The Combinatorial Seller s Bid Double Auction: An Asymptotically Efficient Market Mechanism*

Transcription

1 The Combintoril Seller s Bid Double Auction: An Asymptoticlly Efficient Mret Mechnism* Rhul Jin IBM Wtson Reserch Hwthorne, NY rhul.jin@us.ibm.com Prvin Vriy EECS Deprtment University of Cliforni, Bereley vriy@eecs.bereley.edu We consider the problem of efficient mechnism design for multilterl trding of multiple goods with independent privte types for plyers nd incomplete informtion mong them. The problem is prtly motivted by n efficient resource lloction problem in communiction networs where there re both buyers nd sellers. In such setting, ex post budget blnce nd individul rtionlity re ey requirements, while efficiency nd incentive comptibility re desirble gols. Such mechnisms re difficult if not impossible to design [36]. We propose combintoril mret mechnism which in the complete informtion cse is efficient, budget-blnced, ex post individul rtionl nd lmost dominnt strtegy incentive comptible. In the incomplete informtion cse, it is budget-blnced, ex post individul rtionl nd symptoticlly efficient nd Byesin incentive comptible. Thus, we re ble to chieve efficiency, budget-blnce nd individul rtionlity by compromising on incentive comptibility, chieving only we version of it. History : This version: July 5, Introduction We study multilterl trding problem with multiple indivisible goods nd independent privte types in which ex post budget-blnce is required. The problem is prtly motivted by the need to design mechnisms for efficient resource lloction exchnge between strtegic internet service providers such s AOL nd Comcst who lese trnsmission cpcity (or bndwidth) to form desired routes nd networs nd crriers such s Qwest nd MCI who own cpcity on individul lins. Bndwidth is trded in discrete mounts, sy multiples of 100 Mbps, nd hence is n indivisible good. Thus, the buyers wnt bndwidth on combintions of severl lins vilble in multiples of some indivisible unit. This mes the problem combintoril. We consider the interction in severl settings. (Similr problems lso occur in other settings such s electricity mrets [40] nd spectrum uctions [34]) We propose combintoril sellers bid double uction (c-sebida) mechnism for such settings tht chieves socilly desirble interction mong strtegic gents. The mechnism is combintoril since buyers me bids on combintions of goods, such s severl lins tht form route. However, ech seller offers to sell only single type of good (e.g., bndwidth on single lin). The mechnism mimics competitive mret: it tes ll buy nd sell bids, solves mixed-integer progrm tht mtches bids to mximize the socil surplus, nd nnounces prices t which the mtched (i.e., ccepted) bids re settled. The settlement price for n good is the highest price sed by mtched seller (hence sellers bid uction). As result there is uniform price for ech good. * This reserch ws supported by the Ntionl Science Foundtion grnts ECS nd CNS , nd Fujitsu Lbs, USA. 1

2 2 It is shown tht in the c-sebida uction gme with complete informtion, Nsh equilibrium exists; it is not generlly competitive equilibrium, nor is it unique. Nevertheless, there is n lloctively efficient Nsh equilibrium wherein it is wely dominnt strtegy for ll buyers nd for ll sellers except the mtched seller with the highest-s price to be truthful. Moreover, every Nsh equilibrium in wely rtionlizble strtegies is efficient in the single good cse. In the combintoril cse, every Nsh equilibrium with non-zero trde for ech good is efficient. Since in n uction, plyers usully hve incomplete informtion, following Hrsnyi [15], we then consider the Byesin-Nsh equilibrium of the uction gme. We show tht if the plyers use only ex post individul rtionl (IR) strtegies [32], the semi-symmetric Byesin-Nsh equilibrium strtegies (wherein ll sellers selling the sme good use the sme strtegy) converge to truth-telling s the number of plyers becomes very lrge. Previous Wor nd Our Contribution. The -double uction ws introduced by Chtterjee nd Smuelson [8] s model of bilterl brgining. It ws shown by Myerson nd Stterthwite [36] tht when there is incomplete informtion, there exists no bilterl mechnism which is Byesin incentive comptible, individul rtionl, budget-blnced nd efficient. Thus, the notion of constrined incentive efficiency ws considered by Wilson [50]. The -double uction mechnism ws further generlized to the single-good multilterl cse by Stterthwite nd Willims [45, 46]. In this pper, we consider multilterl trding mechnism for multiple objects. The mechnism my be considered to be generliztion nd modifiction of the -double uction mechnism (plese see remr 1 nd exmple 2 in section 2 for similrities nd differences). A survey of the vst uction theory literture is provided in [26, 49]. Mny re extensions of Vicrey s ides [48]. Recently, [27] introduced generliztion of the VCG mechnism with prticiption costs for multi-dimensionl types nd multiple objects. Also, [9] extends the VCG mechnism to the cse of common vlues, nd shows it is constrined efficient. Some multi-round scending bid uctions [5, 39] chieve the sme outcome s VCG. However, these re single-sided uction mechnisms. A Vicrey double uction mechnism for single goods is proposed in [52] but it is neither (ex post) budget-blnced nor individul rtionl. It ppers very difficult to chieve ex post budget blnce (long with efficiency nd individul rtionlity) in double-sided uction mechnisms [38]. Our interest is in double-sided uction mechnism for multiple goods with independent privte types (nd qusi-liner utility functions). We propose combintoril double uction mechnism which is individul rtionl nd budget-blnced by design, mes smll compromise on incentivecomptibility nd yet is efficient. It is non-vcg-type double-sided uction mechnism for multiple goods. Lie the proposl in [6], our mechnism is lso NP-hrd. But the mechnism s mixed-integer liner progrm structure mes the computtion mngeble for mny prcticl pplictions [23]. The interply between economic, gme-theoretic nd computtionl issues hs spred interest in lgorithmic mechnism design [42, 49]. The generlized Vicrey uction mechnisms for multiple heterogeneous goods re not computtionlly trctble [37, 38]. Thus, mechnisms tht rely on pproximtion of the integer progrm [37, 44] or liner progrmming [7] hve been proposed. The results here lso relte to the recent efforts in the networ pricing literture [29]. There is n ongoing effort to propose mechnisms for divisible resource lloction in networs through uctions [25] nd to understnd the worst cse Nsh equilibrium efficiency loss of such mechnisms when users ct strtegiclly [22]. Optiml mechnisms for single divisible goods tht minimize this efficiency loss hve been proposed [51, 30] though not extended to the incomplete informtion cse nor for multiple goods. Most of this literture regrds the good (in this cse, bndwidth) s divisible, with complete informtion for ll plyers. The cse of combintoril bids on multiple indivisible goods or incomplete informtion cse is hrder.

3 3 The results in this pper re significnt from severl perspectives. It is well nown tht the only nown positive result in the mechnism design theory is the VCG clss of mechnisms [32]. The generlized Vicrey uction (GVA) (with complete informtion) is ex post individul rtionl, dominnt strtegy incentive comptible nd efficient. It is however not budget-blnced. With incomplete informtion, the expected version of GVA (dagva) [2, 4] is Byesin incentive comptible, efficient nd budget-blnced. It is, however, not ex post individul rtionl. Indeed, in the complete informtion setting there cn be no mechnism tht is efficient, budget-blnced, ex post individul rtionl nd dominnt strtegy incentive comptible (Hurwicz impossibility theorem) [16]. In the incomplete informtion setting there is no mechnism which is efficient, budgetblnced, ex post individul rtionl nd Byesin incentive comptible (Myerson-Stterthwite impossibility theorem) [36]. In this pper, we provide non-vcg combintoril (mret) mechnism which in the complete informtion cse is efficient, budget-blnced, ex post individul rtionl nd lmost dominnt strtegy incentive comptible. In the incomplete informtion cse, it is budget-blnced, ex post individul rtionl nd symptoticlly efficient nd Byesin incentive comptible. Thus, we re ble to chieve efficiency, budget-blnce nd individul rtionlity by compromising on incentive comptibility, chieving only we version of it. Moreover, we show tht Nsh equilibrium lloction (sy of networ resource lloction gme) is efficient (zero efficiency loss) nd ny (semi-symmetric) Byesin-Nsh equilibrium lloction is symptoticlly efficient. This wor cn lso be seen s contribution to the brgining gmes literture. The proposed multilterl trding mechnism for multiple indivisible goods yields n symptoticlly efficient lloction even in the cse of incomplete informtion. To our nowledge, this seems to be the only nown generliztion of the Myerson-Stterthwite [36] trding environment for multiple heterogeneous goods. Moreover, we provide positive result: While it is impossible to chieve Byesin incentive comptibility nd efficiency long with ex post budget blnce nd individul rtionlity, it is possible to chieve these properties symptoticlly even in multilterl, multiple good trding environment. The rest of this pper is orgnized s follows. In Section 2 we present the combintoril seller s bid double uction (c-sebida) mechnism. In Section 3 we consider Nsh equilibrium of the complete informtion uction gme. In Section 4 we consider the Byesin-Nsh equilibrium of the incomplete informtion uction gme for multiple goods. 2. The Combintoril Sellers Bid Double Auction A buyer plces buy bids for bundle of goods. A buyer s bid is combintoril: he must receive ll goods in his bundle or nothing. A buy-bid consists of buy-price per unit of the bundle nd mximum demnd, the mximum number of units of the bundle tht the buyer needs. On the other hnd, ech seller mes non-combintoril bids. A sell-bid consists of n s-price nd mximum supply, the mximum number of units the seller offers for sle. The mechnism collects ll nnounced bids, mtches subset of these to mximize the surplus (eqution (1), below) nd declres settlement price for ech good t which the mtched buy nd s bids which we cll the winning bids re trnscted. This constitutes the pyment rule. As will be seen, ech mtched buyer s buy bid is lrger, nd ech mtched seller s s bid is smller thn the settlement price, so the outcome respects individul rtionlity. There is n symmetry: buyers me multi-good combintoril bids, but sellers only offer one type of good. This yields uniform settlement prices for ech good. Plyers bids my not be truthful. They now how the mechnism wors nd formulte their bids to mximize their individul returns. In the combintoril sellers bid double uction (c-sebida), ech plyer plces only one bid. c-sebida is double uction becuse both buyers nd sellers bid; it is sellers bid uction becuse the settlement price depends only on the mtched sellers bids.

4 4 Forml mechnism. There re L goods l 1,, l L, m buyers nd n sellers. Buyer i hs (true) reservtion vlue v i per unit for bundle of goods R i {l 1,, l L }, nd submits buy bid of b i per unit nd demnds up to δ i units of the bundle R i. Thus, the buyers hve qusi-liner utility functions of the form u b i(x; ω, R i ) = v i (x) + ω where ω is money nd { x v i, for x δ i, v i (x) = δ i v i, for x > δ i. Seller j hs (true) per unit cost c j nd offers to sell up to σ j units of l j t unit price of j. Note tht there my be mny sellers j, j, etc., selling the sme good l j = l j = l, etc. Denote L j = {l j }. The sellers lso hve qusi-liner utility functions of the form u s j(x; ω, L j ) = c j (x) + ω where ω is money nd { x c j, for x σ j, c j (x) =, for x > σ j. The mechnism receives ll these bids, nd mtches some buy nd sell bids. The possible mtches re described by integers x i, y j : 0 x i δ i is the number of units of bundle R i llocted to buyer i nd 0 y j σ j is the number of units of good l j sold by seller j. The mechnism determines the lloction (x, y ) s the solution of the surplus mximiztion problem MIP: mx x,y i b ix i j jy j (1) s.t. y j j1(l L j ) x i i1(l R i ) 0, l [1 : L], x i {0, 1,, δ i }, i, y j [0, σ j ], j. MIP is mixed integer progrm: Buyer i s bid is mtched up to his mximum demnd δ i ; Seller j s bid will lso be mtched up to his mximum supply σ j. x i is constrined to be integrl; y j will be integrl due to the demnd less thn equl to supply constrint. The settlement price is the highest s-price mong mtched sellers, ˆp l = mx{ j : y j > 0, l L j }. (2) The pyments re determined by these prices. If no seller of good l is mtched, i.e., good l is not trded, the price of ˆp l is unspecified. Mtched buyers py the sum of the prices of goods in their bundle; mtched sellers receive pyment equl to the number of units sold times the price for the good. Unmtched buyers nd sellers do not get ny lloction nd do not me or receive ny pyments. This completes the mechnism description. If i is mtched buyer (x i > 0), it must be tht his bid b i l R i ˆp l ; for otherwise, the surplus (1) cn be incresed by eliminting the corresponding mtched bid. Similrly, if j is mtched seller (yj > 0), nd l L j, his bid j ˆp l, for otherwise the surplus cn be incresed by eliminting his bid. Thus the outcome of the uction respects individul rtionlity. It is esy to understnd how the mechnism pics mtched sellers. For ech good j, seller with lower s bid will be mtched before one with higher bid. So sellers with bid j < ˆp l sell ll their supply (yj = σ j ). At most one seller with s bid j = ˆp l sells only prt of his totl supply (yj < σ j ). On the other hnd, becuse their bids re combintoril, the mtched buyers re selected only fter solving the MIP. Exmple 1. Consider one good, three buyers ech of whom wnts one unit nd three sellers ech of whom hs one unit to offer. Suppose buyers bid b 1 = 3.1, b 2 = 2.1, b 3 = 1.1 nd sellers bid 1 = 1, 2 = 2, 3 = 3. Then, the reveled socil surplus in MIP (1) is mximized when buyers 1 nd 2, nd sellers 1 nd 2 re mtched. The price then is ˆp = 2. Note tht if bids of buyer 3 nd seller 3 re lso ccepted, this will result in lower reveled socil surplus.

5 Remrs. 1. We designed c-sebida so tht its outcome mimics competitive equilibrium with prticulr interest in the combintoril cse. The single good version SeBiDA resembles the -double uction ( specil cse being clled the buyer s bid double uction [46, 45, 47]). The -DA is defined s follows: Sellers submit offers j, j = 1,, n nd buyers bids b i, i = 1,, n. To determine who trdes, list these offers/bids s s (1) s (2) s (2n) where s (l) denotes the lth order-sttistic. Thus, s (n) could either be buy-bid or sell-offer. Then, for given [0, 1], pic price to be p() = (1 )s (n) + s (n+1). Sell-offers below p nd buy-bids bove p re ccepted. Others re not. For the specil cse of = 1, the -DA mechnism is the sme s the buyer s bid double uction (BBDA) mechnism [45]. The sell-side version would te = 0 with p = s (n). But note tht despite similr nomenclture nd spirit, BBDA nd c-sebida determine prices differently. We illustrte the difference through n exmple. Exmple 2. Consider one good, three buyers ech of whom wnts one unit nd three sellers ech of whom hs one unit to offer. Suppose buyers bid b 1 = 6.1, b 2 = 3.1, b 3 = 1.1 nd sellers bid 1 = 2, 2 = 4, 3 = 5. (i) BBDA: Then, s (3) = 3.1 nd s (4) = 4, nd the price determined by BBDA is p = 4 with one trde between buyer 1 nd seller 1. The sell-side version of BBDA would determine price p = 3.1 with single trde. -DA determines price p [3.1, 4]. (ii) c-sebida: The mechnism proposed in this pper, on the other hnd, determines one trde between buyer 1 nd seller 1 with price p = 2. Thus, the mechnism proposed in this pper is distinct from BBDA [46]. It is lso not cler wht generliztion of the -double uction or BBDA would be to the combintoril cse. 2. The issue of computtionl complexity for such mechnisms becomes very importnt when there re lrge number of plyers. Similr concerns rise in [6] s well. However, the computtionl problem here involves solving mixed liner progrm, for which computtionlly efficient pproximtion lgorithms hve been devised. Developing n pproximtion lgorithm for the prticulr MIP here will be underten in the future. 3. The ties between plyers will be broen by rndomly picing the winners. This hs no effect on the uction s outcome, or its properties unlie other mechnisms. 3. Complete Informtion Nsh Equilibrium Anlysis: c-sebida is Efficient We first focus on how strtegic behvior of plyers ffects price when they hve complete informtion. We will ssume tht plyers don t strtegize over the bundles R i nd the quntities (nmely, δ i, σ j ), which will be considered fixed in the plyers bids. A strtegy for buyer i is buy bid b i, strtegy for seller j is n s bid j. Let θ denote collective strtegy. Given θ, the mechnism determines the lloction (x, y ) nd the prices {ˆp l }. So the pyoff to buyer i nd seller j is, respectively, u b i(θ) = v i (x i ) x i l R i ˆp l, (3) u s j(θ) = y j l L j ˆp l c j (y j ). (4) The bids b i, j my be different from the true vlutions v i, c j, which however figure in the pyoffs. A collective strtegy θ is Nsh equilibrium if no plyer cn increse his pyoff by unilterlly chnging his strtegy [11]. Define socil welfre function for the uction gme s 5 SW (x, y) = i v i x i j c j y j.

6 6 where (x, y) stisfy the fesibility conditions of MIP (1). An uction mechnism is sid to be (lloctively) efficient if every Nsh equilibrium lloction mximizes socil welfre. We sy tht strtegy b i is wely dominted for plyer i if there exists strtegy b i of plyer i such tht u i (b i, b i ) u i ( b i, b i ), b i with strict inequlity for t lest one b i where b i re the strtegies of the other plyers. Such strtegies re considered unliely to be plyed. Strtegies which re not wely dominted will be clled undominted. Strtegies which remin undominted fter iterted elimintion of wely dominted strtegies will be clled wely rtionlizble strtegies [11]. They re so clled becuse it is considered rtionl for plyers to ply only such strtegies when it is common nowledge tht ll plyers re rtionl. We now construct Nsh equilibrium nd show it yields n efficient lloction (Theorem 1). We then show tht when plyers only ply wely rtionlizble strtegies, ll resulting Nsh equilibri re efficient in the single good cse. For simplicity, we ssume tht ech buyer bids for t most one unit, nd ech seller sells t most one unit of the item (so δ i, σ j equl 1 in (3), (4)). Suppose there re m buyers nd n sellers, whose true vlutions nd costs lie in [0, 1]. To void trivil cses of non-uniqueness, ssume ll buyers hve different vlutions nd ll sellers hve different costs. Theorem 1. (i) A Nsh equilibrium (b, ) exists in the c-sebida gme. (ii) There is Nsh equilibrium wherein except for the mtched seller with the highest bid on ech item, ech plyer bids truthfully. (iii) Furthermore, in cse of single good, ny Nsh equilibrium in wely rtionlizble strtegies hs n efficient lloction. (iv) In the combintoril cse, if there is trde for ech good, then every Nsh equilibrium is efficient. Proof: (i) Suppose buyer i demnds the bundle R i with reservtion vlue v i nd the seller (l, j) (the j-th seller offering item l) hs reservtion cost c l,j. Assume without loss of generlity tht c l,1 c l,nl, in which n l is the number of sellers offering item l. We will itertively construct set of strtegies to consider s Nsh equilibrium. Consider the surplus mximiztion problem (1) with true vlutions nd costs. Denote by (x, y ) corresponding efficient (i.e., socilly optiml) lloction. Let B be the set of mtched buyers, B be the set of unmtched buyers, S be the set of mtched sellers, S be the set of unmtched sellers nd l the number of mtched sellers offering item l determined by the MIP. Set l,0 = l,1 for ll l (needed for l,l to be defined for l = 0); b i = v i for ll i; 0 l,j = c l,j. Define γi t := b i l R i t l, l, the reveled surplus of buyer i t stge t 0 nd t current prices t l, l. Define σ l t := min{γi t > 0 : i B, l R i }, the minimum reveled surplus mong mtched buyers on item l, nd σ l t := mx{γi t > 0 : i B, l R i }, the mximum positive reveled surplus mong the unmtched buyers on item l t the current prices. Define L 0 t := {l : l R i, i B, γi t = 0}, the set of items with mtched buyer with zero surplus, L 0 t := {l : l R i, i B, γi t = 0}, the set of items with unmtched buyers who hve zero reveled surplus t the current prices, nd ˆL 0 t := {l : t l, l +1 t l, l = 0}, the set of items where the current price is equl to the the s-bid of the first unmtched seller. (Note tht if l = 0, i.e., there is no mtch on l, l,0 = l,1. Thus, l ˆL 0 t ). Define L + t := {l : l R i, i B, γi t 0}\ L 0 t, the set of items which hve mtched buyers with positive surplus (but none with zero surplus), L + t := {l : l R i, i B, γi t 0}\ L 0 t, the set of items with unmtched buyers who hve positive reveled surplus t the current prices (but none with non-positive surplus), nd ˆL + t := {l : t l, l +1 t l, l 0}\ˆL 0 t, the set of items where the current price is strictly smller thn the the s-bid of the first unmtched seller.

7 7 Now, the lgorithm to compute Nsh equilibrium is the following: (1) If Γ 1 t := L + t L + t \ˆL 0 t, then σ t l := min( σ t l, σ t l ) l Γ 1 t. Pic n ˆl such tht ˆl rg min l {σ t l > 0 : l Γ 1 t }. (5) (2) If Γ 1 t =, nd Γ 2 t := L + t ˆL + t then σ t l := σ t l, l Γ 2 t. Pic n ˆl such tht ˆl rg min l {σ t l > 0 : l Γ 2 t }. (6) (3) If ˆl obtined in steps (1) or (2), updte ˆl,ˆl s t+1 := min{ ṱ, ˆl,ˆl l,ˆl+1 ṱ + σ ṱ }. (7) l l,ˆl else terminte. Denote the resulting bids fter the lgorithm hs terminted by (b, ), nd the corresponding sets of items by L +, L +, ˆL +, etc. First, we rgue tht the lgorithm will converge. Observe tht in step (1), eqution (??) is updted t most M times, i.e., t most once for ech buyer since once the surplus of n unmtched buyer t current prices is non-positive, it is not iterted upon gin. In step (2), eqution (??) is updted t most L times, once for ech item. Thus, the lgorithm performs step (3), eqution (??) t most M + L times, nd hence only finitely mny times. Second, we clim tht if we strt with some efficient lloction (x, y ), then t ech itertion of the lgorithm (steps (1)&(3), or steps (2)&(3)), the lloction does not chnge. First observe tht for ll buyers (mtched nd unmtched) on item l, if the price-determining bid l,l is incremented by ny δ > 0 (smll-enough), then the chnge in the reveled surplus γ i of ll buyers who wnt tht item (i.e., l R i ), is the sme. Thus, the lloction remins the sme, before nd fter such n itertion, s long s the γ i of ll mtched buyers on tht item remins non-negtive. Thus, the lloction remins unchnged in step (1)&(3). In step (2), the mrginl (price-determining) mtched seller s bid is incresed such tht the reveled surplus γ i for mtched buyer i on ny item remins non-negtive, nd the new bid is up to the bid of the lowest unmtched sell bid on tht item. Thus, gin the lloction fter n itertion of steps (2)&(3) remins the sme s before the itertion. Since, we strt with the lloction (x, y ), the lloction when the lgorithm converges is lso (x, y ). Third, we rgue tht the bids computed by the lgorithm re Nsh equilibrium. We show this by showing tht no plyer hs n incentive to devite. First, consider ny unmtched seller offering item l. Becuse of the updte in eqution (??), his reservtion cost is higher thn the bid of the mrginl mtched seller. He hs no incentive to bid lower thn l, l to get mtched since by bidding lower thn his reservtion cost, he my get mtched but his pyoff will be negtive. Next, consider ny non-mrginl mtched seller (l, j) (l, l ) offering item l. By bidding higher or lower he cnnot chnge the price of the item but my end up getting unmtched. Thus, bidding truthfully is best-response of ll sellers other thn the mrginl mtched sellers. Now, consider the mrginl mtched seller (l, l ). Then, either l ˆL 0, in which cse there is n unmtched seller with the sme s bid, or l ˆL + but by step (2), L + ˆL + =, i.e., there is mtched buyer on this item with zero surplus. Now, if he bids lower then l, l, his pyoff will decrese. He could bid higher but then either there is n unmtched seller of the item with the sme s bid, or there is mrginl mtched buyer on tht item whose surplus is zero. So, if he bids higher thn l, l, either he will become unmtched nd the first unmtched seller of the item will become mtched, or the mrginl mtched buyer with zero surplus will become unmtched

8 8 cusing this mrginl mtched seller to be unmtched s well. Thus, l, l is best response of the mrginl seller given tht ll other plyers (except the mrginl sellers of the other items) bid truthfully. Now, let us consider the buyers. Consider ny mtched buyer i. He hs non-negtive pyoff t current prices l, l. He hs no incentive to bid higher, nd by bidding lower he cn lower the prices but only if he becomes unmtched. So, he hs no incentive to devite from his bid b i given the bids of ll the other plyers (b i, ). Thus, it is the best response of ll such buyers to bid truthfully. Now, consider ny unmtched buyer with non-positive surplus t current prices, clerly ny such buyer hs no incentive to devite since by incresing their bid, they might be ble to get mtched t current prices but then their pyoff will be non-positive. Lst, consider ny unmtched buyer i with positive surplus t current prices. Then, for ny l R i, either l ˆL + or l ˆL 0 (recll it includes items with no mtches). Further, by step (1), L + L + ˆL + =. (In fct, by step (2) L 0 L + ˆL + =, but it is not relevnt here.) Thus, for ny item l R i (note such n l L + ), if l / ˆL +, then l ˆL 0. And if l / L +, then l L 0. In other words, for every l R i, either item l is such tht there is n unmtched seller with sme s bid s current price, or there is mtched buyer on the item with zero surplus. Now, suppose tht buyer i increses his bid to very high to mtch. Then, it will cuse some buyers B i on items l R i L 0 ˆL + to be unmtched. Let σ(b i ) denote the (ctul) surplus of such buyers, nd σ i the (ctul) surplus of buyer i t the current prices. Denote by S i the mrginl unmtched sellers on items l R i ˆL 0, nd the sum of their s-bids by ( S i ). And denote by S i, the mrginl mtched sellers on items R i \ˆL 0, nd the sum of their bids by (S i ). Then, it is true tht (otherwise buyer i would hve mtched insted of buyers B i ) σ(b i ) σ i = v i ( S i ) (S i ). But then, σ(b i ) = 0 since ll such buyers demnd items l L 0 R i. Thus, the unmtched buyer cnnot hve strictly positive surplus t current prices, nd so no such buyer exists. Thus, (b, ) is Nsh equilibrium. The corresponding lloction (x, y ) s determined bove is efficient since it mximizes (1) with true vlutions. (iii) We now show tht in cse of single good ny Nsh equilibrium lloction in wely rtionlizble strtegies is efficient. (We will drop the subscript l for sellers). First, observe tht seller s bid below his cost is wely dominted by his bid t cost: Thus, j c j, j. Further, since this elimintion of strtegy spce of the sellers is common nowledge, no buyer will bid below c min = min j c j. Let B mtched nd S mtched denote the set of buyers nd sellers tht re mtched t Nsh equilibrium ( b, ã). It is worth noting tht t n equilibrium, the trnsction price p = min{b i : i B mtched } = mx{ j : j S mtched }. Now suppose z := ( x, ỹ) is n lloction, corresponding to the Nsh equilibrium ( b, ã), which is not efficient. There re two min cses: (1) No Trde is Efficient Cse: Suppose tht the efficient lloction (z := (x, y )) involves no trde, but the lloction z does. This implies tht v i < c j, i, j but there exists some buyer î nd seller ĵ such tht bî cĵ. Then, either bî > vî or ĵ < cĵ. In both cses, one of the buyer î or the seller ĵ hs n incentive to devite. (2) Non-zero Trde is Efficient Cse: () First, suppose tht the efficient lloction z involves trde but the lloction z involves no trde. Let i denote the buyer with highest vlue v i nd j denote the seller with the lest cost c j (c j = c min ). Then, v i c j but c min b i < j. But then this cnnot be Nsh equilibrium since either the buyer or the seller will hve n incentive to devite.

9 (b) Now, suppose tht the efficient lloction z involves trde nd the lloction z involves trde but is not efficient. Then, the two lloctions must differ in one of the following wys s we go from z to z: (i) z nd z differ only mong sellers: A (non-empty) set of sellers S out mtched in z, is no longer mtched in z nd (non-empty) set of sellers S in re now mtched; (ii) z nd z differ only mong buyers: A (non-empty) set of buyers B out mtched in z, is no longer mtched in z nd (non-empty) set of buyers B in re now mtched; (iii) All buyers nd sellers mtched in z remin mtched in z, nd some new buyers B in nd some new sellers S in now get mtched; (iv) No new buyers nd sellers re mtched in z nd some old buyers B out nd some old sellers S out re now not mtched; (v) (Generl Cse ) A set of buyers B out nd set of sellers S out re no longer mtched nd set of buyers B in nd set of sellers S in re now mtched in z. Cse (i) Suppose j 1 S in nd j 2 S out. Then, it must be tht c j1 > c j2 but ã j1 < ã j2. But then either j 1 s pyoff is negtive or j 2 cn lso bid just below j 1 s bid. In either cse z cnnot be Nsh equilibrium lloction. Cse (ii) Suppose i 1 B in nd i 2 B out. Then it must be tht v i1 < v i2 nd b i1 > b i2. But then either i 1 s pyoff is negtive or i 2 cn lso bid just bove i 1 s bid. In either cse z cnnot be Nsh equilibrium lloction. Cse (iii) Denote ĭ := rg mx i Bin bi nd j := rg min j Sin ã j. Then, vĭ < c j nd bĭ ã j. But then t lest one of the two hs negtive pyoff t ( b, ã), nd so will devite, in which cse it cnnot be Nsh equilibrium outcome. Cse (iv) Denote ǐ := rg mx i Bout v i nd ǰ := rg min j Sout c j. And denote the trnsction price with bids ( b, ã) by p. Then, v ǐ c ǰ nd b ǐ < ã ǰ. Now, if ã ǰ < v ǐ, then clerly, buyer ǐ hs n incentive to bid just bove ã ǰ nd mtch. Similrly, if b ǐ > c ǰ, then seller ǰ hs n incentive to bid just below bǐ nd mtch. In either of these cses, the bids under considertion cnnot be Nsh equilibrium. Now, let us consider the cse b ǐ c ǰ v ǐ ã ǰ. There re three sub-cses: if p ( b ǐ, c ǰ ], then buyer ǐ cn rise his bid nd mtch; if p [v ǐ, ã ǰ ), then seller ǰ cn lower his bid nd mtch; nd if p (c ǰ, ã ǐ ), then both the buyer ǐ nd the seller ǰ hve n incentive to devite from their current bids nd mtch. Thus, in none of the bove sub-cses cn the bids under considertion be Nsh equilibrium. Cse (v) Denote î := rg min i Bin bi nd ĵ := rg mx j Sin ã j, nd ǐ := rg mx i Bout bi nd ǰ := rg min j Sout ã j. And denote the trnsction price with bids ( b, ã) by p. Then, bî p ãĵ nd bǐ p ã ǰ. Now, observe tht v ǐ > vî nd c ǰ < cĵ since plyers ǐ nd ǰ re mtched in z, the efficient lloction but plyers î nd ĵ re not. Further, bî p. So, either vî p, in which cse v ǐ p s well nd so buyer ǐ cn increse his bid to mtch; or vî < p, in which cse buyer î hs negtive pyoff nd so it will decrese his bid. Thus, in either cse, the buyer hs n incentive to devite, nd hence the lloction z cnnot correspond to Nsh equilibrium. A similr rgument cn lso be given for sellers. 9

10 10 Thus, for every cse bove, the corresponding bids cnnot be Nsh equilibrium. This proves clim (iii) of the theorem. The proof of prt (iv) is similr in some detils to tht for prt (iii) nd cn be found in the ppendix. Remrs. 1. It is obvious tht if the minimum in step (??) is not unique, the efficient Nsh equilibrium will not be unique. 2. Prts (i) nd (ii) of the bove result still hold when buyers me multiple unit combintoril bids nd sellers me single unit non-combintoril bids. 3. Note tht there re other Nsh equilibri where buyers my not bid their true vlution. Consider the setting of exmple 1. Exmple 3. Consider the bids to be 1 = 2.05, 2 = 2.05, 3 = 3, b 1 = 2.05, b 2 = 2.05 nd b 3 = 1.1. It is esy to chec this is Nsh equilibrium with efficient lloction. But note tht buyer 2 does not bid true vlution. Thus, in c-sebida it is not dominnt-strtegy for buyers or sellers to be truthful. 4. We hve considered Nsh equilibrium in wely rtionlizble strtegies since it is not rtionl for plyers to ply wely dominted strtegies. However, if we do consider ll strtegies, there re no-trde Nsh equilibri which my not be efficient s the following exmples show. Exmple 4. Consider buyer with v = 0.7 nd seller with c = 0.3. Clerly trde is possible nd in fct ny b = [0.3, 0.7] is Nsh equilibrium with n efficient outcome. However, consider the bids b = 0 nd = 1. Clerly, this is Nsh equilibrium with no trde, which is inefficient. But, these strtegies re strictly dominted by other strtegies, e.g., the buyer cn bid nything bove 0.3 nd the seller nything below 0.7. Exmple 5. Consider two goods (A nd B) cse. There is one buyer with v = 0.7 for one unit of both goods, nd zero otherwise. There is one seller who offers good A nd hs c 1 = 0.2 nd nother seller who offers good B nd hs c 2 = 0.3. Clerly, the efficient lloction involves n exchnge between these plyers. Now, consider b = 0.6, 1 = 0.4 nd 2 = 0.5. It is no-trde Nsh equilibrium. In fct, it is esy to chec tht there does not exist n efficient Nsh equilibrium even in wely rtionlizble strtegies. It is interesting to note tht Theorem 2. With multiple unit buy-bids nd single unit sell-bids, i.e., σ j = 1, j, the Nsh equilibrium lloction nd prices ((x, y ), ˆp) is competitive equilibrium. Proof: Consider mtched seller. He supplies exctly one unit t prices ˆp while n unmtched, nonmrginl seller (l, j) for j > l + 1, supplies zero units. The unmtched mrginl seller (l, l ) will supply zero units since ˆp l l,l +1. Now, consider mtched buyer i. At prices ˆp, he demnds up to δ i units of its bundle. If it is the mrginl mtched buyer, its surplus is zero nd it my receive nything up to δ i. If it is non-mrginl mtched buyer, it receives δ i units. An unmtched buyer, on the other hnd, hs zero demnd t prices ˆp. Thus, totl demnd equls totl supply, nd the mret clers. 4. Asymptotic Byesin Incentive Comptibility of c-sebida We now consider the incomplete informtion cse for the combintoril-sebida. Anlysis for the simpler non-combintoril setting cn be found in [20]. We nlyze the c-sebida mret mechnism in the limit of lrge number of plyers. Suppose there re n l sellers of good l, l = 1,, L nd m buyers with m l buyers who wnt good l, i.e., hve l in their bundle.

11 We will consider Byesin gme to model incomplete informtion. Let c l,j nd l,j denote the cost nd s-bid of the jth seller of good l respectively, nd v i nd b i denote the vlution nd buy-bid of the ith buyer with bundle R i respectively. Suppose nture drws c l,1,, c L,nL independently from the probbility distribution U[0, 1] nd drws v 1,, v m independently from probbility distributions, v i U[0, R i ]. Ech plyer is then reveled his own vlution or cost. It is common informtion tht the seller (l, j) s costs re drwn from U[0, 1] nd buyer i s vlutions re drwn from U[0, R i ], his R i being nown to ll. Let α l,j : [0, 1] [0, 1] denote the strtegy of the seller (l, j) nd β i : [0, R i ] [0, R i ] denote the strtegy of the buyer i. Then, the pyoff received by the buyers nd sellers is s defined by equtions (3) nd (4). Let θ = (α 1,1,, α L,nL, β 1,, β m ) denote the collective strtegy of the buyers nd the sellers. A buyer i chooses strtegy β i to mximize E[u b i(θ); β i ], the conditionl expecttion of the pyoff given its strtegy β i. The seller (l, j) chooses strtegy α l,j to mximize E[u s l,j(θ); α l,j ], the conditionl expecttion of the pyoff given its strtegy α l,j. The Byesin-Nsh equilibrium of the gme is then the Nsh equilibrium of the Byesin gme defined bove [11]. We consider semi-symmetric Byesin-Nsh equilibri, i.e., equilibri where ll the sellers of the sme good use the sme strtegy α l while the buyers my use different strtegies β i, since they my demnd bundles of different sizes. Let α l (c) := c nd β i (v) := v denote the truth-telling strtegies. Under the strtegy profile (α 1,, α L, β 1,, β m ), we denote the distribution of sbids l, nd buy-bids b i s F l nd G i respectively. We denote [1 F (x)] by F (x). Under α l nd β i, F l = U[0, 1] nd G i = U[0, R i ]. We will ssume tht plyers re ris-verse nd consider only those bid strtegies which stisfy the ex post individul rtionlity constrint, i.e., α l (c) c nd β i (v) v. Denote X l = {α l : α l (c) c}, X = X 1 X L, α n = (α1 n,, αl) n nd α = ( α 1,, α L ) when there re n sellers of ech good. Also denote Y i = {β i : β i (v) v}, Y = Y 1 Y m nd β n = (β1 n,, βm) n nd β = ( β 1,, β m ) when there re m buyers nd n sellers for ech good. Let m l denote the number of buyers who wnt good l. We will ssume tht m l = O(n). We consider single unit bids nd ssume tht semi-symmetric Byesin-Nsh equilibrium exists. And following Wilson [50, 45, 46, 47, 43], we me the following ssumption: Assumption 1. There exist symmetric Byesin-Nsh equilibri which hve seller s strtegies such tht α n(c) is uniformly bounded in n nd c. Theorem 3. Consider the c-sebida uction gme with (α, β) X Y, i.e., both buyers nd sellers hve ex post individul rtionlity constrint. Let (α n, β n ) be semi-symmetric Byesin Nsh equilibrium with m buyers nd n sellers of ech good. Then, (i) β n i (v) = β i (v) = v for i = 1,, m nd n 2, nd (ii) (α n, β n ) ( α, β) in the sup norm s n, i.e., c-sebida is symptoticlly Byesin incentive comptible. We proceed in three steps nd first prove two lemms. Lemm 1. Consider the c-sebida uction gme with m buyers nd n l sellers for good l. Suppose the sellers use bid strtegy profile α = (α 1,, α L ) with f l (), the pdf of its s-bid under strtegy α l. Then, the best-response strtegy profile of the buyers β n stisfies β n i (v) v for i = 1,, m nd n 2. Remrs. 1. As we noted in the single good cse s well, buyer s strtegy is to bid more thn his true vlue. This t first glnce seems surprising. However, intuitively it mes sense for this mechnism since the prices re determined by the sellers bids lone, nd by bidding higher, buyer only increses his probbility of being mtched. Of course, if he bids too high, he my end up with negtive pyoff. The result implies tht under the ex post individul rtionlity constrint, the buyers lwys use the strtegy profile β n = β. 2. It is lso worth noting tht the result cn be esily extended to the cse when ll the sellers my use different strtegies. The next step is to loo t the best response strtegy of the sellers when the buyers bid truthfully. 11

12 12 Lemm 2. Consider the c-sebida uction gme with n l = n sellers of good l nd m l buyers who wnt the good in their bundle, nd suppose the buyers bid truthfully, i.e., βi n = β i, nd let α n be the sellers best-response strtegy. Then, (α n, β) ( α, β) in the sup norm s n. The conclusion of this lemm is wht we would expect intuitively. If ll buyers bid truthfully, then s the number of sellers increses, incresed competition forces them to bid closer nd closer to their true costs. We cn use the bove two lemms to prove the min result of this section. Proof: (Theorem 3) By Lemm 1, when the sellers use the strtegy profile α = α n, the buyers under the ex post individul rtionlity constrint use the strtegy profile β. By Lemm 2, when the buyers bid truthfully, sellers best-response is α n. Thus, (α n, β) is Byesin-Nsh equilibrium with n sellers on ech good. Further, Lemm 2 shows tht (α n, β n ) = (α n, β) ( α, β) s n, which is the conclusion we wnted to estblish. Thus, under the ex post individul rtionlity constrint, c-sebida is ex nte budget blnced (of course, ex post budget blnced s well), symptoticlly Byesin incentive comptible nd efficient. Unlie in the complete informtion cse when the mechnism is not incentive comptible, yet the outcome is efficient, in the incomplete informtion cse, the mechnism is symptoticlly efficient. Remrs. 1. The bove result holds for ny rbitrry m, the number of buyers, nd in prticulr, for the cse where m increses with n to infinity. When m is finite, W nd W in proposition 1 both converge to zero in probbility. 2. The result bove depends on ssuming Wilson s hypothesis. Such n ssumption hs lso been mde in [50, 47, 43]. In fct, we hve been ble to show tht the seller strtegies we consider bove re strictly incresing nd uniformly continuous on [0,1] for every n. Assuming tht the strtegies re monotoniclly decresing in n (it might be possible to rgue this using results from monotone comprtive sttics [35]), we cn conclude using Dini s theorem [1] tht the strtegies converge uniformly. This yields equicontinuity of the strtegies nd it might be possible then to conclude existence of strtegies tht stisfy Wilson s hypothesis. However, we hve not been ble to completely resolve this open problem s of now. 3. Existence of semi-symmetric pure strtegy Byesin-Nsh equilibri hs been considered in the literture. A very generl result is obtined using fixed point theory on perturbed gmes [10] (see lso [39, 19]). They estblish existence of monotone pure strtegy equilibri in lrge enough uniform-price double-sided uctions. In [24], existence of monotone pure strtegy byesin-nsh equilibrium in mret mechnisms with generl vlues hs been studied using Lttice-theoretic methods [35, 33]. While [47, 28] show existence for prticulr uctions by showing the existence of solutions to the differentil equtions tht describe the equilibri. 4. The mechnism proposed in this pper is relted to the buyer s bid double uction (BBDA) mechnism [45, 47] nd its generliztion for single goods, the -double uction mechnism. For the specil cse of = 1, the -DA mechnism is the sme s BBDA. But note tht despite similr nomenclture nd spirit, BBDA nd SeBiDA determine prices differently. While the spirit of the two mechnisms is the sme (mximizing the efficiency of trding), the prices nd the pyments re different. Plese see remr 1 nd exmple 2 in section 2 for more detils on differences nd similrities. An exmple illustrting tht in c-sebida, neither the buyers nor the sellers hve dominnt strtegy to be truthful ws given in exmple 3 of section 3. This is lso the cse for BBDA s proved by the following counterexmple. Exmple 6. Consider one good type with two buyers who hve vlutions v 1 = 3.1, v 2 = 2.1 nd two sellers who hve costs c 1 = 1, c 2 = 2. Consider the bids b 1 = 2.05, b 2 = 2.05 nd 1 = 2.05, 2 = BBDA then determines price of p = 2.05 with two trdes. Moreover, this is full informtion Nsh equilibrium. But note tht neither the buyers nor the sellers re truthful.

13 13 5. Finlly, the ex post individul rtionlity constrint seems restrictive t first glnce. However, in two humn subject experiments we hve conducted using this mechnism [23], it ws observed tht ll subjects cted ris-verse nd in fct lwys used strtegies tht were ex post individul rtionl. Thus, the predictive power of the result does not seem diminished in rel settings despite the ssumption mde. 5. Conclusions We hve introduced combintoril, sellers bid, double uction (c-sebida). The first result concerned the Nsh equilibri for c-sebida with full informtion. In c-sebida, settlement prices re determined by sellers bids. We showed tht the lloction of c-sebida is efficient. Moreover, there is Nsh equilibrium in undominted strtegies wherein truth-telling is dominnt strtegy for ll plyers except the highest mtched seller for ech good. The second result concerned the Byesin-Nsh equilibrium of the mechnism under incomplete informtion. We showed tht under the ex post individul rtionlity constrint, the semisymmetric Byesin-Nsh equilibrium strtegies converge to truth-telling. Thus, the mechnism is symptoticlly Byesin incentive comptible, nd hence symptoticlly efficient. Thus, we hve proposed n exchnge mechnism for the multilterl Myerson-Stterthwite [36] trding environment with multiple goods. In such n environment it is impossible to chieve ll the four desirble properties of n uction mechnism. Nevertheless, we hve shown tht it is still possible to chieve ex post budget blnce nd individul rtionlity, nd symptotic Byesin incentive comptibility nd efficiency. In [21], we considered more generl setting nd showed tht competitive equilibrium exists in continuum model of n exchnge economy with indivisible goods nd money ( divisible good). There, using results from optiml control, we lso showed tht within the continuum model, c- SeBiDA outcome is competitive equilibrium. This gin suggests tht in the finite setting, the uction outcome is close to efficient. We hve tested the proposed mechnism c-sebida through humn-subject experiments. Those results cn be found elsewhere [23]. Finlly, while our wor ws primrily motivted by mret mechnism design problem, it cn lso be considered s n indirect contribution to the strtegic foundtions of competitive mrets [12]. This body of literture reltes Nsh nd Byesin-Nsh equilibrium with competitive equilibrium. The bsic ide is tht s the economy gets lrge (in our context the number of buyers nd sellers nd quntities of goods ll go to infinity), Nsh equilibrium strtegies should converge to competitive equilibrium strtegies, becuse the mret power diminishes. The reltionship is first investigted in [41]. In lter pper [14], it is shown tht under certin regulrity conditions, sufficiently replicted economy hs n lloction which is incentivecomptible, individully-rtionl nd ex-post ɛ-efficient. Similrly [17] shows tht the demnd functions tht n gent might consider bsed on strtegic considertions converge to the competitive demnd functions. Further, [18] shows tht under certin conditions on beliefs of individul gents, not only do the strtegic behviors of individul gents converge to the competitive behvior but the Nsh equilibrium lloctions lso converge to the competitive equilibrium lloction. The formultion in [50] is buyer s bid double uction with single type of good tht mximizes surplus. It is shown tht with Byesin-Nsh strtegies, the mechnism is symptoticlly incentive efficient, the notion of incentive efficiency being different from tht of incentive comptibility nd efficiency tht we use here. Along different line of investigtion, [13, 46, 43] investigte the rte of convergence of the Nsh equilibri to the competitive equilibri for buyer s bid double uction. Finlly, implementtion nd mechnism design in setting with continuum of plyers is discussed in [31]. We hve provided mret mechnism tht symptoticlly chieves competitive behvior in multilterl, multiple good trding environment with incomplete informtion.

14 14 Appendix A: Proof of Lemm 1 Proof: Set l,0 = c l,0 = 0 nd b 0 = v 0 = L. Fix buyer i with vlution v nd bundle R i. Suppose the sellers use fixed bidding strtegy α nd denote the buyers best-response strtegy profile by β n. Let θ i denote the strtegy of ll the other plyers. Then, there is level U, function of θ i such tht the bid b of i is ccepted if b > U. It is esy to see tht the lloction z(b) = (x(b), y(b)) is some z = (x, y ) for ll b > U. Suppose not: Let z 1 be the lloction for U < U 1 < b < U 2 nd z 2 be the lloction for b > U 2. But clerly, the uction surplus, b U 1 > b U 2 for bids b > U 2 s well. Thus, the lloction z 1 will yield higher uction surplus thn z 2 for b > U 2 s well. Thus, z 2 = z 1 nd the corresponding price Y is the sme for ll b > U. Note tht Y U. Thus, buyer i s pyoff when he bids b is { π i(b) v Y, if b > U = (8) 0, otherwise. The expected pyoff denoted by π i then is given by π i(b) = b u nd the buyer i s best response stisfies the differentil eqution d π i db = 0 b 0 0 (v y)f Y,U (y, u) dydu (9) (v y)f Y,U (y, b) dy = 0 (10) The boundry condition for the differentil eqution is π i(0) = 0. Since the left-hnd side of the eqution bove is lwys non-negtive (nd in fct positive) for ll b v, it is cler tht the best response b = β n i (v) v, n 2. Appendix B: Proof of Lemm 2 Proof: Fix good l (sy =1). Set l,0 = c l,0 = 0, nd b 0 = v 0 = L. Fix seller (l, j) with cost c (in the rest of the proof we will refer to this seller s seller j). Consider the uction gme, denoted G (l,j), in which seller j bids very high nd his bid is not ccepted, nd ll buyers bid truthfully. Let z = (x, y) denote the corresponding lloction. Denote the number of mtched buyers nd sellers on good l by K l, X = l(kl ), the bid of the highest mtched seller, Y = l(kl +1), the bid of the lowest unmtched seller, nd Z = l(kl 1), the bid of the next highest mtched seller. Suppose seller j bids nd let z t = ( x t, ỹ t ) be the corresponding lloction. Let the lloction z t differ from z in the following wy: There is set of buyers B t nd set of sellers S t whose bids re ccepted in z but not in z t. And there is set of buyers B t nd set of sellers S t (excluding j) whose bids re ccepted in z t but not in z. Then, the seller j s bid is ccepted if the uction surplus now is greter, i.e., if v( B t ) ( S t ) > v(b t ) (S t ), (11) Thus, if < W t := (v( B t ) ( S t )) (v(b t ) (S t )), the bids corresponding to lloction z t result in higher uction surplus thn the bids corresponding to the lloction z. Now, for vrious levels of bid, there my be mny lloctions z t, t = 0,, T with corresponding levels W t, t = 0,, T. Observe tht one possible lloction is B t = B t =, S t =, S t = {(l, (K l ))} with (sy) W 0 = X. This is the cse when the only chnge is tht the seller j displces the highest mtched seller (l, (K l )) on the good. Denote W := mx t 1 W t. Note tht out of the vrious levels W t, only the mximum mtters since the bid is ccepted s long s < mx t 0 W t. Further, when tht is true, the resulting lloction will be the one corresponding to t = rg mx t 0 W t.

15 15 Thus, the pyoff of the j-th seller when he bids = α(c) is given by x c, if < Z < X < W, or Z < < X < W ; c, if Z < X < < W, or Z < < W < X, or π j () = Z < W < < X, or W < Z < < X; z c, if < Z < W < X, or < W < Z < X, or W < < Z < X. (12) The pyoff of the seller s his bid vries is shown grphiclly in figure 2. The reder cn convince himself tht the only relevnt quntities for pyoff clcultion re X, Z nd W. Thus, there re three cses: (i) Z < X < W, (ii) Z < W < X nd (iii) W < Z < X. It is esy to verify tht the expected pyoff of seller j, denoted by π j stisfies the differentil eqution d π j () = [P n (A ) + P n (B ) + P n (C )]d ( c)[dp n (D ) + dp n (E )], (13) with the boundry condition π j (1) = 0, where A denotes the event {X < < W }. As is incresed by d, the pyoff to the seller increses by d since seller j is the price-determining seller. Similrly, B denotes the event {Z < < W < X} nd seller j is the price-determining seller. In the sme wy, C denotes the event {mx(z, W ) < < X} nd seller j is the price-determining seller. D denotes the event {X < nd W [, + d]}, so tht the seller j becomes unmtched s it increses its bid from to + d. Similrly, E is the event {W < nd X [, + d]}. And so, s he increses his bid, he becomes unmtched. Figure 2 shows these events grphiclly. Events A, B nd C correspond to vrious cses when the chnge in the bid from to +d, cuses chnge in pyoff of d. Events D nd E correspond to cses when the chnge in the bid from + d, cuses chnge in pyoff of ( c). Given the strtegy profile α used by the sellers, the strtegy profile β used by the buyers, let the probbility distribution of s-bid of seller on good l be F (with pdf f). Note tht α nd F depends on n. We first obtin symptotic upper nd lower bounds on W (here clled W n to stress its dependence on n). Proposition 1. Define W := X 1(K1 ) nd W := X 1(K1 +1). Then, (i) W W n W in probbility, i.e., P (W n W ), P (W W n ) 1 s n. (ii) For ny ɛ > 0 nd lrge enough n, P (W n > ɛ) P (W > ɛ) nd P (W n ɛ) P (W ɛ). Proof: (i) Let B 1 denote the set of buyers who wnt good l = 1, nd whose bids re not ccepted when seller is not present. Consider ny buyer t B 1. Then, W t = [v t (S(L 1t ) S(L 2t ))] + [v(b t ) (S(L 3t ) S(L 4t ))] [v(b t ) (S(L 1t ) S(L 3t ) S(L 5t ))], (14) where S(L) denotes the highest mtched sellers on the set of goods L, S(L) denotes the lowest unmtched sellers on goods L, (S) denotes the sum of bids of the sellers S, B t is the set of buyers (excluding t) whose bids cn get ccepted t seller bid, B t is the set of buyers which become unmtched t new seller bid. Above, L 1t is the set of goods lso demnded by buyer t nd on