Polyhedral Clinching Auctions for Two-sided Markets

Hiroshi Hirai∗ and Ryosuke Sato†

August 15, 2017

Abstract

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

1 Introduction

Mechanism design for auctions in two-sided markets is a challenging and urgent issue, especially forrapidly growing fields of internet advertisement. In ad-exchange platforms, the owners of websiteswant to get revenue by selling their ad slots, and the advertisers want to purchase ad slots. Auctionis 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 exchangesand spectrum license reallocation; see e.g., [2, 5]. Despite its potential applications, auction theoryfor two-sided markets is currently far from dealing with such real-world markets. The main difficultyis that the auctioneer has to consider incentives of buyers and sellers, both possibly strategic, andis confronted with nonexistence theorems of desirable mechanisms achieving both accuracy andefficiency, even in the simplest case of bilateral trade; see [15].

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

(i) Dobzinski et al. [4] presented the first effective framework for budget-constrained markets.Generalizing the celebrated clinching framework by Ausubel [1], they proposed an incentivecompatible, individual rational, and pareto optimal mechanism, called “Adaptive ClinchingAuction”, for markets in which the budget information is public to the auctioneer. This worktriggers subsequent works dealing with more complicated settings [3, 6, 10, 11].

(ii) Goel et al. [11] utilized polymatroid theory to generalize the above result for a broader class ofauction models including previously studied budgeted settings as well as new models for con-temporary auctions such as Adwords Auctions. Here a polymatroid is a polytope associated

∗The University of Tokyo, Tokyo, Japan. hirai@mist.i.u-tokyo.ac.jp†The University of Tokyo, Tokyo, Japan. ryosuke sato@mist.i.u-tokyo.ac.jp

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with a monotone submodular function, and can represent the space of feasible transactionsunder several natural constraints. They presented a polymatroid-oriented clinching mecha-nism, “Polyhedral Clinching Auction,” for markets under polymatroidal environments. Thismechanism enjoys incentive compatibility, individual rationality, and pareto optimality, andcan be implemented via efficient submodular optimization algorithms developed in the liter-ature of combinatorial optimization [8, 16].

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

Our contribution. We present a model and mechanism for auctions in two-sided markets. Ourmarket is modeled as a bipartite graph of buyers and sellers, with transacting divisible and ho-mogeneous goods through the links. Each buyer wants goods under a limited budget. Each sellerconstrains transactions of his goods by a monotone submodular function on the set of edges linkedto him. Namely, possible transactions are restricted to the corresponding polymatroid. In the auc-tion, each buyer reports his bid and budget to the auctioneer, and each seller reports his reservedprice. In our model, the reserved price is assumed to be identical with his true valuation. Theutilities are quasi-linear (within budget) on their valuations and payments/revenues. The goal ofthis auction is to determine transactions of goods, payments of buyers, and revenues of sellers, withwhich all participants are satisfied. In the case of single seller, this model coincides with that ofGoel et al [11]. For this model, we present a mechanism that satisfies the incentive compatibilityof buyers, individual rationality, pareto optimality and strong budget balance.

Our mechanism is a generalization of the polyhedral clinching auction by Goel et al., andis also built on the clinching framework and polymatroid theory. As price clocks increase, eachbuyer clinches a maximal amount of goods not affecting other buyers. Each buyer transacts withmultiple sellers, and hence conducts a multidimensional clinching. We prove in Theorem 3.1 thatfeasible transactions for the clinching forms a polymatroid, which we call the clinching polytope,and moreover the corresponding submodular function can be computed in polynomial time. Thusour mechanism is implementable. We also reveal in Theorem 3.4 that the allocation of buyersobtained by our mechanism is the same as that by the original polyhedral clinching auction appliedto the reduced one-side market, obtained by aggregating all sellers to one seller. This means thatour mechanism achieves the same performance as the original one under a more complex setting ofmultiple sellers.

Our framework is applicable to a wide variety of auction models in two-sided markets, thanks tothe strong expressive power of polymatroids. Examples include two-sided extensions of multiunitauctions [4] and matching markets [6] (for divisible goods), and reservation exchange markets [9].We demonstrate how our framework is applied to auctions for display advertisements.

Related work. Double auction is the simplest auction for two-sided markets, where buyers andsellers have unit demand and unit supply, respectively. The famous Myerson-Satterthwaite impossi-bility theorem [15] says that there is no mechanism which simultaneously satisfies incentive compat-ibility (IC), individual rationality (IR), pareto optimality (PO), and budget balance (BB). McAfee[13] proposed a mechanism which satisfies (IC),(IR), and (BB). Recently, Colini-Baldeschi et al. [2]proposed a mechanism which satisfies (IC),(IR), and (BB), and achieves an O(1)-approximation tothe maximum social welfare.

Goel et al. [9] considered a two-sided market model, called a reservation exchange market,for internet advertisement. They formulated several axioms of mechanisms for this model, andpresented an (implementable) mechanism satisfying (IC) for buyers, (IR), maximum social welfare,and a certain fairness for sellers, called the envy-freeness. This mechanism is also based on the

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clinching framework, and sacrifices (IC) for sellers to avoid the impossibility theorem. Their settingis non-budgeted.

Freeman et al. [7] formulated the problem of wagering as an auction in a special two-sidedmarket, and presented a mechanism, called “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-sidedmarkets.

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. In Section 3, we present and analyze ourmechanism. In Section 4, we give proofs.

Notation. Let R+ denote the set of nonnegative real numbers, and let RE+ denote the set of allfunctions from a set E to R+. For f ∈ RE+, we often denote f(e) by fe, and write as f = (fe)e∈E .

Let us recall theory of polymatroids and submodular functions; see [8, 16]. A monotone sub-modular function on set E is a function f satisfying:

f(∅) = 0,

f(S) ≤ f(T ) (S, T ⊆ E, S ⊆ T ),

f(S ∪ {e})− f(S) ≥ f(T ∪ {e})− f(T ) (S, T ⊆ E,S ⊆ T, e ∈ E \ T ),

where the third inequality is equivalent to

f(S) + f(T ) ≥ f(S ∩ T ) + f(S ∪ T ) (S, T ⊆ E).

The polymatroid P := P (f) associated with monotone submodular function f is defined by

P :={x ∈ RE+ |

∑e∈F

xe ≤ f(F ) (F ⊆ E)},

and the base polytope of f is defined by

B :={x ∈ P |

∑e∈E

xe = f(E)},

which is equal to the set of all maximal points in P . A point in B is obtained by the greedyalgorithm [8] in polynomial time, provided the value of f for each subset F ⊆ E can be computedin polynomial time.

2 Main result

We consider a two-sided market consisting of n buyers and m sellers. Our two-sided market ismodeled as a bipartite graph (N,M,E) of disjoint sets N and M of nodes and edge set E ⊆ N×M ,where N and M represent a set of buyers and sellers, respectively, and buyer i ∈ N and seller j ∈Mare adjacent if and only if i wants the goods of seller j. An edge (i, j) ∈ E is denoted by ij.

For buyer i (resp. seller j), let Ei (resp. Ej) denote the set of edges incident to i (resp. j).In the market, the goods are divisible and homogeneous. Each buyer i has three nonnegative real

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numbers vi, v′i, Bi ∈ R+, where vi and v′i are his valuation and bid, respectively, for per unit of

goods sold by buyers in Ei, and Bi is his budget. Each buyer i acts strategically for maximizinghis utility ui (defined later), and hence his true valuation vi is not necessarily equal to bid v′i. Inthis market, each buyer i reports v′i and Bi to the auctioneer. Each seller j also has a valuation ρj∈ R+ for the unit of his goods, and reports ρj 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 avoidthe impossibility theorem as in [9]). He also has a monotone submodular function fj on Ej , whichconstrains transactions of goods through Ej . The value fj(F ) for F ⊆ Ej means the maximumpossible amount of goods transacted through edge subset F . In particular, fj(Ej) is interpreted ashis stock of goods. Note that these assumptions on sellers are characteristic of our model.

Under this setting, the goal is to design a mechanism determining a reasonable allocation. Anallocation A of the auction is a triple A := (w, p, r) of a transaction vector w = (wij)ij∈E , a paymentvector p = (pi)i∈N , and a revenue vector r = (rj)j∈M , where wij is the amount of transactions ofgoods between buyer i and seller j, pi is the payment of buyer i, and rj is the revenue of seller j.For each j ∈ M , the restriction w|Ej of the transaction vector w to Ej ⊆ E must belong to thepolymatroid Pj :

w|Ej ∈ Pj (j ∈M).

Also the payment pi of buyer i must be within his budget Bi:

pi ≤ Bi (i ∈ N).

A mechanism M is a function that gives an allocation A = (w, p, r) from public informationI := ((N,M,E), {v′i}i∈N , {Bi}i∈N , {ρj}j∈M , {fj}j∈M ) that the auctioneer can access. The truevaluation vi of buyer i is private information that only i can access. We regard I and {vi}i∈N asthe input of our model.

Next we define the utilities of buyers and sellers. For an allocation A = (w, p, r), the utilityui(A) of buyer i is defined by:

ui(A) :=

vi∑ij∈Ei

wij − pi if pi ≤ Bi,

−∞ otherwise.

Intuitively, this is the sum of surplus for their goods if the payment is within his budget, and −∞otherwise. The utility of seller j is defined by:

uj(A) := rj + ρj(fj(Ej)−∑ij∈Ej

wij). (1)

This is the sum of revenue and the total valuation of his remaining goods. In this model, weconsider the following properties of mechanism M.

(ICb) Incentive Compatibility of buyers: For every input I, {vi}i∈N , it holds

ui(M(I)) ≤ ui(M(Ii)) (i ∈ N),

where Ii is obtained from I by replacing bid v′i of buyer i with his true valuation vi. Thismeans that it is the best strategy for buyers to report their true valuation.

(IRb) Individual Rationality of buyers: For each buyer i, there is a bid v′i such that i always obtainsnonnegative utility. If (ICb) holds, then (IRb) is written as

ui(M(Ii)) ≥ 0 (i ∈ N).

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(IRs) Individual Rationality of sellers: The utility of each seller j after the auction is at least theutility ρjfj(Ej) at the beginning:

uj(M(I)) ≥ ρjfj(Ej) (j ∈M).

By (1), (IRs) is equivalent to

rj ≥ ρj∑ij∈Ej

wij . (2)

(SBB) Strong Budget Balance: All payments of buyers are directly given to sellers:∑i∈N

pi =∑j∈M

rj .

(PO) Pareto Optimality: There is no allocation A := (w, p, r) which satisfies∑i∈N

pi ≥∑j∈M

rj and

the following three conditions:

ui(M(I∗)) ≤ ui(A) (i ∈ N),

uj(M(I∗)) ≤ uj(A) (j ∈M),

and at least one of the inequalities holds strictly, where I∗ is obtained from I by replacing{v′i}i∈N with {vi}i∈N . Namely, there is no other allocation superior to that given by M inall aspects, provided all buyers report their true valuation.

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

Theorem 2.1. There exists a mechanism which satisfies all of (ICb),(IRb),(IRs), (SBB), and (PO).

The detail of our mechanism is explained in the next section.

Remark 1. Maximizing the social welfare, the sum of utilities of all participants, is usually set asthe goal in traditional auction theory. However, in budgeted settings, it is shown in [4] that themaximum social welfare and incentive compatibility of buyers cannot be achieved simultaneously.As in the previous mechanisms [3, 4, 6, 10, 11], we take priority over incentive compatibility.

Application to display ad auction of multiple sellers and slots Here we consider followingauction for display advertisements (ads) between advertisers and owners of websites. Each ownerj wants to sell ad slots in his website. Each advertiser i wants to purchase the slots. Namely, theowners are sellers, and advertisers are buyers, where buyer i is linked to seller j if i is interested inthe slots of the website of j. The market is modeled as bipartite graph (N,M,E) as above.

Each seller has sj ad slots. Each slot l ∈ {1, 2, . . . , sj} has a barometer αjl ∈ R+ for the qualityof ads, which is measured by the expected number of views of l over a certain period. The view-impression is a unit of this barometer; namely, slot l has αjl view-impressions. In the market, buyerspurchase view-impressions from sellers, and have bids and valuations for the unit view-impression.After the auction, suppose that buyer i obtains wij view-impressions from j. Seller (owner) jassigns ads of buyers (linked to j) to his slots so that the same ads cannot be displayed in distinctslots at the same time. An ad-slotting is naturally represented by (S, ψ) for a subset S of buyersand an injection ψ from S to the set of his slots {1, 2, . . . , sj}. (The number of slots is at least |S|

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by adding dummy slots of 0 view-impressions.) Seller j displays ads of i according to a probabilitydistribution p on the set of all ad-slottings satisfying

wij =∑(S,ψ)

p(S, ψ)αjψ(i). (3)

Namely the expected number of view-impressions of ads i in the website of j is equal to wij . Theexistence of such a p constrains transactions wij , and is equivalent to the condition that

∑i∈S wij

is at most the sum of |S| highest αjl for each subset S of buyers linked to j. This condition canbe written by a monotone submodular function, and seller j restricts his transactions wij by thecorresponding polymatroid, to conduct the above way of ad-slotting after the auction. Now themarket falls into our model, and our mechanism is applicable, where buyers can have budget limits.

This market is a modification of Adwords auctions in [11] for display ad auctions with multiplewebsites, where the way of ad-slotting according to (3) is based on their idea. Also this market isa special reservation exchange market in the sense of [9]. Our framework is applicable to generalreservation exchange market with budget constraints and polymatroid constraints.

3 Polyhedral Clinching Auctions in Two-sided Markets

3.1 Mechanism

Here we describe our mechanism for Theorem 2.1. First we make the following preprocessing on themarket. Buyers and sellers are numbered as N = {1, 2, ..., n} and M = {1, 2, ...,m}. For each sellerj ∈ M , add to N a virtual buyer n + j corresponding to j, and add to E a new edge connectingn+ j and j; buyer n+ j transacts only with j. The virtual buyer n+ j has sufficiently large budgetBn+j =∞, and reports the valuation ρj of j as bid v′n+j . The valuation vn+j is set as vn+j := v′n+j

(though we do not use vn+j). Submodular function fj is extended by

fj(F ) :=

{fj(Ej) if (n+ j)j ∈ F,fj(F ) otherwise.

(4)

Namely the goods purchased by buyer n+ j is interpreted as the unsold goods of j. The utility ofseller j (in the original market) is the sum of the revenue of seller j and the utility of buyer n+ jafter the auction; then we need not to take ρj into account by this preprocessing.

After the preprocessing, we apply Algorithm 1. The meaning of variables ci, ξij , wij , pi, rj , di, land fixed parameter ε > 0 is explained as follows:

• ci is the price clock of buyer i, which is used as the transaction cost for per unit of goods atthis moment. It starts at 0 and increases by ε in each step.

• ξij is the increase of transactions between buyer i and seller j in the current iteration.

• wij is the total amount of transaction between buyer i and seller j. Transaction vector(wij)ij∈E must belong to the polymatroid

P :=⊕j∈M

Pj = {w ∈ RE+ | w|Ej ∈ Pj (j ∈M)}.

• pi is the payment of buyer i.

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Algorithm 1 Polyhedral Clinching Auction for Two-Sided Market

1: pi := 0, ci := 0, di :=∞ (i ∈ N) and l := 1.2: rj := 0 (j ∈M) and wij := 0 (ij ∈ E).3: while di 6= 0 for some i ∈ N do4: for i = 1, 2, . . . , n+m do5: Clinch a maximal increase (ξij)ij∈Ei not affecting other buyers.6: wij := wij + ξij (ij ∈ Ei).7: pi := pi + ci

∑ij∈Ei

ξij (i ∈ N).

8: di :=

{(Bi − pi)/ci if ci < v′i,

0 otherwise,(i ∈ N).

9: end for10: rj := rj +

∑ij∈Ej

ciξij (j ∈M).

11: cl := cl + ε.12: l := l + 1 mod m+ n.

13: dl :=

{(Bl − pl)/cl if cl < v′l,

0 otherwise.14: end while

• rj is the revenue of seller j.

• di is the demand of buyer i, which is interpreted as the maximum amount of transactionsthat buyer i can get in the future.

• l is the buyer whose price clock is increased in the next iteration.

Algorithm 1 terminates when di = 0 for all buyer i, and outputs (w, p, r) at this moment. Theallocation (w, p, r) obtained by Algorithm 1 includes that for virtual buyers. Thus we transformthis allocation into the allocation without virtual buyers in the following way. For each j, we regardw(n+j)j as the remaining goods of seller j, and update the revenue rj by

rj := rj − pn+j .

After the update, the triple (w, p, r) for non-virtual buyers and sellers is regarded as the allocationin the original market, and as the output of our mechanism.

We explain the details of “not affecting other buyers” in line 5. For transaction vector w anddemand vector d := (di)i∈N , define the remnant supply polytope Pw,d by

Pw,d := {x ∈ RE+ | w + x ∈ P,∑ij∈Ei

xij ≤ di (i ∈ N)}. (5)

In addition, for ξ := (ξij)ij∈Ei ∈ REi+ , define the remnant supply polytope P iw,d(ξ) of remaining

buyers N \ {i} by

P iw,d(ξ) := {u ∈ RN\{i}+ | ∃x ∈ Pw,d, x|Ei = ξ,∑kj∈Ek

xkj = uk (k ∈ N \ {i})}. (6)

The first polytope Pw,d ⊆ P represents the feasible increases of transactions under w and d, and

the second polytope P iw,d(ξ) ⊆ RN\{i}+ represents the possible amounts of goods that buyers except

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i can get in the future, provided i got ξ ∈ REi+ in this iteration. Then the condition for clinching

ξi := (ξij)ij∈E , not to affect other buyers is naturally written as P iw,d(ξ) = P iw,d(0). This motivates

to define the polytope P iw,d by

P iw,d := {ξ ∈ REi+ | P iw,d(ξ) = P iw,d(0)},

which we call the clinching polytope of buyer i. A clinching vector (ξij)ij∈Ei in line 5 is chosen asa maximal vector in the clinching polytope P iw,d. Such a clinching can be done in polynomial timeby the following theorem, which implies that our mechanism is implementable in practice.

Theorem 3.1. The clinching polytope P iw,d is a polymatroid, and the value of the corresponding

submodular function can be computed in polynomial time. In particular, a maximal vector of P iw,dcan be computed in polynomial time by the greedy algorithm.

The proof of Theorem 3.1 is given in Section 4.2.

3.2 Analysis

Here we analyze our mechanism for proving Theorem 2.1. First we show (SBB) for our mechanism.In line 8 and line 10 of Algorithm 1, the total payment of buyers (including virtual buyers) isdirectly given to sellers in each iteration. Also both the total payment and the total revenue aredecreasing by the total payment of virtual buyers in the transformation of the allocation. Thus wehave:

Lemma 3.2. Our mechanism satisfies (SBB).

For the individual rationality of sellers (IRs), we will prove in Section 4.4 that each seller j andnon-virtual buyer i transact only when the price clock ci is at least valuation ρj and virtual buyern+ j vanishes, i.e., dn+j = 0. Consequently we obtain (IRs); see also (2).

Proposition 3.3. Our mechanism satisfies (IRs).

Next we consider other properties (ICb),(IRb), and (PO) in Theorem 2.1. For proving theseproperties, we utilize the framework by Goel et al. [11] on one-sided markets under polymatroidalenvironments. From our two-sided market (including virtual buyers), we construct a one-sidedmarket consisting of N and one seller (= auctioneer) to which their framework is applicable. Thepolymatroidal environment on N is defined by the following polytope P ⊆ RN+ :

P := {y ∈ RN+ | ∃w ∈ P, yi =∑ij∈Ei

wij (i ∈ N)}.

Then P is a polymatroid, which immediately follows from a network induction of a polymatroid [14].An allocation of this market is a pair (y, p) of y, p ∈ RN+ , where yi and pi is the transactionand payment, respectively, of buyer i to the seller. The transaction vector y must belong to thepolymatroid P . For each buyer i, payment pi must be within his budget Bi. The utility of buyer i

is defined in (13) by replacing∑ij∈Ei

wij with yi. Now the original polyhedral clinching auction [11]

is applicable to this one-sided market, and is given in Algorithm 2. The variables pi, ci, di, l andthe parameter ε are the same as in Algorithm 1. The meaning of other variables are:

• yi is the amount of goods of buyer i.

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Algorithm 2 Polyhedral Clinching Auction [11] for One-sided Market.

1: yi := 0, pi := 0, ci := 0, di :=∞ (i ∈ N) and l := 12: while d 6= 0 do3: Clinch a maximal increase (ζi)i∈N not affecting other buyers.4: yi := yi + ζi, pi := pi + ciζi (i ∈ N)5: cl := cl + ε

6: di :=

{(Bi − pi)/ci if ci < v′i,

0 otherwise,(i ∈ N).

7: l := l + 1 mod m+ n8: end while

• ζi is the amount of goods that buyer i clinches in the current iteration.

Again we explain the details of “not affecting other buyers” in line 3 according to [11]. For trans-action vector y and demand vector d, define the remnant supply polytope Py,d by

Py,d := {z ∈ RN+ | y + z ∈ P , yi ≤ di (i ∈ N)}.

Also, for ζ ∈ R+, define the remnant supply polytope of remaining buyers N \ {i} by

P iy,d(ζ) := {σ ∈ RN\{i}+ | ∃z ∈ Py,d, zi = ζ, zk = σk (k ∈ N \ {i})}.

These polytopes correspond to the polytopes (5) and (6) in Algorithm 1, respectively. For each i,clinching ζi in line 3 is chosen as the maximum ζ for which P iy,d(ζ) = P iy,d(0).

Theorem 3.4. Suppose that ε is the same in both Algorithms. At the end of algorithm, (y, p) in

Algorithm 2 is equal to ((∑ij∈Ei

wij)i∈N , p) in Algorithm 1.

The proof of Theorem 3.4 is given in Section 4.3. Goel et al. [11] proved that Algorithm 2satisfies (ICb) and (IRb). Consequently we have:

Corollary 3.5. Our mechanism satisfies (ICb) and (IRb).

Goel et al. [11] proved (PO) for Algorithm 2. We combine Theorem 3.4 with their proof method.and careful considerations on utilities of sellers, and prove (PO) for our mechanism:

Theorem 3.6. Our mechanism satisfies (PO).

The proof is given in Section 4.5.

Remark 2. Our model and mechanism are naturally adapted to the case where utility ui of eachbuyer i is quasi-linear on a convex region defined by a concave non-decreasing function φi : R+ → R+

with φi(0) = 0:

ui(A) :=

vi∑ij∈Ei

wij − pi if pi ≤ φi(∑ij∈Ei

wij),

−∞ otherwise.

Such a concave budget constraint was considered by Goel et al. [10] for one-sided markets. Theyshowed that the polyhedral clinching auction can be generalized to this setting. By using theiridea, our two-sided version can also be adapted to have the same conclusion in Theorem 2.1, whereeach buyer reports the region {(pi, zi) | pi ≤ φ(zi)} to the auctioneer, and the budget feasibility isreplaced by pi ≤ φ(

∑ij∈Ei

wij).

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Remark 3. In our mechanism, feasible increases in clinching form a polymatroid, and hence amaximal increase can be easily obtained by the greedy algorithm. However there is a degree offreedom of choosing a maximal increase. The price of goods is increasing, and each seller obtains(actual) revenues from non-virtual buyers after his virtual buyer vanishes. Hence sellers obtainhigher revenues when their goods are sold in later iterations. This means that the point selection ineach clinching is directly related to sellers’ revenues. Formalizing an appropriate fairness conceptfor sellers and designing a fair clinching rule deserve challenging future work.

4 Proofs

In this section, we use the following abbreviated notation. For φ ∈ RS and X ⊆ S, let

φ(X) :=∑x∈X

φ(X).

In the market (N,M,E), for node subset Z, let EZ denote the set of edges incident to nodes in Z.For edge subset F , let NF denote the set of buyers incident to F .

In the following, we often use minimum-value functions of form

G(X) = minY

H(X,Y ).

By a minimizer Y of G(X) we mean such Y that satisfies G(X) = H(X,Y ).

4.1 Polymatroidal network flow

In the proof, we utilize polymatroidal network flow model by Lawler and Martel [12]. A poly-matroidal network is a directed network (V,E) with source s and sink t such that each node vhas polymatroids P+

v and P−v defined on the sets δ+v and δ−v of edges leaving v and entering v,

respectively.A flow is a function ϕ : E → R+ satisfying

ϕ(δ+v) = ϕ(δ−v) (v ∈ V \ {s, t}),ϕ|δ+v ∈ P+

v ,

ϕ|δ−v ∈ P−v .

Let f+v and f−v be monotone submodular functions corresponding to P+

v and P−v , respectively. Inthe case where the network has edge-capacity c : E → R+, the polymatroid representing capacity-constraint is given by submodular function

F 7→ c(F ) (F ⊆ δ+v (or δ−v )). (7)

The flow-value of a flow ϕ is defined as ϕ(δ+s ) − ϕ(δ−s )(= ϕ(δ−t ) − ϕ(δ+

t )). The following is ageneralization of the max-flow min-cut theorem for polymatroidal networks, and is also a versionof the polymatroid intersection theorem.

Theorem 4.1 (Lawler and Martel [12]). The maximum value of a flow is equal to

minU,A,B

{∑

v∈V \U

f−v (δ−v ∩A) +∑v∈U

f+v (δ+

v ∩B)}, (8)

where U ranges over all node subsets with s ∈ U 63 t and {A,B} ranges over all partitions of edgesleaving U .

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4.2 Proof of Theorem 3.1

Let d be the demand vector, and w the transaction vector in line 4 in the current iteration. Westart to study polytopes Pw,d, P

iw,d(ξ), and P iw,d from viewpoint of polymatroidal network flow. For

each seller j, define polytope Pj,w by

Pj,w := {x ∈ REj

+ | w|Ej + x ∈ Pj}.

Since Pj,w is a contraction of Pj (see; e.g. Fujishige [8, Section 3.1]), Pj,w is a polymatroid, and thecorresponding submodular function fj,w : 2Ej → R+ is given by

fj,w(F ) := minF ′⊇F

{fj(F ′)− w(F ′)} (F ⊆ Ej).

Let us construct a polymatroidal network from the market (N,M,E), demand d, and polyma-troids Pj,w for each j ∈ M . Each edge in E is directed from N to M . For each buyer i, considercopy i′ and edge i′i. The set of the copies of X ⊆ N is denoted by X ′. Add source node s and edgesi′ for each i′ ∈ N ′. Also, add sink node t and edge jt for each j ∈ M . Edge-capacity c is definedby

c(e) :=

{di if e = i′i for i ∈ N,∞ otherwise.

(9)

Two polymatroids of each node in N ∪ N ′ ∪ M \ {s, t} is defined as follows. For j ∈ M , thepolymatroid on δ−j = Ej is defined as Pj,w, and the other polymatroid on δ+

j is defined by capacityas in (9). Both two polymatroids of other nodes are defined by capacity (9). Let N denote theresulting polymatroidal network.

A flow ϕ in N is an edge-weight satisfying the flow-conservation law, the capacity constraint,and polymatroid constraint ϕ|Ej ∈ Pj,w (j ∈M). Now the remnant supply polytope Pw,d is writtenby using flows in the network N :

Pw,d = {x ∈ RE+ | ∃ϕ : flow in N , x = ϕ|E}.

In the proof, the following function fw,d : 2E → R+ plays a key role

fw,d(F ) := maxx∈Pw,d

x(F ) = maxϕ: flow inN

ϕ(F ) (F ⊆ E). (10)

We also define f, fw : 2E → R+ by

f(F ) :=∑j∈M

fj(Ej ∩ F ) (F ⊆ E),

fw(F ) := minF ′⊇F

{f(F ′)− w(F ′)} =∑j∈M

fj,w(Ej ∩ F ) (F ⊆ E).

Note that both f and fw is monotone submodular. In particular, f corresponds to the polymatroidP (by a theorem in McDiarmid [14]). Then we obtain the following:

Theorem 4.2. For F ⊆ E, it holds

fw,d(F ) = minX⊆NF

{fw(EX ∩ F ) + d(NF \X)}, (11)

and fw,d(F ) can be computed in polynomial time, provided the value oracle of each fj is available.

11

Proof. We modify N so that fw,d(F ) is equal to the maximum flow value. Change the capacityof each e ∈ E \ F to 0. Also, for each i ∈ N \ NF , change c(i′i) to 0. Then, the maximum flowvalue is equal to fw,d(F ). By Theorem 4.1, the value fw,d(F ) is equal to (8). Observe that subsetU attaining the minimum can take a form of {s} ∪N ′ ∪X for X ⊆ NF . Then the set δ+

U of edgesleaving U is equal to {i′i | i ∈ NF \ X} ∪ EX . In a partition {A,B} of δ+

U attaining minimum,edge i′i contributes di in both cases of i′i ∈ A and i′i ∈ B. Each edge e in F must be in A (byc(e) =∞), and each edge e in EX \ F can be in B (by c(e) = 0). Thus we have (11).

By the above modification, fw,d(F ) can be computed by solving a maximum polymatroidal flowproblem. There are several polynomial time algorithms for this problem; see Fujishige [8, Section5.5].

Note that fw,d is not necessarily submodular. Let ξ ∈ RE+. Next we consider the remnant supplypolytope of remaining buyers except i

P iw,d(ξ) := {u ∈ RN\{i}+ | ∃x ∈ Pw,d, x|Ei = ξ,∑kj∈Ek

xkj = uk (k ∈ N \ {i})}.

Define a set function hiξ on N \ {i} by

hiξ(Y ) = minG⊆Ei

{fw,d(EY ∪G)− ξ(G)} (Y ⊆ N \ {i}).

Theorem 4.3. hiξ is monotone submodular, and P iw,d(ξ) coincides with the polymatroid associated

with hiξ.

Proof. We utilize polymatroidal network N . Then u ∈ P iw,d(ξ) if and only if there exists a flowϕ in N such that ϕ(ij) = ξ(ij) for ij ∈ Ei, ϕ(i′i) = ξ(Ei), and ϕ(sk′) = uk for k ∈ N \ {i}. Bymodifying N , we provide an equivalent condition as follows. Replace ij ∈ Ei by sj with capacityξij , change capacity of sk′ to uk for k ∈ N \ {i}, delete nodes i′ and i. By ξ(Ei) ≤ di, the abovecondition is equivalent to: the maximum flow value is equal to ξ(Ei) + u(N \ {i}), or equivalently,U = {s} and (A,B) = (∅, δ+

s ) attain the minimum of (8). In the minimum of (8), it suffices toconsider U of form U = {s} ∪ Y ′ ∪ Z for Z ⊆ Y ⊆ N . Then the set δ+

U of edges leaving U is equalto Ei ∪ {sk′ | k ∈ N \ (Y ∪ {i})} ∪ {k′k | k ∈ Y \ Z} ∪ EZ . In any partition {A,B} of δ+

U , edgesk′ contributes uk, and k′k contributes dk. It must hold EZ ⊆ A. Thus it suffices to consider apartition {G,Ei \G} of Ei. From this and Theorem 4.1, u ∈ Pw,d if and only if

ξ(Ei) + u(N \ {i}) ≤ fw(EZ ∪G) + ξ(Ei \G) + u((N \ {i}) \ Y ) + d(Y \ Z)

(G ⊆ Ei, Z ⊆ Y ⊆ N \ {i}).

This is equivalent to

u(Y ) ≤ minZ⊆Y,G⊆Ei

{fw(EZ ∪G) + d(Y \ Z)− ξ(G)} (Y ⊆ N \ {i}). (12)

Define hiξ : 2N\{i} → R+ by the right hand side of (12). It suffices to show that hiξ is monotone

submodular, and equals to hiξ. Here hiξ is viewed as a network induction of fw [14], and is necessarily

monotone submodular; one can also see this fact directly by taking minimizers of hiξ(X) and hiξ(Y ).

Finally we show hiξ(Y ) = hiξ(Y ). Here hiξ is also written as

hiξ(Y ) = minG⊆Ei

{ minZ′⊆NEY ∪G

{fw(EZ′ ∩ (EY ∪G)) + d(NEY ∪G \ Z′)} − ξ(G)}

= minZ′⊆Y ∪{i}, G⊆Ei

{fw(EZ′ \ (Ei \G)) + d(NEY ∪G \ Z′)− ξ(G)}, (13)

12

where we use the fact that if G = ∅, then Z ′ and Z ′ \ {i} have the same value in (13). We considerthe minimum of (13) over all Z ′ ⊆ Y and over all Z ′ = Z ∪ {i} with Z ⊆ Y . The first minimum αis equal to

minZ′⊆Y, G⊆Ei

{fw(E′Z) + d(NEY ∪G \ Z)− ξ(G)}

= minZ′⊆Y, G⊆Ei

{fw(E′Z) + d(Y \ Z) + d(NG)− ξ(G)}

= minZ′⊆Y

{fw(E′Z) + d(Y \ Z)},

where G = ∅ attains the minimum since di ≥ ξ(G). The second minimum β is equal to

minZ⊆Y, G⊆Ei

{fw(EZ∪{i} \ (Ei \G)) + d(NEY ∪G \ (Z ∪ {i}))− ξ(G)}

= minZ⊆Y, G⊆Ei

{fw(EZ ∪G) + d(Y \ Z)− ξ(G)}

= hiξ(Y )

Observe that α ≥ β. Thus hiξ(Y ) = min{α, β} = β = hiξ(Y ) as required.

Finally we consider the clinching polytope P iw,d of buyer i. Define a set function hiw,d on Ei by

hiw,d(F ) := fw,d(EN\{i} ∪ F )− fw,d(EN\{i}) (F ⊆ Ei).

Theorem 4.4. hiw,d is monotone submodular, and P iw,d coincides with the polymatroid associated

with hiw,d.

Theorem 3.1 is obtained as an immediate corollary of Theorem 4.2 and Theorem 4.4. For theproof of Theorem 4.4, we show the following lemma.

Lemma 4.5. (i) Let F ⊆ Ei. The function X 7→ fw,d(EX \ (Ei \ F )) is monotone submodularon N .

(ii) Let X ⊆ N \ {i}. The function G 7→ fw,d(EX ∪G) is monotone submodular on Ei.

Proof. The monotonicity of both functions in (i) and (ii) is immediate from the monotonicity of fw,din Theorem 4.2. Thus we show the submodularity.

(i) The submodularity can directly be shown by using formula (11) and taking minimizers offw,d(EX \ (Ei \F )) and fw,d(EY \ (Ei \F )). We use a flow interpretation. Consider N , and removeall edges in E \ (EX \ (Ei \F )). Then fw,d(EX \ (Ei \F )) is equal to the maximum value of a flowX ′ to t. Thus, X → fw,d(EX \ (Ei \ F )) is a network induction of fw, and submodular.

(ii) Let F,G ⊆ Ei. It suffices to consider the case where F 6⊆ G and G 6⊆ F . In particular, bothF and G are nonempty. Thus NEX∪F = NEX∪G = X ∪ {i}.

Suppose that Y,Z ⊆ X ∪ {i} are minimizers of fw,d(EX ∪ F ) and fw,d(EX ∪ G), respectively.Namely fw,d(EX ∪F ) = fw(EY ∩ (EX ∪F )) + d((X ∪{i}) \Y ) and fw,d(EX ∪G) = fw(EZ ∩ (EX ∪G)) + d((X ∪ {i}) \ Z).

13

Case 1: both Y and Z contain i. Then we directly obtain

fw,d(EX ∪ F ) + fw,d(EX ∪G)

= fw(EY \{i} ∪ F ) + d((X ∪ {i}) \ Y ) + fw(EZ\{i} ∪G) + d((X ∪ {i}) \ Z)

≥ fw((EY \{i} ∪ EZ\{i}) ∪ (F ∪G)) + d((X ∪ {i}) \ (Y ∪ Z))

+ fw((EY \{i} ∩ EZ\{i}) ∪ (F ∩G)) + d((X ∪ {i}) \ (Y ∩ Z))

= fw(EY ∪Z\{i} ∪ (F ∪G)) + d((X ∪ {i}) \ (Y ∪ Z))

+ fw(EY ∩Z\{i} ∪ (F ∩G)) + d((X ∪ {i}) \ (Y ∩ Z))

≥ fw,d(EX ∪ (F ∪G)) + fw,d(EX ∪ (F ∩G)).

Case 2: one of Y and Z does not contain i, say i 6∈ Z. Since G 6= ∅, we have

fw,d(EX ∪G) = minZ⊆X{fw(EZ ∩ (EX ∪G)) + d((X ∪ {i}) \ Z)}

= minZ⊆X{fw(EZ) + d(X \ Z) + di} = fw,d(EX) + d(NF∪G).

By the monotonicity of fw,d and (11), we obtain

fw,d(EX ∪ F ) + fw,d(EX ∪G) = fw,d(EX ∪ F ) + fw,d(EX) + d(NF∪G)

≥ fw,d(EX ∪ (F ∩G)) + fw,d(EX) + d(NF∪G)

≥ fw,d(EX ∪ (F ∩G)) + fw,d(EX ∪ (F ∪G)).

Therefore the function G 7→ fw,d(EX ∪G) is submodular.

Proof of Theorem 4.4. By Theorem 4.3, we have P iw,d(ξ) = P iw,d(0) if and only if hiξ(X) = hi0(X)for each X ⊆ N \ {i}. By the monotonicity of fw,d, the latter condition is equivalent to

minF⊆Ei

{fw,d(EX ∪ F )− ξ(F )} = fw,d(EX) (X ⊆ N \ {i}).

Thus ξ ∈ REi+ belongs to P iw,d if and only if for each F ⊆ Ei, it holds

ξ(F ) ≤ fw,d(EX ∪ F )− fw,d(EX) (X ⊆ N \ {i}) (14)

= minX⊆N\{i}

{fw,d(EX ∪ F )− fw,d(EX)}

= minX⊆N\{i}

{fw,d(EX∪{i} \ (Ei \ F ))− fw,d(EX \ (Ei \ F ))}

= fw,d(EN \ (Ei \ F ))− fw,d(EN\{i} \ (Ei \ F ))

= fw,d(EN\{i} ∪ F )− fw,d(EN\{i})= hiw,d(F ),

where the third equality follows from the submodularity (Lemma 4.5 (i)) of

X 7→ fw,d(EX \ (Ei \ F )) =

{fw,d(EX\{i} ∪ F ) if i ∈ X,fw,d(EX) otherwise,

on N . By Lemma 4.5 (ii), hiw,d is monotone submodular on Ei. This concludes that the clinching

polytope P iw,d is the polymatroid associated with hiw,d, as required.

14

By (14) in the proof, we obtain the following corollary.

Corollary 4.6. Suppose that buyer i clinches ξ ∈ REi+ of goods. For each F ⊆ Ei it holds

ξ(F ) ≤ fw,d(EX ∪ F )− fw,d(EX) (X ⊆ N \ {i}).

Also we can obtain the following.

Corollary 4.7. Buyer i gets ξ(Ei) = fw,d(E)− fw,d(E \ Ei) amount of goods.

4.3 Proof of Theorem 3.4

Here we give a proof of Theorem 3.4. It is known in [11] that P is a polymatroid, and thecorresponding submodular function g is

g(X) = f(EX) (X ⊆ N).

Suppose that transaction vector w ∈ RE+ and demand vector d ∈ RN+ are given. Then Pw,d is apolymatroid, and the corresponding submodular function gw,d is given by

gw,d(X) = minZ⊆X{minZ′⊇Z

{g(Z ′)− w(EZ′)}+ d(X \ Z)}.

The clinch ζi in Algorithm 2 is given as follows:

Theorem 4.8 (Goel et al. [11]). It holds ζi = gw,d(N)− gw,d(N \ {i}) for each i ∈ N .

The goal of this section is to prove following:

Theorem 4.9. It holds fw,d(EX) = gw,d(X) for each X ⊆ N .

Proposition 4.10. The value fw,d(E) − fw,d(E \ Ei) does not change after the clinch of buyerk ∈ N \ {i}.

In particular, the total amount ξ(Ei) of the clinch of each buyer i is determined by w and d atthe beginning of For Loop in line 4 (i.e., independent of the ordering of buyers). By Theorems 4.8and 4.9, we obtain Theorem 3.4, and conclude that in Algorithm 1 each buyer obtains the sametotal amount of goods as in Algorithm 2.

To prove Theorem 4.9, we focus on the minimizer of gw,d(X), where gw,d : 2N → R+ is defined by

gw,d(X) := fw,d(EX) = minZ⊆X{fw(EZ) + d(X \ Z)} (X ⊆ N).

Then the following lemma holds.

Lemma 4.11. For Z ⊆ N , the following conditions are equivalent.

(i) Z is a (unique) minimal minimizer of gw,d(Z).

(ii) Z is a minimal minimizer of gw,d(X) for some X with Z ⊆ X ⊆ N .

Proof. It suffices to show that (ii) implies (i). Suppose that Z is a minimal minimizer of gw,d(X),but Z is not a minimal minimizer of gw,d(Z). Then there exists a set Z ′ ⊂ Z such that

fw(EZ′) + d(Z \ Z ′) ≤ fw(EZ).

Then

fw(EZ) + d(X \ Z) ≥ fw(EZ′) + d(Z \ Z ′) + d(X \ Z)

= fw(EZ′) + d(X \ Z ′),

which contradicts the minimality of Z.

15

Let U be the set of subsets Z ⊆ N satisfying the conditions in Lemma 4.11. Then we obtainthe following proposition.

Proposition 4.12. For X ∈ U , it holds

gw,d(X) = f(EX)− w(EX). (15)

Proof. Since X ∈ U , we have

gw,d(X) = fw(EX)(:= minF⊇EX

{f(F )− w(F )}).

At the beginning of the auction, it holds U = 2N , and (15) holds because of wij = 0 (ij ∈ E) andthe monotonicity of fj (j ∈ M). We are going to show that subsets in U vanish only when dl isrecalculated in line 13. First we show that U does not change, and (15) holds throughout the Forloop (line 4-line 9). In an iteration, for X ∈ U , suppose that (15) holds in the line 4. Let d be thecurrent demand vector, and w the current transaction vector. Then for each proper subset Z ⊂ X,it holds

fw(EX) < fw(EZ) + d(X \ Z). (16)

Suppose that buyer i ∈ X clinches ξ ∈ REi+ of goods. Recall that gw,d is represented by

gw,d(X) := fw,d(EX) = minZ⊆X{ minF⊇EZ

{f(F )− w(F )}+ d(X \ Z)} (X ⊆ N).

For each F ⊇ EX , f(F ) − w(F ) decreases by ξ(Ei) after w is recalculated in line 6. Then theequality

fw(EX) = f(EX)− w(EX). (17)

is keeping after line 6. Also the demand di decreases by ξ(Ei) after di is recalculated in line 8. Notethat nontrivial clinch ξ 6= 0 occurs only if di > 0 and ci < v′i. Thus, min

F⊇EZ

{f(F )−w(F )}+d(X \Z)

decreases by ξ(Ei) for each Z ⊆ X (regardless of whether Z contains i or not). Therefore, X isstill a minimal minimizer of gw,d(X); X ∈ U . By this fact together with (17), we obtain (15), asrequired.

Suppose that i ∈ N \X clinches ξ ∈ REi+ of goods. By Corollary 4.6, it holds that

ξ(F ) ≤ hiw,d(F ) ≤ fw,d(EZ ∪ F )− fw,d(EZ) ≤ fw(EZ ∪ F )− fw,d(EZ) (18)

for each F ⊆ Ei, Z ⊆ N . In this case, demand dk for each k ∈ X is unchanged in line 8. After therecalculation of w and d, gw,d(X) is written as

minZ⊆X{ minF⊇EZ

{f(F )− w(F )− ξ(Ei ∩ F )}+ d(X \ Z)}. (19)

Since it holds

minG⊆Ei

{fw(EZ ∪G)− ξ(G)} = minG⊆Ei

{ minG′⊇EZ∪G

{f(G′)− w(G′)} − ξ(G)}

= minF⊇EZ

{ minF ′⊆Ei

{f(F )− w(F )− ξ((Ei ∩ F ) \ F ′)}} = minF⊇EZ

{f(F )− w(F )− ξ(Ei ∩ F )},

we obtain

minF⊇EZ

{f(F )− w(F )− ξ(Ei ∩ F )} = minG⊆Ei

{fw(EZ ∪G)− ξ(G)} = fw(EZ),

16

where the second equality is obtained by (18). Thus, (19) is equal to

minZ⊆X{fw(EZ) + d(X \ Z)}.

By (16), X is still a minimal minimizer of gw,d(X); X ∈ U . This indicates that gw,d(X) does notchange, and (15) still holds after the clinch of buyer i ∈ N \ X. Therefore, the transactions ofbuyers do not change the set U , and it still holds (15) in each iteration.

Finally we show that no sets are added to U by the calculation of dl in line 13. It is sufficientfor us to consider the case of l ∈ X. Here for each X /∈ U which contains l, there exists a set Z ⊂ Xsuch that

fw(EZ) + d(X \ Z) ≤ fw(EX).

Since fw does not depend on d, and d is non-increasing, this inequality still holds after the recal-culation of dl. Therefore it still holds that X /∈ U .

Proof of Proposition 4.10. Suppose that k clinches ξk ∈ REk+ amount of goods in an iteration. We

prove that both fw,d(E) and fw,d(E\Ei) decrease by ξk(Ek). Suppose that Z is a minimal minimizerof fw,d(E) in line 4. Then it holds

fw(EZ) + d(N \ Z) ≤ fw(EY ) + d(N \ Y ) (Y ⊆ N). (20)

By Proposition 4.11, we obtain Z ∈ U . Thus by Proposition 4.12, it holds

fw,d(E) = fw(EZ) + d(N \ Z) = f(EZ)− w(EZ) + d(N \ Z).

After the recalculation in line 9, f(EZ) − w(EZ) + d(N \ Z) decreases by ξk(Ek) regardless ofwhether Z contains k or not (since k ∈ N).

Let Z ′ be a minimal minimizer of fw,d(E) in line 9. As seen in the proof of Proposition 4.12,the family U is monotone non-increasing throughout Algorithm 1, and hence it holds that Z ′ ∈ Uin line 4. Therefore the value fw(EZ′) + d(N \ Z ′) = f(EZ′) − w(EZ′) + d(N \ Z ′) also decreasesby ξk(Ek) after the recalculation. Thus we obtain fw(EZ) + d(N \Z) = fw(EZ′) + d(N \Z ′) in theFor Loop. We can conclude that fw,d(EN ) decreases by ξ(Ek) through the clinch of ξ(Ek) amountof goods by buyer k. The same argument is true for fw,d(E \ Ei) (since k ∈ N \ {i}).

Proof of Theorem 4.9. By Theorem 4.2,

fw,d(EX) = gw,d(X) = minZ⊆X{fw(EZ) + d(X \ Z)}

= minZ⊆X{ minF⊇EZ

{f(F )− w(F )}+ d(X \ Z)}

≤ minZ⊆X{minZ′⊇Z

{f(EZ′)− w(EZ′)}+ d(X \ Z)}

= minZ⊆X{minZ′⊇Z

{g(Z ′)− w(EZ′)}+ d(X \ Z)} = gw,d(X).

Let Z ⊆ X be a minimal minimizer of gw,d(X). Then by Lemma 4.11, it holds Z ∈ U . Thus byProposition 4.12,

gw,d(Z) = fw(EZ) = minF⊇EZ

{f(F )− w(F )} = f(EZ)− w(EZ).

Since it holds

fw(EZ) = minF⊇EZ

{f(F )− w(F )} ≤ minZ′⊇Z

{g(Z ′)− w(E′Z)} ≤ f(EZ)− w(EZ),

17

we obtain equalities

fw(EZ) = minF⊇EZ

{f(F )− w(F )} = minZ′⊇Z

{g(Z ′)− w(E′Z)} = f(EZ)− w(EZ).

It holds

fw,d(EX) = gw,d(X) = fw(EZ) + d(X \ Z)

= minZ′⊇Z

{g(Z ′)− w(EZ′)}+ d(X \ Z)

≥ gw,d(X).

Therefore it holds fw,d(EX) = gw,d(X).

4.4 Proof of Proposition 3.3

Lemma 4.13. For each seller j, if cn+j < ρj, then wij = 0 (i ∈ Nj \ {n+ j}).

Proof. We consider the transaction of seller j with buyer i ∈ Nj \ {n + j} when cn+j < ρj and

wij = 0 (i ∈ Nj \ {n + j}). By Corollary 4.6, the transaction ξ ∈ REi+ of buyer i ∈ Nj \ {n + j}

through edge subset F ⊆ Ei satisfies

ξ(F ) ≤ fw,d(EX ∪ F )− fw,d(EX) (X ⊆ N \ {i}).

Letting X = {n+ j} and F = {ij}, we obtain

ξij ≤ fw,d({(n+ j)j} ∪ {ij})− fw,d({(n+ j)j}).

By the formula (11) of fw,d, modification (4) of fj , and dn+j =∞,

fw,d({(n+ j)j} ∪ {ij}) = min{fw({(n+ j)j} ∪ {ij}), fw({(n+ j)j}) + di}= min{fj(Ej)− w(n+j)j , fj(Ej)− w(n+j)j + di} = fj(Ej)− w(n+j)j ,

fw,d({(n+ j)j}) = min{fw({(n+ j)j})} = fj(Ej)− w(n+j)j ,

and thus we obtain ξij = 0.

Using this lemma, we prove Proposition 3.3.

Proof of Proposition 3.3. Lemma 4.13 indicates that seller j does not transact with buyers otherthan n + j until cn+j = ρj . Therefore, after transforming the allocation of Algorithm 1 into theallocation without virtual buyers, it holds

rj =∑ij∈Ej

ciwij ≥ cn+jw(Ej) ≥ ρjw(Ej) (j ∈M),

where ci ≥ cn+j always holds for non-virtual buyer i in Algorithm 1. Thus we obtain

uj(M(I)) = rj + ρj(fj(Ej)− w(Ej)) ≥ ρjfj(Ej).

18

4.5 Proof of Theorem 3.6

We denote the set of non-virtual buyers by N∗ and the set of virtual buyers by V . Namely,N = N∗ ∪ V . By (ICb) (Corollary 3.5), we may assume that buyers report their true valuations;v′i = vi (i ∈ N∗). Let A = (w, p, r) be the allocation obtained by Algorithm 1 (including theallocation of virtual buyers). By Theorem 3.4, we can utilize the following proposition obtained inthe proof of Lemma 3.8 by Goel et al. [11]. A subset X ⊆ N is said to be tight with respect to wif w(X) = f(EX)(= g(X)).

Proposition 4.14 (Goel et al. [11]). There exist a chain ∅ = X0 ⊂ X1 ⊂ X2 ⊂ · · · ⊂ Xt = N oftight sets and ik ∈ Xk \Xk−1 for k = 1, 2, . . . , t such that we have vi ≥ vik for each k = 1, 2, . . . , tand i ∈ Xk.

Proof of Theorem 3.6. Suppose to the contrary that there exists an allocation (without virtual

buyers) A′ = (w′, p′, r′) with∑i∈N∗

p′i ≥∑j∈M

r′j such that A′ satisfies

viw(Ei)− pi ≤ viw′(Ei)− p′i (i ∈ N∗), (21)

(rj − pn+j) + ρjw(n+j)j ≤ r′j + ρj(fj(Ej)− w′(Ej)) (j ∈M), (22)

and at least one of the inequalities holds strictly. Here A′ is extended to an allocation with virtualbuyers by:

w′(n+j)j := fj(Ej)− w′(Ej) (j ∈M),

pn+j := 0 (j ∈M).

In the sequel, edges to virtual buyers are included in E and Ej . Then we have

w′(EN ) =∑j∈M

fj(Ej) = f(EN ). (23)

Then (21) and (22) are rewritten as

vi(w(Ei)− w′(Ei)) ≤ pi − p′i (i ∈ N∗), (24)

(rj − r′j)− (pn+j − p′n+j) ≤ ρj(w(n+j)j − w′(n+j)j) (j ∈M). (25)

Consider a chain ∅ = X0 ⊂ X1 ⊂ X2 ⊂ · · · ⊂ Xt = N , and ik ∈ Xk \Xk−1 in Proposition 4.14. By(23), it still holds that

f(EXk) = w(EXk

) (k ∈ {0, 1, 2, . . . , t}),f(EXt) = f(EXN

) = w′(EXN) = w′(EXt).

Also for each k ∈ {1, 2, . . . , t}, we define the following sets:

X∗k : = Xk ∩N∗,Zk : = Xk \Xk−1,

Z∗k : = Zk ∩N∗.

Adding (24) and (25), we prove, by induction, that∑i∈X∗k

(pi− p′i) ≥∑

j :n+j∈Xk∩V((rj − r′j)− (pn+j − p′n+j)) + vik(w(EXk

)−w′(EXk)) (k ∈ {1, 2, . . . , t}).

(26)

19

For each k ∈ {1, 2, . . . , t}∑i∈Z∗k

(pi − p′i) ≥∑i∈Z∗k

vik(w(Ei)− w′(Ei)) = vik(w(EZ∗k )− w′(EZ∗k )), (27)

where the inequality follows from (24) and vi ≥ vik (i ∈ Xk). In the case of k = 1,∑i∈X∗1

(pi − p′i) =∑i∈Z∗1

(pi − p′i) ≥∑i∈Z∗1

vi1(w(Ei)− w′(Ei))

=∑i∈X∗1

vi1(w(Ei)− w′(Ei)) + vi1(w′(EX1)− w(EX1) + w(EX1)− w′(EX1))

= vi1(w′(EX1)− w′(EX∗1 )− w(EX1) + w(EX∗1 )) + vi1(w(EX1)− w′(EX1))

= vi1(w′(EX1∩V )− w(EX1∩V )) + vi1(w(EX1)− w′(EX1))

≥∑

j :n+j∈X1∩V((rj − r′j)− (pn+j − p′n+j)) + vi1(w(EX1)− w′(EX1)),

where the second inequalities follows from (25). Thus (26) holds in k = 1. Suppose that (26) holdsin k = s. We have∑

i∈X∗s+1

(pi − p′i) =∑i∈X∗s

(pi − p′i) +∑

i∈Z∗s+1

(pi − p′i)

+ vis+1(w(EXs+1)− w′(EXs+1) + w′(EXs+1)− w(EXs+1))

≥∑

j :n+j∈Xs∩V((rj − r′j)− (pn+j − p′n+j)) + vis(w(EXs)− w′(EXs))

+ vis+1(w(EZ∗s+1)− w′(EZ∗s+1

)) + vis+1(w(EXs+1)− w′(EXs+1))

+ vis+1(w′(EXs+1)− w(EXs+1)),

where the inequality follows from (26) for k = s and (27). Here we obtain

vis(w(EXs)− w′(EXs)) + vis+1(w(EZ∗k )− w′(EZ∗k )) + vis+1(w′(EXs+1)− w(EXs+1))

≥ vis+1(w(EXs)− w′(EXs) + w(EZ∗s+1)− w′(EZ∗s+1

) + w′(EXs+1)− w(EXs+1))

= vis+1(w′(EZs+1∩V )− w(EZs+1∩V )),

where the inequality follows from vis ≥ vis+1 . Therefore,∑i∈X∗s+1

(pi − p′i) ≥∑

j :n+j∈Xs∩V((rj − r′j)− (pn+j − p′n+j)) + vis+1(w(EXs+1)− w′(EXs+1))

+ vis+1(w′(EZs+1∩V )− w(EZs+1∩V ))

≥∑

j :n+j∈Xs∩V((rj − r′j)− (pn+j − p′n+j))

+∑

j :n+j∈Zs+1∩V((rj − r′j)− (pn+j − p′n+j)) + vis+1(w(EXs+1)− w′(EXs+1))

=∑

j :n+j∈Xs+1∩V((rj − r′j)− (pn+j − p′n+j)) + vis+1(w(EXs+1)− w′(EXs+1)),

where the second inequality follows from (25). Therefore, we obtain (26).

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Since w(EXt) = w′(EXt) = f(EXt) for k = t, we have∑i∈X∗t

(pi − p′i) ≥∑

j :n+j∈Xt∩V((rj − r′j)− (pn+j − p′n+j)). (28)

By (SBB), it holds ∑i∈X∗t

pi =∑i∈N∗

pi =∑j∈M

(rj − pn+j) =∑

j :n+j∈Xt∩V(rj − pn+j).

Substituting this to (28), we obtain∑i∈N∗

p′i =∑i∈X∗t

p′i ≤∑

j :n+j∈Xt∩V(r′j − p′n+j) =

∑j∈M

(r′j − p′n+j).

Since p′n+j = 0 (j ∈M), ∑i∈N∗

p′i ≤∑j∈M

r′j . (29)

Since∑i∈N∗

p′i ≥∑j∈M

r′j , the equality holds in (29). Here (29) was deduced by adding all inequal-

ities in (24) and (25). Thus all inequalities in (24) and (25) hold in equality, which contradicts theassumption of A′ that at least one of inequalities is strict.

Acknowledgements. This work was partially supported by JSPS KAKENHI Grant Numbers25280004, 26330023, 26280004, 17K00029, and by JST ERATO Grant Number JPMJER1201,Japan.

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