Efficient Two-Sided Markets with Limited Information

03/17/2020 ∙ by Paul Dütting, et al. ∙ Google LSE Sapienza University of Rome 0

Many important practical markets inherently involve the interaction of strategic buyers with strategic sellers. A fundamental impossibility result for such two-sided markets due to Myerson and Satterthwaite establishes that even in the simplest such market, that of bilateral trade, it is impossible to design a mechanism that is individually rational, truthful, (weakly) budget balanced, and efficient. Even worse, it is known that the "second best" mechanism-the mechanism that maximizes social welfare subject to the other constraints-has to be carefully tailored to the Bayesian priors and is extremely complex. In light of this impossibility result it is very natural to seek "simple" mechanisms that are approximately optimal, and indeed a very active line of recent work has established a broad spectrum of constant-factor approximation guarantees, which apply to settings well beyond those for which (implicit) characterizations of the optimal (second best) mechanism are known. In this work, we go one step further and show that for many fundamental two-sided markets-e.g., bilateral trade, double auctions, and combinatorial double auctions-it is possible to design near-optimal mechanisms with provable, constant-factor approximation guarantees with just a single sample from the priors! In fact, most of our results in addition to requiring less information also improve upon the best known approximation guarantees for the respective setting.

READ FULL TEXT VIEW PDF
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

Recently, increased attention has turned to the problems that arise in two-sided markets, in which the set of agents is partitioned into buyers and sellers. In contrast to the one-sided setting (where one could say that the mechanism itself initially holds the items), in the two-sided setting the items are initially held by the sellers, who have valuations over the items they hold, and who are assumed to act rationally and strategically. The mechanism’s task is now to decide which buyers and sellers should trade, and at which prices, with the goal of maximizing the social welfare

of the reallocation of the goods. Two-sided markets are usually studied in a Bayesian setting: there is public knowledge of probability distributions, one for each buyer and one for each seller, from which the valuations of the buyers and sellers are drawn.

In two-sided markets, a further important requirement is strong budget balance (SBB), which states that monetary transfers happen only among the agents in the market, i.e., the buyers and sellers are allowed to trade without leaving to the mechanism any share of the payments, and without the mechanism adding money to the market. A weaker version of SBB often considered in the literature is weak budget balance (WBB), which only requires the mechanism not to inject money into the market. However, it is known from the work of [33] that it is generally impossible for an individually rational (IR), Bayesian incentive compatible (BIC), and WBB mechanism to maximise social welfare in such a market, even in the bilateral trade setting, i.e., when there is just one seller and one buyer and public knowledge of the distribution of agents’ values. This is in sharp contrast to the celebrated optimal results available for one-sided markets [43, 32].

A recent line of research has focused on mechanisms that satisfy IR, SBB, and IC (or failing that, the weaker notion of BIC), and that reallocate the items in such a way that the expected social welfare is within some constant fraction of the optimum, where the expectation is taken over the given probability distributions of the agents’ valuations and over the random choices of the mechanism. Such mechanisms circumvent the impossibility result of [33] by weakening the requirement of optimal social welfare to that of approximately optimal social welfare.

Important special cases of two-sided markets that have been considered with the aforementioned goal include: bilateral trade with one buyer and one seller [8, 24]; double auctions in which unit-demand buyers interact with multiple sellers that each hold a single copy of an identical item [13, 19]; and combinatorial double auctions in which the buyers have combinatorial (e.g., fractionally subadditive (XOS)) valuations and the sellers hold non-identical items [14, 7].

We improve on the existing results in two ways. First of all, we give the first two-sided market mechanisms with limited information. More concretely, we show how to obtain mechanisms for two-sided markets that satisfy IR, BB, IC, and that achieve a constant factor approximation of the optimum social welfare, even when the mechanism only knows a single sample from each distribution of the buyers and of the sellers. Secondly, in some cases, we are able to improve over previous bounds obtained with full knowledge of the distributions. Our work is close in spirit to the previous works on one-side mechanism design that obtain approximately optimal revenue with limited information about an existing distribution of bidders’ values [18, 23, 12], and to the previous work on prophet inequalities with limited information from the distributions [2, 16, 38, 17].

1.1 Overview of the Results

This paper studies the problem of designing mechanisms for two-sided markets in the Bayesian setting with limited information: that is we consider the case in which both buyers’ and sellers’ valuations are private information drawn from independent, arbitrary distributions that are known through a limited number of samples. If not specified otherwise, a single sample is available from each distribution. We study this problem in various settings, summarized in Table 1.

1.1.1 Bilateral Trade

As a warm up, we consider the bilateral trade setting with one buyer with valuation and a seller with valuation . We present an IR, IC, SBB mechanism that gives a -approximation using a single sample from as posted price for the agents. We also show that no deterministic IC mechanism (all IC mechanisms are posted price [13]) that uses just a single sample from or just a single sample from can do better. Our result achieves the same approximation bound obtained when is known by posting a price equal to the median of [8]. Our mechanism also matches with just one sample the best possible result that can be obtained by a deterministic mechanism that uses only or only [8]. The work of [8] also gives a randomized mechanism that achieves a -approximation by using full knowledge of . We show a IC, IR, SBB mechanism that gives a -approximation using samples from .

Setting IR+IC SBB WBB Approximation Samples Arrivals Poly
bilateral trade No/Yes Yes
double auction matroid No/Yes Off/Off Yes
double auction -uniform matroid No/Yes OnRa/OnRa Yes
double auction -uniform matroid No/Yes OnFi/OnRa Yes
combinatorial XOS No/Yes Off/Off No
combinatorial submodular/XOS No/Yes Off/Off Yes
combinatorial GS No/Yes Off/Off Yes
combinatorial unit demand No/Yes OnRa/Off Yes
Table 1: Overview of single-sample mechanisms for two-sided markets that we develop in this paper. The samples column indicates whether we need a single sample from the buyer side and seller side. The arrival column specifies the arrival order of the buyers and sellers. Off stands for offline, OnFi for online fixed order, and OnRa for online random order.

1.1.2 Double Auctions

In the double auction setting, there are unit-demand buyers and unit-supply sellers with identical items. Buyer valuations are drawn from , and seller valuations are drawn from , independently. We distinguish between offline and online mechanisms. In offline mechanisms, the agents can trade in any order. For online mechanisms, the agents can trade in online fixed order or in online random order. We also consider additional constraints on the set of buyers that can trade simultaneously in the form of downward closed set systems that captures for example matroid settings. We prove the following main theorem (informal version):

Theorem 1.

Denote by the approximation guarantee of an offline/online one-sided IC, IR, single-sample mechanism for the intersection of a downward closed set system with a -uniform matroid. We give a two-sided single-sample IR, IC, SBB mechanism for double auctions with constraints on the buyers that yields in expectation a approximation to the expected optimal social welfare. The mechanism inherits the same online/offline properties of the one-sided mechanism on the buyer side and it is online random order on the seller side.

For general matroid settings, by applying the optimal offline truthful VCG mechanism on the buyer side, we obtain a -approximate single-sample two-sided mechanism for double auctions for general matroid constraints on the buyers. This improves over the best known approximation guarantee of due to Colini-Baldeschi et al. [14] for this problem obtained with full knowledge of and !

We obtain two single-sample online mechanisms for the two-sided setting with different information requirements and for different online arrival models by using the truthful -competitive secretary algorithm for -uniform matroids due to Kleinberg [25] or the truthful -competitive single-sample prophet inequality for -uniform matroids due to Azar et al. [2]. The resulting approximation guarantees of improve for a large spectrum of values of and over the best known bound of for this problem [14] obtained with full knowledge of the distributions.

Observe that any future advances on -approximate secretary algorithms or single-sample prophet inequalities for the intersection of any matroid (partition matroid, graphical matroid, etc.) with a -uniform matroid constraint would lead to a new -approximate single-sample two-sided mechanism with online arrivals.

1.1.3 Combinatorial Double Auctions

We also consider combinatorial double auctions with buyers having combinatorial valuations for sets of items and unit-supply sellers with non-identical items. Buyer valuation functions are drawn from , . Seller valuations are drawn from , independently. We specifically consider fractionally subadditive (XOS) valuations. We prove the following main theorem (informal version):

Theorem 2.

Denote by the approximation guarantee of any one-sided IR, IC offline/online single-sample mechanism for maximizing social welfare for XOS valuations. We give a two-sided single-sample mechanism for combinatorial double auctions with XOS buyers and unit-supply sellers that is IR, IC, WBB, and provides in expectation a approximation. The two-sided mechanism inherits the offline/online properties of the one-sided mechanism on the buyer side and is offline on the seller side.

This result implies a (non-computational) -approximation for XOS valuations via the VCG mechanism. A -approximate poly-time mechanism in the demand oracle model for XOS valuations follows from the recent breakthrough of [1] and a -approximate poly-time mechanism for Gross Substitute (GS) valuations follows from the LP-based poly-time algorithm of Nisan and Segal [35] or the poly-time algorithms for convolutions of -valuation functions [30, 31] (also see the excellent survey [36] for the latter).

In the special case of unit-demand buyers and unit supply sellers with non-identical items, we obtain a single sample IR, IC, WBB mechanism that yields a -approximation that is online random order on the buyer side and offline on the seller side. The mechanism uses the one-sided online truthful -approximate secretary matching mechanism [37].

These results compare with the IR, IC, SBB -approximation mechanism for XOS valuations and unit supply sellers of [14] with full knowledge of the distributions that is online on the buyer side and offline on the seller side.

Again, any future improvements in the one-sided problem (whether offline or online) will translate into two-sided results through our theorem(s).

1.2 Techniques

Our techniques are very different from those in prior work on revenue-maximizing one-sided mechanisms from samples [18, 23, 12], and also from the prophet inequalities with limited information literature [2, 16, 38, 17].

A first challenge that we encounter, and show how to solve approximately with a single sample, already occurs in the bilateral trade case. The difficulty here is to decide whether any given buyer-seller “couple” with valuations and should trade. Ideally, they would trade whenever . However, as we know from [33] we can’t achieve this with a IR, IC, and BB mechanism; and a constant factor loss is unavoidable [13].

Our solution to this problem is simple (but the analysis requires some care!): Simply draw a single sample from the seller distribution and post this a price, and let the buyer and seller trade if the buyer’s value is above this price and the seller’s value is below. Clearly, this entails some loss, namely whenever the buyer has a higher valuation then the seller but either both are below the price (and so the buyer does not accept) or both are above the price (and so the seller does not accept). However, as we show, the loss is not too bad: posting the seller sample as a price will, in expectation, recover of the optimal social welfare.

Our analysis of this bilateral trade case indeed applies to any fixed value , and we exploit this for our double auction results. Here we show that approximately optimal solutions result from combining one-sided mechanisms that (approximately) choose an optimal set of buyers (the highest buyers in the unconstrained case), and randomly match the tentatively selected buyers to the sellers, maxing the price that the buyers would face in the one-sided mechanism with the single sample from the respective seller’s distribution.

The second—and main—challenge arises when going from single parameter to multi-parameter settings with non-identical items, because here we can’t just randomly match buyers to sellers (as this would jeopardize the approximation guarantee), but if we don’t just match them randomly ensuring truthfulness becomes a very tricky thing!

Our solution to this is to modify any given one-sided mechanism for the buyers by discounting the buyer’s valuations for the different items by the respective seller’s sample valuations. We show that selecting allocations based on this, proposing these (possibly set-wise) trades to the buyers and sellers, increasing the payments asked from the buyers as determined by the one-sided mechanism by the respective seller samples, and offering the sellers to trade at “their” sample, yields an IR, IC, and WBB mechanism. Moreover, and perhaps surprisingly, this same mechanism also ensures near-optimal social welfare!

1.3 Further Related Work

There are two important precursors to the more recent work on approximately optimal simple mechanisms for two-sided markets: The first studies the non-truthful buyer’s bid mechanism in a double auction setting with i.i.d. buyers and i.i.d. sellers, and shows convergence to efficiency as the number of sellers and buyers grows to infinity [40, 39, 41]. The second is work on trade reduction mechanisms [27, 3, 4, 19], which starts from McAfee’s truthful trade reduction mechanism for double auctions, which extracts a fraction of the maximum social welfare, where is the number of traders in the optimal solution.

A number of recent works [42, 9, 28, 10, 15, 5] has considered the related objective of optimizing the gain from trade, which measures the expected increase in total value that is achievable by applying the mechanism, with respect to the initial allocation to the sellers. Gain from trade is harder to approximate than social welfare, and approximations of the optimal Bayesian mechanism are only possible in BIC implementations.

Goldner et al. [21] recently suggested an alternative, resource augmentation approach to gains from trades in two-sided markets, in the spirit of the celebrated result of Bulow and Klemperer [11]. They ask how many buyers (resp. sellers) need to be added into the market so that a variant of McAfee’s trade reduction mechanism yields a gain from trade superior to the optimal gains from trade in the original market. As a side product they obtain a -approximate single-sample mechanism for gains from trade under natural conditions on the distributions.

Another related line of work considers the problem of maximizing revenue in double auction settings, either in static environments [22, 34] or in dynamic environments [6]. A variation where both buyers and sellers arrive dynamically and the mechanism can hold on to items was investigated in [20, 26].

Our work is also related to prior work which has studied how incentive compatible mechanisms can be composed in an incentive compatible manner, such as [29] in the context of one-sided markets and [19] in the context of two-sided markets.

2 Model and Definitions

Two-Sided Markets.

In a two-sided market we are given a set of buyers and a set of sellers . Each seller has a single indivisible item for sale. Every seller has a private valuation for the item she sells. Each buyer has a private valuation function , mapping each set of sellers to a non-negative real. We write and

for the vector of buyer valuations and seller valuations, respectively. Buyer and seller valuations are drawn independently from distributions

for and for .

In our model, sellers have a single indivisible item for sale. We refer to such sellers as unit supply sellers. The valuation functions of the buyers will be constrained to come from some class of functions . Buyers are unit demand if for each buyer and set of sellers , . Buyers have fractionally subadditive (or XOS) valuations if for each buyer and every set of sellers , , where is a set of additive valuation functions.

We say that items are identical if the valuation function of all buyers only depends on the cardinality of the set they receive, i.e., for all and all with we have . Otherwise, items are non-identical.

We also allow for constraints on which buyers can trade simultaneously. We express these constraints through set systems . We require these set systems to be downward closed. That is, whenever and , then also . Of particular importance for our work will be matroids, i.e., downward-closed set systems that additionally satisfy a natural exchange property. Formally: Whenever and then there exists such that . A special case are -uniform matroids where whenever .

An allocation is a partition of the sellers into disjoint sets , i.e., and for all , with the interpretation that buyer for receives the items of the sellers in . An allocation is feasible if the set of buyers that receive a non-empty allocation is admissible (i.e., ). The social welfare of an allocation is given by the sum of the valuations that buyers for have for the items of the sellers in their respective sets plus the valuations of the sellers that are not assigned to any buyer, i.e.,

We use to denote the feasible allocation that maximizes social welfare.

Mechanisms.

A (direct revelation) mechanism receives bids from each buyer and from each seller . The bids of the buyers are constrained to be consistent with the class of functions of their valuations. Bids represent reported valuations, and need not be truthful. In analogy to our notation for valuations, we use and for the vector of bids of all buyers or all sellers, respectively.

A mechanism is defined through an allocation rule and a payment rule .

The mapping from bids to feasible allocations can be randomized, in which case

is a random variable. The payments can also be randomized. We interpret the vector of payments as the payments that the buyers need to make to the mechanism, and that the sellers receive from the mechanism. We use the shorthand

and to refer to the payment from buyer to the mechanism and from the mechanism to seller. We require that if seller keeps her item.

A mechanism is single-sample if the only information it is given about the two sets of distributions for and for is a single sample from each of these distributions.

Utilities.

We assume that buyers and sellers have quasi-linear utilities, and that they are utility maximizers. The utility of buyer with valuation function in mechanism under bids is given by her valuation for the items she receives minus payment. That is,

where the expectation is over the randomness in the mechanism. The utility of a seller in mechanism under bids is the payment she receives if she sells the item and her valuation for her item otherwise. Formally,

where the expectation is over the randomness in the mechanism.

Goals.

We seek to design mechanisms, and specifically single-sample mechanisms, with the following desirable properties:

(1) Individual Rationality. Mechanism is individually rational (IR) if for all and for all .

(2) Incentive Compatibility. Mechanism is (dominant-strategy) incentive compatible (IC) or truthful if for each buyer and each seller , all valuation functions and , all possible bids by the buyers, and all possible bids by the sellers , it holds that

where denotes the set of all the buyer bids but ’s and denotes the set of all the seller bids but ’s.

(3) Budget Balance. A truthful mechanism is weakly budget balanced (WBB) if

and it is strongly budget balanced (SBB) if the above holds with equality.

(4) Efficiency. Finally, a truthful mechanism provides an -approximation to the optimal social welfare, for some , if it holds that

Remark.

Our mechanisms will actually satisfy even stronger IR and BB properties in that they will satisfy these conditions “ex post” (i.e., pointwise).

3 Warm-Up: Bilateral Trade

As a warm up we consider the important special case of bilateral trade where there is only a single seller and a single buyer. Note that in this case it’s optimal to trade whenever the buyer’s value exceeds that of the seller, so .

We show how to get a -approximation with just a single sample! Our mechanism, which we refer to as SamplePrice, (Algorithm 1), couldn’t be simpler. It’s almost trivial: Draw a sample from the seller distribution and use this sample as a posted price.

Our mechanism differs from the strategies employed in prior work in that it does not choose a posted price that corresponds to the median (or other quantiles) of the seller distribution.

Sample
if  and  then
      // Let the seller and the buyer trade
       ,
      
end if
ALGORITHM 1 SamplePrice
Theorem 3.

Algorithm SamplePrice is individually rational, truthful, strongly budget balanced and provides, in expectation over the sample, a -approximation to the optimal social welfare.

Proof.

It is easy to verify that SamplePrice is IR, IC, and SBB. So all we need to show is that it achieves the claimed approximation guarantee.

To this end let , , and denote the random variables that correspond to the seller valuation, the buyer valuation, and the price chosen by SamplePrice; and use , and to denote specific realizations of these random variables. Let denote the random variable that corresponds to the social welfare achieved by SamplePrice.

We will use an insight from [8], namely that it suffices to show the approximation ratio for any fixed buyer value and truncated seller values where values of above are mapped to . This simplifies the benchmarking as under this condition it is always optimal to assign the item to the buyer. That is, we want to show that

SamplePrice transfers the item from the seller to the buyer whenever , otherwise the seller keeps her item. So,

If we condition with respect to the event and , we have:

Let To complete the proof we will show that . This will then show the claim as then

It remains to prove the non-trivial part of this proof, which is that . Recall our notation that for all , and that we defined . Then with and the pseudo-inverse of :

(1)
(2)
(3)

where in (1) we have used the property that is distributed exactly like (see, for example, [44] for a proof), in (2) that and in (3) the fact that is non-decreasing. ∎

In Appendix A we show that it’s not possible to “derandomize” SamplePrice by simply posting the mean of the seller distribution.

Our next theorem shows that our analysis of SamplePrice is tight, and that’s it is best possible among “deterministic” mechanisms that only use a single sample from the buyer or the seller distribution. A proof can be found in Appendix B.

Theorem 4.

Every posted-price mechanism that receives a single sample from the buyer distribution or a single sample from the seller distribution and sets a deterministic price for each sample it may receive has approximation ratio at least .

In Appendix C we propose the SampleQuantile mechanism, which receives samples from and show the following performance guarantee.

Theorem 5.

Mechanism SampleQuantile is individually rational, truthful, and budget balanced. For every , given samples, it provides in expectation the following approximation guarantee:

4 Double Auctions

We now move on to to more general settings with multiple unit-demand buyers and multiple unit-supply sellers with identical items. We present a general technique for turning truthful one-sided mechanisms into truthful two-sided mechanisms.

The one-sided mechanisms we consider are for so-called binary single-parameter problems. In such a problem, an agent can either win or lose, and has a value for winning. The set of agents that can simultaneously win is given by a set system . The social welfare of a feasible set is simply the sum of the winning agents’ valuations.

Given two set systems and on a ground set , we define its intersection to be the set system that contains all sets such that and .

Theorem 6.

Denote by the approximation guarantee of an offline/online one-sided IC, IR, single-sample mechanism for welfare maximization in a binary single-parameter problem whose feasible solutions correspond to the intersection of a downward-closed set system with a -uniform matroid. Then there is a two-sided single sample IR, IC, SBB mechanism for double auctions with constraints on the buyers that yields in expectation a

approximation to the expected optimal social welfare. Moreover, the mechanism inherits the same online/offline properties of the one-sided mechanism on the buyer side and it is online random order on the seller side.

We first provide a formal proof of the following special case, and then argue how to generalize it.

Theorem 7.

Let . There is a IR, IC, SBB, -approximate single-sample mechanism for unconstrained double auctions that approaches the buyers in online fixed order and the sellers in online random order.

We give the mechanism for Theorem 7 in Section 4.1, and a proof of its properties in Section 4.2. We explain how to generalize the construction and the proof in Section 4.3.

4.1 Mechanism for Theorem 7

Our mechanism—Two-Sided Rehearsal (Algorithm 2)—runs the one-sided Rehearsal algorithm due to Azar et al. [2] to select the top buyers. Azar et al. [2] show that the combined value of the buyers that beat their price provides, in expectation, a approximation to the expected value of the highest buyers.

Our twist to this mechanism is that we pair buyers that would be selected by the Rehearsal algorithm with a random seller , offering them to trade at a price that is the max of the respective buyer’s and the respective seller’s sample .

This adds the needed component to take into account the valuations on the seller side of the market, and will serve as an insurance that any good trade we propose has a good chance of actually happening.

Let be the set of the largest buyer samples, together with copies of the -largest buyer sample
Let be the smallest element of
Fix any order on the buyers (or assume buyers arrive online)
for each , in this order do
        if  then
             Delete from the highest value such that
             Pick uniformly at random , delete from (or assume sellers arrive online)
             Propose a trade to for the price of
             if  and both agree then
                   Make the trade at this price
             end if
            
       end if
      
end for
ALGORITHM 2 Two-Sided Rehearsal

4.2 Proof of Theorem 7

We begin by showing that our two-sided mechanism inherits IR and IC from its one-sided counterpart, and that it is strongly budget balanced.

Lemma 1.

The Two-Sided Rehearsal mechanism is IR, IC, and SBB.

Proof.

In Two-Sided Rehearsal no money is ever received by the mechanism itself. The only exchange of money happens between buyer-seller pairs that also exchange an item. The mechanism is therefore strongly budget balanced.

It is also clear that the mechanism is individually rational as buyers and sellers would only accept trades at prices that are lower resp. higher than their respective valuations.

Furthermore, the mechanism is truthful for agents on both sides of the market. Each buyer is presented with a trading opportunity once; and the price depends only on the samples and the valuations of previously considered buyers. She can only accept or reject, but never influence it—and will therefore not profit from reporting a lower or higher value. Sellers, on the other hand, are also guaranteed to be considered only once. They, too, have no means of influencing the price they are presented with, and can only accept or reject. ∎

It remains to show the claimed approximation guarantee. Just as in the case of bilateral trade, we will do the bulk of the work for fixed buyer valuations.

Lemma 2.

Fix and and let denote the approximation guarantee of the Rehearsal algorithm. The Two-Sided Rehearsal mechanism yields in expectation a

approximation to the expected optimal social welfare.

In what follows, we use to refer to the one-sided version of Rehearsal, and to refer to the two-sided version. We use to denote the set of tentative buyers chosen by , we use to denote the set of buyers that end up with an item in , and we use to denote the set of sellers that keep their item in . The expected social welfare achieved by is:

(4)

The key bit in our proof is the following lemma, which relates the performance of the one-sided mechanism to that of the two-sided mechanism.

Lemma 3.

Let denote the set of tentative buyers chosen by the one sided mechanism , let denote the set of buyers that trade in the two-sided mechanism , and let denote the set of sellers that keep their item in the two-sided mechanism . Then,

Proof.

In order to prove the lemma we will show that for any fixed buyer valuations , buyer samples , and corresponding set of tentative buyers , in expectation over the seller valuations , the seller samples , and the randomness in the pairing of buyers and sellers,

The actual claim then follows by taking expectation over buyer valuations , buyer samples , and the corresponding set of tentative buyers .

In order to do the analysis let’s fix any buyer with its value and tentative payment . Let’s denote by the random seller associated to , and by and two independent samples from that seller’s value distribution. Note that from buyer ’s perspective seller is just a uniform random seller from .

The contribution of the couple to the social welfare is if there is a sale and otherwise, and there is a sale when .

To analyze this contribution, we fix some constant which will be set later and we define as the probability that a random draw from the distribution of seller chosen uniformly at random from is at most .

If then the probability to have a sale is at least

So the expected contribution is at least .

Else, if , either there is a sale (and hence the contribution is ) or there is not, and hence the contribution is . So the expected contribution is at least

Putting the two cases together, we have that the contribution of buyer and her random partner to the expected social welfare is at least

(5)

We next show how to lower bound this term. For the sake of simplicity we present here the proof for continuous seller distributions, the argument for general distributions is given in Appendix D. For continuous seller distributions, is continuous and increasing while is continuous and decreasing, hence there exists a solution to

(6)

So a lower bound to (5) is given by

One concise way to express the progress so far is:

(7)

where the randomness is over the choice of the random seller and its values and . The above holds for all buyers , so we can sum up for all buyers in , then use linearity of expectation and the fact that to obtain

as claimed. ∎

We are now ready to prove the performance guarantee of the two-sided mechanism .

Proof of Lemma 2.

Recall that we use to denote the set of buyers that actually do get an item in our two-sided mechanism , and that we use to denote those sellers that do not make any trade and keep their item. With this notation the expected social welfare achieved by our two-sided mechanism is

For a given set of valuations of the buyers, denote by the set of buyers with the highest values. We can upper bound the expected optimal social welfare by the optimal solution for the buyers plus all seller values

(8)

Recall that the one-sided mechanism computes a set of buyers whose accumulated expected values are at least times the expected one-sided optimum. Hence, for the considered buyers ,

(9)

By combining Inequality (9) with our upper bound on the expected optimal social welfare in Inequality (8), we obtain

(10)

First consider the second term on the right hand side of Inequality (10). In our two-sided mechanism sellers trade only if they are matched to a buyer with higher valuation. Therefore, we can replace as follows:

Now consider the first term on the right hand side of Inequality (10). Our two-sided mechanism does not make trades for each buyer in . Despite the fact that generally , as we show in Lemma 3, in expectation, is a lower bound on our mechanism’s social welfare. Using this we obtain

All in all, we get:

as claimed. ∎

4.3 Proof of Theorem 6

For the more general result in Theorem 6 we run the given one-sided mechanism on the intersection of the given feasibility constraint with a -uniform matroid. This gives a set of tentative buyers along with tentative buyer prices . We can then randomly match the tentative buyers to sellers, offering buyer-seller pairs to trade at price .

The proof that the resulting mechanism is IR, IC, and SBB is basically identical to that of Lemma 1. They key is that truthfulness of the one-sided mechanism ensures that tentative buyers want an opportunity to trade, and cannot manipulate the price they face for this opportunity.

For the performance analysis we claim that Lemma 2 applies more generally with being the approximation guarantee of the one-sided mechanism. In fact, the only change to the above proof that is required for this generalization is to redefine as the optimal one-sided solution containing at most buyers.

5 Combinatorial Double Auctions

Up to this point, our results were for unit-demand buyers and unit-supply sellers with identical items. This means that we have focused on so called single-parameter settings. In this section, we give up on this assumption and turn towards multi-parameter versions of our techniques, which implies a whole set of complications that our methods need to handle in addition.

Theorem 8.

Denote by the approximation guarantee of any one-sided IR, IC offline/online single-sample mechanism for maximizing social welfare for XOS valuations. We give a two-sided single-sample mechanism for combinatorial double auctions with XOS buyers and unit-supply sellers that is IR, IC, WBB, and provides in expectation a approximation. The two-sided mechanism inherits the offline/online properties of the one-sided mechanism on the buyer side and is offline on the seller side.

We describe the mechanism (resp. reduction) that achieves the properties claimed in Theorem 8 in Section 5.1, and establish that it actually achieves these properties in Section 5.2.

5.1 The Mechanism

The basic idea behind our mechanism 2XOS (Algorithm 3), is to run the given truthful one-sided mechanism on discounted buyer valuations and on a subset of the sellers. Note that the problem can be viewed as finding a hypermatching in a bipartite hypergraph with hyperedge set defined as all tuples s.t. .

First, given valuations and samples for each seller, we determine a subset of the sellers as follows. For each we put in if . Otherwise, we will drop from our considerations. Next we determine discounted valuations. For a given buyer and a given set of sellers let denote the additive supporting function of buyer for set . We define the discounted valuation that buyer has for the set of sellers as:

We note that adjusting the valuations like this retains the XOS property of the original valuations. A proof can be found in Appendix E. The same holds for the gross substitutes class considered in Table 1: after the adjustment, GS valuations remain GS.

Then, we run the one-sided mechanism on the resulting hypergraph consisting of all buyers, only the sellers in , and hyperedge valuations . This will lead to an allocation and payments for each .

Afterwards, we assign sets to buyer increasing buyer ’s payment relative to the payment in the one-sided mechanism by the sum of the samples for and pay each seller whose item has been sold the respective sample .

The construction given in our mechanism is stated in the value-oracle model, and direct computation of the adjusted valuations would be inefficient. We provide a discussion on how to implement the mechanism efficiently in Section 5.3. For purposes of our analysis, we assume that the one-sided mechanism always assigns each buyer an inclusion-minimal set of items giving the according buyer at least the same utility (this can, e.g., be ensured by employing a simple type of tie-breaking which favors small sets over larger ones).

Set ,
for all  do
       Propose to a price of
       if  accepts then
             Set
            
       end if
      
end for
for all  do
      
      
end for
Let be the assignment on induced by running on the hypergraph , with hyperedge weights , presenting buyers to according to its input requirements (e.g., offline or in random order)
for all  do
       pays price , where is the price charged to by
       gets assigned the items in
       for each  do
             receives a payment of
            
       end for
      
end for
ALGORITHM 3 2XOS

5.2 Proof of Theorem 8

We start by establishing the individual rationality, truthfulness, and budget balance properties of the two-sided mechanism claimed in Theorem 8.

Lemma 4.

Given that an IR and IC one-sided mechanism is used, the two-sided mechanism 2XOS is IC, IR, and WBB.

Proof.

Let’s fix any realization of the valuations. We start by showing truthfulness. Fix a seller : the only interaction has with the algorithm is by accepting or rejecting the posted price , which is independent of , in exchange for his item, which is clearly truthful and individually rational.

Fixing a buyer : the algorithm will ask about his valuation and modify it to

reflecting that any item