Polyhedral Clinching Auctions for Two-sided Markets

08/15/2017 ∙ by Hiroshi Hirai, et al. ∙ 0

In this paper, we present a new model and mechanism for auctions in two-sided markets of buyers and sellers, with budget constraints imposed on buyers. Our mechanism is viewed as a two-sided extension of the polyhedral clinching auction by Goel et al., and enjoys various nice properties, such as incentive compatibility of buyers, individual rationality, pareto optimality, strong budget balance. Our framework is built on polymatroid theory, and hence is applicable to a wide variety of models that include multiunit auctions, matching markets and reservation exchange markets.

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1 Introduction

Mechanism design for auctions in two-sided markets is a challenging and urgent issue, especially for rapidly growing fields of internet advertisement. In ad-exchange platforms, the owners of websites want to get revenue by selling their ad slots, and the advertisers want to purchase ad slots. Auctions are an efficient way of mediating them, allocating ad slots, and determining payments and revenues, where the underlying market is two-sided in principle. Similar situations arise from stock exchanges and spectrum license reallocation; see e.g., [BKLT2016, DRT2014]. Despite its potential applications, auction theory for two-sided markets is currently far from dealing with such real-world markets. The main difficulty is that the auctioneer has to consider incentives of buyers and sellers, both possibly strategic, and is confronted with impossibility theorems to design a mechanism achieving both accuracy and efficiency, even in the simplest case of bilateral trade [M1983].

In this paper, we address auctions for two-sided markets, aiming to overcome such difficulties and provide a reasonable and implementable framework. To capture realistic models mentioned above, we deal with budget constraints on buyers. The presence of budgets drastically changes the situation in which traditional auction theory is not applicable. Our investigation is thus based on two recent seminal works on auction theory of budgeted one-sided markets:

  • Dobzinski et al. [DLN2012] presented the first effective framework for budget-constrained markets. Generalizing the celebrated clinching framework by Ausubel [A2004], they proposed an incentive compatible, individually rational, and pareto optimal mechanism, called the “Adaptive Clinching Auction”, for markets in which the budget information is public to the auctioneer. This work triggered subsequent works dealing with more complicated settings [BCMX2010, BHLS2015, DHS2015, FLSS2011, GMP2013, GMP2014, GMP2015].

  • Goel et al. [GMP2015] utilized polymatroid theory to generalize the above result for a broader class of auction models including previously studied budgeted settings as well as new models for contemporary auctions such as Adwords Auctions. Here a polymatroid is a polytope associated with a monotone submodular function, and can represent the space of feasible transactions under several natural constraints. They presented a polymatroid-oriented clinching mechanism, called “Polyhedral Clinching Auction

    ,” for markets with polymatroidal environments. This mechanism enjoys incentive compatibility, individual rationality, and pareto optimality, and can be implemented via efficient submodular optimization algorithms that have been developed in the literature of combinatorial optimization

    [F2005, S2003].

The goal of this paper is to extend this line of research to reasonable two-sided settings.

Our contribution.

We present a new model and mechanisms for auctions in two-sided markets. Our market is modeled as a bipartite graph of buyers and sellers, with transacting goods through the links. The goods are divisible and common in value. Each buyer wants the goods under a limited budget. Each seller constrains transactions of his goods by a monotone submodular function on the set of edges linked to him. Namely, possible transactions are restricted to the corresponding polymatroid. In the auction, each buyer reports his bid and budget to the auctioneer, and each seller reports his reserved price. In our model, the reserved price is assumed to be identical with his true valuation; this assumption is crucial for avoiding impossibility theorems. The utilities are quasi-linear (within budget) on their valuations and payments/revenues. The goal of this auction is to determine transactions of goods, payments of buyers, and revenues of sellers, with which all participants are satisfied. In the case of a single seller, this model coincides with that of Goel et al [GMP2015].

For this model, we present two mechanisms that satisfy the incentive compatibility of buyers, individual rationality, pareto optimality and strong budget balance. Our mechanisms are built on and analyzed via polymatroidal network flow model by Lawler and Martel [LM1982]. This is a notable feature of our technical contribution. It is the first to apply polymatroidal network flow to mechanism design.

The first mechanism is a “reduce-and-recover” algorithm via a one-sided market: The mechanism constructs “the reduced one-sided market” by aggregating all sellers to one seller, applies the original clinching auction of Goel et al. [GMP2015]

to determine a transaction vector, payments of buyers, and the total revenue of the seller. The transaction vector of the original two-sided market is recovered by computing a polymatroidal network flow. The total revenue is distributed to the original sellers arbitrarily so that incentive rationality of sellers and strong budget balance are satisfied. We prove in Theorem

LABEL:mechanism_1 that this mechanism satisfies the desirable properties mentioned above. Also this mechanism is implementable by polymatroid algorithms. The characteristic of this mechanism is to determine statically transactions and revenues of sellers at the end. This however can cause unfair revenue sharing, which we will discuss in Section LABEL:sec:discussion.

The second mechanism is a two-sided generalization of the polyhedral clinching auction by Goel et al. [GMP2015], which determines dynamically transactions and revenues, and improves fairness on the revenue sharing. The mechanism works as the original clinching auction: As price clocks increase, each buyer clinches a maximal amount of goods not affecting other buyers. Here each buyer transacts with multiple sellers, and hence conducts a multidimensional clinching. We prove in Theorem LABEL:greedy an intriguing property that feasible transactions of sellers for the clinching forms a polymatroid, which we call the clinching polytope, and moreover the corresponding submodular function can be computed in polynomial time. Thus this mechanism is also implementable. We reveal in Theorem LABEL:relation that the allocation to the buyers obtained by this mechanism is the same as that by the original polyhedral clinching auction applied to the reduced one-side market. This means that the second mechanism achieves the same performance for buyers as that in the original one, and also improves the first mechanism in terms of the revenues sharing on sellers.

Our framework captures a wide variety of auction models in two-sided markets, thanks to the strong expressive power of polymatroids. Examples include two-sided extensions of multiunit auctions [DLN2012] and matching markets [FLSS2011] (for divisible goods), and a version of reservation exchange markets [GLMNP2016]. We demonstrate how our framework is applied to auctions for display advertisements. In addition, our model can incorporate with concave budget constraints in Goel et al. [GMP2014]. Also our result can naturally extend to concave budget settings (Remark LABEL:rem:concave). Thus our framework is applicable to more complex settings occurring in the real world auctions, such as average budget constraints.

Related work.

Double auction is the simplest auction for two-sided markets, where buyers and sellers have unit demand and unit supply, respectively. The famous Myerson-Satterthwaite impossibility theorem [M1983] says that there is no mechanism which simultaneously satisfies incentive compatibility (IC), individual rationality (IR), pareto optimality (PO), and budget balance (BB). McAfee [MA1992] proposed a mechanism that satisfies (IC),(IR), and (BB). Recently, Colini-Baldeschi et al. [BKLT2016] proposed a mechanism that satisfies (IC), (IR), and strong budget balance (SBB), and achieves an -approximation to the maximum social welfare.

Goel et al. [GLMNP2016] considered a two-sided market model, called a reservation exchange market, for internet advertisement. They formulated several axioms of mechanisms for this model, and presented an (implementable) mechanism satisfying (IC) for buyers, (IR), maximum social welfare, and a fairness concept for sellers, called -envy-freeness. This mechanism is also based on the clinching framework, and sacrifices (IC) for sellers to avoid the impossibility theorem. Their setting is non-budgeted.

Freeman et al. [FPW2017] formulated the problem of wagering as an auction in a special two-sided market, and presented a mechanism, called the “Double Clinching Auctions”, satisfying (IC), (IR), and (BB). They verified by computer simulations that the mechanism shows near-pareto optimality. This mechanism is regarded as the first generalization of the clinching framework to budgeted two-sided settings, though it is specialized to wagering.

Our results in this paper provide the first generic framework for auctions in budgeted two-sided markets.

Organization of this paper.

The rest of this paper is organized as follows. In Section 2, we introduce our model and present the main result and applications. In Section LABEL:sec:mechanisms, we present and analyze our mechanisms. In Section LABEL:sec:discussion, we discuss our mechanisms and raise future research issues. In Section LABEL:sec:proof, we give proofs.

Notation.

Let denote the set of nonnegative real numbers, and let denote the set of all functions from a set to . For , we often denote by , and write as . Also we denote by , and denote by . For , let denote the restriction of to . Also let denote the sum of over , i.e., .

Let us recall theory of polymatroids and submodular functions; see [F2005, S2003]. A monotone submodular function on set is a function satisfying:

(1)

where the third inequality is equivalent to the submodularity inequality:

The polymatroid associated with monotone submodular function is defined by

and the base polytope of is defined by

which is equal to the set of all maximal points in . A point in is obtained by the greedy algorithm in polynomial time, provided the value of for each subset can be computed in polynomial time.

2 Main result

We consider a two-sided market consisting of buyers and sellers. Our market is modeled as a bipartite graph of disjoint sets , of nodes and edge set , where and represent the sets of buyers and sellers, respectively, and buyer and seller are adjacent if and only if wants the goods of seller . An edge is denoted by .

For buyer (resp. seller ), let (resp. ) denote the set of edges incident to (resp. ). In the market, the goods are divisible and homogeneous. Each buyer has three nonnegative real numbers , where and are his valuation and bid, respectively, for one unit of the goods, and is his budget. Each buyer acts strategically for maximizing his utility (defined later), and hence his bid is not necessarily equal to the true valuation . In this market, each buyer reports and to the auctioneer. Each seller  also has a valuation for one unit of the goods, and reports to the auctioneer as the reserved price of his goods, the lowest price that he admits for the goods. In particular, he is assumed to be truthful (to avoid the impossibility theorem, as in [GLMNP2016]). He also has a monotone submodular function on , which controls transactions of goods through . The value for means the maximum possible amount of goods transacted through edge subset . In particular, is interpreted as his stock of goods. These assumptions on sellers are characteristic of our model.

Under this setting, the goal is to design a mechanism determining a reasonable allocation. An allocation of the auction is a triple of a transaction vector , a payment vector , and a revenue vector , where is the amount of transactions of goods between buyer and seller , is the payment of buyer , and is the revenue of seller . For each , the restriction of the transaction vector to must belong to the polymatroid corresponding to :

Also the payment of buyer must be within his budget :

A mechanism is a function that gives an allocation from public information , , that the auctioneer can access. The true valuation of buyer is private information that only can access. We regard and as the input of our model.

Next we define the utilities of buyers and sellers. For an allocation , the utility of buyer is defined by:

(2)

Namely the utility of a buyer is the valuation of obtained goods minus the payment. The utility of seller is defined by:

(3)

This is the sum of revenues and the total valuation of his remaining goods. In this model, we consider the following properties of mechanism .

  • Incentive Compatibility of buyers: For every input , it holds

    where is obtained from by replacing bid of buyer with his true valuation . This means that it is the best strategy for each buyer to report his true valuation.

  • Individual Rationality of buyers: For each buyer , there is a bid such that always obtains nonnegative utility. If (ICb) holds, then (IRb) is written as

  • Individual Rationality of sellers: The utility of each seller after the auction is at least the utility at the beginning:

    By (3), (IRs) is equivalent to

    (4)
  • Strong Budget Balance: All payments of buyers are directly given to sellers:

  • Pareto Optimality: There is no allocation which satisfies and the following three conditions:

    and at least one of the inequalities holds strictly, where is obtained from by replacing with . Namely, there is no other allocation superior to that given by for all buyers and sellers, provided all buyers report their true valuation.

They are desirable properties that mechanisms should have. The main result is:

Theorem 2.1.

There exists a mechanism that satisfies all of (ICb),(IRb),(IRs), (SBB), and (PO).

The details of our mechanisms are explained in Section 2.

Remark 1.

Maximizing the social welfare, the sum of utilities of all participants, is usually set as the goal in traditional auction theory. However, in budgeted settings, it is shown in [DLN2012] that the maximum social welfare and incentive compatibility of buyers cannot be achieved simultaneously. As in the previous works [BCMX2010, BHLS2015, DLN2012, DHS2015, FLSS2011, GMP2013, GMP2014, GMP2015], we give priority to incentive compatibility.

2.1 Application to display ad auction of multiple sellers and slots

We present applications of our results to display advertisements (ads) between advertisers and owners of websites. Each owner wants to sell ad slots in his website. Each advertiser wants to purchase the slots. Namely, the owners are sellers, and advertisers are buyers, where buyer is linked to seller if is interested in the slots of the website of . The market is modeled as a bipartite graph as above. We consider the following two types of ad auction, which are viewed as reservation exchange markets in the sense of [GLMNP2016].

2.1.1 Page-based ad auction

The website of seller consists of pages , where each page has ad slots. Each buyer purchases at most one slot from each page (so that the same advertisement cannot be displayed simultaneously). To control transaction of each seller , set as

(5)

Then is actually a monotone submodular function. It turns out that is appropriate for our purpose. Hence the market falls into our model, and our mechanisms are applicable to obtain transaction .

We first explain the way of ad-slotting at owner from the obtained , in which the meaning of will be made clear. We consider the following network . Let denote the set of buyers linked to . Consider buyers in and pages of as nodes in . Add a directed edge from each buyer to each page with unit capacity. Add source node and edge for each with capacity . Also, add sink node and edge for each page with capacity . Consider a maximum flow in the resulting network . From the max-flow min-cut theorem, one can see that is nothing but the maximum value of a flow from to . Hence, transaction satisfies the polymatroid constraint of if and only if every maximum flow in attains capacity bound on each source edge .

From a maximum flow , the owner conducts ad-slotting according to values . Consider first an ideal situation where is integer-valued, i.e., . Then the owner  naturally assigns the ad of buyer at page if , since is the sum of over (by flow conservation law and ), and at most buyers purchase page . Consider the usual case where is not integer-valued. The value

will be interpreted as the probability that the ad of

is displayed at page . Add dummy buyers to and define so that (if necessary). Consider probability over the set of all -element subsets of satisfying

(6)

The owner selects a -element subset with probability , and displays the ads of at page . By (6), the ad of advertiser is displayed with probability , as desired.

The probability with (6) can be constructed by the following algorithm, where .

  • Let , and let for all -element subsets of .

  • If , then output .

  • Sort , and let .

  • Define as the maximum with for and for .

  • Let for , and go to 1.

Let us sketch the correctness of the algorithm. In step 2, it always holds , and . Then it holds in step 3 that for , and for . Thus is positive. After steps 3 and 4, is zero for some or holds for some . For the latter case, is always contained in for the subsequent iterations. If for all , then for . Thus the algorithm terminates after iterations. By construction, it holds and .

2.1.2 Quality-based ad auction

Each seller has one page with ad slots. Each slot has a barometer for the quality of ads, which can be thought of as the expected number of views of over a certain period. Suppose that . The view-impression is the unit of this barometer; namely, slot has view-impressions. In the market, buyers purchase view-impressions from sellers, and have bids and valuations for the unit view-impression. After the auction, suppose that buyer obtains units of view-impressions from . Seller assigns ads of buyers (linked to ) to his slots so that the same ads cannot be displayed in distinct slots at the same time. An ad-slotting is naturally represented by for a set of buyers and a bijection from to the slots with the highest quality. Here the number of slots is assumed at least by adding dummy slots of view-impressions. Seller displays ads of

according to a probability distribution

on the set of all ad-slottings satisfying

(7)

Namely the expected number of view-impressions of ads in the website of is equal to . The existence of such a constrains transactions , and is equivalent to the condition that for each set of buyers linked to , is at most the sum of highest . This condition can be written by the following monotone submodular function :

Then seller restricts his transactions by the corresponding polymatroid , to conduct the above way of ad-slotting after the auction. Again the market falls into our model, and is a modification of Adwords auctions in [GMP2015] for display ad auctions with multiple websites, where the way of ad-slotting according to (7) is based on their idea.

Notice that an extreme point of is precisely a vector such that for some subset and bijection , it holds if , or otherwise. In particular, (7) is viewed as a convex combination of extreme points of . Therefore, the required probability distribution is obtained by expressing as a convex combination of extreme points of .

The page-based ad auction in Section 2.1.1 can incorporate the quality-based formulation. Suppose that each slot (including dummy slots) on the page has a barometer for the quality of ads, and that for each . Replace in (5) by

(8)
(9)

This is also monotone submodular, and our mechanisms are applicable. The obtained transaction is distributed to view-impression per page so that , and satisfies polymatroid constraint by . Such a distribution can easily be obtained via polymatroidal network flow, introduced in the next section. The owner conducts, at each page, the ad-slotting in the same way as above.