Combinatorial Auctions with Interdependent Valuations: SOS to the Rescue
Alon Eden, Michal Feldman, Amos Fiat, Kira Goldner, Anna R. Karlin
CCombinatorial Auctions with Interdependent Valuations:SOS to the Rescue ∗ Alon Eden † Michal Feldman ‡ Amos Fiat § Kira Goldner ¶ Anna R. Karlin (cid:107)
June 4, 2019
Abstract
We study combinatorial auctions with interdependent valuations. In such settings, eachagent i has a private signal s i that captures her private information, and the valuation functionof every agent depends on the entire signal profile, s = ( s , . . . , s n ). The literature in economicsshows that the interdependent model gives rise to strong impossibility results, and identifiesassumptions under which optimal solutions can be attained. The computer science literatureprovides approximation results for simple single-parameter settings (mostly single item auctions,or matroid feasibility constraints). Both bodies of literature focus largely on valuations satisfyinga technical condition termed single crossing (or variants thereof).We consider the class of submodular over signals (SOS) valuations (without imposing anysingle-crossing type assumption), and provide the first welfare approximation guarantees formulti-dimensional combinatorial auctions, achieved by universally ex-post IC-IR mechanisms.Our main results are: ( i ) 4-approximation for any single-parameter downward-closed settingwith single-dimensional signals and SOS valuations; ( ii ) 4-approximation for any combinatorialauction with multi-dimensional signals and separable -SOS valuations; and ( iii ) ( k + 3)- and(2 log( k ) + 4)-approximation for any combinatorial auction with single-dimensional signals, with k -sized signal space, for SOS and strong-SOS valuations, respectively. All of our results extendto a parameterized version of SOS, d -SOS, while losing a factor that depends on d . Maximizing social welfare with private valuations is a solved problem. The classical Vickrey-Clarke-Grove (VCG) family of mechanisms [Vickrey, 1961; Clarke, 1971; Groves, 1973], of whichthe Vickrey second-price auction is a special case, are dominant strategy incentive-compatible andguarantee optimal social welfare in general social choice settings. ∗ The work of A. Eden and M. Feldman was partially supported by the European Research Council under theEuropean Unions Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement number 337122, by theIsrael Science Foundation (grant number 317/17), and by an Amazon research award. The work of A. Eden andA. Fiat was partially supported by ISF 1841/14. The work of A. Karlin and K. Goldner was supported by NSFgrants CCF-1420381 and CCF-1813135. The work of K. Goldner was also supported by a Microsoft Research PhDFellowship. † Tel Aviv University ( [email protected] ) ‡ Tel Aviv University ( [email protected] ) § Tel Aviv University ( [email protected] ) ¶ University of Washington ( [email protected] ) (cid:107) University of Washington ( [email protected] ) a r X i v : . [ c s . G T ] J un n this paper, we consider combinatorial auctions, where each agent has a value for every subsetof items, and the goal is to maximize the social welfare, namely the sum of agent valuations fortheir assigned bundles. As a special case of general social choice settings, the VCG mechanismsolves this problem optimally, as long as the values are independent .There are many settings, however, in which the independence of values is not realistic. If theitem being sold has money-making potential or is likely to be resold, the values different agentshave may be correlated, or perhaps even common. A classic example is an auction for the right todrill for oil in a certain location [Wilson, 1969]. Importantly, in such settings, agents may havedifferent information about what that value actually is. For example, the value of an oil leasedepends on how much oil there actually is, and the different agents may have access to differentassessments about this. Consequently, an agent might change her own estimate of the value of theoil lease given access to the information another agent has. Similarly, if an agent had access to theresults of a house inspection performed by a different agent, that might change her own estimateof the value of a house that is for sale.The following model due to Milgrom and Weber [1982], described here for single-item auctions,has become standard for auction design in such settings. These are known as interdependent valuesettings (IDV) and are defined as follows: • Each agent i has a real-valued, private signal s i . The set of signals s = ( s , s , . . . , s n ) maybe drawn from a (possibly) correlated distribution.The signals summarize the information available to the agents about the item. For example,when the item to be sold is a house, the signal could capture the results of an inspection andprivately collected information about the school district. In the setting of oil drilling rights,the signals could be information that each companies’ engineers have about the site based ongeologic surveys, etc. • The value of the item to agent i is a function v i ( s ) of the signals (or information) of all agents.A typical example is when v i ( s ) = s i + β (cid:80) j (cid:54) = i s j , for some β ≤
1. This type of valuationfunction captures settings where an agent’s value depends both on how much he likes theitem ( s i ) and on the resale value which is naturally estimated in terms of how much otheragents like the item ( (cid:80) j (cid:54) = i s j ) [Myerson, 1981].In the economics literature, interdependent settings have been studied for about 50 years (withfar too many papers to list; for an overview, see [Krishna, 2009]). Within the theoretical com-puter science community, interdependent (and correlated) settings have received less attention (seeSection 1.4 for further discussion and references). Consider the goal of maximizing social welfare in interdependent settings. Here, a direct revelationmechanism consists of each agent i reporting a bid for their private signal s i , and the auctioneerdetermining the allocation and payments. (It is assumed that the auctioneer knows the form of thevaluation functions v i ( · ).) See also [Krishna, 2009; Milgrom, 2004].
2n interdependent settings, it is not possible to design dominant-strategy incentive-compatibleauctions, since an agent’s value depends on all of the signals, so if, say, agent i misreports hissignal, then agent j might win at a price above her value if she reports truthfully. The nextstrongest equilibrium notion one could hope for is to maximize efficiency in ex-post equilibrium:bidding truthfully is an ex-post equilibrium if an agent does not regret having bid truthfully, giventhat other agents bid truthfully. In other words, bidding truthfully is a Nash equilibrium forevery signal profile. A strong impossibility result due to Jehiel and Moldovanu [2001] shows thatwith multi-dimensional signals, maximizing welfare is generically impossible even in Bayes-Nashequilibrium. For single-item auctions with single-dimensional signals, a characterization of ex-post incentivecompatibility in the IDV setting is known, analogous to Myerson’s characterization for the inde-pendent private values model (e.g., Roughgarden and Talgam-Cohen [2016]). The characterizationsays that there are payments that yield an ex-post incentive-compatible mechanism if and only ifthe corresponding allocation rule is monotone in each agent’s signal, when all other signals are heldfixed. Maximizing efficiency in ex-post equilibrium is also provably impossible unless the valuationfunctions v i ( s ) satisfy a technical condition known as the single-crossing condition [Milgrom andWeber, 1982; d’Aspremont and G´erard-Varet, 1982; Maskin, 1992; Ausubel, 1999; Dasgupta andMaskin, 2000; Athey, 2001; Bergemann, Shi, and V¨alim¨aki, 2009; Chawla, Fu, and Karlin, 2014;Che, Kim, and Kojima, 2015; Li, 2016; Roughgarden and Talgam-Cohen, 2016]. I.e., the influenceof agent i ’s signal on his own value is at least as high as its influence on other agents’ values, whenall other signals s − i are held fixed . When the single-crossing condition holds, there is a general-ization of VCG that maximizes efficiency in ex-post equilibrium. (See [Cr´emer and McLean, 1985,1988; Krishna, 2009].)Unfortunately, the single crossing condition does not generally suffice to obtain optimal so-cial welfare in settings beyond that of a single item auction with single-dimensional signals. It isinsufficient in fairly simple settings, such as two-item, two-bidder auctions with unit-demand val-uations (see Section A), or single-parameter settings with downward-closed feasibility constraints(see Section B).Moreover, there are many relevant single-item settings where the single-crossing condition doesnot hold. For example, suppose that the signals indicate demand for a product being auctioned,agents represent firms, and one firm has a stronger signal about demand, but is in a weaker positionto take advantage of that demand. A setting like this could yield valuations that do not satisfy thesingle crossing condition. For a concrete example, consider the following scenario given by [Maskin,1992] and [Dasgupta and Maskin, 2000]. Example 1.1.
Suppose that oil can be sold in the market at a price of 4 dollars per unit andtwo firms are competing for the right to drill for oil. Firm 1 has a fixed cost of 1 to produceoil and a marginal cost of 2 for each additional unit produced, whereas firm 2 has a fixed costof 2 and a marginal cost of 1 for each additional unit produced. In addition, suppose that firm Except perhaps in degenerate situations. Note that, of course, every ex-post equilibrium is a Bayes-Nash incentive compatible equilibrium, but notnecessarily vice versa, and therefore ex-post equilibria are much more robust: they do not depend on knowledge ofthe priors and bidders need not think about how other bidders might be bidding. This increases our confidence thatan ex-post equilibrium is likely to be reached. For more details on this and other related work, see Section 1.4. This implies that given signals s − i , if agent i has the highest value when s i = s ∗ , then agent i continues to havethe highest value for s i > s ∗ . This is precisely the monotonicity needed for ex-post incentive compatibility. does a private test and discovers that the expected size of the oil reserve is s units. Then v ( s , s ) = (4 − s − s − , whereas v ( s , s ) = (4 − s − s − . These valuationsdon’t satisfy the single-crossing condition since firm 1 needs to win when s is low and lose when s is high. This paper addresses the following two issues related to social welfare maximization in the inter-dependent values model:1. To what extent can the optimal social welfare be approximated in interdependent settingsthat do not satisfy the single-crossing condition?2. How far beyond the single item, single-dimensional setting can we go?Given the impossibility result of Jehiel and Moldovanu [2001], we ask if it is possible to approximately maximize social welfare in combinatorial auctions with interdependent values ?The first question was recently considered by Eden, Feldman, Fiat, and Goldner [2018] who gavetwo examples pointing out the difficulty of approximating social welfare without single crossing.Example 1.2 shows that even with two bidders and one signal, there are valuation functions forwhich no deterministic auction can achieve any bounded approximation ratio to optimal socialwelfare.
Example 1.2 (No bound for deterministic auctions Eden et al. [2018]) . A single item is for sale.There are two players, A and B , only A has a signal s A ∈ { , } . The valuations are v A (0) = 1 v B (0) = 0 v A (1) = 2 v B (1) = H, where H is an arbitrary large number. If A doesn’t win when s A = 0 , then the approximation ratiois infinite. On the other hand, if A does win when s A = 0 , then by monotonicity, A must also winat s A = 1 , yielding a /H fraction of the optimal social welfare. The next example can be used to show that there are valuation functions for which no random-ized auction performs better (in the worst case) than allocating to a random bidder ( i.e. , a factor n approximation to social welfare), even if a prior over the signals is known. Example 1.3 ( n lower bound for randomized auctions Eden et al. [2018]) . There are n bidders , . . . , n that compete over a single item. For every agent i , s i ∈ { , } , and v i ( s ) = (cid:89) j (cid:54) = i s j + (cid:15) · s i for (cid:15) → that is, agent i ’s value is high if and only if all other agents’ signals are high simultaneously. Whenall signals are 1, then in any feasible allocation, there must be an agent i which is allocated withprobability of at most /n . By monotonicity, this means that the probability this agent is allocatedwhen the signal profile is s (cid:48) = ( − i , i ) is at most /n as well. Therefore, the achieved welfare atsignal profile s (cid:48) is at most /n + ( n − · (cid:15) , while the optimal welfare is , giving a factor n gap . Eden et al. [2018] show that there exists a prior for which the n gap still holds, even if the mechanism knowsthe prior. some assumption is needed if we are to get good approximations to social welfare.The approach taken by Eden et al. [2018] was to define a relaxed notion of single-crossing that theycalled c -single crossing and then provide mechanisms that approximately maximize social welfare,where the approximation ratio depends on c and n , the number of agents.In this paper, we go in a different direction, starting with the observation that in Example 1.3,the valuations treat the signals as highly-complementary–one has a value bounded away from zeroonly if all other agent’s signals are high simultaneously. This suggests that the case where thevaluations treat the signals more like “substitutes” might be easier to handle.We capture this by focusing on s ubmodular over signals (SOS) valuations. This means that forevery i and j , when signals s − j are lower, the sensitivity of the valuation v i ( s ) to changes in s j ishigher. Formally, we assume that for all j , for any s j , δ ≥
0, and for any s − j and s (cid:48)− j such thatcomponent-wise s − j ≤ s (cid:48)− j , it holds that v i ( s j + δ, s − j ) − v i ( s j , s − j ) ≥ v i ( s j + δ, s (cid:48)− j ) − v i ( s j , s (cid:48)− j ) . Many valuations considered in the literature on interdependent valuations are SOS (though thisterm is not used) Milgrom and Weber [1982]; Dasgupta and Maskin [2000]; Klemperer [1998]. Thesimplest (yet still rich) class of SOS valuations are fully separable valuation functions , where thereare arbitrary (weakly increasing) functions g ij ( s j ) for each pair of bidders i and j such that v i ( s ) = n (cid:88) j =1 g ij ( s j ) . A more general class of SOS valuation functions are functions of the form v i ( s ) = f ( (cid:80) nj =1 g ij ( s j )),where f is a weakly increasing concave function.We can now state the main question we study in this paper: to what extent can social welfarebe approximated in interdependent settings with SOS valuations? Unfortunately, Example 1.2 itselfdescribes SOS valuations, so no deterministic auction can achieve any bounded approximation ratio,even for this subclass of valuations. Thus, we must turn to randomized auctions.
All of our positive results concern the design of randomized, prior-free, universally ex-post incentive-compatible (IC), individually rational (IR) mechanisms . Prior-free means that the rules of themechanism makes no use of the prior distribution over the signals, thus need not have any knowledgeof the prior.Our first result provides approximation guarantees for single-parameter downward-closed set-tings. An important special case of this result is single-item auctions, which was the focus of Edenet al. [2018].
Theorem 4.1 (See Section 4): For every single-parameter downward-closed setting, if the valuationfunctions are SOS, then the
Random Sampling Vickrey auction is a universally ex-post IC-IRmechanism that gives a 4-approximation to the optimal social welfare. This type of valuation function is ubiquitous in the economics literature on inderdependent settings; often withthe function simply assumed to be a linear function of the signals (see, e.g., Jehiel and Moldovanu [2001]; Klemperer[1998]). n − i has a signal s iT for each subset T of items, and a valuation function v iT := v iT ( s T , s T , . . . , s nT ). For this setting, it is not at all clear under what conditions it mightbe possible to maximize social welfare in ex-post equilibrium. However, rather surprisingly (see the related work section below), for the case of separable SOS valuations , we are able to extend the 4-approximation guarantee to combinatorial auctions. Theorem 5.1 (See Section 5): For every combinatorial auction, if the valuation functions areseparable-SOS, then the
Random Sampling VCG auction is a universally ex-post IC-IR mechanismthat gives a 4-approximation to the optimal social welfare.Finally, we consider combinatorial auctions where each agent i has a single-dimensional signal s i , but where the valuation function v iT for each subset of items T is an arbitrary SOS valuationfunction v iT ( s , . . . , s n ). For this case, we show the following: Theorems 6.1 and 6.5 (See Sections 6.1 and 6.2): Consider combinatorial auctions with single-dimensional signals, where each signal takes one of k possible values. If the valuation functions areSOS, then there exists a universally ex-post IC-IR mechanism that gives a ( k + 3)-approximation tothe optimal social welfare. If the valuations are strong-SOS , the approximation ratio improvesto O (log k ).All of the above results, as well as our lower bounds, are summarized in Table 1. In addition, allof the results in this paper generalize easily, with a corresponding degradation in the approximationratio, to the weaker requirement of d -SOS valuations . The fundamental tension in settings with interdependent valuations that is not present in theprivate values setting is the following. Consider, for example, a single item auction setting whereagent 1’s truthful report of her signal increases agent 2’s value . Since, this increases the chance thatagent 2 wins and may decrease agent 1’s chance of winning, it might motivate agent 1 to strategizeand misreport.Our approach is to simply prevent this interaction. Without looking at the signals, our mecha- See the related work and also Lemmas A.2 and A.3, which show that under one natural generalization of single-crossing to the setting of two items and two agents that are unit demand, single crossing is not sufficient for fullefficiency. A valuation is separable-SOS if the valuation for an agent can be split into two parts, an SOS function of allother signals and an arbitrary function of the agents’ own signal. Such valuations generalize the fully separable casediscussed above. See definition 2.16 See definition 2.14. A valuation function is d -SOS if for all j , for all δ >
0, and for any s − j and s (cid:48)− j such that component-wise s − j ≤ s (cid:48)− j , it holds that d · ( v i ( s j + δ, s − j ) − v i ( s j , s − j )) ≥ v i ( s j + δ, s (cid:48)− j ) − v i ( s j , s (cid:48)− j ) . : potential winners and certain losers. Losers neverreceive any allocation. When estimating the value of a potentially winning agent i , we use onlythe signals of losers and i ’s own signal(s). Thus, potential winners can not impact the estimatedvalues and hence allocations of other potential winners. This resolves the truthfulness issue. Theremaining question is: can we get sufficiently accurate estimates of the agents’ values when weignore so many signals?The key lemma ( Lemma 3.1
Section 3) shows that we can do so, when the valuations are SOS.Specifically, for any agent i , if all agents other than i are split into two random sets A (losers) and B (potential winners), and the signals of agents in the random subset B are “zeroed out”, then theexpected value agent i has for the item is at least half of her true valuation. That is, E A [ v i ( s i , s A , B )] ≥ v i ( s ) . Dealing with combinatorial settings is more involved as the truthfullness characterization is lessobvious, but the key ideas of random partitioning and using the signals of certain losers remain atthe core of our results.
While this paper deals entirely with welfare maximization, our results have significance for theobjective of maximizing the seller’s revenue. Eden et al. [2018] give a reduction from revenue max-imization to welfare maximization in single-item auctions with SOS valuations. Thus, the constantfactor approximation mechanism presented in this paper implies a constant factor approximationto the optimal revenue in single-item auctions with SOS valuations. We note that this is the firstrevenue approximation result that does not assume any single-crossing type assumption ([Chawlaet al., 2014; Eden et al., 2018; Roughgarden and Talgam-Cohen, 2016; Li, 2016] require singlecrossing or approximate single crossing).Finally, one can easily verify that, based on Yao’s min-max theorem, the existence of a random-ized prior-free mechanism that gives some approximation guarantee (in expectation over the coinflips of the mechanism) implies the existence of a deterministic prior-dependent mechanisms thatgives the same approximation guarantee (in expectation over the signal profiles).
As discussed above, in single-parameter settings, there is an extensive literature on mechanismdesign with interdependent valuations that considers social welfare maximization, revenue maxi-mization and other objectives. However, the vast majority of this literature assumes some kind ofsingle-crossing condition and, in the context of social welfare, focuses on exact optimization.There are two papers that we are aware of that study the question of how well optimal so-cial welfare can be approximated in ex-post equilibrium without single-crossing. The first is theaforementioned paper [Eden et al., 2018] on single item auctions with interdependent valuations.They defined a parameterized version of single-crossing, termed c -single crossing, where c > c -singlecrossing valuations, they provide a number of results including a lower bound of c on the approxi-mation ratio achievable by any mechanism, a matching upper bound for binary signal spaces, and as in [Goldberg, Hartline, and Wright, 2001]. ≥ / ∀ mech . ≤ / k -sized Signal Space ≥ / ( k + 3) ∀ mech . ≤ / k -sized Signal Space ≥ / (log( k ) + 2) ∀ mech . ≤ / ≥ / ∀ mech . ≤ / d -approximate SOS/Strong-SOSvaluations, while losing a factor that depends on d . All positive results are obtained with universallyex-post IC-IR randomized mechanisms.mechanisms that achieve approximation ratios of ( n − c and 2 c / √ n (the first is deterministicand the second is randomized).Ito and Parkes [2006] also consider approximating social welfare in the interdependent setting.Specifically, they propose a greedy contingent-bid auction (a la [Dasgupta and Maskin, 2000]) andshow that it achieves a √ m approximation to the optimal social welfare for m goods, in the specialcase of combinatorial auctions with single-minded bidders.For multidimensional signals and settings, the landscape is sparser (and bleaker) and, to ourknowledge, focuses on exact social welfare maximization. Maskin [1992] has observed that, ingeneral, no efficient incentive-compatible single item auction exists if a buyer’s valuation dependson a multi-dimensional signal.Jehiel and Moldovanu [2001] consider a very general model in which there is a set K of possiblealternatives, and a multidimensional signal space, where each agent j has a signal s jki for eachoutcome k and other agent j . In their model the valuation function of an agent i for outcome k islinear in the signals, that is, v i ( k ) := (cid:80) j a jki s jki . Thus, their valuation functions are, in one sense,a special case of our separable valuation functions. On the other hand, they are more general inthat all quantities depend on the outcome k . Thus, there are allocation externalities. Their mainresult is that, generically, there is no Bayes-Nash incentive compatible mechanism that maximizessocial welfare in this setting. However, they do give an ex-post IC mechanism that maximizes socialwelfare with both information and allocation externalities if the signals are one-dimensional, thevaluation functions are linear in the signals, and a single-crossing type condition holds.Jehiel, Meyer-ter Vehn, Moldovanu, and Zame [2006] go on to show that the only deterministicsocial choice functions that are ex-post implementable in generic mechanism design frameworkswith multidimensional signals, interdependent valuations and transferable utilities, are constantfunctions.Finally, Bikhchandani [2006] considers a single item setting with multidimensional signals butno allocation externalities and shows that there is a generalization of single-crossing that allowssome social choice rules to be implemented ex-post.For further analysis and discussion of implementation with interdependent valuations, see e.g.,Bergemann and Morris [2005] and McLean and Postlewaite [2015].8or further literature in computer science on interdependent and correlated values, see [Ro-nen, 2001; Constantin, Ito, and Parkes, 2007; Constantin and Parkes, 2007; Klein, Moreno, Parkes,Plakosh, Seuken, and Wallnau, 2008; Papadimitriou and Pierrakos, 2011; Dobzinski, Fu, and Klein-berg, 2011; Babaioff, Kleinberg, and Paes Leme, 2012; Abraham, Athey, Babaioff, and Grubb, 2011;Robu, Parkes, Ito, and Jennings, 2013; Kempe, Syrgkanis, and Tardos, 2013; Che et al., 2015; Li,2016; Chawla et al., 2014]. In Section 4, we will consider single-parameter settings with interdependent valuations and downward-closed feasibility constraints. In these settings, a mechanism decides which subset of agents 1 , . . . , n are to receive “service” (e.g., an item). The feasibility constraint is defined by a collection
I ⊆ [ n ] of subsets of agents that may feasibly be served simultaneously. We restrict attention to downward-closed settings , which means that any subset of a feasible set is also feasible. A simple example isa k -item auction, where I is the collection of all subsets of agents of size at most k .For these settings, we use the interdependent value model of Milgrom and Weber [1982]: Definition 2.1 (Single Dimensional Signals, Single Parameter Valuations) . Each agent j has aprivate signal s j ∈ R + . The value agent j gives to “receiving service” v j ( s ) ∈ R + , is a functionof all agents’ signals s = ( s , s , . . . , s n ) . The function v j ( s ) is assumed to be weakly increasing ineach coordinate and strictly increasing in s i . (Deterministic Single Parameter Mechanisms) . A deterministic mechanism M =( x, p ) in the downward closed setting is a mapping from reported signals s = ( s , . . . , s n ) to alloca-tions x ( s ) = { x i ( s ) } ≤ i ≤ n and payments p ( s ) = { p i ( s ) } ≤ i ≤ n , where x i ( s ) ∈ { , } indicates whetheror not agent i receives service and p i ( s ) is the payment of agent i . It is required that the set ofagents that receive service is feasible, i.e., { i | x i ( s ) = 1 } ∈ I . (The mechanism designer knows theform of the valuation functions but learns the private signals only when they are reported.) Definition 2.3 (Agent utility) . Given a deteministic mechanism ( x, p ) , the utility of agent i whenher true signal is s i , she reports s (cid:48) i and the other agents report s − i is u i ( s (cid:48) i , s − i | s i ) = x i ( s (cid:48) i , s − i ) v i ( s i , s − i ) − p i ( s (cid:48) i , s − i ) . Agent i will report s (cid:48) i so as to maximize u i ( s (cid:48) i , s − i | s i ) . We use u i ( s ) to denote the utility when shereports truthfully, i.e., u i ( s i , s − i | s i ) . Definition 2.4 (Deterministic ex-post incentive compatibility (IC)) . A deterministic mechanism M = ( x, p ) in the interdependent setting is ex-post incentive compatible (IC) if, irrespective of thetrue signals, and given that all other agents report their true signals, there is no advantage to anagent to report any signal other than her true signal. In other words, assuming that s − i are thetrue signals of other bidders, u i ( s (cid:48) i , s − i | s i ) is maximized by reporting s i truthfully. efinition 2.5 (Deterministic ex-post individual rationality (IR)) . A deterministic mechanism inthe interdependent setting is ex-post individually rational (IR) if, irrespective of the true signals,and given that all other agents report their true signals, no agent gets negative utility by participatingin the mechanism.
If a deterministic mechanism is both ex-post IR and ex-post IR we say that it is ex-post IC-IR.
Definition 2.6.
A deterministic allocation rule x is monotone if for every agent i , every signalprofile of all other agents s − i , and every s i ≤ s (cid:48) i , it holds that x i ( s i , s − i ) = 1 ⇒ x i ( s (cid:48) i , s − i ) = 1 . Proposition 2.1. [Roughgarden and Talgam-Cohen, 2016] For every deterministic allocation rule x for single parameter valuations, there exist payments p such that the mechanism ( x, p ) is ex-postIC-IR if and only if x i is monotone for every agent i . A randomized mechanism is a probability distribution over deterministic mecha-nisms.
Definition 2.8 (Universal ex-post IC-IR) . A randomized mechanism is said to be universally ex-post IC-IR if all deterministic mechanisms in the support are ex-post IC-IR.
Sections 5 and 6 focus on combinatorial auctions, where there are n agents and m items. In thesesettings, a mechanism is used to decide how the items are partitioned among the agents. Weconsider two models for the interdependent valuations: Definition 2.9 (Single Dimensional Signals, Combinatorial Valuations) . Each agent i has a signal s i ∈ R + . The value agent i gives to subset of items T ⊆ [ m ] , which we denote by v jT ( s ) , is afunction of s = ( s , s , . . . , s n ) . Definition 2.10 (Multidimensional Combinatorial Signals, Combinatorial Valuations) . Here, eachagent has a signal for each subset of items; for any agent i , we use s iT to denote agent i ’s signalfor subset of items T ⊆ [ m ] . The value agent i gives to set T is denoted by v iT ( s T ) where s T =( s T , s T , . . . , s nT ) ∈ R + n . We use s to denote the set of all signals { s T } T ⊆ m . In both cases, each v iT ( · ) is assumed to be a weakly increasing function of each signal andstrictly increasing in s i (or s iT respectively), and known to the mechanism designer.We give subsequent definitions only for multidimensional combinatorial signals, as single di-mensional signals can be viewed as a special case of multi-dimensional signals where s iT = s i forall T . (Deterministic mechanisms for combinatorial settings) . A deteministic mechanism M = ( x, p ) is a mapping from reported signals s to allocations x = { x iT } (where each x iT ∈ { , } )and payments p = { p iT } for all ≤ i ≤ n and T ⊂ { , . . . , m } such that: For other types of signals and interdependent valuation models, see, e.g., Jehiel and Moldovanu [2001]. Agent j is allocated the set T iff x jT ( s ) = 1 ; • For each agent j , there is at most one T for which x jT ( s ) = 1 ; • The sets allocated to different agents do not intersect. • The payment for agent j when her allocation is set T is p jT ( s ) . Definition 2.12 (Agent Utility) . The utility of agent i when her signals are s i = { s iT } T ⊂ m , shereports s (cid:48) i and the other agents report s − i is u i ( s (cid:48) i , s − i | s i ) = (cid:88) T ⊆ m x iT ( s (cid:48) i , s − i )[ v iT ( s iT , s − iT ) − p iT ( s (cid:48) i , s − i )] . Given a mechanism M = ( x, p ) , agent i will report s (cid:48) i so as to maximize u i ( s (cid:48) i , s − i | s i ) . We use u i ( s ) to denote the utility when she reports truthfully, i.e., u i ( s i , s − i | s i ) . The definitions of ex-post incentive compatibility (IC) and ex-post individually rationality (IR)for deterministic mechanisms for combinatorial settings are the same as the appropriate definitionsfor single parameter mechanisms (Definitions 2.4 and 2.5 with the obvious modifications).
As with single parameter mechanisms, a randomized mechanism for a combinatorial setting is aprobability distribution over deterministic mechanisms for the combinatorial setting, and a ran-domized mechanism is said to be universally ex post IC-IR if all deterministic mechanisms in thesupport are themselves ex-post IC-IR.
As discussed in the introduction, our results will rely on an assumption about the valuation functionsthat we call submodularity over signals or SOS. The SOS (resp. strong-SOS) notion we use is thesame as the weak diminishing returns (resp. strong diminishing returns) submodularity notionin [Bian, Levy, Krause, and Buhmann, 2017; Niazadeh, Roughgarden, and Wang, 2018] . SOSwas also used in [Eden et al., 2018], generalizing a similar notion in [Chawla et al., 2014]. Definition 2.13 ( d -approximate submodular-over-signals valuations ( d -SOS valuations)) . A valu-ation function v ( s ) is a d -SOS valuation if for all j , s j , δ ≥ , s − j = ( s , . . . , s j − , s j +1 , . . . , s n ) and s (cid:48)− j = ( s (cid:48) , . . . , s (cid:48) j − , s (cid:48) j +1 , . . . , s (cid:48) n ) such that s (cid:48)− j is smaller than or equal to s − j coordinate-wise, it holds that d · (cid:0) v ( s (cid:48)− j , s j + δ ) − v ( s (cid:48)− j , s j ) (cid:1) ≥ v ( s − j , s j + δ ) − v ( s − j , s j ) (1) If v satisfies this condition with d = 1 , we say that v is an SOS valuation function. Weak diminishing returns submodularity was introduced in [Soma and Yoshida, 2015], where it’s termed “di-minishing returns submodularity”. efinition 2.14 ( d -approximate strong submodular-over-signals valuations ( d -strong-SOS valua-tions)) . The valuation function v ( s ) is a d strong-SOS valuation if for any j , δ ≥ , s = ( s , . . . , s n ) and s = ( s (cid:48) , . . . , s (cid:48) n ) such that s (cid:48) is smaller than or equal to s coordinate-wise, it holds that d · (cid:0) v ( s (cid:48)− j , s (cid:48) j + δ ) − v ( s (cid:48)− j , s (cid:48) j ) (cid:1) ≥ v ( s − j , s j + δ ) − v ( s − j , s j ) (2) If v satisfies this condition with d = 1 , we say that i ’s valuation functions are “strong-SOS”. Definition 2.15 (SOS-valuations settings) . We say that a mechanism design setting with in-terdependent valuations is an
SOS-valuations setting or, equivalently, that the agents have SOS-valuations, in each of the following cases: • Single parameter valuations (as in definition 2.1): for every i , the valuation function v i ( s ) isSOS. • Combinatorial valuations with single-parameter signals (as in definition 2.9): for every i and T , the valuation function v iT ( s ) is SOS; • Combinatorial valuations with multi-parameter signals (as in definition 2.10): for every i and T , v iT ( s T ) is SOS, where s T = ( s T , . . . , s nT ) .Similar definitions can be given for d -SOS valuation settings and d -strong-SOS valuation settings. Finally, in section 5, we will specialize to the case of separable
SOS valuations.
Definition 2.16 (Separable SOS valuations) . We say that a set of valuations as in Definition 2.10are separable SOS valuations if for every agent i and subset T of items, v iT ( s T ) can be written as v iT ( s T ) = g − iT ( s − iT ) + h iT ( s iT ) , where g − iT ( · ) and h iT ( · ) are both weakly increasing and g − iT ( s − iT ) is itself an SOS valuationfunction. Observation 2.2.
A separable SOS valuation function is itself an SOS valuation function.
We can similarly define separable d -SOS valuations. Lemma 2.3.
Let v : R + n → R + be a d -SOS function. Let A ⊆ [ n ] and B = [ n ] \ A . For any s A , y A ∈ R + | A | , and s B , s (cid:48) B ∈ R + | B | such that s B is smaller than s (cid:48) B coordinate wise, d · ( v ( s A + y A , s B ) − v ( s A , s B )) ≥ v ( s A + y A , s (cid:48) B ) − v ( s A , s (cid:48) B ) . Proof.
Let i , i , . . . , i | A | be the elements of A . For 1 ≤ j ≤ | A | , let s j and s (cid:48) j denote the vectors s j = (cid:16) ( s i + y i ) , . . . , ( s i j + y i j ) , s i j +1 , . . . , s i | A | , s B (cid:17) , s (cid:48) j = (cid:16) ( s i + y i ) , . . . , ( s i j + y i j ) , s i j +1 , . . . , s i | A | , s (cid:48) B (cid:17) . s | A | = ( s A + y A , s B ), and s (cid:48)| A | = ( s A + y A , s (cid:48) B ).It follows from the d -SOS definition that for every 1 ≤ j ≤ | A | , d · (cid:0) v ( s j ) − v ( s j − ) (cid:1) ≥ v ( s (cid:48) j ) − v ( s (cid:48) j − ) , (3)where s = ( s A , s B ) and s (cid:48) = ( s A , s (cid:48) B ).Summing Equation (3) for j = 1 , , . . . , | A | proves the claim. The following is a key lemma which is used for both single parameter and combinatorial settings.
Lemma 3.1.
Let v i : R + n → R + be a d -SOS function. Let A be a uniformly random subset of [ n ] \ { i } , and let B := ([ n ] \ { i } ) \ A . It now holds that E A [ v i ( s A , B , s i )] ≥ d + 1 v i ( s ) , where the expectation is over the random choice of A .Proof. We consider two equiprobable events, • A = S ⊂ [ n ] \ { i } is chosen as the random subset. • A = T = ([ n ] \ { i } ) \ S is chosen as the random subset.Normalize the valuations so that v i ( s ) = 1 and define α, β ∈ [0 ,
1] such that v i ( s S , T , s i ) = α, v i ( S , s T , s i ) = β. It follows that β = v i ( S , s T , s i ) ≥ v i ( S , s T , s i ) − v i ( S , T , s i ) ≥ ( v i ( s S , s T , s i ) − v i ( s S , T , s i )) /d = (1 − α ) /d, where the first inequality follows from non-negativity of v i ( S , T , s i ), and the second inequalityfollows from v i being d -SOS and Lemma 2.3.Similarly, we have that α ≥ (1 − β ) /d ⇒ β ≥ − αd ;It follows that α + β ≥ max (cid:18) α + 1 − αd , α + 1 − αd (cid:19) . Solving for equality of the two terms, we get that α = 1 / ( d + 1) which implies that α + β ≥ d + 1 . S, T ) that partition [ n ] \ { i } . For every such ( S, T ) pair, itfollows that v i ( s S , T , s i ) + v i ( S , s T , s i ) = α + β ≥ d +1 .We conclude with the following, where the third line follows from the fact that there are 2 n − / S, T ) pairs that partition [ n ] \ { i } : E A (cid:2) v i ( s A , B , s i ) (cid:3) = (cid:88) A ⊆ [ n ] \{ i } Pr[ A ] · v i ( s A , B , s i )= 12 n − · (cid:88) A ⊆ [ n ] \{ i } v i ( s A , B , s i ) ≥ n − · n − · d + 1 = 1 d + 1 , as desired. In this section we describe the
Random Sampling Vickrey (RS-V) mechanism that achieves a4-approximation for single-parameter downward-closed environments with SOS valuations and a2( d + 1)-approximation for d -SOS valuations. We then give a lower bound of 2 and √ d for SOSand d -SOS valuations respectively, even in the case of selling a single item.Let I ⊆ [ n ] be a downward-closed set system. We present a mechanism that serves only setsin I and gets a 2( d + 1)-approximation to the optimal welfare. Random Sampling Vickrey (RS-V): • Elicit bids ˜s from the agents. • Partition the agents into two sets, A and B , uniformly at random. • For i ∈ B , let w i = v i ( ˜s A , ˜ s i , B \{ i } ). • Allocate to a set of bidders in argmax S ∈I : S ⊆ B (cid:40)(cid:88) i ∈ S w i . (cid:41) Theorem 4.1.
For agents with SOS valuations, and for every downward-closed feasibility constraint I , RS-V is an ex-post IC-IR mechanism that gives -approximation to the optimal welfare. For d -SOS valuations, the mechanism gives a d + 1) -approximation to the optimal welfare.Proof. We first show the allocation is monotone in one’s signal, and hence, by Proposition 2.1, themechanism is ex-post IC-IR. Fix a random partition (
A, B ). • Agents in A are never allocated anything and thus their allocation is weakly monotone intheir signal. 14 For an agent i ∈ B , increasing ˜ s i can only increase w i , whereas it leaves w j unchanged forall j ∈ B \ { i } . Thus, this only increases the weight of feasible sets (subsets of B in I ) that i belongs to. Therefore, increasing s i can only cause i to go from being unallocated to beingallocated.For approximation, consider a set S ∗ ∈ argmax S ∈I (cid:80) i ∈ S v i ( s ) that maximizes social welfare.For every i ∈ S ∗ , from the Key Lemma 3.1, we have that E B [ w i · i ∈ B ] = E B [ v i ( s i , s A , B − i ) | i ∈ B ] · P r ( i ∈ B ) ≥ v i ( s ) d + 1 · . (4)For every set B , the fact that I is downward-closed implies that S ∗ ∩ B ∈ I . Therefore, S ∗ ∩ B is eligible to be selected by RS-V as the allocated set of bidders. We have that the values of thebidders we allocate to are at least E B (cid:34) max S ∈I : S ⊆ B (cid:88) i ∈ S w i (cid:35) ≥ E B (cid:34) (cid:88) i ∈ S ∗ ∩ B w i (cid:35) = E B (cid:34) (cid:88) i ∈ S ∗ w i · i ∈ B (cid:35) = (cid:88) i ∈ S ∗ E B [ w i · i ∈ B ] ≥ (cid:88) i ∈ S ∗ v i ( s )2( d + 1) , as desired. Since the allocated bidders’ true values at s are only higher than the proxy values w i ,this continues to hold.We note that for the case of downward-closed feasibility constraints, even if the valuationssatisfy single-crossing, there can be an n − √ d ) for SOS and d -SOS valuations respec-tively.The lower bounds apply to arbitrary randomized mechanisms . Theorem 4.2.
No ex-post IC-IR mechanism (not necessarily universal) for selling a single itemcan get a better approximation than(a) a factor of 2 for SOS valuations.(b) a factor of Ω( √ d ) for d -SOS valuations.Proof. Let x i ( s ) be the probability agent i is allocated at signal profile s . Notice that for every s , (cid:80) i x i ( s ) ≤
1, otherwise the allocation rule is not feasible.(a) Consider the case where there are two agents, 1 and 2, s ∈ { , } and agent 2 has no signal.The valuations are v (0) = 1, v (1) = 1 + (cid:15) , v (0) = 0 and v (1) = H for H (cid:29) (cid:29) (cid:15) . It iseasy to see the valuations are SOS.In order to get better than a 2-approximation at s = 0, we must have x (0) > /
2. Bymonotonicity, this forces x (1) > / x (1) < / A randomized mechanism takes as input the set of signals s and produces as output x i ( s ) and p i ( s ) for eachagent i , where x i ( s ) is the probability that agent i wins and p i ( s ) is agent i ’s expected payment. Such a mechanismis ex-post IC (but not necessarily universally so) if and only if x i ( s i , s − i ) is monotonically increasing in s i . s = 1 is x (1) v (1) + x (1) v (1) < H/ s = 1 is H . For a large H , this approaches a 2-approximation.Note that this lower bound applies even given a known prior distribution on the signals inthe event that we have a prior on the signals that satisfies: Pr[ s = 0] · s = 1] · H .(b) Consider the case where there are n = √ d agents and s i ∈ { , } for every agent i . Thevaluation of agent i is v i ( s ) = (cid:40)(cid:80) j (cid:54) = i s j + (cid:15) · s i ∃ j (cid:54) = i : s j = 0 d + (cid:15) · s i s j = 1 ∀ j (cid:54) = i, where (cid:15) → d -SOS, notice that whenever a signal s j changes from 0 to 1,the valuation of agent i (cid:54) = j increases by 1 unless all other signals beside i ’s are already set to1, in which case the valuation increases by d − √ d + 2 < d . Consider valuation profiles s i =(0 i , − i ). Note that by monotonicity, for every truthful mechanism, it must be the case that x i ( s i ) ≤ x i ( ). Since any feasible allocation rule must satisfy (cid:80) √ di =1 x i ( ) ≤
1, then it must bethe case there exists some agent i such that x i ( ) ≤ √ d , which by monotonicity implies that x i ( s i ) ≤ √ d . However, at profile s i , v i ( s i ) = d while v j ( s i ) = √ d − < √ d for all j (cid:54) = i , so weget that the expected welfare of the mechanism at s i is at most x i ( s i ) · d +(1 − x i ( s i )) ·√ d ≤ √ d, while the optimal welfare is d . Again, the lower bound also applies to the setting with knownpriors on the signals using a prior that satisfies: Pr[ s i ] = Pr[ s j ] = √ d for all i and j . In this section we present an ex-post IC-IR mechanism that gives 1 / Random-sampling VCG auction is a natural extension of the
Random-Sampling Vickrey (RS-V) auction presented in Section 4. Note that unlike
RS-V , herewe need to explicitly define payments so that the obtained mechanism is ex-post IC-IR. We de-rive VCG-inspired payments which align the objective of the mechanism with that of the agents.Separability is used here, as without it, the payment term would have been affected by the agent’sreport (while with separability, only the allocation is affected by it).
Random-Sampling VCG (RS-VCG): • Agents report their signals ˜ s . • Partition the agents into two sets A and B uniformly at random. • For each agent j ∈ B and bundle T ⊆ [ m ], let w jT := v jT (˜ s jT , ˜ s AT , B − j T ) = g − jT (˜ s AT , B − j T ) + h jT (˜ s jT ) . Let the allocation be { T i } i ∈ B ∈ argmax { S i } i ∈ B (cid:88) i ∈ B w iS i ; i.e., { T i } i ∈ B is the allocation that maximizes the “welfare” using w iT ’s. • Set the payment for a winning agent i ∈ B receiving set of goods T i to be: p i (˜ s ) := g − iT i (˜ s − iT i ) − g − iT i (˜ s AT i , B − i T i ) − (cid:88) j ∈ B \{ i } w jT j + w − i , where w − i = max partitions { T (cid:48) j } (cid:88) j ∈ B \{ i } w jT (cid:48) j , that is, w − i is the weight of the best allocation without agent i .Since the w jT ’s do not depend on agent i ’s report (since i is in B ), w − i doesn’t depend on agent i ’s report. Therefore, we can (and will) ignore this term when considering incentive compatibilitybelow.Note also that since the maximal partition guarantees that w − i ≥ (cid:80) j ∈ B \{ i } w jT j , and mono-tonicity of valuations in signals guarantees that g − iT i (˜ s − i ) ≥ g − iT i (˜ s A , B − i ). Therefore, the pay-ments p i (˜ s ) are always nonnegative. Theorem 5.1.
Random-Sampling VCG is an ex-post IC-IR mechanism that gives a 4-approximationto the optimal social welfare for any combinatorial auction setting with separable SOS valuations.Proof.
First we show that if the agents bid truthfully, then the mechanism gives a 4-approximationto social welfare. For every agent i and bundle T , E B [ w iT · i ∈ B ] = E B [ v iT ( s iT , s AT , B − i T ) | i ∈ B ] · P r ( i ∈ B ) ≥ v iT ( s T )2 · , (5)where the inequality follows by applying Lemma 3.1 with d = 1.Let S ∗ , . . . , S ∗ n be the true welfare maximizing allocation. Then, E B (cid:34) max partitions { T i } (cid:88) i ∈ B w iT i (cid:35) ≥ E B (cid:34)(cid:88) i w iS ∗ i · i ∈ B (cid:35) = (cid:88) i E B [ w iS ∗ i · i ∈ B ] ≥ (cid:88) i v iS ∗ i ( s S ∗ i ) , where the last inequality follows by substituting S ∗ i in T in Equation (5) for every i . Since v iT ( s )is always at least w iT , this proves the approximation ratio.Next, we show that RS-VCG is universally ex-post IC. Fix a random partition (
A, B ). Supposethat when all agents bid truthfully { T ∗ j } j ∈ B = argmax partitions { T j } (cid:88) j ∈ B w jT j . i ∈ B bid truthfully and i bids s (cid:48) i instead of his true signal vector s i . Let { T (cid:48) j } j ∈ B be the resulting allocation. Therefore, agent i ’s utility when reporting s (cid:48) i (afterdisregarding the w − i term as mentioned above) is: v iT (cid:48) i ( s ) − p i ( s (cid:48) i , s − i ) = g − iT (cid:48) i ( s − iT (cid:48) i ) + h iT (cid:48) i ( s iT (cid:48) i ) − p i ( s (cid:48) i , s − i )= g − iT (cid:48) i ( s − iT (cid:48) i ) + h iT (cid:48) i ( s iT (cid:48) i ) − g − iT (cid:48) i ( s − iT (cid:48) i ) − g − iT (cid:48) i ( s AT (cid:48) i , B − i T (cid:48) i ) − (cid:88) j ∈ B \{ i } w jT (cid:48) j = h iT (cid:48) i ( s iT (cid:48) i ) + g − iT (cid:48) i ( s AT (cid:48) i , B − i T (cid:48) i ) + (cid:88) j ∈ B \{ i } w jT (cid:48) j = w iT (cid:48) i + (cid:88) j ∈ B \{ i } w jT (cid:48) j = (cid:88) j ∈ B w jT (cid:48) j ≤ (cid:88) j ∈ B w jT ∗ j , where (cid:80) j ∈ B w jT ∗ j is i ’s utility for bidding truthfully.Finally, we show that the mechanism is ex-post IR. Indeed, from above, agent i ’s utility whenreporting truthfully (and without disregarding the w − i term) is v iT ∗ i ( s T ∗ i ) − p i ( s ) = (cid:88) j ∈ B w jT ∗ j − w − i = (cid:88) j ∈ B w jT ∗ j − max partitions { T (cid:48) j } (cid:88) j ∈ B \{ i } w jT (cid:48) j ≥ . In the case of separable d -SOS valuations, the Random-Sampling VCG is an ex-post IC-IR mech-anism that gives 2( d + 1)-approximation to the social welfare. The proof is identical to Theorem5.1, except that Equation (5) is changed to E B [ w iT · i ∈ B ] ≥ v iT ( s T )2( d + 1) , since we apply Lemma 3.1 with an arbitrary d . Remark 5.2.
Theorem 5.1 is clearly analogous to the VCG mechanism for combinatorial auctionswith private values. As with VCG for private values, in many cases, there is unlikely to be apolynomial time algorithm to compute allocations and payments. Exceptions include settings weknow and love such as unit-demand auctions, additive valuations, etc.
In this section we consider combinatorial valuations (general combinatorial auctions) with single-dimensional signals (as given by Definition 2.9).When the signal space of each agent is of size at most k , we present a mechanism that gets( k +3)-approximation for SOS valuations (see Section 6.1), and a mechanism that gets (2 log k +4)-approximation for strong-SOS valuations (Definition 2.14, see Section 6.2 for details regarding themechanism). For d -SOS and d -strong-SOS valuations, the mechanism generalizes to give O ( dk )-and O ( d log k )-approximations respectively, as shown in Section C.18e first decompose the optimal welfare into two parts, OTHER and
SELF . Each part will becovered by a corresponding mechanism. Let T ∗ = { T ∗ i } i ∈ [ n ] be a welfare-maximizing allocation atsignal profile s , and let W ∗ ( s ) be the social welfare of T ∗ at s . Consider the following decomposition: W ∗ ( s ) = (cid:88) i v iT ∗ i ( s )= (cid:88) i v iT ∗ i ( s − i , i ) + (cid:88) i : s i > (cid:0) v iT ∗ i ( s ) − v iT ∗ i ( s − i , i ) (cid:1) ≤ (cid:88) i v iT ∗ i ( s − i , i ) + (cid:88) i : s i > (cid:0) v iT ∗ i ( − i , s i ) − v iT ∗ i ( ) (cid:1) (6) ≤ (cid:88) i v iT ∗ i ( s − i , i ) (cid:124) (cid:123)(cid:122) (cid:125) OTHER + k − (cid:88) (cid:96) =1 (cid:88) i : s i = (cid:96) v iT ∗ i ( − i , s i ) (cid:124) (cid:123)(cid:122) (cid:125) SELF , (7)where Equation (6) follows from the definition of submodularity (and therefore, also follows thedefinition of strong-submodularity). The last inequality follows from the non-negativity of v iT ∗ i ( ).The first term in the decomposition represents the contribution of others’ signals to one’s valuefrom his allocated bundle, while the second term represents one’s contribution to his own value.Each of these terms will be targeted using a different mechanism. Whereas the OTHER term willbe targeted using the same mechanism in both the SOS and strong-SOS cases, the
SELF term willbe treated differently. ( k + 3) -approximation for SOS valuations Suppose s i ∈ { , , . . . , k − } for all i . The mechanism is as follows:Mechanism k signals High-Low ( k -HL): With probability p RT = k − k +3 , run Random Threshold ; otherwise, run
Random Sampling , as de-scribed below:Mechanism
Random Threshold • Choose a random threshold (cid:96) uniformly in { , . . . , k − } . • Let N ≥ (cid:96) = { i : s i ≥ (cid:96) } be the “high” agents; i.e., agents with signal at least (cid:96) , and let N <(cid:96) = [ n ] \ N ≥ (cid:96) be the “low” agents. • For every high agent i ∈ N ≥ (cid:96) and bundle T , let ¯ v iT := v iT ( s N <(cid:96) , (cid:96) N ≥ (cid:96) ) • For every low agent i ∈ N <(cid:96) and bundle T , let ¯ v iT := 0. • Let the allocation be ¯ T ∈ argmax S = { S i } i ∈ N ≥ (cid:96) (cid:88) i ∈ N ≥ (cid:96) ¯ v iS i . ( i.e. , the allocation that maximizes the “welfare” of high agents using values ¯ v iT .) • Agent i that receives bundle ¯ T i pays v i ¯ T i ( s − i , s i = (cid:96) − Random Sampling • Split the agents into sets A and B uniformly at random. • For each i ∈ B and bundle T , let ˜ v iT := v ij ( s A , B ). • For each i ∈ A and bundle T , let ˜ v iT := 0. • Let the allocation be ˜ T ∈ argmax S = { S i } i ∈ B (cid:88) i ∈ B ˜ v iS i . ( i.e. , the allocation that maximizes the “welfare” of agents in B using values ˜ v iT .) • Charge no payments.The k-HL mechanism is a random combination of two mechanisms:
Random Threshold approx-imates the welfare contribution of the bidders’ signals to their own value (the
SELF term);
RandomSampling approximates the welfare contributions of the bidders’ signals to other bidders’ values(the
OTHER term). We wish to prove the following theorem.
Theorem 6.1.
For every combinatorial auction setting with SOS valuations, single-dimensionalsignals, and signal space of size k , i.e. s i ∈ { , , . . . , k − } ∀ i , mechanism k -HL is an ex-post IC-IRmechanism that gives ( k + 3) -approximation to the optimal social welfare. We first argue that the mechanism is ex-post IC-IR.
Proof of ex-post IC-IR.
Random Sampling is ex-post IC-IR since the agents that might receiveitems (agents in B ) cannot change the allocation since their signals are ignored (and they paynothing).As for Random Threshold , consider a threshold (cid:96) chosen by the mechanism. If the agent’ssignal is below (cid:96) and the agent reports (cid:96) or above, then his payment, if allocated bundle T is v iT ( s − i , s i = (cid:96) − ≥ v iT ( s ); i.e., the agent’s utility is non-positive. Bidding a different value below (cid:96) will grant the agent no items. If his value is (cid:96) or above, then bidding a different signal above (cid:96) will result in the same outcome, since the sets N ≥ (cid:96) and N <(cid:96) remain the same. If he bids a signalbelow (cid:96) , then he won’t receive any item, and his utility will be 0, while bidding his true signal willresult in non-negative utility.In Lemma 6.3, we prove that Random Sampling covers the
OTHER component of the socialwelfare, and in Lemma 6.2, we show that
Random Threshold covers the
SELF component.
Lemma 6.2.
For SOS valuations, the
Random Threshold mechanism gives a ( k − -approximationto the SELF component of the optimal social welfare.Proof.
Consider a threshold (cid:96) ∈ { , . . . , k − } chosen in Random Threshold . Whenever (cid:96) is chosen,we have that (cid:88) i : s i = (cid:96) ¯ v iT ∗ i = (cid:88) i : s i = (cid:96) v iT ∗ i ( s N <(cid:96) , (cid:96) N ≥ (cid:96) ) ≥ (cid:88) i : s i = (cid:96) v iT ∗ i ( − i , s i ) . Since
Random Threshold chooses an allocation ¯ T = { ¯ T i } i ∈ N ≥ (cid:96) that maximizes the welfare under¯ v iT ’s, the value of the allocation is only larger than the left expression above. Because v i ¯ T i ( s ) ≥ ¯ v i ¯ T i ,20e get that if (cid:96) was chosen, which happens with probability k − , the welfare achieved is at least (cid:80) i : s i = (cid:96) v iT ∗ i ( − i , s i ) . Therefore, the welfare from running
Random Threshold is at least k − (cid:88) (cid:96) =1 k − (cid:88) i : s i = (cid:96) v iT ∗ i ( − i , s i ) , ≥ SELF k − . Lemma 6.3.
For SOS valuations, the
Random Sampling mechanism gives a -approximation tothe OTHER component of the optimal social welfare.Proof.
Consider a set T . Using an application of the Key Lemma 3.1 with respect to v iT ( s − i , i ),we see that E A,B [˜ v iT ] ≥ Pr[ i ∈ B ] · E A,B [˜ v iT | i ∈ B ] = 12 E A,B \ i [˜ v iT | i ∈ B ] ≥ v iT ( s − i , i ) . (8)Therefore, the expected weight of the allocation { T ∗ i } i ∈ [ n ] using weights ˜ v iT ’s is E A,B (cid:20) (cid:88) i ˜ v iT ∗ i (cid:21) = (cid:88) i E A,B (cid:20) ˜ v iT ∗ i (cid:21) ≥ (cid:88) i v iT ∗ i ( s − i , i ) = OTHER . Since the mechanism chooses the optimal allocation according to the ˜ v iT ’s, its weight can only belarger. Moreover, since ˜ v iT = v iT ( s − i , ≤ v iT ( s ), the welfare achieved by the mechanism is atleast OTHER , as desired.We conclude by proving the claimed approximation ratio. Proof of approximation.
According to Lemma 6.2,
Random Threshold approximates
SELF to afactor of k −
1. According to Lemma 6.3 that
Random Sampling approximates
OTHER to a factorof 4. Therefore, running
Random Threshold with probability p RT and Random Sampling withprobability 1 − p RT yields a welfare of p RT SELF k − − p RT ) OTHER k − k + 3 · SELF k − k + 3 · OTHER SELF + OTHER k + 3 ≥ W ∗ ( s ) k + 3 , where the inequality follows Equation (7). O (log k ) -Approximation with Strong-SOS Valuations Strong-SOS valuations means the effect on the valuation is concave in one’s own signal. This allowsus to use a bucketing technique in order to give an O (log k )-approximation to the SELF componentin the decomposition depicted by Equation (7). 21onsider the
SELF term in Equation (7). We can bound this term as follows:
SELF = k − (cid:88) (cid:96) =1 (cid:88) i : s i = (cid:96) v iT ∗ i ( − i , s i )= log k (cid:88) (cid:96) =1 (cid:88) i : 2 (cid:96) − ≤ s i < (cid:96) v iT ∗ i ( − i , s i ) ≤ log k (cid:88) (cid:96) =1 (cid:88) i : 2 (cid:96) − ≤ s i < (cid:96) v iT ∗ i ( − i , (cid:96) − i ) , (9)where the inequality follows the definition of strong-SOS valuations.We introduce mechanism Random Bucket to give an O (log k )-approximation to the upper boundin Equation (9).Mechanism Random Bucket : • choose (cid:96) uniformly in { , . . . , log k } . • Let N B (cid:96) = { i : such that s i ≥ (cid:96) − } be the agents with signal at least 2 (cid:96) − and N ¬ B (cid:96) =[ n ] \ N B (cid:96) . • For i ∈ N B (cid:96) and bundle T , let ¯ v iT := v iT ( s N ¬ B(cid:96) , (cid:96) − N B(cid:96) ) (and ¯ v iT := 0 for i ∈ N ¬ B (cid:96) ). • Let the allocation be ¯ T ∈ argmax S = { S i } i ∈ NB(cid:96) (cid:88) i ∈ N B(cid:96) ¯ v iS i . ( i.e. , the allocation that maximizes the “welfare” of high agents using values ¯ v iT .) • Agent i that receives bundle ¯ T i pays v i ¯ T i ( s − i , s i = 2 (cid:96) − − Random Bucket . Lemma 6.4.
For strong-SOS valuations, the
Random Bucket mechanism is ex-post IC-IR and givesa k approximation to the SELF component of the optimal social welfare.Proof.
The proof of ex-post IC-IR is identical to that of mechanism
Random Threshold , as bothare threshold-based mechanisms. The proof of the approximation guarantee is also very similar tothat of
Random Threshold .Consider a threshold 2 (cid:96) − for (cid:96) ∈ { , . . . , k − } chosen in Random Bucket . Whenever (cid:96) is chosen,we have that (cid:88) i : 2 (cid:96) − ≤ s i < (cid:96) ¯ v iT ∗ i = (cid:88) i : 2 (cid:96) − ≤ s i < (cid:96) v iT ∗ i ( s N ¬ B(cid:96) , (cid:96) − N B(cid:96) ) ≥ (cid:88) i : 2 (cid:96) − ≤ s i < (cid:96) v iT ∗ i ( − i , (cid:96) − i ) . Since
Random Bucket chooses an allocation that maximizes the ¯ v iT ’s, the value of the allocationis only larger. Because v i ¯ T i ( s ) ≥ ¯ v i ¯ T i , we get that if (cid:96) was chosen, which happens with probability22 log k , the welfare achieved is at least (cid:80) i : 2 (cid:96) − ≤ s i < (cid:96) v iT ∗ i ( − i , s i ) . Therefore, the welfare from running
Random Bucket is at least log k (cid:88) (cid:96) =1 k (cid:88) i : 2 (cid:96) − ≤ s i < (cid:96) v iT ∗ i ( − i , s i ) , ≥ SELF k . Mechanism k -signals Strong-SOS ( k -SS ) runs Random Bucket with probability p RB = log klog k +2 and mechanism Random Sampling with probability 1 − p RB . Theorem 6.5.
For every combinatorial auction with single-dimensional signals with strong-SOSvaluations and signal space of size k , i.e. s i ∈ { , , . . . , k − } ∀ i , mechanism k -SS is ex-post IC-IR,and gives (2 log k + 4) -approximation to the optimal social welfare.Proof. We already established that both
Random Bucket and
Random Sampling are ex-post IC-IR, hence k -SS is ex-post IC-IR as well. As for the approximation, according to Lemma 6.4,with probability p RB we get 2 log k -approximation to SELF , and according to Lemma 6.3, withprobability 1 − p RB we get a 4-approximation to OTHER . Overall, the expected welfare is at least p RB SELF k + (1 − p RB ) OTHER
SELF + OTHER k + 4 ≥ W ∗ k + 4 , as desired. Our analysis and results suggest many open problems: • For combinatorial auctions with multi-dimensional signals: is separability a necessary condi-tion for achieving constant approximation to welfare? This problem is open even for single-dimensional signals, and even for “simple” combinatorial valuations, such as unit-demand. • For single-parameter SOS valuations, downward closed feasibility, and single-dimensional sig-nals, closing the gap between 1 / / • The exact same gap applies for combinatorial, separable-SOS valuations with multi-dimensionalsignals. • How does the distinction between SOS and strong-SOS affect the problems above, if at all? • When considering the relaxation of SOS valuations to d -SOS valuations, there is a gap betweenthe positive and negative results with respect to the dependence on d .23ore generally, what other classes of valuations give rise to approximately efficient mechanisms insettings with interdependent valuations? Acknowledgements
We gratefully thank an anonymous referee who pointed out that many ofthe proofs in this paper, hold, with minor adjustments, for subadditive over signals valuations.Surprisingly, submodular over signals valuations are not a special case of subadditive over signalsvaluations. However, strong submodular over signals valuations are so. The actual situation israther subtle and we will address this issue in a subsequent version of this paper.
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Unit-Demand Valuations with Single-Crossing
Whereas single-crossing is a strong enough condition to implement the fully efficient mechanismin a variety of single-parameter environments, generalizations of this condition fail even in thesimplest multi-parameter environments. We consider the case where bidders are unit demand andeach bidder has a scalar as a signal. We define single-crossing for this setting as follows.
Definition A.1 (Single-crossing for unit-demand valuations) . A valuation profile v is said to besingle crossing if for every agent i , signals s − i , item j and agent (cid:96) , ∂∂s i v ij ( s − i , s i ) ≥ ∂∂s i v (cid:96)j ( s − i , s i ) . (10)In this section, we show that in the case two non-identical items are for sale, and the valuationsare unit demand and satisfy single-crossing as defined in Equation (10), any truthful mechanism isbounded away from achieving full efficiency.In order to give the lower bound, we first give a characterization of ex-post IC and IR mecha-nisms in multi-dimensional environments in interdependent values settings (Section A.1). We thenturn to prove the lower bound (Section A.2). A.1 Cycle Monotonicity
In the IPV model, Rochet [1987] introduced cycle monotonicity as a necessary and sufficient con-dition on the allocation to be implementable in dominant strategies (DSIC) for multidimensionalenvironments. It was noticed that a straightforward analogue holds for the IDV value model, forex-post implementability (EPIC) (in Vohra [2007], this fact is stated without a proof).Fix a feasible allocation rule x = { x i } i ∈ [ n ] , where x iT ( s ) is the probability agent i receives abundle T under bid profile s . For each agent i , consider the graph G x i where there is a vertex foreach signal profile s , and there is a directed edge from s to t if s − i = t − i . The weight of edge ( s , t )is w ( s , t ) = E T ∼ x i ( s ) [ v iT ( s )] − E T ∼ x i ( t ) [ v iT ( s )] = (cid:88) T ⊆ [ m ] x iT ( s ) v iT ( s ) − (cid:88) T ⊆ [ m ] x iT ( t ) v iT ( s ) . The following theorem states that a necessary and sufficient condition for ex-post implementabil-ity of x is that for every agent i , every directed cycle in G x i is non-negative. The proof is a straight-forward adjustment of the original proof in Rochet [1987], and is given below for completeness. Theorem A.1.
The allocation rule x is implementable by an ex-post IC mechanism if and only iffor every agent i , all directed cycles in G x i have non-negative weight.Proof. We first show that if the allocation rule is implementable, then there are no negative cycles.Fix some payment rule p = { p i } i ∈ [ n ] , where p i ( s ) is the payment of agent i under bid profile s . Let s − i be the real signals of all bidders except i , and consider a cycle s → s → . . . → s (cid:96) → s in G x i ,where s t = ( s − i , s i = ζ t ) for t ∈ [ (cid:96) ]. Since ( x , p ) is an ex-post IC mechanism, for every true signal s i = s , agent i is at least as well off bidding s than any other bid s (cid:48) . We get that E T ∼ x i ( s ) [ v iT ( s )] − p i ( s ) ≥ E T ∼ x i ( s ) [ v iT ( s )] − p i ( s )... E T ∼ x i ( s (cid:96) − ) [ v iT ( s (cid:96) − )] − p i ( s (cid:96) − ) ≥ E T ∼ x i ( s (cid:96) ) [ v iT ( s (cid:96) − )] − p i ( s (cid:96) ) E T ∼ x i ( s (cid:96) ) [ v iT ( s (cid:96) )] − p i ( s (cid:96) ) ≥ E T ∼ x i ( s ) [ v iT ( s (cid:96) )] − p i ( s )27umming over the above inequalities and using the convention that (cid:96) + 1 = 1, we get that (cid:96) (cid:88) j =1 E T ∼ x i ( s j ) [ v iT ( s j )] − (cid:96) (cid:88) j =1 p i ( s j ) ≥ (cid:96) (cid:88) j =1 E T ∼ x i ( s j +1 ) [ v iT ( s j )] − (cid:96) (cid:88) j =1 p i ( s j ) ⇐⇒ (cid:96) (cid:88) j =1 (cid:0) E T ∼ x i ( s j ) [ v iT ( s j )] − E T ∼ x i ( s j +1 ) [ v iT ( s j )] (cid:1) ≥ , where the LHS of the last inequality is exactly the weight of the cycle.We now show how to compute payments that implement a given allocation rule x that inducesno negative cycles for any i and G x i . Given G x i , one can compute payments as follows. • Add a dummy node d with edges of weight 0 to all nodes in G x i . • For every node s of G x i , let δ ( s ) be the distance of the shortest path from d to s . • Set p i ( s ) = − δ ( s ).Fix signals of the other players s − i . Let s be player i ’s true signal and s (cid:48) be some other possiblesignal for i . Denote s = ( s − i , s ) and s (cid:48) = ( s − i , s (cid:48) ). Consider the nodes s and s (cid:48) in G x i . Since δ ( s (cid:48) )is the length of the shortest path from d , it must be that δ ( s (cid:48) ) ≤ δ ( s ) + w ( s , s (cid:48) ) , where w ( s , s (cid:48) ) is the weight of the edge from s to s (cid:48) . Substituting w ( s , s (cid:48) ) = E T ∼ x i ( s ) [ v iT ( s )] − E T ∼ x i ( s (cid:48) ) [ v iT ( s )], p i ( s ) = − δ ( s ), and p i ( s (cid:48) ) = − δ ( s (cid:48) ), we get E T ∼ x i ( s ) [ v iT ( s )] − p i ( s ) ≥ E T ∼ x i ( s (cid:48) ) [ v iT ( s )] − p i ( s (cid:48) ) , as desired. A.2 Lower Bounds for Deterministic and Randomized Mechanisms
Lemma A.2.
There exists a setting with two items and two agents with unit-demand and singlecrossing valuations, such that no deterministic truthful mechanism achieves more than / of theoptimal welfare. Figure 1: An instance with unit-demand single-crossing valuations where no deterministic truthfulallocation achieves more than a half of the optimal welfare.28 roof.
Consider the setting depicted in Figure 1, with two agents, 1 and 2, and two items, a and b . s ∈ { , } and s is fixed. The values at s = 0 are v a (0) = 1 , v b (0) = 0 , v a (0) = 0 , v b (0) = 1 , and at s = 1 are v a (1) = 1 + H + (cid:15), v b (1) = H, v a (1) = H, v b (1) = 1 , for some arbitrarily large H and a sufficiently small (cid:15) . One can easily verify that the valuations sat-isfy Equation (10), and hence single crossing; indeed, when agent 1’s signal increases, the valuationof agent 1 for each one of the item increases by more than the change in agent 2’s valuation.We show that no deterministic truthful mechanism can get better than 2-approximation. Inorder to get better than 2-approximation, the mechanism must allocate item a to agent 1 and item b to bidder 2 at signal s = 0. At s = 1, allocating item b to agent 1 and item a to agent 2 obtainsa welfare of 2 H , while any other allocation obtains at most a welfare of H + 2 + (cid:15) . Since H can bearbitrarily large, one must allocate item b to agent 1 and item a to agent 2 at signal s = 1 in orderto get an approximation ratio better than 2. Consider such an allocation rule x , and the graph G x .This graph has one cycle, with one edge from s = 0 to s = 1 and one edge from s = 1 to s = 0.The weight of this cycle is( v a (0) − v b (0)) + ( v b (1) − v a (1)) = (1 −
0) + ( H − ( H + 1 + (cid:15) )) = − (cid:15) < . Based on Theorem A.1, this implies that this allocation rule is not implementableFigure 2: An instance with unit-demand single-crossing valuations where no randomized truthfulallocation achieves more than √ of the optimal welfare. Lemma A.3.
There exists a setting with two items and two agents with unit-demand and singlecrossing valuations, such that no randomized truthful mechanism achieves more than √ of theoptimal welfare. roof. Consider the setting depicted in Figure 2, with two agents, 1 and 2, and two items, a and b . s ∈ { , } and s ∈ { , } . The values are v a (0 ,
0) = 1 , v b (0 ,
0) = 0 , v a (0 ,
0) = 0 , v b (0 ,
0) = 1 ,v a (1 ,
0) = 1 + √ H, v b (1 ,
0) =
H, v a (1 ,
0) =
H, v b (1 ,
0) = 1 ,v a (0 ,
1) = 1 , v b (0 ,
1) =
H, v a (0 ,
1) =
H, v b (0 ,
1) = 1 + √ H,v a (1 ,
1) = 1 + √ H, v b (1 ,
1) =
H, v a (1 ,
1) =
H, v b (1 ,
1) = 1 + √ H, for an arbitrarily large H . One can easily verify that the valuations are single crossing. We claimthat the following equalities hold with respect to the allocation rule of the optimal randomizedmechanism:(a) For every s , s , x a ( s , s ) = x b ( s , s ) and x a ( s , s ) = x b ( s , s ).(b) For some q ∈ [0 , x a (0 ,
0) = x b (0 ,
0) = q and x ∅ (0 ,
0) = x ∅ (0 ,
0) = 1 − q .(c) For some p ∈ [0 , x a (0 ,
1) = p and x b (0 ,
1) = 1 − p .We next prove the above equalities.(a) Consider some implementable allocation rule ¯ x , and consider the allocation rule ˜ x where˜ x a ( s , s ) = ¯ x b ( s , s ) and ˜ x a ( s , s ) = ¯ x b ( s , s ) for every s , s . Note that the valuationsare symmetric; i.e., the role of item a (resp. b ) for agent 1 is the same as the role of items b (resp. a ) for agent 2. By symmetry, ¯ x is implementable if and only if ˜ x is implementable,and both allocation rules have the same approximation guarantee. Clearly, an allocationrule x that applies allocation rules ¯ x and ˜ x , with probability each, maintains the sameapproximation guarantee. Moreover, this allocation rule satisfies the desired property.(b) The optimal mechanism gains nothing from assigning any positive probability for allocatingitem b to agent 1 under signal profile (0 , b grants no value toagent 1, and in terms of incentives, it can only incentivize agent 1 to misreport his signalat signal profile (1 , a to agent 2 under signal profile (0 , x a (0 ,
0) = x b (0 ,
0) = q for some q ∈ [0 , b and agent 2 has some probability to get item a ).(c) Consider G x and the cycle C = (0 , → (1 , → (0 ,
0) in G x . This is the only cycle thatcontains the node (1 ,
0) in G x . Assume x ∅ (1 , >
0. Transferring z ∈ (0 ,
1] probability from x ∅ (1 ,
0) to x a (1 ,
0) decreases the weight of the edge (0 , → (1 ,
0) by z , and increases theweight of the edge (1 , → (0 ,
0) by z (1 + √ H ) > z . Therefore, its net effect on the weightof C is positive. Transferring z ∈ (0 ,
1] probability from x ∅ (1 ,
0) to x b (1 ,
0) does not affectthe weight of the edge (0 , → (1 , , → (0 ,
0) by zH . Therefore, its net effect on the weight of C is positive. Since transferring x ∅ (1 ,
0) to x a (1 ,
0) and x b (1 ,
0) increases welfare and does not violate cycle monotonicity, the optimalmechanism clearly assigns no probability to x ∅ (1 , x { a,b } (1 , >
0. By Moving this probability to x a (1 , , C does not change. Therefore, wemay also assume the mechanism does not assign positive utility to x { a,b } (1 , C must benon-negative . This translates to the following condition. (cid:0) E T ∼ x (0 , [ v T (0 , − E T ∼ x (1 , [ v T (0 , (cid:1) − (cid:0) E T ∼ x (1 , [ v T (1 , − E T ∼ x (0 , [ v T (1 , (cid:1) = ( q − p ) + (cid:16) p (1 + √ H ) + (1 − p ) H − q (1 + √ H ) (cid:17) ≥ ⇒ q ≤ p (cid:18) − √ (cid:19) + 1 √ . In the optimal mechanism, q will be as large as possible in order to maximize the expected welfareat signal profile (0 , q = p (cid:16) − √ (cid:17) + √ . Therefore, the approximationratio at profile (0 ,
0) is at most q = p (cid:16) − √ (cid:17) + √ . At profile (0 , a is allocated to agent 1(which happens with probability p ), the welfare of the mechanism is at most 2 + √ H , while thewelfare of the optimal allocation is 2 H . As H can be arbitrarily large, this approximation ratio tendsto √ . Therefore, the approximation ratio at profile (1 ,
0) is at most p √ + (1 − p ) = 1 − p (cid:16) − √ (cid:17) .The optimal mechanism would balance between the approximation ratio at (0 ,
0) and at (1 , p that solves p (cid:18) − √ (cid:19) + 1 √ − p (cid:18) − √ (cid:19) . Solving for p , we get p = . This leads to an approximation ratio of at most √ , as promised. B n − Lower Bound for Deterministic Mechanisms with Single-Crossing SOS Valuations.
We show that for downward-closed environments, even if valuations satisfy a single-crossing con-dition and are SOS, any deterministic mechanism cannot obtain a better approximation to theoptimal welfare than n − Theorem B.1.
There exists a downward-closed environment with valuations that satisfy single-crossing for which no deterministic mechanism more than a n − fraction of the optimal welfare.Proof. Consider a set of n bidders, where I = { } ∪ P ( { , . . . , n } ), where P ( { , . . . , n } ) is thepower set of the set { , . . . , n } . Only agent 1 has a signal s ∈ { , } , and other players do not havesignals. The valuations are: v (0) = 1 v (1) = 1 + Hv i (0) = 0 v i (1) = H ∀ i ∈ { , . . . , n } for an arbitrary large value H (cid:29)
1. Once can easily verify these valuations satisfy single-crossingand SOS.Any deterministic mechanism that wants to get any approximation to the social welfare mustallocate to agent 1 when s = 0. In addition, if a deterministic mechanism wants to get a betterapproximation than n − s = 1.31therwise, none of the bidders in { , . . . n } can get allocated because the only set in I that containsagent 1 is the singleton set. Therefore, if agent 1 is allocated at s = 1, the achieved welfare is1 + H , whereas the optimal welfare is ( n − · H (when serving all agents in { , . . . , n } ). For anarbitrary large H This ratio approaches n − s = 0 and not serving agent 1 at s = 1 is violatesmonotonicity. Remark B.2.
The n − factor is tight for single-crossing valuations. If [ n ] ∈ I , then the mechanismcan always allocate all agents. Otherwise, one can always allocate only to the highest valued agent,which is monotone because of single crossing. Since the largest feasible set is of size at most n − in this case, allocating to the highest valued agent yields an approximation ratio of n − . C Results for d -SOS We now extend the results in Section 6 to the case of combinatorial d -SOS and combinatorial d -strong-SOS valuations with single-dimensional signals. We first note that if we consider d -SOSvaluations, then Equation (6) in the decomposition becomes W ∗ ≤ (cid:88) i v iT ∗ i ( s − i , i ) + (cid:88) i : s i > d · (cid:0) v iT ∗ i ( − i , s i ) − v iT ∗ i ( ) (cid:1) ≤ (cid:88) i v iT ∗ i ( s − i , i ) (cid:124) (cid:123)(cid:122) (cid:125) OTHER + k − (cid:88) (cid:96) =1 (cid:88) i : s i = (cid:96) d · v iT ∗ i ( − i , s i ) (cid:124) (cid:123)(cid:122) (cid:125) SELF , (11)We now show the extension of Theorem 6.1 to d -SOS valuations. Theorem C.1.
For every combinatorial auction with d -SOS valuations over single-dimensionalsignals, and signal space of size k , i.e., s i ∈ { , , . . . , k − } ∀ i , there exists a truthful mechanismthat gives d ( k + 1) + 2 -approximation to the optimal social welfare.Proof. The mechanism is identical to k -HL , but runs (Random Threshold) with probability p RT = ( k − dd ( k +1)+2 and (Random Sampling) With probability 1 − p RT . The mechanism was already provedto be truthful in Section 6.1. Random Threshold now gives a d ( k − SELF term. The proof isthe same as of Lemma 6.2, but the extra factor of d comes from the fact the the new SELF term is d times larger. Random Sampling gives a 2( d + 1)-approximation to the OTHER term. While this term is thesame for d -SOS, the new factor is due to the fact that when applying Lemma 3.1 in the proof ofLemma 6.3, we get that E A,B [˜ v iT ] ≥ d +1) v iT ( s − i , i ) instead of the bound we get in Equation (8).The new approximation guarantee follows from the new decomposition, the new approxima-tion guarantees the various mechanisms get for the terms of the decomposition, and the updatedprobability p RT .We next extend Theorem 6.5. Theorem C.2.
For every combinatorial auction with d -strong-SOS valuations over single-dimensionalsignals, and signal space of size k , i.e., s i ∈ { , , . . . , k − } ∀ i , there exists a truthful mechanismthat gives ( d ( d + 1) log k + 2( d + 1)) -approximation to the optimal social welfare. roof. The mechanism is identical to mechanism k -SS from Section 6.2, but runs Random Bucket with probability p RB = d log kd log k +2 and (Random Sampling) With probability 1 − p RB .The SELF term from Equation (9) is now bounded via the following:
SELF = k − (cid:88) (cid:96) =1 (cid:88) i : s i = (cid:96) d · v iT ∗ i ( − i , s i )= log k (cid:88) (cid:96) =1 (cid:88) i : 2 (cid:96) − ≤ s i < (cid:96) d · v iT ∗ i ( − i , s i ) ≤ log k (cid:88) (cid:96) =1 (cid:88) i : 2 (cid:96) − ≤ s i < (cid:96) d ( d + 1) · v iT ∗ i ( − i , (cid:96) − i ) , (12)where the inequality follows the definition of d -strong-SOS valuations.The new bound changes the guarantee of Random Bucket to get a d ( d + 1) log k -approximationto the SELF term, where the proof is identical to that of Lemma 6.4.As stated in Theorem C.1,
Random Sampling approximates the
OTHER term to a factor 2( dd