1 Introduction
Maximizing social welfare with private valuations is a solved problem. The classical VickreyClarkeGrove (VCG) family of mechanisms [Vickrey, 1961; Clarke, 1971; Groves, 1973], of which the Vickrey secondprice auction is a special case, are dominant strategy incentivecompatible and guarantee optimal social welfare in general social choice settings.
In this paper, we consider combinatorial auctions, where each agent has a value for every subset of items, and the goal is to maximize the social welfare, namely the sum of agent valuations for their assigned bundles. As a special case of general social choice settings, the VCG mechanism solves 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 the item being sold has moneymaking potential or is likely to be resold, the values different agents have may be correlated, or perhaps even common. A classic example is an auction for the right to drill for oil in a certain location [Wilson, 1969]
. Importantly, in such settings, agents may have different information about what that value actually is. For example, the value of an oil lease depends on how much oil there actually is, and the different agents may have access to different assessments about this. Consequently, an agent might change her own estimate of the value of the oil lease given access to the information another agent has. Similarly, if an agent had access to the results of a house inspection performed by a different agent, that might change her own estimate of the value of a house that is for sale.
The following model due to Milgrom and Weber [1982], described here for singleitem auctions, has become standard for auction design in such settings. These are known as interdependent value settings (IDV) ^{1}^{1}1 See also [Krishna, 2009; Milgrom and Milgrom, 2004]. and are defined as follows:

Each agent has a realvalued, private signal . The set of signals may be 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 and privately 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 on geologic surveys, etc.

The value of the item to agent is a function of the signals (or information) of all agents.
In the economics literature, interdependent settings have been studied for about 50 years (with far too many papers to list; for an overview, see [Krishna, 2009]). Within the theoretical computer science community, interdependent (and correlated) settings have received less attention [Ronen, 2001; Ito and Parkes, 2006; Constantin et al., 2007; Constantin and Parkes, 2007; Klein et al., 2008; Papadimitriou and Pierrakos, 2011; Dobzinski et al., 2011; Babaioff et al., 2012; Abraham et al., 2011; Robu et al., 2013; Kempe et al., 2013; Che et al., 2015; Roughgarden and TalgamCohen, 2016; Li, 2016; Chawla et al., 2014; Eden et al., 2018].
1.1 Maximizing Social Welfare
Consider the goal of maximizing social welfare in interdependent settings. Here, a direct revelation mechanism consists of each agent reporting a bid for their private signal , and the auctioneer determining the allocation and payments. (It is assumed that the auctioneer knows the form of the valuation functions .)
In interdependent settings, it is not possible^{2}^{2}2 Except perhaps in degenerate situations. to design dominantstrategy incentivecompatible auctions, since an agent’s value depends on all of the signals, so if, say, agent misreports his signal, then agent might win at a price above her value if she reports truthfully. The next strongest equilibrium notion one could hope for is to maximize efficiency in expost equilibrium: bidding truthfully is an expost equilibrium if an agent does not regret having bid truthfully, given that other agents bid truthfully. ^{3}^{3}3 Note that, of course, every expost equilibrium is a BayesNash incentive compatible equilibrium, but not necessarily vice versa, and therefore such equilibria are much more robust: they do not depend on knowledge of the priors and bidders need not think about how other bidders might be bidding. This increases our confidence that an expost equilibrium is likely to be reached. In other words, bidding truthfully is a Nash equilibrium for every signal profile.
For singleitem auctions, a characterization of expost incentive compatibility in the IDV setting is known [Roughgarden and TalgamCohen, 2016], analogous to Myerson’s characterization for the independent private values model. The characterization says that there are payments that yield an expost incentivecompatible mechanism if and only if the corresponding allocation rule is monotone in each agent’s signal, when all other signals are held fixed. Maximizing efficiency in expost equilibrium is also provably impossible unless the valuation functions satisfy a technical condition known as the singlecrossing condition [Milgrom and Weber, 1982; d’Aspremont and GérardVaret, 1982; Maskin, 1992; Ausubel et al., 1999; Dasgupta and Maskin, 2000; Athey, 2001; Bergemann et al., 2009; Chawla et al., 2014; Che et al., 2015; Li, 2016; Roughgarden and TalgamCohen, 2016]. I.e., the influence of agent ’s signal on his own value is at least as high as its influence on other agents’ values, when all other signals are held fixed ^{4}^{4}4This implies that given signals , if agent has the highest value when , then agent continues to have the highest value for . This is precisely the monotonicity needed for expost incentive compatibility.. When the singlecrossing condition holds, there is a generalization of VCG that maximizes efficiency in expost equilibrium. (See [Crémer and McLean, 1985, 1988; Krishna, 2009].)
Unfortunately, the single crossing condition does not generally suffice to obtain optimal social welfare in settings beyond that of a single item auction. It is insufficient even in very simple settings, such as twoitem, twobidder auctions with unitdemand valuations (see Section A), or singleparameter settings with downwardclosed feasibility constraints (see Section B).
Moreover, there are many relevant singleitem settings where the singlecrossing condition does not 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 position to take advantage of that demand. A setting like this could yield valuations that do not satisfy the single crossing condition. For a concrete example, consider the following scenario given by Dasgupta and Maskin [Dasgupta and Maskin, 2000].
Example 1.1.
Suppose that oil can be sold in the market at a price of 4 dollars per unit and two firms are competing for the right to drill for oil. Firm 1 has a fixed cost of 1 to produce oil and a marginal cost of 2 for each additional unit produced, whereas firm 2 has a fixed cost of 2 and a marginal cost of 1 for each additional unit produced. In addition, suppose that firm 1 does a private test and discovers that the expected size of the oil reserve is units. Then , whereas . These valuations don’t satisfy the singlecrossing condition since firm 1 needs to win when is low and lose when is high.
1.2 Research Problems
This paper addresses the following two issues related to social welfare maximization in the interdependent values model:

To what extent can the optimal social welfare be approximated in interdependent settings that do not satisfy the singlecrossing condition?

How far beyond the single item setting can we go? Is it possible to approximately maximize social welfare in combinatorial auctions with interdependent values?
The first question was recently considered by Eden et al. [2018] who gave two 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 for which no deterministic auction can achieve any bounded approximation ratio to optimal social welfare.
Example 1.2 (No bound for deterministic auctions Eden et al. [2018]).
A single item is for sale. There are two players, and , only has a signal . The valuations are
where is an arbitrary large number. If doesn’t win when , then the approximation ratio is infinite. On the other hand, if does win when , then by monotonicity, must also win at , yielding a fraction of the optimal social welfare.
The next example can be used to show that there are valuation functions for which no randomized auction performs better (in the worst case) than allocating to a random bidder (i.e., a factor approximation to social welfare), even if a prior over the signals is known.
Example 1.3 ( lower bound for randomized auctions Eden et al. [2018]).
There are bidders that compete over a single item. For every agent , , and
that is, agent ’s value is high if and only if all other agents’ signals are high simultaneously. When all signals are 1, then in any feasible allocation, there must be an agent
which is allocated with probability of at most
. By monotonicity, this means that the probability this agent is allocated when the signal profile is is at most as well. Therefore, the achieved welfare at signal profile is at most , while the optimal welfare is , giving a factor gap ^{5}^{5}5 Eden et al. [2018] show that there exists a prior for which the gap still holds, even if the mechanism knows the prior..Therefore, 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 singlecrossing that they called single crossing and then provide mechanisms that approximately maximize social welfare, where the approximation ratio depends on and , 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 highlycomplementary–one has a value bounded away from zero only if all other agent’s signals are high simultaneously. This suggests that the case where the valuations treat the signals more like “substitutes” might be easier to handle.
We capture this by focusing on submodular over signals (SOS) valuations. This means that for every and , when signals are lower, the sensitivity of the valuation to changes in is higher. Formally, we assume that for all , for any , , and for any and such that componentwise , it holds that
Many valuations considered in the literature on interdependent valuations are SOS (though this term is not used) Milgrom and Weber [1982]; Dasgupta and Maskin [2000]; Klemperer [1998]. The simplest (yet still rich) class of SOS valuations are fully separable valuation functions ^{6}^{6}6 This type of valuation function is ubiquitous in the economics literature on inderdependent settings; often with the function simply assumed to be a linear function of the signals (see, e.g., Jehiel and Moldovanu [2001b]; Klemperer [1998])., where there are arbitrary (weakly increasing) functions for each pair of bidders and such that
A more general class of SOS valuation functions are functions of the form , where is a weakly increasing concave function.
We can now state the main question we study in this paper: to what extent can social welfare be approximated in interdependent settings with SOS valuations? Unfortunately, Example 1.2 itself describes 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.
1.3 Our Results and Techniques
All of our positive results concern the design of randomized, priorfree, universally expost incentivecompatible (IC), individually rational (IR) mechanisms. Priorfree means that the rules of the mechanism makes no use of the prior distribution over the signals, thus need not have any knowledge of the prior.
Our first result provides approximation guarantees for singleparameter downwardclosed settings. An important special case of this result is singleitem auctions, which was the focus of Eden et al. [2018].
Theorem 4.1 (See Section 4): For every singleparameter downwardclosed setting, if the valuation functions are SOS, then the Random Sampling Vickrey auction is a universally expost ICIR mechanism that gives a 4approximation to the optimal social welfare.
Interestingly, no deterministic mechanism can give better than an approximation for arbitrary downwardclosed settings, even if the valuations are single crossing, and this is tight. Recall that for a single item auction, or even multiple identical items, with single crossing valuations, the deterministic generalized Vickrey auction obtains the optimal welfare Maskin [1992]; Ausubel et al. [1999].
We then turn to multidimensional settings. In the most general combinatorial auction model that we consider, each agent has a signal for each subset of items, and a valuation function . For this setting, it is not at all clear under what conditions it might be possible to maximize social welfare in expost equilibrium.^{7}^{7}7 See the related work and also Lemmas A.2 and A.3, which show that under one natural generalization of singlecrossing to the setting of two items and two agents that are unit demand, single crossing is not sufficient for full efficiency.
However, rather surprisingly (see the related work section below), for the case of separable SOS valuations^{8}^{8}8 A valuation is separableSOS if the valuation for an agent can be split into two parts, an SOS function of all other signals and an arbitrary function of the agents’ own signal. Such valuations generalize the fully separable case discussed above. See definition 2.16, we are able to extend the 4approximation guarantee to combinatorial auctions.
Theorem 5.1 (See Section 5): For every combinatorial auction, if the valuation functions are separableSOS, then the Random Sampling VCG auction is a universally expost ICIR mechanism that gives a 4approximation to the optimal social welfare.
Finally, we consider combinatorial auctions where each agent has a singledimensional signal , but where the valuation function for each subset of items is an arbitrary SOS valuation function . For this case, we show the following:
Theorems 6.1 and 6.5 (See Sections 6.1 and 6.2): Consider combinatorial auctions with singledimensional signals, where each signal takes one of possible values. If the valuation functions are SOS, then there exists a universally expost ICIR mechanism that gives a approximation to the optimal social welfare. If the valuations are strongSOS ^{9}^{9}9See definition 2.14., the approximation ratio improves to .
All of the above results, as well as our lower bounds, are summarized in Table 1. In addition, all of the results in this paper generalize easily, with a corresponding degradation in the approximation ratio, to the weaker requirement of SOS valuations ^{10}^{10}10 A valuation function is SOS if for all , for all , and for any and such that componentwise , it holds that .
1.3.1 Intuition for results
The fundamental tension in settings with interdependent valuations that is not present in the private values setting is the following. Consider, for example, a single item auction setting where agent 1’s truthful report of her signal increases agent 2’s value. Since, this increases the chance that agent 2 wins and may decrease agent 1’s chance of winning, it might motivate agent 1 to strategize and misreport.
Our approach is to simply prevent this interaction. Without looking at the signals, our mechanism randomly divides the agents into two sets^{11}^{11}11 as in [Goldberg et al., 2001].: potential winners and certain losers. Losers never receive any allocation. When estimating the value of a potentially winning agent , we use only the signals of losers and ’s own signal(s). Thus, potential winners can not impact the estimated values and hence allocations of other potential winners. This resolves the truthfulness issue. The remaining question is: can we get sufficiently accurate estimates of the agents’ values when we ignore 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 , if all agents other than are split into two random sets (losers) and (potential winners), and the signals of agents in the random subset are “zeroed out”, then the expected value agent has for the item is at least half of her true valuation. That is,
Dealing with combinatorial settings is more involved as the truthfullness characterization is less obvious, but the key ideas of random partitioning and using the signals of certain losers remain at the core of our results.
1.3.2 Additional remarks
While this paper deals entirely with welfare maximization, our results have significance for the objective of maximizing the seller’s revenue. Eden et al. [2018] give a reduction from revenue maximization to welfare maximization in singleitem auctions with SOS valuations. Thus, the constant factor approximation mechanism presented in this paper implies a constant factor approximation to the optimal revenue in singleitem auctions with SOS valuations. We note that this is the first revenue approximation result that does not assume any singlecrossing type assumption ([Chawla et al., 2014; Eden et al., 2018; Roughgarden and TalgamCohen, 2016; Li, 2016] require single crossing or approximate single crossing).
Finally, one can easily verify that, based on Yao’s minmax theorem, the existence of a randomized priorfree mechanism that gives some approximation guarantee (in expectation over the coin flips of the mechanism) implies the existence of a deterministic priordependent mechanisms that gives the same approximation guarantee (in expectation over the signal profiles).
Setting  Approximation Guarantees  












1.4 Additional Related Work
As discussed above, in singleparameter settings, there is an extensive literature on mechanism design with interdependent valuations that considers social welfare maximization, revenue maximization and other objectives. However, the vast majority of this literature assumes some kind of singlecrossing 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 social welfare can be approximated in expost equilibrium without singlecrossing. The first is the aforementioned paper [Eden et al., 2018] on single item auctions with interdependent valuations. They defined a parameterized version of singlecrossing, termed single crossing, where is a parameter that indicates how close is the valuation profile to satisfy singlecrossing. For single crossing valuations, they provide a number of results including a lower bound of on the approximation ratio achievable by any mechanism, a matching upper bound for binary signal spaces, and mechanisms that achieve approximation ratios of and (the first is deterministic and the second is randomized).
Ito and Parkes [2006] also consider approximating social welfare in the interdependent setting. Specifically, they propose a greedy contingentbid auction (a la [Dasgupta and Maskin, 2000]) and show that it achieves a approximation to the optimal social welfare for goods, in the special case of combinatorial auctions with singleminded bidders.
Beyond that, for multidimensional signals and settings, the landscape is sparser (and bleaker) and, to our knowledge, focuses on exact social welfare maximization. Maskin [1992] has observed that, in general, no efficient incentivecompatible single item auction exists if a buyer’s valuation depends on a multidimensional signal. Jehiel and Moldovanu [2001a], building on earlier work of Dasgupta and Maskin [2000], consider a very general model in which there is a set of possible alternatives, and a multidimensional signal space, where each agent has a signal for each outcome and other agent . In their model the valuation function of an agent for outcome is linear in the signals, that is, . Thus, their valuation functions are, in one sense, a special case of our separable valuation functions. On the other hand, they are more general in that all quantities depend on the outcome . Thus, there are allocation externalities. Their main result is that, generically, there is no BayesNash incentive compatible mechanism that maximizes social welfare in this setting. However, they do give an expost IC mechanism that maximizes social welfare with both information and allocation externalities if the signals are onedimensional, the valuation functions are linear in the signals, and a singlecrossing type condition holds.
Jehiel et al. [2006] go on to show that the only deterministic social choice functions that are expost implementable in generic mechanism design frameworks with multidimensional signals, interdependent valuations and transferable utilities, are constant functions.
Finally, Bikhchandani [2006] considers a single item setting with multidimensional signals but no allocation externalities and shows that there is a generalization of singlecrossing that allows some social choice rules to be implemented expost.
2 Model and Definitions
2.1 Single Parameter Settings
In Section 4, we will consider singleparameter settings with interdependent valuations and downwardclosed feasibility constraints. In these settings, a mechanism decides which subset of agents are to receive “service” (e.g., an item). The feasibility constraint is defined by a collection of subsets of agents that may feasibly be served simultaneously. We restrict attention to downwardclosed settings, which means that any subset of a feasible set is also feasible. A simple example is a item auction, where is the collection of all subsets of agents of size at most .
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 has a private signal . The value agent gives to “receiving service” , is a function of all agents’ signals . The function is assumed to be weakly increasing in each coordinate and strictly increasing in .
2.1.1 Deterministic Mechanisms
Definition 2.2 (Deterministic Single Parameter Mechanisms).
A deterministic mechanism in the downward closed setting is a mapping from reported signals to allocations and payments , where indicates whether or not agent receives service and is the payment of agent . It is required that the set of agents that receive service is feasible, i.e., . (The mechanism designer knows the form of the valuation functions but learns the private signals only when they are reported.)
Definition 2.3 (Agent utility).
Given a deteministic mechanism , the utility of agent when her true signal is , she reports and the other agents report is
Agent will report so as to maximize . We use to denote the utility when she reports truthfully, i.e., .
Definition 2.4 (Deterministic expost incentive compatibility (IC)).
A deterministic mechanism in the interdependent setting is expost incentive compatible (IC) if, irrespective of the true signals, and given that all other agents report their true signals, there is no advantage to an agent to report any signal other that her true signal. In other words, assuming that are the true signals of other bidders. is maximized by reporting truthfully.
Definition 2.5 (Deterministic expost individual rationality (IR)).
A deterministic mechanism in the interdependent setting is expost 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 participating in the mechanism.
If a deterministic mechanism is both expost IR and expost IR we say that it is expost ICIR.
Definition 2.6.
A deterministic allocation rule is monotone if for every agent , every signal profile of all other agents , and every , it holds that .
Proposition 2.1.
[Roughgarden and TalgamCohen, 2016] For every deterministic allocation rule for single parameter valuations, there exist payments such that the mechanism is expost ICIR if and only if is monotone for every agent .
2.1.2 Randomized Mechanisms
Definition 2.7.
A randomized mechanism is a probability distribution over deterministic mechanisms.
Definition 2.8 (Universal expost ICIR).
A randomized mechanism is said to be universally expost ICIR if all deterministic mechanisms in the support are expost ICIR.
2.2 Combinatorial Valuations with Interdependent Signals
Sections 5 and 6 focus on combinatorial auctions, where there are agents and items. In these settings, a mechanism is used to decide how the items are partitioned among the agents. We consider two models for the interdependent valuations: ^{12}^{12}12 For other types of signals and interdependent valuation models, see, e.g., Jehiel and Moldovanu [2001a].
Definition 2.9 (Single Dimensional Signals, Combinatorial Valuations).
Each agent has a signal . The value agent gives to subset of items , which we denote by , is a function of .
Definition 2.10 (Multidimensional Combinatorial Signals, Combinatorial Valuations).
Here, each agent has a signal for each subset of items; for any agent , we use to denote agent ’s signal for subset of items . The value agent gives to set is denoted by where . We use to denote the set of all signals .
In both cases, each is assumed to be a weakly increasing function of each signal and strictly increasing in (or respectively), and known to the mechanism designer.
We give subsequent definitions only for multidimensional combinatorial signals, as single dimensional signals can be viewed as a special case of multidimensional signals where for all .
2.2.1 Deterministic Mechanisms
Definition 2.11 (Deterministic mechanisms for combinatorial settings).
A deteministic mechanism is a mapping from reported signals to allocations (where each ) and payments for all and such that:

Agent is allocated the set iff ;

For each agent , there is at most one for which ;

The sets allocated to different agents do not intersect.

The payment for agent when her allocation is set is .
Definition 2.12 (Agent Utility).
The utility of agent when her signals are , she reports and the other agents report is
Given a mechanism , agent will report so as to maximize . We use to denote the utility when she reports truthfully, i.e., .
2.2.2 Randomized Mechanisms
As with single parameter mechanisms, a randomized mechanism for a combinatorial setting is a probability distribution over deterministic mechanisms for the combinatorial setting, and a randomized mechanism is said to be universally ex post ICIR if all deterministic mechanisms in the support are themselves expost ICIR.
2.3 Submodularity over signals (SOS)
As discussed in the introduction, our results will rely on an assumption about the valuation functions that we call submodularity over signals or SOS. The SOS (resp. strongSOS) notion we use is the same as the weak diminishing returns (resp. strong diminishing returns) submodularity notion in [Bian et al., 2017; Niazadeh et al., 2018]^{13}^{13}13 Weak diminishing returns submodularity was introduced in [Soma and Yoshida, 2015], where it’s termed “diminishing returns submodularity”. . SOS was also used in [Eden et al., 2018], generalizing a similar notion in [Chawla et al., 2014].
Definition 2.13 (approximate submodularoversignals valuations (SOS valuations)).
A valuation function is a SOS valuation if for all , , ,
such that is smaller than or equal to coordinatewise, it holds that
(1) 
If satisfies this condition with , we say that is an SOS valuation function.
Definition 2.14 (approximate strong submodularoversignals valuations (strongSOS valuations)).
The valuation function is a strongSOS valuation if for any , ,
such that is smaller than or equal to coordinatewise, it holds that
(2) 
If satisfies this condition with , we say that ’s valuation functions are “strongSOS”.
Definition 2.15 (SOSvaluations settings).
We say that a mechanism design setting with interdependent valuations is an SOSvaluations setting or, equivalently, that the agents have SOSvaluations, in each of the following cases:

Single parameter valuations (as in definition 2.1): for every , the valuation function is SOS.

Combinatorial valuations with singleparameter signals (as in definition 2.9): for every and , the valuation function is SOS;

Combinatorial valuations with multiparameter signals (as in definition 2.10): for every and , is SOS, where .
Similar definitions can be given for SOS valuation settings and strongSOS 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 2.10 are separable SOS valuations if for every agent and subset of items, can be written as
where and are both weakly increasing and is itself an SOS valuation function.
Observation 2.2.
A separable SOS valuation function is itself an SOS valuation function.
We can similarly define separable SOS valuations.
2.4 A useful fact about SOS valuations
Lemma 2.3.
Let be a SOS function. Let and . For any , and such that is smaller than coordinate wise,
3 The Key Lemma
The following is a key lemma which is used for both single parameter and combinatorial settings.
Lemma 3.1.
Let be a SOS function. Let be a uniformly random subset of , and let . It now holds that
where the expectation is over the random choice of .
Proof.
We consider two equiprobable events,

is chosen as the random subset.

is chosen as the random subset.
Normalize the valuations so that and define such that
It follows that
where the first inequality follows from nonnegativity of , and the second inequality follows from being SOS and Lemma 2.3.
Similarly, we have that
It follows that
Solving for equality of the two terms, we get that which implies that
Partition the event space into pairs that partition . For every such pair, it follows that .
We conclude with the following, where the third line follows from the fact that there are such pairs that partition :
as desired. ∎
4 SingleParameter Valuations
In this section we describe the Random Sampling Vickrey (RSV) mechanism that achieves a 4approximation for singleparameter downwardclosed environments with SOS valuations and a approximation for SOS valuations. We then give a lower bound of 2 and for SOS and SOS valuations respectively, even in the case of selling a single item.
Let be a downwardclosed set system. We present a mechanism that serves only sets in and gets a approximation to the optimal welfare.
Random Sampling Vickrey (RSV):

Elicit bids from the agents.

Partition the agents into two sets, and , uniformly at random.

For , let .

Allocate to a set of bidders in
Theorem 4.1.
For agents with SOS valuations, and for every downwardclosed feasibility constraint , RSV is an expost ICIR mechanism that gives approximation to the optimal welfare. For SOS valuations, the mechanism gives a approximation to the optimal welfare.
Proof.
We first show the allocation is monotone in one’s signal, and hence, by Proposition 2.1, the mechanism is expost ICIR. Fix a random partition .

Agents in are never allocated anything and thus their allocation is weakly monotone in their signal.

For an agent , increasing can only increase , whereas it leaves unchanged for all . Thus, this only increases the weight of feasible sets (subsets of in ) that belongs to. Therefore, increasing can only cause to go from being unallocated to being allocated.
For approximation, consider a set that maximizes social welfare. For every , from the Key Lemma 3.1, we have that
(4) 
For every set , the fact that is downwardclosed implies that . Therefore, is eligible to be selected by RSV as the allocated set of bidders. We have that the values of the bidders we allocate to are at least
as desired. Since the allocated bidders’ true values at are only higher than the proxy values , this continues to hold.
∎
We note that for the case of downwardclosed feasibility constraints, even if the valuations satisfy singlecrossing, there can be an gap between the optimal welfare and the welfare that the best deterministic mechanism can get. This is stated in Theorem B.1 in Section B.
The following lower bounds, Theorem 4.2 show that even for a single item setting, one cannot hope to get a better approximation than and for SOS and SOS valuations respectively.The lower bounds apply to arbitrary randomized mechanisms^{14}^{14}14 A randomized mechanism takes as input the set of signals and produces as output and for each agent , where is the probability that agent wins and is agent ’s expected payment. Such a mechanism is expost IC (but not necessarily universally so) if and only if is monotonically increasing in ..
Theorem 4.2.
No expost ICIR mechanism (not necessarily universal) for selling a single item can get a better approximation than

a factor of 2 for SOS valuations.

a factor of for SOS valuations.
Proof.
Let be the probability agent is allocated at signal profile . Notice that for every , , otherwise the allocation rule is not feasible.

Consider the case where there are two agents, 1 and 2, and agent 2 has no signal. The valuations are , , and for . It is easy to see the valuations are SOS.
In order to get better than a 2approximation at , we must have . By monotonicity, this forces as well, and hence by feasibility. This implies that the expected welfare when is , while the optimal welfare when is . For a large , this approaches a 2approximation. Note that this lower bound applies even given a known prior distribution on the signals in the event that we have a prior on the signals that satisfies: .

Consider the case where there are agents and for every agent . The valuation of agent is
where .
To see that the valuations are SOS, notice that whenever a signal changes from 0 to 1, the valuation of agent increases by 1 unless all other signals beside ’s are already set to , in which case the valuation increases by . Consider valuation profiles . Note that by monotonicity, for every truthful mechanism, it must be the case that . Since any feasible allocation rule must satisfy , then it must be the case there exists some agent such that , which by monotonicity implies that . However, at profile , while for all , so we get that the expected welfare of the mechanism at is at most while the optimal welfare is . Again, the lower bound also applies to the setting with known priors on the signals using a prior that satisfies: for all and .
∎
5 Combinatorial Auctions with Separable Valuations
In this section we present an expost ICIR mechanism that gives of the optimal social welfare in any combinatorial auction setting with separable SOS valuations (as in Definition 2.16). The mechanism, that we call the Randomsampling VCG auction is a natural extension of the RandomSampling Vickrey (RSV) auction presented in Section 4. Note that unlike RSV, here we need to explicitly define payments so that the obtained mechanism is expost ICIR. We derive VCGinspired 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’s report (while with separability, only the allocation is affected by it).
RandomSampling VCG (RSVCG):

Agents report their signals .

Partition the agents into two sets and uniformly at random.

For each agent and bundle , let

Let the allocation be
i.e., is the allocation that maximizes the “welfare” using ’s.

Set the payment for a winning agent receiving set of goods to be:
where
that is, is the weight of the best allocation without agent .
Since the ’s do not depend on agent ’s report (since is in ), doesn’t depend on agent ’s report. Therefore, we can (and will) ignore this term when considering incentive compatibility below.
Note also that since the maximal partition guarantees that , and monotonicity of valuations in signals guarantees that . Therefore, the payments are always nonnegative.
Theorem 5.1.
RandomSampling VCG is an expost ICIR mechanism that gives a 4approximation to 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 4approximation to social welfare. For every agent and bundle ,
(5) 
where the inequality follows by applying Lemma 3.1 with .
Let be the true welfare maximizing allocation. Then,
where the last inequality follows by substituting in in Equation (5) for every . Since is always at least , this proves the approximation ratio.
Next, we show that RSVCG is universally expost IC. Fix a random partition . Suppose that when all agents bid truthfully
Suppose that all agents but bid truthfully and bids instead of his true signal vector . Let be the resulting allocation. Therefore, agent ’s utility when reporting (after disregarding the term as mentioned above) is: