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 nd Prvin Vriy EECS Deprtment University of Cliforni, Bereley (rjin,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 [34]. We propose combintoril mret mechnism which in the complete informtion cse is lwys 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. This seems to be the only nown combintoril mret mechnism with these properties. History : This version: Februry 14, Introduction The generl equilibrium model of competitive mrets [3] ignores the strtegic behvior of mret prticipnts. Thus, estblishing the strtegic foundtions of competitive mrets hs become centrl problem in economic theory [12]. Often, competitive mret mechnisms re modelled s lrge double uctions with incomplete informtion [22]. At the sme time, efficient double uction mechnisms re of independent interest for mny multilterl trding environments. Such trding environments vry in type specifiction for the plyers: Plyers my hve vlues/types which re either common or privte, nd in cse privte, independent or correlted. The type spce cn be one-dimensionl or multi-dimensionl. The clssicl Vicrey-Clre-Groves (VCG) mechnism is Byesin incentive-comptible, individul rtionl nd (lloctively) efficient with independent privte vlues mong plyers for multiple goods. Extensions hve lso been proposed for plyers with common nd correlted privte vlues [9, 25]. A Vicrey double uction for single good hs lso been proposed [50]. However, the VCG clss of double uction mechnisms re not ex post budget-blnced, which is ey requirement for mny multilterl trding environments. In this pper 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 * The first uthor would lie to thn Chris Shnnon, Hl Vrin, nd Jen Wlrnd for some very helpful discussions. Significnt contribution to this wor ws lso mde by Chris Ksiris through his experimentl wor bsed on this pper [21]. This reserch ws supported by the Ntionl Science Foundtion grnts ECS nd xx123, nd Fujitsu Lbs, USA. 1

2 2 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 [38] nd spectrum uctions [32], see section 6 for discussion of other pplictions). 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 item (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 item is the highest price sed by mtched seller (hence sellers bid uction). As result there is uniform price for ech item. 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, (surprisingly) every Nsh equilibrium is (lloctively) efficient. Moreover, there is Nsh equilibrium in undominted strtegies 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. 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 [30], the semi-symmetric Byesin-Nsh equilibrium strtegies (wherein ll sellers selling the sme item 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 [34] 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 [48]. The -double uction mechnism ws further generlized to the single-item multilterl cse by Stterthwite nd Willims [43, 44]. 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. A survey of the vst uction theory literture is provided in [24, 47]. Mny re extensions of Vicrey s ides [46]. Recently, [25] 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, 37] chieve the sme outcome s VCG. However, these re single-sided uction mechnisms. A Vicrey double uction mechnism for single goods is proposed in [50] 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 [36]. 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. This ppers to be the first such double-uction mechnism for multiple goods which is not of VCG-type. 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 [21]. The interply between economic, gme-theoretic nd computtionl issues hs spred interest in lgorithmic mechnism design [40, 47]. The generlized Vicrey uction mechnisms for multiple heterogeneous goods re not computtionlly trctble [35, 36]. Thus, mechnisms tht rely on

3 3 pproximtion of the integer progrm [35, 42] or liner progrmming [7] hve been proposed. The results here lso relte to the recent efforts in the networ pricing literture [27]. There is n ongoing effort to propose mechnisms for divisible resource lloction in networs through uctions [23] nd to understnd the worst cse Nsh equilibrium efficiency loss of such mechnisms when users ct strtegiclly [20]. Optiml mechnisms for single divisible goods tht minimize this efficiency loss hve been proposed [49, 28] 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. 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 [30]. 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) [34]. In this pper, we provide non-vcg combintoril (mret) mechnism which in the complete informtion cse is lwys 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 ny Nsh equilibrium lloction (sy of networ resource lloction gme) is lwys efficient (zero efficiency loss) nd ny (semi-symmetric) Byesin-Nsh equilibrium lloction is symptoticlly efficient. This seems to be the only nown combintoril double-uction mechnism with these properties. 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 [34] 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 mechnism proposed is mret mechnism. Thus, we lso me n indirect contribution to the theory of strtegic foundtions of competitive mrets (see section 7 for discussion). 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 5 we consider the Byesin-Nsh equilibrium of the incomplete informtion uction gme for multiple goods. Section 4 provides simplified proof in cse of single good. Section 6 discusses pplictions. 2. The Combintoril Sellers Bid Double Auction A buyer plces buy bids for bundle of items. A buyer s bid is combintoril: he must receive ll items 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

4 4 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 item 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-item combintoril bids, but sellers only offer one type of item. This yields uniform settlement prices for ech item. 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. Forml mechnism. There re L items l 1,, l L, m buyers nd n sellers. Buyer i hs (true) reservtion vlue v i per unit for bundle of items 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. 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 item 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. Mtched buyers py the sum of the prices of items in their bundle; mtched sellers receive pyment equl to the number of units sold times the price

5 for the item. 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 item 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 item, 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. Remrs. 1. The proposed mechnism resembles the -double uction mechnism [43]. We designed c-sebida so tht its outcome mimics competitive equilibrium with prticulr interest in the combintoril cse. The single item version SeBiDA resembles the -double uction ( specil cse being clled the buyer s bid double uction [44, 45]). 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 [43]. But note tht despite similr nomenclture nd spirit, SeBiDA nd c-sebida determine prices differently. It is not cler wht generliztion of the -double uction 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. 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 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. 5

6 6 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 SW (x, y) = i v i x i j c j y j. 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 construct Nsh equilibrium, nd show it yields n efficient lloction (Theorem 1). 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 in undominted strtegies wherein except for the mtched seller with the highest bid on ech item, ech plyer bids truthfully. (iii) Furthermore, ny Nsh equilibrium lloction is efficient. Proof: 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. Set l,0 = c l,0 = 0, b 0 = v 0 = 1. Consider the surplus mximiztion problem (1) with true vlutions nd costs. Let I be the set of mtched buyers nd l the number of mtched sellers offering item l determined by the MIP. Set b i = v i for ll i; 0 l,j = c l,j ; γ t i = b i l R i t l, l, the reveled surplus of mtched buyer i t stge t 0, nd ˆl rg min { min γ t i : γ t i > 0}, (5) l i I:l R i the item with the smllest surplus mong the mtched buyers t stge t, with ech l being piced only once. Denote the corresponding surplus by γ ṱ. We will denote the corresponding minim by l γ ṱ. Now, define l t+1 := min{ ṱ, ˆl,ˆl l,ˆl+1 ṱ + γ ṱ }, (6) l l,ˆl which is the strtegy of seller (ˆl, ˆl) t the t-th stge: His s bid is incresed to decrese the surplus of the mtched buyer with the smllest surplus up to the s bid of the unmtched seller with the lowest bid. For ll other (l, j) (ˆl, ˆl), the s bid remins the sme, t+1 l,j = t l,j. This procedure is repeted until the strtegies converge such tht ech l is piced only once. In fct, it is repeted t most L times. Observe tht t ech stge, the mtches nd the lloctions from the MIP using the current bids (b, t ) do not chnge. Let denote the seller s bids when the procedure converges. We prove tht (b, ) is Nsh equilibrium, by showing tht no plyer hs n incentive to devite. First, n unmtched seller offering item l hs no incentive to bid lower thn l, l : Becuse his reservtion cost is higher thn tht, by bidding lower thn his reservtion cost, it my get mtched but his pyoff will be negtive. Next, consider 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, it is the dominnt strtegy of ll sellers except the mrginl seller (l, l ) to bid truthfully. Now, consider this mrginl mtched seller (l, l ). If he bids lower then l, l, his pyoff will decrese. He could bid higher but becuse of (6), either there is n unmtched seller of the item with the sme s bid, or there is mrginl buyer whose surplus hs been mde zero by (6).

7 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 buyer with zero surplus will become unmtched cusing this mrginl seller to be unmtched s well. Thus, l, l is Nsh strtegy of the mrginl seller given tht ll other plyers (except the mrginl sellers of the other items) bid truthfully. Now, consider the buyers. First, n unmtched buyer i hs no incentive to bid lower thn b i since he wouldn t mtch nywy. And if he bids higher, he my become mtched but his pyoff will become negtive. Next, mtched buyer with positive pyoff hs no incentive to bid lower since by bidding lower he cn lower the prices but only when he becomes unmtched. Also, he certinly hs no incentive to bid higher since by so doing he will not be ble to lower the price. Lstly, consider the mrginl mtched buyers with zero pyoff: Clerly, if they bid higher, their pyoff will become negtive; nd if they bid lower, they will become unmtched. Thus, it is the dominnt strtegy of ll buyers to bid truthfully. The Nsh equilibrium lloction (x, y ) s determined bove is efficient since it mximizes (1) with true vlutions. We now show tht ny Nsh equilibrium lloction is efficient. Suppose ( x, ỹ) is nother Nsh equilibrium lloction which is not efficient. Either there is buyer or seller which goes from being mtched in (x, y ) to being unmtched in ( x, ỹ), or vice-vers. If there is seller tht goes from being mtched to unmtched then either there is mtched seller in (x, y ) replced by nother seller in ( x, ỹ) selling the sme item (cse (i)), or some unmtched sellers in (x, y ) re mtched in ( x, ỹ) with the set of mtched sellers in (x, y ) remining mtched. In this cse, some unmtched buyer must lso become mtched (cse (ii)). The rest of the cses cn be rgued similrly. Thus, the two Nsh equilibrium lloctions would differ in one of the five cses s we go from (x, y ) to ( x, ỹ). (i) A mtched seller (l, j 1 ) is mde unmtched nd unmtched seller (l, j 2 ) is mde mtched; (ii) An unmtched buyer i demnding R i is mde mtched nd set of unmtched sellers J such tht {l : (l, j l ) J} = R i re mde mtched; (iii) A mtched buyer i demnding R i is mde unmtched nd set of mtched sellers J such tht {l j : j J} = R i re mde unmtched; (iv) A set mtched buyers i I demnding R i re mde unmtched nd set of unmtched buyers J with j J demnding R j such tht j J R j = i I R i re mde mtched; Cse (i) We must hve c l,j1 < c l,j2 nd the new bids must stisfy ã l,j2 < ã l,j1. But then either (l, j 2 ) s pyoff is negtive or (l, j 1 ) cn lso bid just bove (l, j 2 ) s bid. In either cse ( x, ỹ) cnnot be Nsh equilibrium lloction. Cse (ii) We must hve v i < (l,j l ) R i c l,jl nd the new bids must stisfy b i > (l,j l ) R i ã l,l with ã l,jl < ã l,l. This mens tht either the buyer or t lest one seller hs negtive pyoff. Thus, ( x, ỹ) cnnot be Nsh equilibrium lloction. Cse (iii) The rgument for this cse is similr to cse (ii). Cse (iv) We must hve v i i > v j J j nd the new bids must stisfy b i i < b j J j. But then either there exists buyer j J whose pyoff is negtive or ll i I cn bid high enough to outbid ll j J. In either cse ( x, ỹ) cnnot be Nsh equilibrium lloction. Thus, the every Nsh equilibrium lloction is efficient. This proves (iii). 7 Remrs. 1. It is obvious tht if the minimum in step (5) is not unique, the Nsh equilibrium will not be unique. However, ny Nsh equilibrium lloction will still be efficient. Furthermore, if

8 8 there is unique efficient lloction, the Nsh equilibrium is lso unique. 2. The bove result still holds when buyers me multiple unit combintoril bids nd sellers me single unit non-combintoril bids. 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, non-mrginl 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. SeBiDA is Asymptoticlly Byesin Incentive Comptible We now consider the incomplete informtion cse. We nlyze the SeBiDA mret mechnism 1 in the limit of lrge number of plyers. We ssume tht the number of buyers nd the number of sellers is the sme, n 2. The results cn be extended to the cse when the number of buyers nd sellers re different. We will consider Byesin gme to model incomplete informtion. Suppose nture drws c 1,, c n independently from probbility distribution U 1 nd drws v 1,, v n independently from probbility distribution U 2, which re such tht the corresponding pdfs u 1 nd u 2 hve full support on [0, 1]. Ech plyer is then told his own vlution or cost. It is common informtion tht the seller costs re drwn from U 1 nd buyer vlutions re drwn from U 2. Let α j : [0, 1] [0, 1] denote the strtegy of the seller j nd β i : [0, 1] [0, 1] 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,, α n, β 1,, β n ) 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 j chooses strtegy α j to mximize E[u s j(θ); α j ], the conditionl expecttion of the pyoff given its strtegy α j. The Byesin-Nsh equilibrium of the gme is then the Nsh equilibrium of the Byesin gme defined bove [11]. We consider symmetric Byesin-Nsh equilibri, i.e., equilibri where ll buyers use the sme strtegy β nd ll sellers use the sme strtegy α. Let α(c) := c nd β(v) := v denote the truthtelling strtegies. Under strtegies α nd β, we denote the distribution of s-bids nd buy-bids b s F nd G respectively. We denote [1 F (x)] by F (x). Under α nd β, F = U 1 nd G = U 2. We consider only those bid strtegies which stisfy the ex post individul rtionlity (IR) constrint, i.e., α(c) c nd β(v) v. Denote X = {α : α(c) c} nd Y = {β : β(v) v}. We consider single unit bids nd ssume tht symmetric Byesin-Nsh equilibrium exists. (See [44] for rguments of existence of Byesin-Nsh equilibri in BBDA.) Theorem 3. Consider the 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 symmetric Byesin Nsh equilibrium with n buyers nd n sellers. Then, (i) β n (v) = β(v) = v n 2, nd (ii) (α n, β n ) ( α, β) in sup norm s n, i.e., SeBiDA is symptoticlly Byesin incentive comptible. 1 This section only trets the non-combintoril cse. The combintoril cse is presented in the next section. The proof in both cses hs the sme philosophy though differs in some detils. We provide it here to ese understnding of the next section.

9 We will first prove two lemms. Lemm 1. Consider the SeBiDA uction gme with n buyers nd n sellers. Suppose the sellers use bid strtegy α with f(), the pdf of its s-bid under strtegy α. Then, every best-response strtegy of the buyers β n stisfies β n (v) v for ll n 2. Proof: Set 0 = c 0 = 0, b 0 = v 0 = 1. Fix buyer j with vlution v. Suppose sellers use fixed bidding strtegy α nd denote the buyers best-response bidding strtegy by β n. Consider the gme denoted G j, where ll plyers except buyer j prticipte nd bid truthfully. Denote the number of mtched buyers nd sellers by K = sup{ : () b () }, which is rndom vrible. Here () denotes the order sttistics incresing with over the s-bids of the prticipting sellers nd b () the order sttistics decresing with over the buy-bids of the prticipting buyers. Denote X = (K), the s-bid of the mtched seller with the highest bid, Y = (K+1), the s-bid of the unmtched seller with the lowest bid nd U = b (K), the buy-bid of the mtched buyer with the lowest bid. It is esy to chec tht when buyer j lso prticiptes nd bids b = β(v), he gets positive pyoff { π j(b) v X, if X < U < b nd U < Y ; = (7) v Y, if X < Y < b nd Y < U. The pyoff of the buyer s function of its bid b is shown grphiclly in figure 1. The reder cn convince himself tht the only relevnt quntities for pyoff clcultion re X, Y nd U. Thus, there re only two possible cses: (i) X < Y < U nd (ii) X < U < Y. Figure 1 (i) shows the cse of (i) nd the pyoffs s b vries. As b increses bove the dotted line, the pyoff chnges from zero to v b. Similrly, s b increses bove the dotted line in figure 1 (ii), the pyoff chnges from zero to v x. The expected pyoff denoted by π j stisfies the differentil eqution where d π j db = P n (A b,b )nf(b)(v b) + =0 b n 1 n 1 P n (A x,y ) = F (x) F n 1 (x) 0 P n (B x,b )nf(x)(n 1)g(b)(v x)dx, (8) ( n 1 ) Ḡ (y)g n 1 (y) is the probbility of the event tht X = x nd Y = y with x < y, mong n 1 sellers nd n 1 buyers. Similrly, n 1 n 1 P n (B x,y ) = F 1 (x) 1 F n 2 n (x) Ḡ 1 (y)g n 1 (y) 1 =1 is the probbility of the event tht X = x nd Y = y with x < y, mong n 1 sellers nd n 2 buyers. The boundry condition for the differentil eqution is π j(0) = 0. The first term bove rises from the chnge in pyoff when b is incresed by b nd U > Y > b > X, nd b + b > Y s shown in figure 1(i). Similrly, the second term is the chnge in pyoff when Y > U > b > X nd b + b > U s shown in figure 1(ii). It is cler from (8) tht for b v, d π j db 9 > 0. Given tht the sellers ply strtegy α, the best-response strtegy of the buyers β n is such tht b = β n (v) nd d π j db = 0. From this it is cler tht b = β n (v) v, n 2. (9)

10 10 The bove conclusion t first glnce seems surprising. A buyer s strtegy is to bid more thn his true vlue. 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 buyer lwys uses the strtegy β n = β. Now, we loo t the best response strtegy of the sellers when the buyers bid truthfully. Lemm 2. Consider the SeBiDA uction gme with n buyers nd n sellers nd suppose buyers bid truthfully, i.e., β n = β, nd let α n be the sellers best-response strtegy. Then, (α n, β) ( α, β) s n. Proof: Set 0 = c 0 = 0, b 0 = v 0 = 1. Fix seller i with cost c. Consider the uction gme, denoted G i, in which seller i does not prticipte nd ll prticipting buyers bid truthfully. As before, denote the number of mtched buyers nd sellers by K = sup{ : () b () }, U = b (K), the bid of the lowest mtched buyer, W = b (K+1), the bid of the highest unmtched buyer, X = (K), the bid of the highest mtched seller, Y = (K+1), the bid of the lowest unmtched seller, nd Z = (K 1), the bid of the next highest mtched seller. Consider the pyoff of the i-th seller when he prticiptes s well. His pyoff 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 i () = Z < W < < X, or (10) W < Z < < X; z c, if < Z < W < X, or < W < Z < X, or W < < Z < X. 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. The expected pyoff denoted by π i stisfies the differentil eqution d π i () d = [P n (A ) + P n (B ) + P n (C )] [ng()p n (D ) + (n 1)f()P n (E )]( c), (11) with the boundry condition π i (1) = 0 where A denotes the event tht there re n 1 sellers nd n buyers nd X < < W. As is incresed by, the pyoff to the seller lso increses by since seller i is the price-determining seller. Similrly, B denotes the event tht there re n 1 sellers nd n buyers nd Z < < W < X nd seller i is the price-determining seller. In the sme wy, C denotes the event tht there re n 1 sellers, n buyers nd mx(z, W ) < < X nd seller i is the price-determining seller. D denotes the event tht there re n 1 sellers nd n 1 buyers, X < (with the n-th buyer bidding ) nd W [, + ] so tht the seller i becomes unmtched s it increses its bid. Similrly, E is the event tht there re n 2 sellers, n buyers, W < (with the (n 1)-th seller bidding ) nd X [, + ]. 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, cuses chnge in pyoff of. Events D nd E correspond to cses when the chnge in the bid from +, cuses chnge in pyoff of ( c).

11 The following cn then be obtined: n 1 n 1 P n (A ) = F () F n n 1 () Ḡ +1 ()G n (+1) () + 1 =0 n 1 n 1 P n (B ) = F 1 () 1 F n n () Ḡ +1 ()G n (+1) () + 1 =1 n 1 n 1 P n (C ) = F 1 () 1 F n n () Ḡ ()G n () =1 n 1 n 1 P n (D ) = F () F n 1 n 1 () Ḡ ()G n 1 () =0 n 1 n 2 P n (E ) = F 1 () 1 F n n 1 () Ḡ ()G n (). (12) =1 Let = α n (c) be the best-response strtegy of the sellers. Then, d π i = 0 t = α d n(c). For ny < c, d π i > 0 from (11). Thus, d = α n (c) c, n 2. (13) 11 If > c, setting (11) equl to zero nd rerrnging, we get from which we obtin f() = [P n (A ) + P n (B ) + P n (C )] ng()p n (D )( c) (n 1)P n (E )( c) α n (c) c [P n (A ) + P n (B ) + P n (C )] ng()p n (D ) 1 n 1 ( n 1 ) 2 z =0 (+1) g() n 1 ( n 1 ) 2z Ḡ g() =0 n 1 ( n 1 ) 2 z =1 (n ) 2 n 1 ( n 1 ) 2z =0 n 1 ( n 1 ) 2 z =1 (n ) n 1 ( n 1 ) 2z =0 F, F 0, Ḡ F F (14) where z = F ()Ḡ() F ()G(). Observe tht the terms Ḡ(), Ḡ() F () nd F () in the numertor re upperbounded by one, nd the term F () in the denomintor is lower-bounded by F (c). It cn now be shown tht ech of the terms converges to zero for ll z > 0 s n 0. Thus, (α n, β) ( α, β). 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. Proof: (Theorem 3) By Lemm 3 when the sellers use strtegy α n, the buyers under the ex post individul rtionlity constrint use strtegy β. By Lemm 4, when the buyers bid truthfully, sellers best-response is α n. Thus, (α n, β) is Byesin-Nsh equilibrium with n plyers on ech side of the mret. Further, Lemm 4 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, SeBiDA is ex nte budget blnced, 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 only symptoticlly efficient.

12 12 5. Asymptotic Byesin Incentive Comptibility of c-sebida We now consider the incomplete informtion cse for the combintoril-sebida. 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. 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 [48, 43, 44, 45, 41], 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 4. 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 will first prove two lemms. Lemm 3. Consider the c-sebida uction gme with m buyers nd n l sellers for item 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. 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))

13 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 = (15) 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 (v y)f Y,U (y, u) dydu (16) (v y)f Y,U (y, b) dy = 0 (17) 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. 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. Now, we loo t the best response strtegy of the sellers when the buyers bid truthfully. Lemm 4. 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. 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 ), (18) 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.

14 14 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. Thus, the pyoff of the j-th seller when he bids = α(c) is given by π j () = x c, c, z c, if < Z < X < W, or Z < < X < W ; if Z < X < < W, or Z < < W < X, or Z < W < < X, or W < Z < < X; if < Z < W < X, or < W < Z < X, or W < < Z < X. 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 )], (20) 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 ))], (21) (19)

15 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 which highest mtched sellers remin mtched; L 2t is the set of goods lso demnded by buyer t where formerly unmtched sellers become mtched; L 3t is the set of goods demnded by buyers B t where highest mtched sellers remin mtched; L 4t is the set of goods demnded by buyers B t where formerly unmtched sellers become mtched; nd L 5t is the set of goods demnded by B t which now become unmtched. The first term in squre brcets in eqution (21) represents the contribution to the uction surplus when buyer t is mtched; the third term represents the contribution to the uction surplus by buyers B t which is being lost when seller is introduced; the second term is the contribution to the uction surplus by buyers B t whose cceptnce becomes possible since buyers B t re now unmtched. Thus, the sets L 1t,, L 5t re disjoint nd do not include l = 1. Thus, bid cn be ccepted if W t > for some t B 1, i.e., if W := mx t B1 W t >. Clerly, the third term in the squre brcets of eqution (21) is greter thn the second term in the squre brcets, otherwise the bids of B t, S(L 3t ), S(L 4t ) would hve been ccepted before insted of bids of plyers B t, S(L 1t L 3t L 5t ). Thus, W t v t (S(L 1t ) S(L 2t )) v t (S(L 1t ) S(L 2t )), where the second inequlity is obvious. Suppose buyer t wnts only good l = 1. Then, W t v t X 1(K1 +1), the bid of the lowest unmtched seller of good 1, where K 1 is the number of mtches for good l = 1. Next consider buyer t who wnts goods l = 1, 2. Then, W t v t X 2(K2 ) where K 2 is the number of mtches on good 2. Further note tht v t must be smller thn X 1(K1 +1) + X 2(K2 +1), otherwise buyer t could hve mtched with the lowest unmtched sellers on the two goods. Thus, we hve W t X 1(K1 +1) + (X 2(K2 +1) X 2(K2 )). Defining l () = (X l(+1) X l() ), we see tht in generl for buyer t who wnts goods R t (including l = 1), W n := mx W t X 1(K1 +1) + l (K l ) =: W n.s. (22) t B 1 l 1 Now, s n, l (K l ) P 0 (convergence in probbility) for every l. This implies tht Thus, for n W P n W := X 1(K1 +1). P (W n W ) 1. Let us now consider eqution (21) to obtin lower bound. W t [v t (S(L 1t L 2t ))] [(S(L 3t )) (S(L 3t ))] since v(b t ) < (S(L 1t L 2t ) S(L 5t )) (otherwise the set of buyers B t could still mtch). Also, note tht the second term in the squre brcets is l L 3t l (K l ) P 0 s n. Now, if buyer t wnts only one good l = 1, then L 1t, L 2t = nd W t v t X 1(K1 ) otherwise it cnnot mtch. If buyer wnts two goods (sy 1 nd 2), then v t > X 1(K1 ) + X 2(K2 ) otherwise it cnnot mtch. Thus, W t X 1(K1 ) 2 (K 2 ) l L 3t l (K l ). 15

16 16 And, in generl, we hve W n := mx W t X 1(K1 ) l () l (K l ) =: W n. (23) t B 1 l 1 l L 3t Since l (K l ) P 0 s n nd for ll l, we hve which implies for n, W n P W := X 1(K1 ), P (W n W ) 1. (ii) We will prove only the first prt. We now tht W n W n.s. nd W n W i.p. Thus, for some n nd 0 < δ < ɛ, we hve P (W n > ɛ) = P (W n > ɛ, W n W + δ) + P (W n > ɛ, W n < W + δ) P (W n W + δ) + P (W > ɛ δ) nd we get tht lim sup P (W n > ɛ) P (W > ɛ δ) n since lim sup n P (W n W + δ) = 0. Since, the inequlity bove is vlid for ny 0 < δ < ɛ, we hve tht for lrge enough n, P (W n > ɛ) P (W > ɛ). W t cn be interpreted s the effective bid of n unmtched buyer t (who wnts good 1) on good 1. W is the highest such effective bid. As long s is smller thn W, bid cn be ccepted. The proposition bove shows tht W in fct lies between X = X 1(K1 ) nd Y = X 1(K1 +1) when n becomes lrge (we will drop the subscript 1 for good l = 1 below). For single good cse, W = b K+1, the highest unmtched buy-bid on the good, which is smller thn Y, nd cn only be ccepted upon introducing nother seller with bid if it is bigger thn X. Now, observe tht P n (A ) = P (X < W >, K = )P (W >, K = ) P (X () < < X (+1) )P (W >, K = ) P (X () < < X (+1) )P (X (+1) > ) n 1 n 1 = F ()F n 1 n 1 () F i ()F n 1 i () (24) i =0 The first equlity follows from conditioning nd Byes rule nd uses proposition 1. The second inequlity holds symptoticlly (for lrge n). The lst equlity is obtined using order sttistics rguments. In the sme wy, we cn obtin the following: i=0 P n (B ) = P (X ( 1) < < X () )P (X (+1) > ) n 1 F 1 () 1 F n () n 1 F i ()F n 1 i (), (25) i i=0