1 w t ≥ CS n . Since the scheme S chooses the Rank t with largest value, we now have the following corollary: Corollary 1. Consumer surplus of S is an O (log n ) -approximation to CS . .1.2 O ( K ) -Approximation of CS For proving this part, we will assume n > 5. For smaller values of n , the analysis in the previoussection already yields a constant-approximation to CS . Using Lemma 4, we know CS = P · E [ Z ].Setting x = 0, we therefore have: E [ Z ] = K (cid:88) k =1 Pr [ Z ≥ x k ] · ( x k − x k − ) . Therefore, there is a k ∗ ∈ { , , . . . , K} such that E [ Z ] ≤ K · Pr [ Z ≥ x k ∗ ] · ( x k ∗ − x k ∗ − ). Wefix this k ∗ , and deduce: CS ≤ K · P · Pr [ Z ≥ x k ∗ ] · ( x k ∗ − x k ∗ − ) . Therefore, if we show that CS ( S ) ≥ Ω(1) · P · Pr [ Z ≥ x ∗ k ] · ( x ∗ k − x k ∗ − ), then we have an O ( K )-approximation to CS .In order to show the former statement, we need the following probability lemma: Lemma 6. Given n independent Bernoulli random variables, X i ∈ { , } for i ∈ { , , . . . , n } , let N = (cid:80) i X i . Then there exists a value j ∈ { , , . . . , n } such that: min (cid:18) , E [ N ] j (cid:19) − Pr [ N ≥ j + 1] ≥ Pr [ N ≥ . Proof. Let α q be the probability that N = q . At least one of the following holds: • If E [ N ] < 5, then for j = 5, we have:min (cid:18) , E [ N ]5 (cid:19) − Pr [ N ≥ 6] = 15 E [ N ] − Pr [ N ≥ ≥ n (cid:88) q =1 qα q − α ≥ · n (cid:88) q =1 α q = Pr [ N ≥ . • If 5 ≤ E [ N ] ≤ n , then we set j = 2 · (cid:100) E [ N ] (cid:101) . It is easy to see that for this setting, we havemin (cid:16) , E [ N ] j (cid:17) ≥ . 4. By a standard application of Chernoff bounds we have: Pr [ N ≥ j + 1] < e − < . . Therefore we have: min (cid:18) , E [ N ] j (cid:19) − Pr [ N ≥ j + 1] ≥ . ≥ Pr [ N ≥ . • If E [ N ] ≥ n , then we set j = n , so thatmin (cid:18) , E [ N ] n (cid:19) − Pr [ N ≥ n + 1] ≥ ≥ Pr [ N ≥ . We now complete the proof of the O ( K )-approximation in the lemma below.16 emma 7. There exists a value t (cid:48) such that CS ( Rank t (cid:48) ) ≥ Ω(1) · P · Pr [ Z ≥ x k ∗ ] · ( x k ∗ − x k ∗ − ) .Proof. Recall that Y i is the distribution of D i conditioned on being strictly below the reserve price r i . We draw one random variable independently from each Y i . We define G k to be the randomvariable corresponds to the number of draws with value at least x k ∗ among those n draws. UsingLemma 3, we have: CS ( Rank t ) ≥ P · E (cid:34) (cid:80) ti =1 Z i t − Z t +1 (cid:35) ≥ P · ( x k ∗ − x k ∗ − ) · (cid:18) min (cid:18) , E [ G k ∗ ] t (cid:19) − Pr [ G k ∗ ≥ t + 1] (cid:19) . We define a Bernoulli random variable X i that is 1 when Y i ≥ x k ∗ and zero otherwise. ApplyingLemma 6, there exists a value of t (cid:48) between 1 and n such that the following holds: (cid:18) min (cid:18) , E [ G ∗ k ] t (cid:48) (cid:19) − Pr [ G ∗ k ≥ t (cid:48) + 1] (cid:19) = Ω(1) · Pr [ G ∗ k ≥ 1] = Ω(1) · Pr [ Z ≥ x k ∗ ] . This when combined with the previous inequality, completes the proof.According to the previous lemma, there is a t (cid:48) such that consumer surplus of Rank t (cid:48) is an O ( K )-approximation and CS ( S ) ≥ CS ( Rank t (cid:48) ) for any t (cid:48) . Therefore, we have the following corollary: Corollary 2. Consumer surplus of S is an O ( K ) -approximation to CS . Combining Corollaries 1 and 2 completes the proof of Theorem 6. Finally, we prove Theorem 3 when D i are not all identical. From Eq. (4), this requires approx-imating CS i and (cid:99) CS k for any i and k . Recall CS i is the surplus lost when a non highest-valuebidder i wins the auction. Assuming we break ties in favor of the higher valued buyer, we have: CS i = (cid:88) (cid:126)v : i = argmax j ( ϕ D j ( v j )) Pr [ D = (cid:126)v ] · (cid:18) max j ( v j ) − v i (cid:19) . (5)Similarly, (cid:99) CS k is the surplus lost when the item is allocated to a buyer with value x k . Againassuming ties are broken in favor of higher-valued buyers, we have: (cid:99) CS k = (cid:88) (cid:126)v : x k = max j ( ϕ D j ( v j )) Pr [ D = (cid:126)v ] · (cid:18) max j ( v j ) − x k (cid:19) . (6)To approximate these quantities, we run the signaling scheme for approximating CS on a modified product distribution. In this new scheme, we fix a cut-off value c , and reveal the identityof all the buyers with realized value strictly greater than c . Let a denote the largest realized valuethat is at most c , and T denote the set of buyers with values strictly bigger than a . We first modifythe distribution D i for i ∈ T as follows: Recall that D i | >c denotes the distribution of V i conditionedon V i > c ; let X i denote the corresponding random variable. We change the distribution to be thatof X i − a , that we denote D j | >c − a . We now subtract a from all the valuations v i , i ∈ T , and run17he signaling scheme S in this instance. The details of the scheme can be found in Algorithm 5,where c is the cutoff parameter, and where we denote by (cid:126)v T the | T | dimensional vector made bychoosing the indices in T from (cid:126)v . Note that c could be different from a when there is no buyerwhose value coincides with c . Algorithm 5: Trunc ( (cid:126)v, D , c ) a ← max v i ≤ c ( v i ); and T ← { j : v j > a } D T,a ← (cid:81) j ∈ T i ( D j | >c − a ) // Modified distributions for j ∈ T . (cid:126)v (cid:48) ← (cid:126)v T − a // a is a vector with all elements equal to a . s ← S ( (cid:126)v (cid:48) , D T,a ) // s is the signal returned by S . return ( a, T i , s ) as final signal Approximating Opt for small n . We choose i ∗ ∈ { , , . . . , n } that maximizes CS i . By Eq. (4),to get an O ( n log n )-approximation to Opt , it suffices to demonstrate an O (log n )-approximation to CS i ∗ . The scheme S chooses i ∗ ∈ { , , . . . n } that maximizes CS i and returns Trunc ( (cid:126)v, D , v i ∗ ). Theorem 7. The consumer surplus of S is an O (log n ) -approximation to CS i ∗ .Proof. Recall that the final scheme S chooses i ∗ ∈ { , , . . . n } that maximizes CS i as defined inEq. (5) and returns Trunc ( (cid:126)v, D , v i ∗ ) as defined in Algorithm 5. Applying Corollary 1 from Sec-tion 5.1 to the modified distribution D T,a defined in Algorithm 5, the consumer surplus of S is atleast: 1ln n + 1 (cid:88) T,a Pr ( T, a ) CS ( D T,a )where Pr ( T, a ) is the probability of the event that v i ∗ = a and the set of all the buyers with valuestrictly larger than a is T .In order to prove the lemma, we need to show the following: CS i ∗ ≤ (cid:88) T,a Pr ( T, a ) CS ( D T,a ) . From Eq. (5), we have: CS i ∗ = (cid:88) (cid:126)v : i ∗ = argmax j ( ϕ D j ( v j )) Pr [ D = (cid:126)v ] · (cid:18) max j ( v j ) − v i ∗ (cid:19) . Let D (cid:48) j = D j | >a − a and let its reserve price be r (cid:48) j . Then, Pr ( T, a ) CS ( D T,a ) = (cid:88) (cid:126)v (cid:48) : v (cid:48) j Any n -dimensional product distribution D admits O ( n log n ) -approximate signaling. Approximating Opt for small K . Similar to the previous section, consider k ∗ ∈ { , , . . . , K} that maximizes (cid:99) CS k . The scheme, denoted by S executes Trunc ( (cid:126)v, D , x k ∗ ). Note that there maybe no bidder with valuation x k ∗ , which motivates the way Algorithm 5 is presented. Theorem 8. The consumer surplus of S is an O ( K ) -approximation to (cid:99) CS k ∗ .Proof. The proof is similar to the proof of Theorem 7 and we omit details. First, by applyingCorollary 2 from Section 5.1 to the modified distribution D T,x k ∗ , the consumer surplus of S definedin Section 5.2 is at least: Ω (cid:18) K (cid:19) (cid:88) T Pr ( T ) CS ( D T,x k ∗ )where Pr ( T ) is the probability that the set of all the buyers with value strictly larger than x k ∗ is T . Furthermore using Eq. (6) and by replacing v i ∗ by x k ∗ in the proof in Theorem 7, we have (cid:99) CS k ∗ ≤ (cid:88) T Pr ( T ) CS ( D T,x k ∗ ) . Combining the above two bounds proves the theorem.Since there are K possible values of k ∗ , the previous theorem directly implies: Corollary 4. Any n -dimensional product distribution D admits O ( K ) -approximate signaling. Combining Corollaries 3 and 4 completes the proof of Theorem 3 for the non-i.i.d. case. The immediate and challenging open questions involve tightening our bounds. First, we do nothave any lower bound for the i.i.d. case, since the bounds in Theorems 1 and 2 both require non-i.i.d. distributions. Indeed, for two-valued i.i.d. distributions, the upper bound on approximationratio is also one. Improving our lower bounds will require a better characterization of the consumersurplus generated by signals that correlate buyer valuations, beyond the Cr´emer-McLean charac-terization [7]. Next, the space of signals we have considered for the upper bounds builds on the n = 1 case [1], and it is conceivable that signals that modify the valuation distribution of multiplebuyers at once may lead to improved bounds. Such signals are challenging to analyze for reasons wedescribe in Section 1.2. Finally, it is quite likely there is a separation between existence results andcomputational results, i.e., there could be an existence result showing a better approximation via anexponential complexity signaling scheme. Such separations are known for the Bayesian persuasionproblem [11], and it would be interesting to derive such results for our multi-agent setting.19oing beyond our specific setting, it would be interesting to explore the equilibria in optimalauctions when the intermediary can send different signals to the seller and to the buyers, muchlike in [2, 17]. At an even higher level, our work can be considered a special case of a largerproblem of information intermediaries for multi-agent mechanisms. As mentioned before, in ourcase, the optimal auction is the mechanism, and the intermediary can change the information tothis mechanism in order to achieve “fairness” between producer and consumer surplus. 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