Combinatorial Auctions with Interdependent Valuations: SOS to the Rescue

03/20/2019 ∙ by Alon Eden, et al. ∙ Tel Aviv University University of Washington 0

We study combinatorial auctions with interdependent valuations. In such settings, each agent i has a private signal s_i that captures her private information, and the valuation function of every agent depends on the entire signal profile, =(s_1,...,s_n). Previous results in economics concentrated on identifying (often stringent) assumptions under which optimal solutions can be attained. The computer science literature provided approximation results for simple single-parameter settings (mostly single item auctions, or matroid feasibility constraints). Both bodies of literature considered only valuations satisfying a technical condition termed single crossing (or variants thereof). Indeed, without imposing assumptions on the valuations, strong impossibility results exist. We consider the class of submodular over signals (SOS) valuations (without imposing any single-crossing type assumption), and provide the first welfare approximation guarantees for multi-dimensional combinatorial auctions, achieved by universally ex-post IC-IR mechanisms. Our main results are: (i) 4-approximation for any single-parameter downward-closed setting with single-dimensional signals and SOS valuations; (ii) 4-approximation for any combinatorial auction with multi-dimensional signals and separable-SOS valuations; and (iii) (k+3)- and (2(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 extend to a parameterized version of SOS, d-SOS, while losing a factor that depends on d.

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1 Introduction

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 which the Vickrey second-price auction is a special case, are dominant strategy incentive-compatible 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 money-making 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 single-item auctions, has become standard for auction design in such settings. These are known as interdependent value settings (IDV) 111 See also  [Krishna, 2009; Milgrom and Milgrom, 2004]. and are defined as follows:

  • Each agent has a real-valued, 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.

    A typical example is when , for some . This type of valuation function captures settings where an agent’s value depends both on how much he likes the item () and on the resale value which is naturally estimated in terms of how much other agents like the item ()  [Myerson, 1981; Klemperer, 1998].

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 Talgam-Cohen, 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 possible222 Except perhaps in degenerate situations. to design dominant-strategy incentive-compatible 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 ex-post equilibrium: bidding truthfully is an ex-post equilibrium if an agent does not regret having bid truthfully, given that other agents bid truthfully. 333 Note that, of course, every ex-post equilibrium is a Bayes-Nash 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 ex-post equilibrium is likely to be reached. In other words, bidding truthfully is a Nash equilibrium for every signal profile.

For single-item auctions, a characterization of ex-post incentive compatibility in the IDV setting is known  [Roughgarden and Talgam-Cohen, 2016], analogous to Myerson’s characterization for the independent private values model. The characterization says that there are payments that yield an ex-post incentive-compatible 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 ex-post equilibrium is also provably impossible unless the valuation functions satisfy a technical condition known as the single-crossing condition  [Milgrom and Weber, 1982; d’Aspremont and Gérard-Varet, 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 Talgam-Cohen, 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 444This 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 ex-post incentive compatibility.. When the single-crossing condition holds, there is a generalization of VCG that maximizes efficiency in ex-post 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 two-item, two-bidder auctions with unit-demand valuations (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 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 single-crossing 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:

  1. To what extent can the optimal social welfare be approximated in interdependent settings that do not satisfy the single-crossing condition?

  2. 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 555 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 single-crossing 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 highly-complementary–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 component-wise , 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 666 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, prior-free, universally ex-post incentive-compatible (IC), individually rational (IR) mechanisms. Prior-free 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 single-parameter downward-closed settings. An important special case of this result is single-item auctions, which was the focus of Eden et al. [2018].

Theorem 4.1 (See Section 4): For every single-parameter downward-closed setting, if the valuation functions are SOS, then the Random Sampling Vickrey auction is a universally ex-post IC-IR mechanism that gives a 4-approximation to the optimal social welfare.

Interestingly, no deterministic mechanism can give better than an -approximation for arbitrary downward-closed 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 multi-dimensional 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 ex-post equilibrium.777 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 full efficiency.

However, rather surprisingly (see the related work section below), for the case of separable SOS valuations888 A valuation is separable-SOS 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 4-approximation guarantee to combinatorial auctions.

Theorem 5.1 (See Section 5): For every combinatorial auction, if the valuation functions are separable-SOS, then the Random Sampling VCG auction is a universally ex-post IC-IR mechanism that gives a 4-approximation to the optimal social welfare.

Finally, we consider combinatorial auctions where each agent has a single-dimensional 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 single-dimensional signals, where each signal takes one of possible values. If the valuation functions are SOS, then there exists a universally ex-post IC-IR mechanism that gives a -approximation to the optimal social welfare. If the valuations are strong-SOS 999See 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 101010 A valuation function is -SOS if for all , for all , and for any and such that component-wise , 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 sets111111 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 single-item auctions with SOS valuations. Thus, the constant factor approximation mechanism presented in this paper implies a constant factor approximation to the optimal revenue in single-item auctions with SOS valuations. We note that this is the first revenue approximation result that does not assume any single-crossing type assumption ([Chawla et al., 2014; Eden et al., 2018; Roughgarden and Talgam-Cohen, 2016; Li, 2016] require single crossing or approximate single crossing).

Finally, one can easily verify that, based on Yao’s min-max theorem, the existence of a randomized prior-free mechanism that gives some approximation guarantee (in expectation over the coin flips of the mechanism) implies the existence of a deterministic prior-dependent mechanisms that gives the same approximation guarantee (in expectation over the signal profiles).

Setting Approximation Guarantees
Single Parameter SOS valuations
Downward Closed Feasibility
Single-Dimensional Signals
    (Section 4)
Arbitrary Combinatorial SOS valuations
Single-Dimensional Signals, -sized Signal Space
    (Section 6.1)
Arbitrary Combinatorial, Strong-SOS Valuations
Single-Dimensional Signals, -sized Signal Space
    (Section 6.2)
Combinatorial, Separable-SOS Valuations
Multi-Dimensional Signals
    (Section 5)
Table 1: The table shows the approximation factors achievable for social welfare maximization with SOS and strong-SOS valuations. Similar results hold for -approximate SOS/Strong-SOS valuations, while losing a factor that depends on . All positive results are obtained with universally ex-post IC-IR randomized mechanisms.

1.4 Additional Related Work

As discussed above, in single-parameter 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 single-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 social welfare can be approximated in ex-post equilibrium without single-crossing. The first is the aforementioned paper [Eden et al., 2018] on single item auctions with interdependent valuations. They defined a parameterized version of single-crossing, termed -single crossing, where is a parameter that indicates how close is the valuation profile to satisfy single-crossing. 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 contingent-bid 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 single-minded 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 incentive-compatible single item auction exists if a buyer’s valuation depends on a multi-dimensional 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 Bayes-Nash incentive compatible mechanism that maximizes social welfare in this setting. However, they do give an ex-post IC mechanism that maximizes social welfare with both information and allocation externalities if the signals are one-dimensional, the valuation functions are linear in the signals, and a single-crossing type condition holds.

Jehiel et al. [2006] go on to show that the only deterministic social choice functions that are ex-post 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 single-crossing that allows some 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].

2 Model and Definitions

2.1 Single Parameter Settings

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 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 downward-closed 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 ex-post incentive compatibility (IC)).

A deterministic mechanism in the interdependent setting is ex-post 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 ex-post individual rationality (IR)).

A deterministic mechanism in the 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 participating in 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 is monotone if for every agent , every signal profile of all other agents , and every , it holds that .

Proposition 2.1.

[Roughgarden and Talgam-Cohen, 2016] For every deterministic allocation rule for single parameter valuations, there exist payments such that the mechanism is ex-post IC-IR 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 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.

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: 121212 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 multi-dimensional 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., .

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 definitions for single parameter mechanisms (Definitions 2.4 and  2.5 with the obvious modifications).

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 IC-IR if all deterministic mechanisms in the support are themselves ex-post IC-IR.

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. strong-SOS) 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]131313 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 submodular-over-signals valuations (-SOS valuations)).

A valuation function is a -SOS valuation if for all , , ,

such that is smaller than or equal to coordinate-wise, it holds that

(1)

If satisfies this condition with , we say that is an SOS valuation function.

Definition 2.14 (-approximate strong submodular-over-signals valuations (-strong-SOS valuations)).

The valuation function is a strong-SOS valuation if for any , ,

such that is smaller than or equal to coordinate-wise, it holds that

(2)

If satisfies this condition with , we say that ’s valuation functions are “strong-SOS”.

Definition 2.15 (SOS-valuations settings).

We say that a mechanism design setting with interdependent 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 , the valuation function is SOS.

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

  • Combinatorial valuations with multi-parameter signals (as in definition 2.10): for every and , is SOS, where .

Similar definitions can be given for -SOS valuation settings and -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 2.10 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,

Proof.

Let be the elements of . For , let and

denote the vectors

Note that , and .

It follows from the -SOS definition that for every ,

(3)

where and .

Summing Equation (3) for proves the claim. ∎

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 non-negativity 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 Single-Parameter Valuations

In this section we describe the Random Sampling Vickrey (RS-V) mechanism that achieves a 4-approximation for single-parameter downward-closed 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 downward-closed set system. We present a mechanism that serves only sets in and gets a -approximation to the optimal welfare.

Random Sampling Vickrey (RS-V):

  • 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 downward-closed feasibility constraint , RS-V is an ex-post IC-IR 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 ex-post IC-IR. 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 downward-closed implies that . Therefore, is eligible to be selected by RS-V 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 downward-closed feasibility constraints, even if the valuations satisfy single-crossing, 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 mechanisms141414 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 ex-post IC (but not necessarily universally so) if and only if is monotonically increasing in ..

Theorem 4.2.

No ex-post IC-IR mechanism (not necessarily universal) for selling a single item can get a better approximation than

  1. a factor of 2 for SOS valuations.

  2. 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.

  1. 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 2-approximation 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 2-approximation. 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: .

  2. 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 ex-post IC-IR 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 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, here we need to explicitly define payments so that the obtained mechanism is ex-post IC-IR. We derive 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’s report (while with separability, only the allocation is affected by it).

Random-Sampling VCG (RS-VCG):

  • 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.

Random-Sampling VCG is an ex-post IC-IR mechanism that gives a 4-approximation 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 4-approximation 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 RS-VCG is universally ex-post 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: