Analysis of Decentralized Decision Processes in Competitive Markets: Quantized Single and Double-Side Auctions

Transcription

1 Analyss of Decentralzed Decson Processes n Compettve Marets: Quantzed Sngle and Double-Sde Auctons Peng Ja and Peter E. Canes Abstract In ths paper two decentralzed decson processes for compettve marets are analyzed under quantzed prcng; these proposed decentralzed processes have toy models whch are smlar to those for maret models n such areas as electrcty systems [1] [5], and communcaton networs [] [8]. Frst, quantzed dynamcal auctons for supply marets (.e., only sellers are assumed to have maret power) are presented to allocate a dvsble resource target among arbtrary populatons of supplers. Both rapd convergence and approxmate socal optma are acheved. Second, the quantzed mechansm s extended to a double aucton crcumstance where competton of both sellers and buyers s consdered. Under the nondscrmnatory prcng assumpton (.e., chargng the same prce for dfferent agents), the aforementoned mechansm s shown to have rapd convergence and effcency performance (.e., maxmum of socal welfare) as n the sngle-sde dynamcal aucton case. Index Terms decentralzed decson, dynamcal auctons, mult-agent systems, compettve marets, non-lnear dynamcs I. INTRODUCTION Desgnng decentralzed decson mang rules to substtute for tradtonal centralzed regulaton has been performed or consdered n several felds of the real world, for example, poneerng electrcty ndustry deregulatons occurred n the Unted Kngdom, New Zealand, Germany, North Amerca, to name but a few, allocaton and prcng protocol desgn n communcaton networs to tae place the flat-rate prcng (ndependent of usage), and so on. The ey motvaton for such restructurng s that the true margnal costs of ndvdual agents can be revealed by decentralzed decsons, and n addton, socal welfare wll be mproved by the competton among the agents. Auctons and ther assocated maret prcng mechansms consttute an mportant and hghly developed framewor wthn whch such proposals may be evaluated for ths problem (see [1] [5] for power systems and [] [1] for communcaton systems). In [8], a so-called Unque-lmt Quantzed Progressve Second Prce (UQ-PSP) aucton mechansm was frst presented for dynamc maret-prcng n communcaton networs and socal networs. Ths mechansm dynamcally allocates a dvsble resource to agents capable of exertng maret power to generate successve prce and quantty bds. Subject to a quantzaton assumpton on the bd prce set, t was proved n [8], [11] that ths Vcrey-Clare-Groves Ths wor was supported by NSERC Dscovery Grants. P. Ja and P. E. Canes are wth the Centre for Intellgent Machnes (CIM) and the Department of Electrcal and Computer Engneerng, McGll Unversty, Montreal, Canada {pja; peterc}@cm.mcgll.ca (VCG)-le mechansm has the followng desrable propertes: (1) quantzed strateges are ɛ-ncentve compatblty (.e., a truth-tellng strategy s domnant up to a quantzed level); (2) the assocated dynamcal aucton systems rapdly converge to an (ɛ-nash) equlbrum f all agents apply quantzed strateges teratvely; () the lmt prce s unque and ndependent of ntal data; (4) the lmt resource allocaton s effcent (n the sense of the optmzaton of the summed ndvdual valuaton functons) up to a quantzed level under mld assumptons of demand functons. Frst, n ths paper, we apply the UQ-PSP aucton framewor to supply marets by consderng the competton among sellers (for example, power generators n electrcty systems or sensor bases n wreless networs), explcatng the fact that ths can be formulated as a dual problem to the quantzed PSP auctons on the demand sde n [8], [11]. As a result, specfed supply-sde dynamcal auctons are shown to have the followng smlar desrable features: () maret partcpants (power generators) have an ncentve to ensure maret effcency (.e., socal optma) and () acceptable prcng can be acheved smply and rapdly. Second, we consder a double aucton case, where both sellers and buyers n the maret are assumed to have maret power, and ther strateges not only nfluence ther peers behavours but also nfluence the dynamcs of the other sde of the maret (see [12] [14]). Subject to the quantzaton (meanngful n practce [15]) and non-dscrmnatory prcng (.e., chargng the same prce to dfferent agents) assumptons, the dynamcal double aucton system, consstng of recursve quantzed bds and maret nformaton from both demand and supply sdes, s shown to converge to an ordertwo orbt. The orbt conssts of the two quantzed prces defnng the smallest prce set approxmatng the compettve equlbrum prce under the quantzed framewor. Socal effcency s hence acheved modulo to a quantzed level. The paper s organzed as follows. In Secton II, the applcaton of UQ-PSP auctons to prcng n a compettve supply maret model s presented and dual results are acheved for convergence and effcency analyss. Secton III consders quantzed double auctons, where dynamcal behavours of both sdes are descrbed. Fnally, numercal smulatons are gven n Secton IV, where t s shown that the quantzed dynamcal aucton systems (for both supply auctons and double auctons) converge rapdly and the lmt prces approxmate the correspondng socal optma.

2 II. SELLER-SIDE AUCTIONS WITH QUANTIZED STRATEGIES In ths secton, a so called UQ-PSP aucton mechansm (see [8], [11]) wll be appled to a supply maret model where competton only occurs on the seller sde. We note that (1) unle the mult-part bds used n the power pool model (see [1], [17]), sellers here compete for nfntely dvsble resource wthn a sngle perod and teratve bds can be observed throughout the maret; (2) the seller-sde aucton consdered here can be formulated as a dual problem to the buyer-sder auctons n [8], [11]. A. Problem formulaton We present the seller-sde Progressve Second Prce (PSP) aucton model as follows: In a non-cooperatve maret, N supplers provde a dvsble resource to satsfy an amount of demand C. Each suppler S, 1 N, maes a two-dmensonal bd s = (ps, qs ) to a maret operator, where qs s the quantty the suppler desres to provde and ps s the unt-prce at whch the suppler would le to sell qs. Here we assume ps p c, for all 1 N, where p c > s defned to be a threshold prce,.e., the hghest prce the maret could bear. The bddng profle s s denoted as s = [s ] and the opponents bddng profle s = s \ {s }. Gven a total quantty C, the Seller Maret-prce Functon (SMF) of S s a left-contnuous (.e., contnuous from the left) functon defned to be: P s (z, s ) = sup{p c y : C qs z}, (1) y ps <y, whch s nterpreted as the maxmum bd prce the seller ass n order to supply the quantty z gven the opponents profle s, the demand quantty C and the threshold prce p c. The functon s only meanngful for z >. Smlarly we defne ts nverse functon Qs as follows: Qs (y, s ) = [C qs ] +, (2) ps <y, whch means the mnmum supply quantty at a bd prce of y gven s. The supply functons σ : R + R +, 1 N, of sellers are non-decreasng and contnuous. Denote Σ (q) = q σ (t)dt, 1 N, as the producton cost functons and I = σ 1, 1 N, as the nverse supply functons. The potental valuaton functon s defned to be V (q) = p c q Σ (q), 1 N. Hypothess 2.1: (Elastcty) In the followng dscusson, we assume that each σ satsfes an elastcty assumpton []: there exsts γ >, such that for all z z C, σ (z ) > mples σ (z) σ (z ) > γ(z z ). The allocaton rule for sellers s defned as follows, a (s) = mn{qs, qs Qs (ps, s ) :ps =ps qs }; () c (s) = j (p c ps j ) [a j (; s ) a j (s ; s )], (4) where a denotes the quantty Seller S sells at a bd prce ps gven the opponents bddng profle s,.e., the mnmum of S s bd quantty qs and the avalable maret quantty at the bd prce ps. The cost to S shall nclude two parts: the producton cost Σ (a (s)) and the opportunty cost c. c s ntroduced based upon the concept of the excluson compensaton prncple [] and the Vcrey-Clare-Groves (VCG) mechansm [18]; t represents the potental dfference n revenue between that contrbuted by all the other sellers dstnct from S when () S s absent from the aucton and () S partcpates n the aucton. Seller S s utlty s defned to be u (s) = p c a (s) Σ (a (s)) c (s), = V (a (s)) c (s), (5).e., the potental valuaton mnus the opportunty cost at the allocated quantty a (s). For the sae of smplcty, here we assume that the sellers n the maret do not have budget constrants. Then gven the opponents bddng profle s, the best reply (.e., maxmzng u (s)) of a seller under the framewor above can be chosen as follows: Lemma 2.2: Subject to Hypothess 2.1, gven s, Seller S s best response s obtaned by s = (ws, vs ), where vs = nf {z : σ (z) > P s (z, s )} ; ws = σ (vs ). () The proof of Lemma 2.2 s smlar to that presented n [] n whch the dual problem on a demand-sde aucton s studed. & & $ *(! Fg. 1. #!! - "#$ ( % - "( $ ( %,!,!! "!!!#!! $!!!!! +(! $!!! '(! ) & &( $, Maret nformaton and the best strategy. %

3 The relatonshp between Seller S s supply functon, ts maret prce functon and ts best strategy gven n () s shown n Fgure 1, where purely for smplcty of portrayal, t s assumed that σ s lnear. To accelerate the convergence of the PSP algorthm n [], a quantzed PSP mechansm called Unque Lmt Quantzed PSP (UQ-PSP) was developed n [11] for demand auctons. Due to the desrable propertes of the UQ-PSP mechansm,.e., rapd convergence and socal effcency up to a quantzed level, and the fact that quantzaton s a practcally meanngful constrant, we apply the UQ-PSP scheme to the supply PSP aucton descrbed above. Specfcally, based upon Lemma 2.2, we mae the quantzed assumptons on sellers behavors. Quantzed Strateges n Supply Auctons: Gven s, C and an ntal quantzed prce set B p, the Unque-lmt Quantzed PSP (UQ-PSP) strategy s = (ps, qs ) s specfed by ps = T 1 (vs, s) { P s (vs, s ), f ws = σ (vs ) = p mn ; max{p j Bp; p j < P s (vs, s )}, otherwse, qs = I (ps ), (7) where p mn s denoted as the mnmum prce n Bp. That s to say, all sellers bd prces are assumed to be chosen from a quantzed prce set Bp and the quantzed strategy (ps, qs ) gven n (7) s a quantzed approxmaton of the best strategy (ws, vs ). Hypothess 2.: All sellers apply the quantzed strateges specfed above. The prevous dscusson represents the statc model of a seller aucton. In the followng we wll assume that all sellers n the aucton update ther strateges progressvely and smultaneously. In partcular, subject to Hypothess 2., the recursve aucton system s defned as follows. Dynamcal Supply Aucton State-Space System: P s +1 (z, s ) = sup{p c y : C z + qs j } ps j <y,j vs +1 = nf { z : σ (z) > P s +1 (z, s ) } ws +1 = σ (vs +1 ) ps +1 ( = T 1 vs +1, s ) qs +1 = I (ps +1 ), (8) wth the ntal condtons ps B p, qs = I (ps ), <, 1 N. In the followng secton, we wll study the dynamcal convergence of the recursve system (8) and the effcency (.e., maxmum of socal welfare) of the lmt bddng profle. B. Convergence and Effcency Results Hypothess 2.4: There exsts a seller S, addtonal to the seller set, who bds s = (ps, C), 1, wth mn{p B ps p {p c } : p > ps }, = f I C and Bp = 1; () max Bp, otherwse, where I = qs. Subject to ths hypothess, the dual convergence and socal effcency arguments wth respect to the auctons on the demand sde [11] are specfed as follows: (The proofs are not shown here due to space constrants.) Theorem 2.5: Subject to Hypotheses 2. and 2.4, the dynamcal supply aucton system (8) converges to a unque quantzed prce p n teratons, where p and satsfy p = max{p Bp : I (p) C}; (1) {p B p : I (p) > C} + 1, (11) and are ndependent of the ntal bddng profle s. The effcency (.e., maxmum of socal welfare) of the lmt bddng profle can also be generated n the smlar ways as n the demand-sde auctons. Lemma 2.: Subject to Elastcty Hypothess 2.1, the lmt bddng profle s specfed n Theorem 2.5 s a δ-nash Equlbrum (.e., the utlty of each agent cannot be mproved more than δ wth unlateral strategy change) where δ = max u (s ) u ((p, I (p )), s ) ; p = max{p : p < p, p Bp}. If the supply functons also satsfy σ (z) σ (z ) < κ(z z ), for all z > z, then the lmt socal welfare at s,.e., the lmt of V (s ), satsfes ( V (s )) max{ V (s)} V (s ) s S = O( δ), where S represents all possble bddng profles. Furthermore, max p p δ, (12) where p, p are any two adjacent quantzed prces n B p. In fact, f the supply functons satsfy the condtons n Lemma 2., t can be shown that the ntersecton between the aggregate supply functon σ (q) and the total quantty q = C corresponds to the socal optmum, that s to say, the prce such that I (p) = C s a socal optmal bd prce. (The proof s smlar to the analogous argument for the maret clearng prce of the demand PSP aucton n [1].) III. DOUBLE AUCTIONS WITH QUANTIZED STRATEGIES In Secton II we assumed that buyers n the compettve supply maret are prcng-tang and they do not nfluence the lmt prce of the dynamcal supply aucton. We wll relax that assumpton here and study an alternatve stuaton where compettve behavours on both the supply-sde and the demand-sde of the maret are consdered. Ths problem s formulated as a double aucton where both buyers and sellers mae two-dmensonal bds smultaneously. Buyers and sellers recursvely update ther strateges to maxmze

4 ther own utlty by observng the current maret nformaton. In contrast to the sngle-sded aucton, each buyer s (respectvely, seller s) bd not only nfluences ts opponents behavours, but also nfluences the dynamcs n the supply (respectvely, demand) sde, and hence have an mpact on the maret total quantty C and the lower-bound (respectvely, upper-bound) of bd prces. Consstent wth our overall theoretcal framewor and for practcal meanngfulness, we retan the basc quantzaton assumpton on bd prces. A. Problem formulaton The quantzed demand PSP aucton was analyzed n [7], where a fxed amount of resource s provded by one seller. As stated above, we shall assume that the total supply C s substtuted by the bddng profle s of the group of sellers defned n Secton II-A. After ntroducng the buyer and seller bddng profles nto one model, the notatons of matched prces and quanttes and potental quantty are presented. Then competton n the double aucton wll be formulated as two dependent sngle-sded auctons usng these jont maret quanttes and prce constrants. That s to say, we descrbe such competton as a non-cooperatve game and ntroduce the couplng parameters between both sdes of the maret n a statc model. Next, n Secton III-B, we construct a dynamcal quantzed double aucton system whch conssts of two coupled recursve subsystems. Specfcally, n a non-cooperatve double aucton, N supplers provde a dvsble resource to satsfy the requrement of M buyers: Bd of Buyer B : b = (pb, qb ), 1 M. Bd of Seller S : s j = (ps j, qs j ), 1 j N. Buyer bddng profle: b [b ] ; and the bddng profle of B s opponents: b b \ {b }. The seller bddng profle and ts opponents bddng profle are respectvely s and s as n Secton II-A. (We wll not repeat the dual terms defned n Secton II-A). The Buyer bd Prce Functon (BPF) s a rghtcontnuous functon defned to be: B(z, b) = nf{y : qb z}. (1) pb >y The Seller bd Prce Functon (SPF) s a rghtcontnuous functon defned to be: S(z, s) = sup{p c y : qs j z}, (14) ps j<y where p c s any gven prce such that p c max B p. The matched prces pb and ps are respectvely the lowest buyer bd prce and the hghest seller bd prce such that pb (b, s) = B(z, b), (15) ps (b, s) = S(z, s). (1) where z = sup q {B(q, b) S(q, s)},.e., the superor quantty such that the BPF functon s not less than the SPF functon. Buyers bddng a prce hgher than or equal to ps and sellers bddng a prce lower than or equal pb are called matched agents. One may chec that pb ps. Defne the matched quanttes Cb (b, s) = qb ; (17) Cs (b, s) = pb ps (b,s) ps pb (b,s) then the potental quantty C are such that qs, (18) C = C(b, s) mn {Cb (b, s), Cs (b, s)}, f pb = = ps, and B p = 1; max {Cb (b, s), Cs (b, s)}, otherwse. (1) Here C s specfed n two dfferent stuatons: (1) when the bd prces of the sellers do not agree wth those of the buyers,.e., pb > ps, or pb = ps and B p > 1, the total avalable quantty consdered by both sdes at the next teraton s chosen as the larger matched quantty. Ths assumpton reflects the maret potentalty for hgher matched quantty level and t encourages more resource to be produced or requred n the maret before the agreement s acheved; (2) when there exsts a unque matched prce,.e., pb = ps and B p = 1, C s assumed to be the smaller accumulated bd quantty on both sdes, whch corresponds to the actual matched quantty f the aucton stops at ths teraton. Fgure 2 descrbes the relatonshp between bd prce functons (BPF and SPF) and the matched nformaton (prces pb, ps and quantty C).!"#$% 45 5 Fg. 2.,'+%"-,#.-!"#$%-/')$*#)- 1%22%"-,#.-!"#$%-/')$*#)- 7 &'()*#*+ Publc bd nformaton n a quantzed double aucton. Gven a potental quantty C, the Buyer Maret-prce Functon (BMF) of B s: P b (z, b ) = nf{y : C qb j z}, pb j>y,j where C s determned by (1) and t was a fxed value for demand auctons n [11]. Its nverse functon s: Qb (y, b ) = [C qb j ] +. pb j>y,j

5 Demand functons δ ( ), 1 N, of buyers are decreasng and contnuous. Denote D = δ 1 as the nverse demand functon and Z (x) = x δ (z)dz as the valuaton (or reward) functons of agents. Hypothess.1: (Elastcty Assumpton []) We assume that Z () =, 1 N, and for any C z > z, there exsts γ such that δ (z) > mples δ (z) δ (z ) < γ(z z ). The allocaton rule (frst presented n [2]) for buyers s defned as follows, a (b) = mn{qb, qb Qb (pb, b ) j:pb j=pb qb j }; (2) c (b) = j pb j [a j (; b ) a j (b ; b )], (21) where a (b) denotes the quantty Buyer B obtans by a bd prce pb when the opponent buyers bd b and the potental quantty s C, and the charge to Buyer B s denoted c (b). Buyer B s utlty [] s defned to be u (b) = Z (a (b)) c (b). (22) B. Quantzed strateges and dynamcal aucton system By usng the defntons above, one may chec that the best strateges (vs, ws ) n the supply-sde specfed n Lemma 2.2 and the best strateges (vb, wb ) n the demand-sde gven n [] are stll domnant wth respect to the gven nformaton {b, s} (and C(b, c)). In the followng we wll ntroduce the quantzed strateges derved from those best strateges for the double aucton n a statc model. Quantzed Strateges for Double Auctons: Gven the bddng profles {b, s}, the supply functon σ, the potental quantty C(b, s) and a quantzed prce set B p, the quantzed strategy s = (ps, qs ) for Seller S s as follows, where ps = T s (b, s) = mn{mn{pb j ps }, T 2 (b, s)}; j qs = I (ps ) = σ 1 (ps ), (2) T 2 (b, s) mn { } p j B p; p j ws, f 1 N qs C; max { p j Bp; p j < P s (vs, s, C) }, f ws p mn and qs (24) > C; p mn, f ws = p mn and qs > C, p mn mn B p, and the best strategy (vs, ws ) s gven n Lemma 2.2. Smlarly, gven {b, s, B p, δ }, we fnd the quantzed strategy b = (pb, qb ) = (T b (b, s), D (pb )) for Buyer B n a dual way. The ey features of (sellers ) quantzed strateges n the double auctons are as follows (and buyers strateges are consdered smlarly): 1. Equaton (2) specfes the upper-bound of sellers bd prces, whch should be less than p u = mn j {pb j ps },.e., the mnmum bd prce of matched buyers (otherwse, zero utlty results for an unmatched seller wth a prce greater than p u ). 2. As presented n (24), when the aggregated demand or aggregated supply s less than the potental quantty C, the agents bd a defensve quantzed prce wth respect to the best bd prce wb or ws, but f the aggregated demand or supply s greater than C, the agents bd an aggressve quantze prce compared wth wb or ws. Hypothess.2: All agents n the consdered double auctons apply the quantzed strateges specfed above. Lemma.: Subject to Hypotheses 2.1 and.1, the quantzed strateges specfed n (2) are ncentve-compatble up to a quantzed level. Setch of proof: The quantzed strateges specfed n (2) approxmate the correspondng best reples up to a quantzed level. One may chec that one agent s utlty wth quantzed strateges s hence maxmzed up to a quantzed level. The ncentve-compatble property (.e., a truth-tellng strategy s domnant) s nherted from the VCG-le allocaton rule. Now we ntroduce the assocated dynamcal system for the double aucton specfed above. Assume that all buyers and sellers do not have budget constrants and update ther strateges smultaneously and recursvely. Then gven B p, {σ j } 1 j N and {δ }, the recursve dynamcal system for the double aucton above conssts of two subsystems: Dynamcal Double Aucton State-Space System: Buyer-sde Dynamcal Aucton Subsystem vb +1 = sup { z : δ (z) > P b +1 (z, b ) } wb +1 = δ (vb +1 ) pb +1 = T b (b, s ) qb +1 = D (pb +1 ), (25a) Seller-sde Dynamcal Aucton Subsystem vs +1 j = nf { z : σ (z) > P s +1 j (z, s j) } ws +1 j = σ (vs +1 j ) ps +1 j = T s (b, s ) qs +1 j = I j (ps +1 j ), (25b) Matched Informaton Update pb +1 = B +1 (z +1, b +1 ) (25c) ps +1 = S +1 (z +1, s +1 ) (25d) C +1 = C +1 (b +1, s +1 ), (25e) wth z +1 = sup q {B +1 (q) S +1 (q)}, the ntal condtons pb B p, qb = D (pb ), 1 M, ps j B p, qs j = I j(ps j ), 1 j N, <. C. Convergence analyss In ths secton, we study the convergence property of the dynamcal double aucton system (25). The convergence s establshed based upon the followng three observatons:

6 (1) n order to acheve postve utltes (.e., match guaranteed), buyers bd prces should be greater than or equal to sellers bd prces (see Clams (a) and (b) n the proof of Lemma. and Lemma A); (2) before an agreement s acheved, the potental quantty C s chosen as the maxmum of the matched quanttes on both sdes (see (1)) whch consequentally ncreases the matched quanttes n the next teraton under the condton n (1). The quantzed prce set also shrns (see Clams (c) and (d) n the proof of Lemma.); () the mnmum matched quantty s upper-bounded by the quantty correspondng to the ntersecton of the aggregate demand functon and the aggregate supply functon (see Theorem.5). We also show n Theorem.5 that the lmt bd profles of the dynamcal double aucton approxmate the assocated compettve equlbrum, whch mples the optmzaton of socal welfare under mld constrants on demand functons and supply functons. Hypothess.4: We assume that the ntal quantzed prce set, the aggregate demand functon of all buyers, and the aggregate supply functon of all sellers are such that for any two dfferent quantzed prces p 1, p 2 Bp, D (p 1 ) I (p 2 ). (2) Ths calculaton hypothess excludes a deadloc due to the quantzaton. Theorem.5: Subject to Hypotheses.2 and.4, for any ntal bddng profle (b, s ) such that C >, the dynamcal double aucton system (25) converges at some <, to a non-trval order-two orbt such that (1) at Iteraton, 1, all sellers have the bd (p 2, I j (p 2)), 1 j N, and all buyers have the bd of (p 1, D (p 1)), 1 M; (2) at Iteraton ( +1), all sellers and buyers have the bd pars (p, I j (p )), 1 j N and (p, D (p )), 1 M, respectvely; () at Iteraton ( + 2), agents repeat ther behavours at the th teraton, where p 1, p 2, p satsfy p 1 = mn{p Bp : D (p) I (p)}; (27) p 2 = max{p B p : p = arg max p {p 1,p }(mn( 2 D (p) D (p), I (p)}; (28) I (p))), (2) and are ndependent of the ntal bddng profles s and b. In partcular, f p 1 = p 2 and D (p 1) = I (p 2), then the system converges to a unque lmt quantzed prce. Setch of proof: Based upon the assumptons on demand functons and supply functons, there exsts a unque prce p e satsfyng D (p e ) = I (p e ), () whch s called the compettve equlbrum prce. One may chec that p 1 s the smallest quantzed prce n the ntal bd prce set B p greater than p e, and p 2 s the largest quantzed prce n B p less than p e. In Lemma. we wll show that, subject to Hypothess.2 and.4, the dynamcal double aucton system (25) wll converge to a two-order orbt, where two lmt adjacent prces satsfy (27) and (28) and C mn = mn{cb, Cs } acheves the maxmum f the system (25) converges. The specal case where p 1 = p 2 = p e, s analyzed n Appendx C.!"#$% " = = " = Fg.. " <! " > =,--"%-(*%./%()1.2')$*#).,--"%-(*%.4'55+.2')$*#). # = 75%*#*#8%. :'##;"#' &'()*#*+ Quantzed approxmaton of a compettve equlbrum. Fgure demonstrates the approxmate compettve equlbrum nature of the lmt order-two orbt for the quantzed double aucton. Lemma.: Subject to Hypothess.2 and.4, for any ntal bddng profle such that C >, the dynamcal double auctons specfed n Theorem.5 converges n tme of order O( Bp ). Proof: See Appendx B. In the order-two orbt, an auctoneer can easly choose a deal prce p {p 1, p 2} to brea the bddng oscllaton, where p gven n (2) s ndependent of the ntal bddng profles s and b. One may chec that, the mnmum matched quantty Cmn acheve the maxmum value at p, whch also corresponds to the maxmum possble socal utlty subject to the quantzed bd prce set Bp. On the other hand, f D (p 1) = I (p 2), the maxmum socal utlty s acheved by ether p 1 or p 2. A. Seller-sde auctons IV. NUMERICAL SIMULATIONS A smple toy model for a power system s presented here to llustrate the results n Secton II. We assume that N power generators n a maret compete for a supply quantty C. Supply functons σ, 1 N, are modelled as σ (q) = max ( ( q + a 2 a 1 )/a, ). An example of a dynamcal UQ-PSP supply aucton s presented n Fgure 4, whch apples the data n Tables I and II. It taes sx steps for the aucton to converge n ths crcumstance.

7 TABLE I TOTAL QUANTITY, NUMBER OF GENERATORS, INITIAL BID PRICES K=1 K=2 C(MW ) N Bp ($/MW ) 5 8 { } Unt Prce Unt Prce TABLE II SUPPLY FUNCTIONS G # N=8 a a a K= Unt Prce K=4 Unt Prce B. Double auctons Two numercal examples are shown for the competton occurrng on both sdes of marets, where all players are assumed to apply the quantzed strateges specfed n Hypothess.2. Fgures 5 and show the convergence of the potental quantty C and the order-two oscllatons whch correspond to the best quantzed approxmatons (from top and from bottom) for the prces assocated wth the compettve equlbra. Also the convergence tme for each example s less than the cardnalty of the ntal quantze prce set,. REFERENCES [1] S. Derajangpetch and G. B. Shebl, Structures and formulatons for electrc power auctons, Electrc Power Systems Research, vol. 54, no., pp , June 2. [2] J. Ncolasen, V. Petrov, and L. Tesfatson, Maret power and effcency n a computatonal electrcty maret wth dscrmnatory double-aucton prcng, IEEE Transactons on Evolutonary Computaton, vol. 5, no. 5, pp , October 21. [] D. Post, S. Coppnger, and G. Sheble, Applcaton of auctons as a prcng mechansm for the nterchange of electrc power, IEEE Transactons on Power Systems, vol. 1, no., pp , August 15. [4] D. J. Swder and C. Weber, Bddng under prce uncertanty n multunt pay-as-bd procurement auctons for power systems reserve, European Journal of Operatonal Research, vol. 181, no., pp , September 27. [5] C. D. Wolfram, Strategc bddng n a multunt aucton: An emprcal analyss of bds to supply electrcty n England and Wales, The RAND Journal of Economcs, vol. 2, no. 4, pp , 18. [] N. Semret, R. R.-F. Lao, A. T. Campbell, and A. A. Lazar, Prcng, provsonng and peerng: Dynamc marets for dfferentated nternet servce and mplcatons for networ nterconnectons, IEEE Journal on Selected Areas n Communcatons, vol. 18, no. 12, pp , December 2. [7] P. Ja, C. W. Qu, and P. E. Canes, On the rapd convergence of a class of decentralzed decson processes: Quantzed progressve second prce auctons, IMA Journal of Mathematcal Control and Informaton, vol. 2, no., pp , 2. [8] P. Ja and P. E. Canes, Auctons on networs: Effcency, consensus, passvty, rate of convergence, n Proc. 48th IEEE Conf. Decson and Control. Shangha, Chna, December 2, pp [] A. A. Lazar and N. Semret, Desgn and analyss of the progressve second prce aucton for networ bandwdth sharng, Telecommuncaton Systems, Specal Issue on Networ Economcs, to appear, 1. [1] P. Malle and B. Tuffn, Multbd auctons for bandwdth allocaton n communcaton networs, n INFOCOM 24. Twenty-thrd AnnualJont Conference of the IEEE Computer and Communcatons Socetes, vol. 1. Hong Kong, March , pp Unt Prce K= Unt Prce K= Fg. 4. Convergence of a power aucton wth 8 agents (as n Table I). The green lnes correspond to sellers supply functons and red lnes are seller maret prce functons Fg. 5. Quantzed prce (Quantty/2) Barganng processes Potental Quantty C Buyers Matched Prce pb * Sellers Matched Prce ps * Number of Iteratons Convergence of a double aucton wth buyers and 5 sellers. [11] P. Ja and P. E. Canes, Analyss of a class of decentralzed dynamcal systems: Rapd convergence and effcency of dynamcal quantzed auctons, under revson for IMA Journal of Mathematcal Control and Informaton, submtted September, 2. [12] R. Wlson, Incentve effcency of double auctons, Econometrca, vol. 5, no. 5, pp , September 185. [1] L. H, A steady-state model of the contnuous double aucton, Quanttatve Fnance, vol., no. 5, pp , 2. [14] S. Gjerstad and J. Dchaut, Prce formaton n double auctons, Journal of Games and Economc Behavor, vol. 22, no. 1, pp. 1 2, January 18. [15] Sotheby s, [1] M. Madrgal and V. Quntana, Exstence and determnaton of compettve equlbrum n unt commtment power pool auctons: prce settng and schedulng alternatves, IEEE Transactons on Power

8 15 Barganng processes Potental Quantty C B. Setch Proof of Lemma. Frst, we defne two quantzed prce sets Fg.. Quantzed prce (Quantty/2) 1 5 Buyers Matched Prce pb * Sellers Matched Prce ps * Number of Iteratons Convergence of a double aucton wth buyers and 25 sellers. Systems, vol. 1, no., pp. 8 88, August 21. [17] A. Motto, F. Galana, A. Conejo, and J. Arroyo, Networ-constraned multperod aucton for a pool-based electrcty maret, IEEE Transactons on Power Systems, vol. 17, no., pp. 4 5, August 22. [18] L. Maows and J. M. Ostroy, Vcrey-clare-groves mechansms and perfect competton, Journal of Economc Theory, vol. 42, no. 2, pp , 187. [1] P. Malle, Maret clearng prce and equlbra of the progressve second prce mechansm, Operatons Research, vol. 41, no. 4, pp , 27. [2] B. Tuffn, Revsted progressve second prce auctons for chargng telecommuncaton networs, Telecommuncaton Systems, vol. 2, no., pp , 22. A. Lemma A: APPENDIX Consder the dynamcal double aucton system (25). Subject to Hypothess.2, f for some, then mn {pb } max {ps }, (1) mn {pb+t } max { } ps +t, (2) for all t. Proof: The result s drectly from Equaton (2) and ts dual equaton n demand sde. Precsely, f (1) holds, we have pb ps max and ps pb mn. Then for any buyer, pb +1 and for any seller ps +1 max{ps j pb } = ps max, j mn{pb j ps } = pb j mn, from (2). Furthermore, when (2) holds, C = max{ qb, qs }. If C = qb, pb+1 pb mn for any buyer, 1 M, and then (2) holds. If C = qs, we can chec n a smlar way that (2) holds. where pb mn = pb max = max N p [ pb mn, ps max) B p ; () M p [ ps mn, pb max] B p, (4) { } mn pb, ps mn = mn { pb }, ps max = max { ps }, { ps }. These notatons can be found n the example n Fgure 2. Then we clam that (a). For any ntal state such that C >, Np 1 =, that s to say, all agents n the aucton are matched (.e. have postve potental allocated quanttes) after the frst teraton; (b). When Np =, Np +1 =, that s to say, the agents recursve strateges guarantee ther matched status remans unchanged n the dynamcal double aucton; (c). When Np =, M +1 p < M p, M p > 2, (5).e., the set of agents bd prces monotoncally shrns untl t conssts of at most two prces, f all agents are matched; (d). When Mp = 2 or Mp = 1, the dynamcal double aucton converges n at most 2 Bp teratons. Proof of Clam (a). Frst we assume that C = Cb (b, s ). Subject to the quantzed strateges specfed n Hypothess.2, each buyer B, 1 M, wll bd a prce pb 1 mn{pb ; pb ps } (snce all ntersectons between demand functons and buyer maret prce functons should be on or above p = mn{pb ; pb ps }, see [11]), and each seller S j, 1 j N, wll bd a prce pb 1 j mn{pb ; pb ps } from (2). Then we have pb 1 mn ps1 max, whch mples that Np 1 =. If C = Cs (b, s ), Np 1 = s also acheved n the smlar way. Proof of Clam (b). See Lemma A. Proof of Clam (c). When Np =, t mples that pb mn ps max. Together wth the defnton of C, we have C = max{ qb, 1 j N qs j }. By checng the quantzed strateges gven n Hypothess.2, we see that (I) pb max pb +1 max, and furthermore (II) pb max > pb +1 max f ether (II.1) pb max > pb mn ; or (II.2) pb max = pb mn and qb < 1 j N qs j. The smlar argument holds for the seller-sde. Therefore, subject to Hypothess.4 and Mp > 2, n each teraton ether the upper bound of Mp decreases, or the lower bound of Mp ncreases, or both happen. That mples (5). Proof of Clam (d). When Mp = 2 or Mp = 1, t s the case that all buyers bd the same prce p 1 and all sellers bd the same prce p 2,

9 where p 1 p 2 are two adjacent quantzed bd prces. (If p 1 = p 2, Mp = 1, otherwse, Mp = 2). By checng the correspondng dynamcs, p 1 and p 2 wll move toward the compettve equlbrum prce p e together n the followng teratons. Precsely, f p 2 > p e, we may verfy (from the defntons of p e n (), demand functons and supply functons n Secton III-A) that I (p 2) > D (p 1). On the buyer-sde, pb +1 = p 2 for all 1 M; and on the seller sde, f p 1 > p 2, ps +1 = p 2 for all 1 N, or f p 1 = p 2, ps +1 = max{p Mp : p < p 2 } for all 1 N. Smlarly, f p 1 < p e, all buyers or all sellers wll bd a hgher prce teratvely untl p 1 p e and p 2 p e, at whch stage the two-order orbt oscllaton wll begn (we can chec that p 1 = p 1 and p 2 = p 2 as defned n (27) and (28)). The last step s to chec that once all agents are n the two-order orbt, they wll stay there for all the followng teratons. Defne Cmn = mn{cb, Cs }. When pb = p 1 and ps = p 2, f Cmn = Cb, subject to Hypothess.4, we have C = Cs > Cb, whch mples that all buyers wll bd pb +1 = p 2 and all sellers wll stay at p 2 at the ( +1)th teraton. We can verfy that p 2 = p as defned n (2). At the ( + 1)th teraton, we can chec C +1 mn = Cs+1 = qs+1 and C +1 = C +1 mn from B+1 p = {p 2} and (1). As a result, all buyers wll bd aggressvely and choose p 1 as ther bd prce at the ( + 2)th teraton, but all sellers wll stll stay at p 2. Therefore, the dynamcal aucton system wll oscllate between the two states (.e., the state at the th teraton and the state at the ( + 1)th teraton). If Cmn = Cs, the same result can be acheved. Overall, snce n every two teratons the upper bound (or the lower bound) of Mp decreases (or ncreases) one prce level untl the dynamcs converge, the convergence tme s bounded by 2 Bp. Furthermore, n the worst case, M p = B p. Based upon Clams (a)-(d), one may chec that the tme for the dynamcal double aucton system (25) to converge to the two-order orbt s of order O( B p ). C. Convergence analyss for the specal case p 1 = p 2 = p e For the specal case that p 1 = p 2 = p e,.e., D (p 1) = I (p 2), we can chec that all buyers and sellers wll stay n p = p e followng Clam (d) n the proof of Lemma.: snce p 1 p e and p 2 p e for a converged bd set Mp = {p 1, p 2 }, f p 1 = p 2 = p e, C = Cmn = qb = 1 j N qs j, and by checng the quantzed strateges defned n Hypothess.2, we have all buyers and sellers wll bd p e at the next teraton. Hence the system (25) converges to p e n ths case.