On the Relationship between the VCG Mechanism and Market Clearing

Transcription

1 On the Relatonshp between the VCG Mechansm and Market Clearng Takash Tanaka 1 Na L 2 Kenko Uchda 3 Abstract We consder a socal cost mnmzaton problem wth equalty and nequalty constrants n whch a central coordnator allocates nfntely dvsble goods to self-nterested N frms under nformaton asymmetry. We consder the Vckrey- Clarke-Groves (VCG mechansm and study ts connecton to an alternatve mechansm based on market clearng-prce. Under the consdered set up, we show that the VCG payments are equal to the path ntegrals of the vector feld of the market clearng prces, ndcatng a close relatonshp between the VCG mechansm and the clearng-prce mechansm. We then dscuss ts mplcatons for the electrcty market desgn and also explot ths connecton to analyze the budget balance of the VCG mechansm. I. INTRODUCTION Market power montorng and mtgaton are essental aspects of today s electrcty market operatons [1]. The effcency due to market manpulatons by strategc frms are not only predcted n theory [2, Ch. 12], but also documented as real ncdents [3]. Whle the current practce of market power montorng reles on relatvely smple metrcs (e.g., Herfndahl Hrschman Index [4], further deregulaton and emergence of new markets (e.g., ISO-aggregator and aggregator-prosumer nterfaces necesstate advanced market power analyss [5] and approprate market desgn. The effcency n olgopolstc electrcty markets has been actvely studed n recent lterature. Among others, a popular venue of research (e.g., [6] [9] s based on the supply functon bddng model by Klemperer and Meyer [1], n whch the market clears accordng to the producers (resp. consumers submtted supply (resp. demand functons. In [1], t s observed that f there s no demand uncertanty and arbtrary supply functon bddng s allowed, numerous multplcty of equlbra emerges. Johar and Tstskls [8] observed that as the strategc flexblty granted to frms ncrease, ther temptaton to msdeclare ther cost ncreases as well, and consdered a smple (scalar-parametrzed supply functon model wth N partcpants and showed that the prce of anarchy (effcency s upper bounded by 1 + 1/(N 2. In [11], t s observed that the worst case effcency depends on the transmsson network topology when there exst transmsson constrants. The effcency under transmsson constrants s further studed n [12] under a slghtly dfferent supply functon bddng model. Whle the market analyss va the supply functon bddng model has been frutful, there are several lmtatons as 1 Department of Aerospace Engneerng and Engneerng Mechancs, Unversty of Texas at Austn, Austn, TX, USA. ttanaka@utexas.edu. 2 School of Engneerng and Appled Scences, Harvard Unversty, Cambrdge, MA, USA. nal@seas.harvard.edu. 3 School of Advanced Scence and Engneerng, Waseda Unversty, Tokyo, Japan. kuchda@waseda.p. well. Frst, t may not be easy to guarantee the exstence of supply functon equlbra n realstc electrcty markets. As demonstrated by [13], a pure Nash equlbrum may fal to exst even n a relatvely smple supply functon bddng model. Second, even f the supply functon equlbra are known to exst, t could be a dffcult task for the frms to fnd them effcently [14]. Fnally, unlke the mechansm desgn approach (dscussed next, the effcency s nevtable. The mechansm desgn s an alternatve approach towards the market power mtgaton. For nstance, the references [15] [18] consder applcatons of the Vckrey-Clarke- Groves (VCG mechansm [2, Ch. 23] to several market desgn problems n power system operatons. The VCG mechansm provdes an attractve market model snce t attans zero effcency and ncentve compatblty (and often ndvdual ratonalty [19]. It s also attractve snce the exstence of equlbra s guaranteed (truth-tellng s an equlbrum and t can be trvally found by the partcpants. However, a maor shortcomng of the VCG mechansm s the lack of budget balance [8], whch s a promnent reason why t s rarely used n practce. The VCG mechansm has other drawbacks such as vulnerablty to coaltons, hgh computatonal complexty, and lack of prvacy [19]. For a successful electrcty market desgn n the future, an ntegrated dscusson on these dfferent approaches are desred n the common framework, so that ther pros and cons are systematcally understood. Therefore, n ths paper, we dscuss dfferent market mechansms, namely the clearngprce mechansm (the underlyng mechansm for the supply functon bddng model and the VCG mechansm, n a common set up. As a common set up, we consder a general socal cost mnmzaton problem wth equalty and nequalty constrants. Our goal s to provde a unfed vew on these seemngly dfferent market mechansms, whch would hopefully be valuable n the future studes on electrcty market desgn. The contrbuton of ths paper s two-fold. Frst, we show a connecton between the VCG mechansm and the clearngprce mechansm by provng that the VCG payment s equal to the path ntegral of the vector felds of market clearng prces under the consdered set up (Theorem 2. It also follows from ths observaton that tax values calculated by the clearng-prce mechansm and the VCG mechansm are smlar when ndvdual partcpants market power s neglgble. Ths result ndcates a close connecton between the VCG and the clearng-prce mechansms. Second, n order to address the budget balance ssue of the VCG mechansm, we apply the above observaton to analyze the worst case budget n the VCG mechansm. Wth some smplfed

2 assumptons, we outlne how the budget can be analyzed usng the man result of ths paper. Ths paper s formulated as follows. In Secton II, a general socal cost mnmzaton problem to be consdered s formulated. The clearng-prce mechansm and the VCG mechansm are descrbed n Secton III. The man techncal result (path ntegral characterzaton of the VCG mechansm s presented n Secton IV. The budget of the VCG mechansm s studed n Secton V. II. PROBLEM FORMULATION Suppose that there exst N partcpatng frms ndexed by = 1, 2,..., N and a sngle auctoneer. Consder a socal cost mnmzaton problem wth equalty and nequalty constrants: P : mn f(x f x (x (1a s.t. g(x g (x (1b h(x h (x =. (1c For each = 1, 2,..., N, x can be nterpreted as producton by the -th frm, or as consumpton f t s negatve. The functon f : R n R s nterpreted as a producton cost f the frm s a producer, or as a negated utlty functon f the frm s a consumer. The aggregaton f(x s referred to as the socal cost functon (negated socal welfare functon. Functons g : R n R k and h : R n R l are nterpreted as the -th frm s contrbuton to the global equalty/nequalty constrants such as the power flow balance constrants, lne flow lmt constrants, local capacty constrants. Constrants (1b and (1c are mposed entry-wse. Remark 1: We remark here that n the problem settng (1, we only consder separable obectve functon and constrants functon. Ths s usually true for electrcty market, especally when DC power flow model s used where all the constrants are n a lnear form [2]. We assume that functons g and h for each = 1, 2,..., N are publcly known. However, the functon f F s assumed to be prvate,.e., known to the -th frm only. The famly F of functons are assumed to be known a pror. An optmal soluton to P wll be denoted by x = (x 1;...; x N 1. A. Bddng and payment model Consder a sngle-round aucton mechansm n whch each frm s frst gven an opportunty to report hs/her prvate functon f F to the auctoneer. In ths step, frms are allowed to msreport ther nformaton and thus f f n general. In the next step, the auctoneer makes a socal decson x = ( x 1;...; x N based on the reported nformaton accordng to a certan decson rule. Fnally, the auctoneer calculates payments π = (π 1 ;...; π N R N accordng to a certan payment rule. Throughout ths paper, we consder π as the payment from the -th frm to the auctoneer, whch can be ether postve or negatve. Here π can be thought of as a tax by whch the -th frm s ncentve s manpulated. 1 We assume x s a column vector. We wrte (x; y = [x y ]. The net cost for the -th frm s C = f ( x + π. An aucton mechansm s characterzed by a partcular choce of a decson rule and a payment rule. We say that an aucton mechansm s (domnant strategy ncentve compatble f for each frm truth-tellng ( f = f s an optmal strategy to mnmze hs/her net cost regardless of the other agents reportng strategy. B. Cost functon bddng vs. supply functon bddng The cost functon bddng model descrbed above s closely related to the supply functon bddng models [1]. In a smple case where a producton frm s producng a sngle commodty (e.g., electrcty, a supply functon s (p descrbes the desred level of producton as a functon of prce p at whch the product can be exchanged n the market. Mathematcally, a producton cost functon f (x and a supply functon s (p are related by s (p = argmax x px f (x. For consumng frms, there s a smlar relatonshp between utlty functons and demand functons. From ths relatonshp (c.f., Legendre transformaton, t can be seen that a cost functon bddng model s equvalent to supply functon bddng model under mld assumptons (e.g., convexty of f. For nstance, for a >, f (x = ax 2 f and only f s (p = 1 2ap. Thus, assumng a lnear supply functon s equvalent to assumng a quadratc cost functon. Alternatvely, suppose that F s the set of non-decreasng, contnuously dfferentable, and strongly convex functons from [, to [, such that f ( = and f ( =. Then f F f and only f s S, where S s the set of non-decreasng contnuous functon from [, to [, such that s ( =. For the rest of ths paper, we focus on the cost functon bddng model only. III. CLEARING-PRICE AND VCG MECHANISMS In ths secton, we formally ntroduce the clearng-prce mechansm and the VCG mechansm for the socal cost mnmzaton problem (1. A. The clearng-prce mechansm The clearng-prce mechansm s characterzed by the followng decson and payment rules. Decson rule: Based on reported functons f, = 1, 2,..., N, the auctoneer formulates a problem P, whch s dentcal to P n (1 except that reported functons are used, and solve for the prmal-dual optmal soluton ( x, µ, λ satsfyng the KKT condton: 2 f ( x + µ g ( x + λ h ( x = (2a g ( x, h ( x = (2b µ T g ( x = (2c µ, (2d 2 When a functon f : R N R k s dfferentable at x R N, we defne f 1 f 1 x 1 x N the gradent matrx of f at x as f(x =... f k x 1 f k x N

3 where (2a s mposed for each = 1,..., N. A socal decson s determned by x = ( x 1;...; x N. Payment rule: The auctoneer computes a vector of the market clearng prces p = µ g ( x + λ h ( x for each = 1,..., N. The payment (from the frm to the auctoneer s determned by π clearng = p x. (3 The clearng-prce mechansm and ts varants are wdely used n practce of power system operatons because of ts smplcty. It s also used as an underlyng market model n the exstng market analyses (e.g., [6] [9]. A drawback of ths mechansm s that sometmes the exstence and unqueness of the soluton satsfyng the KKT condton are not guaranteed especally when a nonconvex problem formulaton s used [21]. Also, the clearng-prce mechansm s not ncentve compatble n general, as demonstrated n the next smple example. Example 1: Consder a model of power supply market wth N power generatng frms. Let x be the producton by the -th frm, and suppose that the total power supply must meet the total demand N. Assume that the true generaton cost functon s gven by f (x = x 2 for every = 1, 2,..., N. The optmal allocaton x that mnmzes the socal cost s obtaned by solvng mn f x (x (4a s.t. x = N. (4b Clearly, the optmal soluton s x = (1; 1; ; 1. Suppose that the functon f (x s publcly unknown besdes the fact that t belongs to the set F = { a x 2 : a > }. Thus, each frm s requred to report hs/her prvate parameter a to the auctoneer. Suppose that the clearng-prce mechansm s appled to Example 1. Based on the reported functons f (x = a x 2, the auctoneer solves the KKT condton x f ( x + λ = = 1, 2,, N (5a x = N (5b for a socal decson x and the market-clearng prce λ. The auctoneer computes the payments for ndvdual frms by π clearng = λ x. (Note that πclearng s negatve n ths example because we follow the conventon that the payment s from the frm to the auctoneer by default. Assumng all generators submt ther cost functons truthfully (.e., f (x = x 2, then x = (1; 1; ; 1 and λ = 2 satsfy the KKT condton. Hence, the reward that each generator receves s calculated as λ x = 2, and the net cost for the frm s C = f ( x + πclearng = 1 2 = 1 (proft s +1. To demonstrate that the clearng-prce mechansm s not ncentve compatble, suppose that the frst frm msreports C 1 = N 2 (1 2a 1 /(a 1 N a N=2 N=3 N=5 N=1 N= a 1 Fg. 1. The frst generator s report a 1 and the resultng cost C 1. (a 1 1, whle other frms reman truthful (a = 1, = 2, 3,, N. The KKT condton (5 s 2a 1 x 1 + λ =, 2 x + λ = = 2, 3,, N x = N from whch we have λ 2a 1 N = a 1 N a 1 + N, N x 1 = a 1 N a 1 + N x a 1 N = = 2, 3,, N. a 1 N a 1 + N Now, the net cost for the frst frm s calculated as C 1 = f 1 ( x 1 + π clearng 1 = N 2 (1 2a 1 (a 1 N a Fgure 1 plots C 1 as a functon of a 1 for varous N. It can be seen that the frst generator can acheve the mnmum net cost C 1 = 1 1 N 2 1 by reportng a 1 = N 1. The net cost n ths case s smaller than that follows from the truthful report (whch s 1. Thus we conclude that truthful report s not a Nash equlbrum (and thus not a domnant strategy. It s also worth mentonng that as the number of partcpants ncreases (N +, the optmal reportng strategy tends to be the truthful one (a 1 1. If all partcpants are truthful n the clearng-prce mechansm, the payment for the -th partcpant s π = p x (6 wth p = µ g (x + λ h (x where (x, µ, λ s the prmal-dual optmal soluton to P. B. The VCG mechansm The VCG mechansm, also known as the pvot mechansm, chooses the payment rule n such a way that the -th frm s payment s equal to ts externalty. To ths end, the mechansm evaluates the mpact of each frm s presence n the market by ntroducng the followng auxlary optmzaton

4 problem for each = 1, 2,, N: P (x : mn x f(x, x s.t. g(x, x h(x, x f (x +f (x g (x +g (x (7a (7b h (x +h (x =. (7c Decson varables n (7 are x = (x 1 ;...; x 1 ; x +1 ;...; x N. Notce that n P (x, an optmal decson x for the frms excludng needs to be found whle x s fxed. Denote by x (x = ( x1 (x ;...; x 1 (x ; x +1 (x ;...; x N (x an optmal soluton to the problem P (x. Note that x (x s parametrzed by x. We nterpret x ( as an optmal socal decson for the frms other than when the -th frm s absent from the market. 3 The VCG mechansm s characterzed by the followng decson and payment rules. Decson rule: Based on the reported functons f, = 1, 2,..., N, the auctoneer formulates optmzaton problems P and P ( ( = 1,, N, whch are dentcal to P and P ( except that reported functons are used. The auctoneer determnes a socal decson x as an optmal soluton to P. The auctoneer also computes an optmal soluton x ( to P ( for every = 1,, N. Payment rule: Payment for the frm s calculated by = f ( x f ( x (. (8 The next Theorem s a straghtforward applcaton of the well-known result (e.g., [2, Ch. 23] [19, Ch.1] [23] to the socal cost mnmzaton problem (1. Theorem 1: The VCG mechansm s domnant strategy ncentve compatble. Proof: Although ths s a mnor varaton of the standard results, we provde a proof for the sake of completeness. We frst assume that there exsts an ndex and reportng strateges f for such that the frm attans a smaller net cost C by msreportng (.e., f f, and then derve a contradcton. Denote by ˆP and ˆP (x the problems (1 and (7 when functons ( f 1,..., f 1, f, f +1,..., f N are reported. Let ˆx and ˆx (x be the optmal solutons to ˆP and ˆP (x respectvely. The net cost for the -th frm s Ĉ = f (ˆx + ˆπ VCG = f (ˆx + f (ˆx f (ˆx (. Also, denote by P and P (x the problems (1 and (7 when functons ( f 1,..., f 1, f, f +1,..., f N are reported. 3 In ths paper, we nterpret the -th frm beng absent from the market as x =. See [22] for related dscussons. Let x and x (x be the optmal solutons to P and P (x respectvely. The net cost n ths case s C = f ( x + π VCG = f ( x + f ( x Now, notce that ˆx f ( x (. ( = x (,.e., the -th frm s reportng strategy does not alter the soluton to the auxlary problem. Thus C < Ĉ mples f ( x + f ( x < f (ˆx + f (ˆx. However, ths s a contradcton to our hypothess that ˆx s an optmal soluton to ˆP. In vew of Theorem 1, the VCG mechansm s sad to mplement an effcent socal choce functon n domnant strateges [23]. Assumng all reports are truthful, the VCG payment becomes 4 = f (x f (x (. (9 It s nstrumental to notce the dfference between the VCG payment (9 and that of the clearng-prce mechansm (6 wth truthful partcpants. To quantfy the dfference, consder an applcaton of the VCG mechansm to the scenaro n Example 1. For each = 1,, N, the VCG payment s computed as = f (x f (x ( = f (1 f N ( N 1 = 2 1 N 1 (1 In other words, each frm receves N 1 for producng x = 1. Recall that the payment determned by the clearngprce mechansm when all reports were truthful was 2. Therefore, n ths example, we have π > π VCG for all, even though n both cases the producton by the frms are the same. Also, notce that the dfference dmnshes as N +. IV. PATH INTEGRAL REPRESENTATIONS OF VCG PAYMENTS In ths secton, we establsh a mathematcal relatonshp between the clearng-prce and the VCG mechansms n a general settng. We frst recall some basc results from nonlnear programmng. A feasble vector x for the problem P (x s sad to be regular [24] f the gradent matrx [ ] x g(x, x x h(x, x has lnearly ndependent rows. Here, g(x, x s the collecton of all actve constrants n g(x, x,.e., the collecton of ndces k such that g k (x, x =. If x s regular, there exsts a unque set of Lagrange multplers 4 We do not dstngush (8 and (9 snce they are equal under a realstc assumpton that all reports are truthful.

5 µ R k, λ R l such that the followng KKT condtons are satsfed [24, Proposton 3.3.1]. x f (x + µ g (x +λ h (x = (11a ( µ g (x + g (x = (11b µ. (11c For notatonal smplcty, n what follows, we often suppress the dependency of (x, µ, λ on x. For each = 1, 2,, N, consder a smooth path γ : [, 1] R n such that γ ( = and γ (1 = x. Notce that the orgn of the path corresponds to the case where the -th frm s absent from the market (x = and the termnal of the path corresponds to the case where the -th frm s assgned an optmal soluton to (1 (x = x. For each pont x γ on the path, as before, we wrte x (x to denote an optmal soluton to P (x. Whenever x (x s regular, we also denote by (µ (x, λ (x the correspondng unque set of Lagrange multplers. Notce that for each = 1,..., N, (x (x, µ (x, λ (x = (x, µ, λ. The next theorem shows that VCG payment π VCG s equal to the path ntegral of the vector feld of the market clearng prces along γ. Theorem 2: Assume f, g and h are contnuously dfferentable, and P admts an optmal soluton x. Let γ : [, 1] R n be any smooth path such that γ ( =, γ (1 = x, and the followng condtons are satsfed: (A P (γ (t admts an optmal soluton x (γ (t for every t [, 1]. (B x (γ (t s contnuously dfferentable n t, and s regular almost everywhere n [, 1]. (C µ (γ (t and λ (γ (t are dfferentable almost everywhere n [, 1]. Then, the followng equalty holds: = γ p (x dx ( = p (γ (t dγ (12 where p (x = µ (x g (x + λ (x h (x. Proof: Snce x (γ (t s a feasble soluton to P (x, t must satsfy the equalty constrant h (x + h (γ (t = for every t [, 1]. Dfferentatng wth respect to t, we obtan dx h (x =. (13 Next, snce the left hand sde of (11b s dfferentable at almost every t [, 1], ( dµ ( + µ ( + h (γ (t dγ g (x + g (γ (t g (x dx + g (γ (t dγ = (14 almost everywhere n [, 1]. We clam that the frst term of (14 s zero. To see ths, suppose that the l-th component of g (x + g (γ (t s strctly negatve at some t [, 1]. Due to contnuty, t s always possble to choose an nterval [t ɛ, t + ɛ] on whch the l-th component of g (x + g (γ (t s strctly negatve. By the complementary slackness condton (11b, the l-th component of µ (γ (t s zero on ths nterval, and so s dµ. Thus (14 becomes g (x dx + g (γ (t dγ = (15 µ ( almost everywhere n [, 1]. Now, the rght hand sde of (12 becomes = = [ µ (γ (t g (γ (t + λ (γ (t h (γ (t ] dγ [ µ (γ (t g (x + λ (γ (t h (x f (x ]dx (16 dx (17 = f (x (x f (x ( (18 = f (x f (x ( (19 =. (2 In (16, we have used (13 and (15. In (17, the KKT condton (11a was used. Snce f (x dx s contnuous n t, the fundamental theorem of calculus s applcable n (18. Remark 2: A specal case of (12 can be found n [2, Ch. 23.C]. Theorem 2 also extends [25, Proposton 4] to general class of problems of the form P. A. Example: Smple power supply market We consder agan the power supply market n Example 1. Recall that x = 1 for every s the optmal soluton. So defne a smooth path γ : [, 1] R by γ (t = t so that γ ( = and γ (1 = x. Snce h (x = x, we have x h (x = 1. There s no nequalty constrant. Therefore, (12 becomes = λ (x dx. (21 Equaton (21 can also be verfed as follows. Note that P (x s mn x x2 + x 2 s.t. x = N x.

6 λ ( It s easy to verfy that the optmal soluton s regular and the correspondng Lagrange multpler s λ (x = 2(x N N 1. Thus the rght hand sde of (21 s computed as λ (x dx = 2(x N N 1 dx = 2 1 N 1. Ths result concdes wth obtaned n (1. B. Graphcal nterpretaton of Theorem 2 In ths subsecton, we ntroduce a graphcal nterpretaton of the formula (12 for the cases wth n = 1. In Fgure 2, let p (x (defned n Theorem 2 be the market clearng prce wrtten as a functon of x. In partcular, p ( s the market clearng prce when the -th frm s consumpton (producton s zero and p = p (x s the market clearng prce when t s x (.e., the optmal soluton to P. The formula (12 shows that the VCG tax (the monetary payment from the frm to the auctoneer s equal to the area below the curve of p (x. On the other hand, by defnton (6, the tax mposed by the clearng-prce mechansm when all reports by the frms are truthful s equal to the area of the rectangle shown n Fgure 2. Fgure 2 shows that the dscrepancy between π VCG and π s closely related to the market power of the -th frm. The next result holds for general cases wth n 1. Lemma 1: Let the optmzaton problem P and the smooth path γ : [, 1] R n defned by γ (t = x t satsfy the condtons n Theorem 2. Suppose, n addton, that there exsts a constant ɛ > such that p (x p ɛ (22 for each pont x on γ, where s the Eucldean norm on R n. Then π VCG π ɛ x. Proof: Due to the Cauchy-Schwarz nequalty, we have ɛ x p (γ (tx p x ɛ x for all t [, 1]. Integratng each sde wth respect to t, and usng the formula (12: = p (γ (t dγ = p (γ (tx, we obtan ɛ x πvcg π ɛ x. Lemma 1 shows that f the prce markup by frm s bounded n the sense of (22 (graphcally, the functon p (x n Fgure 2 s nearly constant, then π VCG s close to π. V. BUDGET LOSS ANALYSIS Although the VCG mechansm mplements effcent socal choce functons n domnant strateges, budget balance property s not guaranteed n general. Ths s a promnent reason why the VCG mechansm s rarely used n practce. In ths secton, we apply Theorem 2 to analyze the budget of the VCG mechansm. Let B VCG be πvcg the budget n the VCG mechansm, and B truthful πclearng, be the budget n the clearngprce mechansm under the assumpton that all players are truthful. Market clearng prces p ( p Fg. 2. π VCG = න A. Quadratc cost functons x p (x dx (Area below the curve π = p x (Rectangle area x (Optmal soluton to P VCG vs clearng-prce mechansms. p (x Consder a specal case of (1 n whch each cost functon s of the form f (x = a x 2 wth a > and the only constrant s x = c. The next result shows that the quantty (B VCG B /B can be explctly computed n terms of ndvdual frm s share s x c. Theorem 3: For each = 1,..., N, assume that the cost functon has the form f (x = a x 2 wth a > and the only constrant s x = c. Then B VCG B s 2 = B 2(1 s. Proof: From the KKT condton, for each = 1,..., N, we have λ (x = 2 ( 1 a 1 (x c. Thus = x ( λ (x dx =2 1 a x 1 ( 1 2 x 2 cx. 1 From the KKT condton, we also have a = 2x Substtutng ths nto the above equalty, we have λ x λ x 2 c. Thus, B VCG 2(x B clearng B clearng = ( = = πvcg x /c x 2(1 x /c c s 2 2(1 s. λ. = + λ c /( λ c In partcular, f s = 1/N for all, we have (B VCG B /B as N. Remark 3: Although Theorem 3 shows B VCG > B, t should not be concluded that the budget n the VCG mechansm s larger than that of the clearngprce mechansm. Ths s because B s evaluated wth the assumpton that all players are truthful, and does not reflect the realstc budget when the players are strategc n the clearng-prce mechansm.

7 B. Bounded market power In general, t s dffcult to provde a bound on the budget of the VCG mechansm. Here, we present a drect consequence of Lemma 1. Theorem 4: Suppose that P and the smooth path γ : [, 1] R n defned by γ (t = x t satsfy the condtons n Theorem 2 for each = 1, 2,..., N. Suppose there exsts a constant ɛ > such that p (x p ɛ for every pont x on γ and for each. Then B ɛ B VCG Proof: By Lemma 1, we have ɛ x x. π ɛ x for each. Thus the clam follows mmedately. Theorem 4 can be appled to the followng supply-demand matchng problem wth transmsson constrants: mn s,d s.t. c (s u (d S D s = d, s, d S D b H G s H L d b. (23a (23b (23c In (23, S s the set of generators, D s the set of consumers, c, S are generaton cost functons, u, D are demand utlty functons, H G s the generaton shft factor matrx, H L s the load shft factor matrx, and b s the transmsson constrant. Wth x = (s, d, the optmzaton problem (23 can be wrtten n the standard form (1. Notce that the quantty x n Theorem 4 s twce the total generaton (consumpton n the system (23. Therefore, f the premses of Theorem 4 are satsfed, t s concluded from Theorem 4 that the excess budget of the VCG mechansm over the clearng-prce mechansm wth truthful partcpants s upper bounded n terms of the bound ɛ on the prce markup and the total generaton (consumpton n the system. VI. CONCLUSION In ths paper, we consdered a fundamental connecton between the clearng-prce mechansm (whch s commonly used n the effcency analyss and the VCG mechansm. As the man techncal result, we showed that the VCG payment (tax s equal to the path ntegral of the vector felds of market clearng prces, from whch we showed that the clearng-prce mechansm and the VCG mechansm have smlar tax rules when ndvdual partcpants market power s neglgble. We also outlned how ths connecton can be exploted to analyze the budget of the VCG mechansm. 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