Mechanism Design for Double Auctions with Temporal Constraints

Transcription

1 Proceedngs of the Twenty-Second Internatonal Jont Conference on Artfcal Intellgence Mechansm Desgn for Double Auctons wth Temporal Constrants Dengj Zhao 1,2 and Dongmo Zhang 1 Intellgent Systems Lab Unversty of Western Sydney, Australa {dzhao,dongmo}@scm.uws.edu.au Laurent Perrussel 2 IRIT Unversty of Toulouse, France laurent.perrussel@unv-tlse1.fr Abstract Ths paper examnes an extended double aucton model where market clearng s restrcted by temporal constrants. It s found that the allocaton problem n ths model can be effectvely transformed nto a weghted bpartte matchng n graph theory. By usng the augmentaton technque, we propose a Vckrey-Clarke-Groves (VCG) mechansm n ths model and demonstrate the advantages of the payment compared wth the classcal VCG payment (the Clarke pvot payment). We also show that the algorthms for both allocaton and payment calculaton run n polynomal tme. It s expected that the method and results provded n ths paper can be appled to the desgn and analyss of dynamc double auctons and futures markets. 1 Introducton A double aucton market allows multple buyers and sellers to trade commodtes smultaneously. Most modern exchange markets, e.g. the New York Stock Exchange, use double aucton mechansms. In a typcal double aucton market, buyers submt bds (buy orders) to the auctoneer (the market maker) offerng the hghest prces they are wllng to pay for a certan commodty, and sellers submt asks (sell orders) to set the lowest prces they can accept for sellng the commodty. The auctoneer collects the orders and tres to match them usng certan market clearng polces n order to make transactons. Although prce s the major concern of market clearng n most double aucton markets, other factors, such as quantty, qualty and temporal constrants, are equally mportant n some market stuatons. For nstance, a futures contract normally specfcs not only the prce of the underlyng commodty but also quantty, qualty and settlement date. Nevertheless, most real-world exchange markets are purely prcedrven and most studes on double auctons are lmted to a sngle-valued doman [Wlson, 1985; McAfee, 1992]. One reason s that some factors, e.g. quantty and qualty, can be elmnated by standardsng exchange commodtes. However, those attrbutes wth a contnuous range or large number of varetes, are hard to standardse. Ths research was supported by the Australan Research Councl through Dscovery Project DP Ths paper consders an extenson of the sngle-valued double aucton model that allows traders to specfy temporal constrants n ther orders. Roughly speakng, an order s wrtten n the form (p, t,t ), where p stands for the order prce and [t,t ] represents the tme perod when the commodty can be exchanged (not for the order tself). In ths extenson, a bd and an ask s matchable f and only f the bd prce s no lower than the ask prce and the ntersecton of ther tme constrants s non-empty. We found that the market clearng problem under ths extenson can be transformed nto a weghted bpartte matchng. Ths allows us to use some standard technques n graph theory, such as augmentaton, for the desgn and analyss of the mechansms. We prove that an allocaton for the double aucton s effcent f and only f t corresponds to a maxmum weghted bpartte matchng of the graph encodng the ncomng orders. Based on that, we develop an effcent and domnant-strategy ncentve-compatble double aucton mechansm,.e. a VCG mechansm [Groves, 1973]. Remarkably, our payment can be mplemented much faster than the classcal VCG payment, known as Clarke pvot payment, whle resultng n the same payments, because t drectly uses the abrdgng and replacng paths generated durng the allocaton process rather than recall the allocaton algorthm as Clarke pvot payment does. It s worth mentonng that durng the last decade many researchers started to look at the mechansm desgn problem for dynamc envronments where traders are arrvng and departng dynamcally, referred to as onlne mechansm desgn [Parkes, 2007]. To model the dynamcs, temporal nformaton s also used. Although the meanng of the temporal nformaton of a trader s type n the onlne settng s dfferent from that n our settng, a trader s type s modelled n the same way n both settngs [Blum et al., 2006; Bredn et al., 2007]. Therefore, the mechansm n our model also provdes an optmal (offlne) soluton for a correspondng dynamc market. Such an optmal soluton can be drectly used for calculatng the compettve rato of an onlne marketclearng algorthm. Moreover, although orders arrve dynamcally, the alternatng paths are relatvely stable and therefore can be used, for example, to dentfy potental good orders to fnd more effcent allocatons n an onlne settng. Ths paper s organsed as follows. In Secton 2 we brefly ntroduce our market model and related concepts. In Secton 3, we ntroduce a graphc representaton for market stua- 472

2 tons and transfer the market clearng problem nto a weghted bpartte matchng. In Secton 4, we concentrate on the desgn of an allocaton algorthm and a payment algorthm, and prove ther desrable propertes. We conclude n Secton 5 wth a bref dscusson for the future work. 2 The Model Consder a double aucton market, n whch a set B of buyers and a set S of sellers trade one commodty smultaneously. Buyers and sellers are traders. Let T = B S and assume that the traders are ndependent and B S =. We also assume that each seller and each buyer supples and demands a sngle unt of the commodty. Each trader T has a prvately observed type θ = (v,s,e ), where v, s and e are non-negatve real numbers, v s the trader s valuaton of a sngle unt of the commodty, and s and e are the startng pont and the endng pont of the tme constrant [s,e ]. If trader s a buyer, obtans utlty v p f receves a unt of the commodty wthn the tme nterval [s,e ] and pays p; obtans zero utlty f pays nothng and does not receve the commodty wthn the tme perod. Smlarly, f s a seller, obtans utlty p v f successfully sells a unt of the commodty wthn the tme perod [s,e ] and receves payment p; obtans zero utlty f fals to sell the commodty wthn the tme perod and no payment s made. Let θ =(θ ) T denote the type profle where θ s the type of trader. θ means the type profle of all traders except trader. Note that we treat a type profle as a vector of types rather than a set of types. Let Θ be the set of all possble types of trader, and we wrte Θ=(Θ ) T. Snce we wll focus on drect-revelaton mechansms, we assume that traders drectly report ther types to the auctoneer as ther orders [Myerson, 2008]. Traders do not necessarly truthfully report ther types but no early-start and no lateend msreports are permtted. Formally, let θ =(v,s,e ) be trader s type and ˆθ =(ˆv, ŝ, ê ) be the trader s report. We assume that [ŝ, ê ] [s,e ]. The ntuton behnd the assumpton s that no trader would report a temporal constrant that mght gve hm negatve utlty. Let R(θ ) be the set of all permtted reports from trader gven hs type θ, R(Θ )= θ Θ R(θ ) be the set of all possble reports from, and R(Θ) = (R(Θ )) T. Gven traders reports θ R(Θ), anask θ =(v,s,e ) (means S) and a bd θ j =(v j,s j,e j ) (means B) are matchable f and only f v v j and [s,e ] [s j,e j ]. That s, the bd s valuaton s no less than the ask s valuaton, and the ntersecton of ther tme constrants s not empty. An allocaton polcy π =(π ) T s a functon that assgns 0 or 1 to each trader, gven traders reports ˆθ R(Θ). For a trader, fπ (ˆθ) =1we say wns; otherwse loses. An allocaton determnes whose order s granted for a transacton. A payment polcy x =(x ) T s a functon that assgns a real number to each trader gven an nput of traders reports ˆθ R(Θ),.e. x (ˆθ) R for all T. Defnton 1. A double aucton mechansm on Θ s a par (π, x), where π s an allocaton polcy and x s a payment polcy. Followng the standard defnton, we say that an aucton mechansm (π, x) s effcent f π maxmzes v + v. B&π (θ)=1 S&π (θ)=0 for any type profle θ =((v,s,e )) T. We say that an aucton mechansm s domnant-strategy ncentve-compatble,.e. truthful, f for each trader, reportng hs true type s hs domnant strategy. There are a number of alternatves to characterse truthfulness n an aucton mechansm. We wll use one of them n ths paper based on [Parkes, 2007; Nsan, 2007]. To descrbe t, we need the followng two auxlary concepts [Parkes, 2007]. For each trader, we defne a partal order on R(Θ ): ˆθ ˆθ ff { v v &[s,e ] [s,e v v &[s,e ] [s,e ], ], f S f B where ˆθ =(v,s,e ) and ˆθ =(v,s,e ) R(Θ ). We say that an allocaton polcy π s monotonc f, for each T, π (ˆθ, ˆθ )=1mples π (ˆθ, ˆθ )=1whenever ˆθ ˆθ. Defnton 2. Gven a monotonc polcy π and traders reports ˆθ R(Θ), the crtcal value of trader of type θ = (v,s,e ) s defned as sup{v : (v,s,e ) R(θ ) c(θ, ˆθ π )= ((v,s,e ), ˆθ )=1}, f S nf{v : (v,s,e ) R(θ ) π ((v,s,e ), ˆθ )=1}, f B It s undefned f no such v exsts. Now we are ready to descrbe a charactersaton of truthfulness, whch wll be used n Secton 4. Theorem 1 s based on Theorem 9.36 n [Nsan, 2007] for a sngle-valued doman and on [Parkes, 2007] for a sngle-valued onlne doman. The proof s omtted here as t s smlar to the above mentoned theorems. Theorem 1. A double aucton mechansm (π, x) s domnantstrategy ncentve-compatble f and only f: π s monotonc. every wnnng seller (buyer) s pad (pays) hs crtcal value, and the payment s 0 for losng traders. 3 Graph Representaton As assumed n the prevous secton, each trader has only one unt of a commodty to sell or buy. Transacton must be made n pars: a seller can only sell hs good to a unque buyer, assumng ther orders are matchable. Ths means that to allocate the goods n a double aucton s to fnd matchngs between buy orders and sell orders. In such a case we can transform the allocaton problem nto a matchng problem n graph theory. As a result an effcent allocaton corresponds to a maxmum-weghted bpartte matchng. We wll frst revew some concepts related to bpartte matchng [West, 2000], encode ncomng orders n a bpartte graph, and then show some specal propertes related to the encodng. 473

3 Defnton 3. A graph G =(V,E) s a bpartte graph f the vertex set V conssts of two dsjont subsets X and Y, and no edge has both end ponts n the same subset. For explctness, we denote the graph as G =((X, Y ),E). Defnton 4. Gven a traders report θ R(Θ), we call G θ =((S θ, B θ ), E) a bpartte graph generated from θ f S θ = {θ : S} and B θ = {θ : B}, E = {(θ,θ j ):θ and θ j s matchable}. Defnton 5. Gven a graph G, a matchng MnGsaset of par-wse non-adjacent edges,.e. no two edges share a common vertex. The sze of M s denoted by M. A vertex s matched f t s ncdent to an edge n the matchng. Otherwse the vertex s free. Gven a matchng M, an M-alternatng path s a path n whch the edges belong alternatvely to M and not to M. an M-augmentng path s an M-alternatng path whose endponts are free. an M-abrdgng path s an M-alternatng path whose frst edge and last edge are n M. an M-replacement path s an M-alternatng path where one of the endponts s free and one of the endng edges s n M. A path s smple f t has no repeated vertces. In the rest of ths paper, we wll only consder smple paths. Fgure 1 shows an example of bpartte representaton of eght dfferent type reports. Lnes and dashed lnes ndcate matched edges and free edges respectvely, and dots and crcles denote matched vertces and free vertces respectvely. The value besde each vertex s ts valuaton. Temporal nformaton s not shown n the graph. It s clear that path (3,10,2,9) s an augmentng path, path (2,10,4,7) s an abrdgng path, and path (2,10,4,7,5) s a replacement path. Fgure 1: Example of Alternatng Paths Gven a matchng M, we can use an M-augmentng path p to augment M by changng all matched edges n p to be free and all the free edges to be matched. By contrast an M- abrdgng path can be used n the same way to abrdge M. Consequently, M wll ncrease (decrease) by one wth one augmentng (abrdgng) process. An M-replacement path can be used to replace a bd or an ask n M wthout changng the status of the other vertces. Defnton 6. An allocaton polcy π s feasble f for any traders reports θ R(Θ), there s a matchng M n the bpartte graph generated from θ such that M exactly covers {θ : π (θ) =1}. It follows that any matchng n a bpartte graph generated from traders reports unquely determnes a feasble allocaton. In the rest of ths paper, we wll only consder feasble allocaton polces. Defnton 7. Gven bpartte graph G θ, an edge e between θ = (v,s,e ) and θ j = (v j,s j,e j ), where S and j B, we defne the weght of e as w(e) =v j v. For any set of edges E E, the weght of E s defned as w(e )= e E w(e). The weght ncrease of an M-alternatng path p s the total weght of free edges n p mnus that of matched edges n p: Δ(p) =w(p M) w(p M), where P s the set of all edges n p. If p s an M-augmentng, M-abrdgng, or M-replacement path, then Δ(p) s the net change n the weght of the matchng after augmentng, abrdgng, or replacng by p: w(m p) =w(m)+δ(p), where M p M P, P s the set of all edges n p, and s the symmetrc dfference operator on sets: A B =(A B) \ (A B). Lemma 1. Gven G θ, a matchng M n G θ, and an M- alternatng path p, Δ(p) s equal to the valuaton of the bd mnus that of the ask of the endponts of p,fps an augmentng path. the valuaton of the ask mnus that of the bd of the endponts of p,fps an abrdgng path. the valuaton of the free (matched) endpont mnus that of the matched (free) endpont of p when the endponts are bds (asks), f p s a replacement path. We wll not provde the proof of Lemma 1 whch follows the weght defnton of the edges. 4 Effcent and Truthful Polcy Desgn In order to desgn a double aucton that s both effcent and truthful, by Theorem 1, we need to fnd an effcent and monotonc allocaton polcy, and a payment polcy that calculates the crtcal value of each wnnng trader. Inspred by the smlarty between ths allocaton problem and the weghted matchng n a bpartte graph, we frst transform the model nto a bpartte graph. Wthn ths graph, we show how to effcently use the well establshed methods from bpartte matchng n the allocaton polcy, and how to calculate crtcal payment wthout runnng the allocaton polcy agan. 4.1 Effcent & Monotonc Allocaton Polcy Wth the above graph encodng of traders reports, we desgned an effcent allocaton polcy by adoptng the maxmum-weghted bpartte matchng that constructs a maxmum-weghted matchng by begnnng wth the empty matchng and repeatedly performng augmentatons usng augmentng paths of maxmum weght ncrease untl a maxmum-weghted matchng s acheved [Tarjan, 1983; Kozen, 1991]. The resultng allocaton polcy s called Maxmum-weghted Bpartte Matchng Allocaton (MBM Allocaton), whch seeks an allocaton that maxmses socal 474

4 welfare of any reports θ, by frst representng θ n a bpartte graph G θ, and then applyng modfed maxmum-weghted bpartte matchng to get a maxmum-weghted matchng M whch determnes all wnnng reports. We added a more detaled path selecton rule n the maxmum-weghted bpartte matchng n order to acheve the monotoncty property. The rule s based on the order p defned for augmentng paths. Let a sequence of vertces θ 1... θ n denote an augmentng path of length n, whch starts from ask θ 1 and ends n bd θ n. We defne p on all augmentng paths based on ther endponts: θ 1... θ n p θ 1... θ m ff (v 1,v n,s 1,e 1,s m,e n ) s (v 1,v m,s 1,e 1,s n,e m), where s s the lexcographc order of two equal length sequences of real numbers: (r1, 1..., rn) 1 s (r1, 2..., rn) 2 ff 1 j n (rj 1 r2 j 1 k<j(rk 1 = r2 k )). We wll use p n MBM Allocaton to dstngush augmentng paths that have the same weght ncreases. Maxmum-weghted Bpartte Matchng Allocaton: Intalzaton: Encode reports θ n bpartte graph G θ. Set the result matchng M = for G θ. Recurson: AugP ath = {p : Δ(p) > 0 and p s an M- augmentng path}. MaxAugP ath =argmax p AugP ath Δ(p). If MaxAugP ath =, stop recurson. Otherwse, let ˆp MaxAugP ath s.t. p p ˆp for any p MaxAugP ath, and M = M ˆp. Output: All reports covered by M wn and all the rest lose. Theorem 2. Maxmum-weghted Bpartte Matchng Allocaton s effcent. We prove Theorem 2 n the Appendx. Here we show one essental lemma used n the proof. In the rest of ths paper, π denotes MBM Allocaton. Lemma 2. Maxmum-weghted Bpartte Matchng Allocaton s effcent f and only f the maxmum-weghted bpartte matchng maxmzes the weght of the matchng. Proof. The weght of the matchng s π v (θ)=1 B π v (θ)=1 S, whch s equal ( to π(θ)=1 B v + π(θ)=0 S ) v S v. S v s fxed, so f the weght of the matchng s maxmsed, then π v (θ)=1 B + π v (θ)=0 S s also maxmzed, and vce versa. Theorem 3. Maxmum-weghted Bpartte Matchng Allocaton s monotonc. Although we added a specfc path selecton rule based on p to avod randomsaton of MBM Allocaton n most cases, there s stll one stuaton where p cannot help. When two types are the same and two augmentng paths of maxmum postve weght ncrease start from them and end n the same vertex, then p cannot separate these two paths clearly,.e. both of them have a chance of beng selected but none of them are guaranteed. Thus we assume that all type reports of sellers (buyers) are dfferent. Note that there mght be more than one augmentng path wth the same endponts, but ths does not affect the determnstc property of MBM Allocaton. The proofs of all the theorems are gven n the Appendx. 4.2 Truthful Payment Polcy We have found an effcent allocaton polcy, MBM Allocaton, and proved ts monotoncty property whch s one of the two ff condtons to satsfy truthfulness. What s left s to calculate the crtcal value for each wnnng trader. Obvously, t s not practcal to calculate the crtcal value as t s defned n Defnton 2. Here we propose another approach whch s nspred by the reverse of MBM Allocaton. A type θ s matched because there s an augmentng path of maxmum postve weght ncrease endng wth θ n some round of the matchng procedure. Therefore, f a type does not satsfy the above condton, t wll not be matched. Ths s the bass of our payment polcy whch s seekng the least volaton condton for each wnnng type,.e. the edge condton between wnnng and losng. Gven traders reports θ, fπ (θ) = 1, the payment for trader, x (θ), s defned n terms of abrdgng and replacement paths startng from θ n the followng, whch s called Mn-Max Payment (MM Payment). x (θ) =0f π (θ) =0. Mn-Max Payment: x (θ) = { mnp D R v(endng(p)), f S max p D R v(endng(p)), f B where D s a set of all abrdgng paths startng from θ, R s a set of all replacement paths startng from θ, and v(endng(p)) s the valuaton of the endng vertex, the endpont other than θ, of path p. For each wnnng ask, MM Payment gves the mnmum valuaton such that, f the ask s valuaton were greater than or equal to that mnmum, t can be removed from the matchng to (weakly) ncrease the weght of the matchng, whle for each wnnng bd, the payment s the correspondng maxmum. The set D gves all possble ways to remove θ by abrdgng, whle the set R gves all possble ways to substtute a free vertex for θ. Note that set D does not necessarly contan the path that was used to match θ, as the path can be changed wth other augmentatons after addng θ. Theorem 4. Gven bpartte graph G θ and a wnnng type θ =(v,s,e ) determned by MBM Allocaton, Mn-Max Payment x (θ) s equal to crtcal value c(θ,θ ). 475

5 Another appealng property of Mn-Max payment s ts ndependence from the allocaton algorthm. We show that Mn-Max payment results n the same payments as the most desrable VCG payment (Clarke pvot payment), but t does not requre the recall of the allocaton algorthm. Clarke pvot payment s defned as x (θ) = V π (θ ) V π (θ), where V π (θ) s the socal value gven traders report profle θ and the allocaton polcy π, and V π (θ) s the socal value wthout countng trader. Lemma 3. Gven traders report θ and effcent and monotonc allocaton polcy π, for each trader, Mn-Max payment (θ) s equal to Clarke pvot payment x C (θ). x MM Proof sketch. We need to prove that for each wnnng type θ f we remove θ from the maxmum-weghted matchng M of bpartte graph G θ by usng the path p that gves x MM (θ), the result matchng M s also maxmum-weghted n G θ. By contradcton, assume that M s not maxmum-weghted, we wll conclude ether M s not maxmum-weghted or path p contradcts the defnton of Mn-Max payment. Corollary 1. Double aucton mechansm (MBM Allocaton, MM Payment) s effcent, domnantstrategy ncentve-compatble and ndvdual-ratonal,.e. traders never get negatve utlty. Fgure 2 shows an example of the double aucton we have defned, where the number besde each vertex s the valuaton of the vertex and the value nsde parentheses s the payment. Fgure 2: MBM Allocaton and MM Payment 4.3 Computatonal Complexty We further show that both our allocaton polcy and payment polcy can be mplemented n polynomal tme and, more mportantly, our payment can be mplemented much faster than Clarke pvot payment. Theorem 5. Let n be the number of traders reports. MBM Allocaton can be mplemented n tme O(n 3 ), and Mn-Max Payment can be mplemented n tme O(n 3 ). Ths result s sgnfcant because, to the best of our knowledge, the mplementatons of Clarke pvot payment cannot avod the recall of the allocaton algorthm [Nsan and Ronen, 1999; Sandholm, 2003]. In other words, for each wnnng report θ, π needs to search another allocaton on the remanng reports θ. Therefore, t wll take O(n) tmes of the allocaton tme n ths model,.e. O(n 4 ) wth MBM Allocaton. 5 Concluson We have developed an effcent and truthful double aucton mechansm (.e. a VCG mechansm) n a model where each trader s type conssts of a valuaton of a commodty and a tme perod that constrans when the commodty can be exchanged. Ths mechansm s charactersed by an allocaton polcy and a payment polcy. By encodng the model n a bpartte graph, we effcently adapted the maxmum-weghted bpartte matchng to get an effcent and monotonc allocaton polcy. We also developed a truthful payment polcy that can be mplemented faster than Clarke pvot payment whle resultng n the same payments as Clarke pvot payment. Myerson et al. [1983] proved that there s no effcent, ncentve-compatble and ndvdual-ratonal blateral trade wthout outsde subsdes,.e. a market wth our mechansm wll run n defct. To avod ths defct, we need to compromse between effcency and truthfulness. There are two possble remedes: ether relaxng effcency, or gvng up ncentve compatblty, as nvestgated by McAfee [1992] and Wurman et al. [1998] n sngle-valued domans. Fndng how these compromses can lead to a realstc mechansm under our model s worth further nvestgaton. Appendx: Proofs of Theorems Proof of Theorem 2: In order to prove Theorem 2, by Lemma 2, we shall prove that the maxmum-weghted bpartte matchng ndeed gves a maxmum-weghted matchng. To do that, we need the two verfed propertes of the maxmum-weghted bpartte matchng gven n Clam 1 and 2 [Tarjan, 1983]. Ths s one of the advantages we ganed by encodng the model n a graph. We wll skp the proofs of the followng two clams. Clam 1. Gven graph G, let M be a matchng of sze k of maxmum weght among all matchngs of sze k n G. If we augment M by an augmentng path of maxmal weght ncrease, then we obtan a matchng of sze k +1of maxmum weght among all matchngs of sze k +1n G. Clam 2. The maxmum-weghted bpartte matchng wll augment along augmentng paths of successvely nonncreasng weght ncrease. By Clam 1, the maxmum-weghted bpartte matchng wll gve a matchng M k of sze k of maxmum weght among all matchngs of sze k after k augmentatons. By Clam 2, M k s also maxmumweghted among all matchngs of sze at most k f the weght ncrease at the k-th augmentaton s postve. Therefore, the matchng t gves untl there s no augmentng path of postve weght ncrease s maxmum-weghted among all matchngs. Proof of Theorem 3: By contradcton, wthout loss of generalty, assume that π (θ) =1and π (θ,θ ) =0for some bds θ θ. Let θ be matched n round k of π(θ),.e some augmentng path endng wth θ s of maxmal weght ncrease n round k. Snce θ and θ are both not matched before round k, so the matchngs are the same n both π(θ) and π(θ,θ ) after any round <k. Let θ m... θ be the augmentng path of maxmal weght ncrease selected n round k of π(θ). Snce θ θ, θ m... θ s an augmentng path n round k of π(θ,θ ) and θ m... θ p θ m... θ. Moreover, n round k, all augmentng paths n π(θ,θ ), except those that end wth θ, are also augmentng paths n π(θ). Thus, n round k of π(θ,θ ), for any augmentng path p that does not end wth θ, p p θ m... θ, and all the rest end wth θ. Therefore, an augmentng path endng wth θ should be selected n round k of π(θ,θ ), whch contradcts the assumpton. Proof of Theorem 4: The proof needs the followng two clams whch can be found n [Kozen, 1991; Blum et al., 2006]. 476

6 Clam 3. Gven two matchngs M and M, M M conssts of a collecton of vertex-dsjont alternatng paths and even length cycles. Clam 4. Gven two matchngs M and M, a vertex v s an endpont of a path n M M f and only f t s matched n ether M or M but not both. Now we are ready to prove Theorem 4. Wthout loss of generalty, assume θ =(v,s,e ) s a wnnng ask, and let x = x (θ) and c = c(θ,θ ). To prove x = c, by the defnton of c, we need to show that for any θ =(v,s,e ): 1. θ :π (θ,θ )=1(v x ). 2. x δ<v <x(π(θ,θ ) =1)for any δ>0. Let M be the matchng of π(θ) and M be that of π(θ,θ ). We wll prove these two condtons one by one blow. Part I: By contradcton, assume that π (θ,θ ) = 1 and v > x. Let A M (B M ) be all the matched asks (bds) n M, and A M (B M ) be all the matched asks (bds) n M. Snce π s monotonc and M and M are maxmum-weghted, t follows that all matched asks n M except for θ must be matched n M,.e. A M \{θ } A M, and all the matched bds n M must be matched n M,.e. B M B M. Thus nequaltes A M 1 < A M and B M B M hold. Moreover, A M = B M and A M = B M, so we get M = M, B M = B M, and A M \{θ } = A M \{θ }. Therefore, by Clam 3 and 4, there s only one alternatng path p only = θ... θ n M M, and all the rest are cycles. If all vertces reachable from θ through M-abrdgng or M-replacement paths are also reachable from θ through M - abrdgng or M -replacement paths, then, snce v >x, there s at least one M -abrdgng or M -replacement path of postve weght ncrease by whch we can remove θ to ncrease the weght of the matchng, whch contradcts the choce of M. Let us prove that the above reachablty condton holds. (1). For any vertex v except for θ (θ )np only, the path between θ (θ ) and v s ether an abrdgng or a replacement path wth respect to M (M ). (2). Any vertex v not n p only that s reachable from θ by an abrdgng or replacement path p s also reachable from θ through the same type of path p. Snce p must be connected wth p only and for any edge e p and e p only,fe M and e M, there must be an even length cycle that contans e n M M, and vce versa,.e. f e connects vertces v 1 and v 2 n p, there s always a correspondng edge or path connectng v 1 and v 2 n p. For nstance, Fgure 3 shows one alternatng path (a,b,c,d,e) and a cycle (h,,j,k) of M M : thn lnes and thck lnes belong to M and M respectvely, whle the double lne between f and g s n both matchngs and dashed lnes are free. It s easy to see that all vertces reachable from a through a M-augmentng or M-replacement path s also reachable from e by a correspondng path wth respect to M. Part II: To prove the second condton, we wll prove π (θ,θ ) = 1 for any v < x. By contradcton, assume that v < x and π (θ,θ ) = 0. By Clam 4 there s a path p θ M M startng from θ and endng wth θ n n ether M or M. Snce x v n by the defnton of x and v <x as we assumed, we can substtute θ for θ n p θ to get an M -alternatng path p θ.ifθ n s matched n M, then p θ s an M -augmentng path and by Lemma 1 Δ(p θ )=v n v > 0, whch contradcts the choce of M. Thus θ n s a matched ask n M, and p θ s an M - replacement path. Snce M s a maxmum-weghted, by Lemma 1 Δ(p θ )=v n v 0. Put all results together, we get contradcton v n v <x v n. Fgure 3: Reachablty Example Proof of Theorem 5: Bpartte graph representaton of the reports takes at most n 2 /4 tme by checkng each par of ask and bd, so there wll be at most m = n 2 /4 edges. Accordng to [Gall, 1986], fndng an augmentng path of maxmal weght ncrease can be solved by Djkstra s algorthm takng O(m+n log n) tme. There are at most n/2 rounds, so MBM Allocaton can be mplemented n tme O(n 3 ). For each wnnng type, MM Payment can be done by depth-frst or breadth-frst search whch takes O(n + m) tme. 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