The Spending Constraint Model for Market Equilibrium: Algorithmic, Existence and Uniqueness Results

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1 The Spendng Constrant Model for Market Equlbrum: Algorthmc, Exstence and Unqueness Results Nkhl R. Devanur [Extended Abstract] College of Computng, Georga Insttute of Technology. 801, Atlantc Dr., Atlanta, GA Vjay V. Vazran ABSTRACT The tradtonal model of market equlbrum supports mpressve exstence results, ncludng the celebrated Arrow- Debreu Theorem. However, n ths model, polynomal tme algorthms for computng (or approxmatng) equlbra are known only for lnear utlty functons. We present a new, and natural, model of market equlbrum that not only admts exstence and unqueness results parallelng those for the tradtonal model but s also amenable to effcent algorthms. Categores and Subject Descrptors: F.2 [Analyss of Algorthms and Problem Complexty]: General; General Terms: Algorthms, Economcs. 1. INTRODUCTION The mathematcal modellng of an economy by Walras [17], followed by the proof of exstence of market equlbra by Arrow and Debreu [2] are consdered of central mportance n mathematcal economcs. However, the ssue of computng an equlbrum has not had smlar success (e.g., see Scarf [15]). Recently, Papadmtrou [14] and Deng, Papadmtrou, and Safra [9] have brought ths queston to the forefront wthn the theoretcal computer scence communty, and ponted out the paucty of polynomal tme algorthms for such questons. [9] gave polynomal tme algorthms for computng equlbra for lnear utlty functons provded the number of goods or buyers s bounded, and left the queston of extendng ths to unbounded goods and utltes. A partal answer to ther queston was provded n [7], who gave a polynomal tme algorthm for the lnear verson of Fsher s problem [15]. [13, 5, 11] gave approxmaton Ths paper s based on [16] and [6]. Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. STOC 04 June 13 15, 2004, Chcago, Illnos, USA. Copyrght 2004 ACM /04/ $5.00. schemes for the lnear utltes case of the Arrow and Debreu model; the frst two are based on [7] and the last s an aucton-based algorthm. In the 1950 s Esenberg and Gale [10] had gven a convex program for computng equlbrum prces for the lnear verson of Fsher s model. Hence, usng the ellpsod algorthm, equlbrum prces can be approxmated for ths case. An exact polynomal tme algorthm follows from a corollary of [7] that equlbrum prces have small denomnators. More recently, Jan [12] gves a convex program whch can be used for computng equlbrum prces for the lnear verson of the Arrow-Debreu model usng the ellpsod algorthm and dophantne approxmaton, leavng open the problem of obtanng a combnatoral algorthm. In ths paper we defne a new class of utlty functons to deal wth the followng two defcences of lnear utlty functons. Frst, snce lnear utlty functons are addtvely separable, a buyer may end up spendng all her money on a sngle tem. Non-separable utlty functons, such as those used by Arrow and Debreu, do not suffer from ths problem; however, such utlty functons are not easy to deal wth computatonally. To deal wth ths ssue, we wll generalze lnear utlty functons by specfyng a lmt on the amount of money buyer can spend on good j; buyer s total utlty functon s stll addtvely separable over goods. Second, lnear utlty functons do not capture the mportant condton of buyers gettng satated wth goods, e.g., as done by concave utlty functons. To capture ths, we generalze further: buyer has several lnear utlty functons for good j, each wth a specfed spendng lmt. We may assume that these functons are sorted by decreasng rates, and hence capture the condton that buyer derves utlty at decreasng rates on gettng more and more of good j. As shown below, these functons can be wrtten as one deceasng step functon. More generally, spendng constrant utltes wll be defned va arbtrary decreasng functons (rather than decreasng step functons). Such utlty functons are natural typcally, people do have rough estmates, ether mplct or explct, on the fracton of ther budget they are wllng to spend on dfferent tems, say mlk, meat, and chocolates. The exact amounts spent on these tems wll of course depend on ther relatve prces. Sometmes spendng constrants are also mposed from outsde, e.g., mortgage companes mpose a lmt on the amount of loan made so that the monthly payment s no more than a certan fracton of one s ncome.

2 Let us extend Fsher s model wth decreasng step spendng constrant utlty functons. Addtonally, we enhance the model by assumng that each buyer has utlty for money, also specfed va a step functon. Now, at specfed prces, a buyer may prefer to spend only part of her money. The noton of equlbrum s generalzed approprately: all goods need to be sold but not all money needs to be spent. The analogous enhancement to the lnear case s called quaslnear n economcs. Our man algorthmc result s a polynomal tme algorthm for computng (the unque) equlbrum prces for ths model. We show exstence of equlbrum prces for the spendng constrant extenson of both Fsher and Arrow-Debreu models wth contnuous, decreasng utlty functons. For the former, Brouwer s fxed pont theorem suffces, however for the latter we need full power of Kakutan s fxed pont theorem. In the former case (Fsher s model), equlbrum prces are unque and n the latter case (Arrow-Debreu) they are not. Unqueness of equlbra has been consdered mportant, snce t s another ndcaton of stablty wthn markets, e.g., see [4]. Usng our algorthm for step decreasng utlty functons as a subroutne, we gve an FPTAS for arbtrary decreasng utlty functons for Fsher s model. We also use ths algorthm as a subroutne to gve an FPTAS for decreasng step utlty functons n the spendng constrant extenson of Arrow-Debreu model. Further extensons yeld an FPTAS for arbtrary decreasng utlty functons n ths model as well. Smlar to [7], our algorthm for step decreasng utlty functons operates by monotoncally rasng prces untl equlbrum prces are reached. Such a procedure s supported by the fact that these utlty functons support weak gross substtutablty,.e., rasng the prce of good j cannot decrease the demand for good j. The key pont of departure wth [7] s the followng: In the lnear case, at any gven prces, each buyer has a set of most desrable goods. Any allocatons made from ths set of goods makes equally happy; t s not essental to allocate any partcular good from ths set. Indeed, the algorthm of [7] explots ths freedom fully t does not need to commt to any allocatons as the prces are beng rased; allocatons are made only at the end, after equlbrum prces have been computed. In our settng, at any prces the optmal bundle of buyer wll nvolve forced allocatons,.e., at these prces, buyer necessarly wants to spend a certan amount of her money on certan goods. However, as prces change, some of the forced allocatons may become undesrable to buyer and need to be deallocated. The man new dea of the algorthm s a way of carryng out ths backtrackng n a way that stll leads to polynomal runnng tme. If utlty functons are assumed strctly concave, there s a unque optmal bundle of goods for each buyer at any gven prces. Therefore, f market clearng prces are announced, each buyer can compute her optmal bundle n a dstrbuted manner and buy goods accordngly, and the market wll clear. Such utlty functons are consdered mportant n Economcs; however, attempts at extendng the algorthm of [7] to the case of concave, or even pecewselnear and concave utltes, have so far faled. Let us outlne the man dffculty nvolved. The algorthm of [7] s nspred by the prmal-dual schema, wth prces playng the role of dual varables n ths prmaldual-type algorthm. Almost all known prmal-dual algorthms work by monotoncally rasng dual varables; very few such algorthms use more sophstcated mechansms of ncreasng and decreasng dual varables n order to obtan a better dual soluton. Indeed, the latter s extremely dffcult to arrange. We observe that pecewse-lnear utltes do not satsfy weak gross substtutablty. As a result, rasng the prce of j may requre lowerng the prce of j. Hence t s reasonable to assume that an teratve algorthm for pecewse-lnear utlty functons wll need to ncrease and decrease prces (dual varables) a rather dffcult matter to arrange, as mentoned above. Our algorthm for step decreasng utltes may help fnesse ths dffculty va the followng teratve procedure: Let f j be the pecewse-lnear utlty functon of buyer for good j and let g j be ts dervatve. Observe that g j s a decreasng step functon. Observe that f the prce of good j s known, say p j, then the functon g j(x jp j) gves the rate at whch derves utlty per unt of j receved as a functon of the amount of money spent on j,.e., the utlty functon requred n the spendng constrant model. Now consder the followng procedure. Start wth an ntal prce vector so that the sum of prces of all goods adds up to the total money possessed by buyers. Usng these prces, convert the gven pecewse-lnear utlty functons nto spendng constrant utlty functons and run the algorthm of the current paper on ths nstance to obtan a new prce vector. Repeat untl the prce vector does not change,.e., a fxed pont s obtaned. It s easy to see that prces at the (unque) fxed pont are equlbrum prces for the gven pecewse-lnear utlty functons. We formally defne the spendng constrant model, for Fsher and Arrow-Debreu settngs, n Secton 2. Exstence and unqueness results are presented n Secton 3. The polynomal tme algorthm for step utlty functons for the Fsher settng s presented n Secton 4, followed by approxmaton algorthms for contnuous deceasng functons (Secton 5). The results of Secton 4 are from [16], and the results of Sectons 3 and 5 are from [6]. The spendng constrant models n the Fsher and Arrow-Debreu settngs were ntroduced n [16] and [6], respectvely. 2. THE SPENDING CONSTRAINT MODEL Fsher s model conssts of a market wth buyers and goods. Each buyer has a specfed amount of money and her utlty functon for goods s specfed. The problem s to fnd prces for goods so that after each buyer s gven her optmal bundle of goods (relatve to these prces), the market clears exactly,.e., there s no defcency or surplus of any good. We extend ths model va spendng constrant utlty functons: Let A be a set of dvsble goods and B a set of buyers, A = n, B = n. Assume that the goods are numbered from 1 to n and the buyers are numbered from 1 to n. Each buyer B comes to the market wth a specfed amount of money, say e() Q + dollars, and we are specfed the quantty, b j Q + of each good j A. For B and j A, let f j : [0, e()] R + be the rate functon of buyer for good j; t specfes the rate at whch derves utlty per unt of j receved, as a functon of the amount of her budget spent on j. If the prce of j s fxed at p j per unt amount of j, then the functon f j /p j gves the rate at whch derves utlty per dollar spent, as a functon of the amount of her

3 budget spent on j. Defne g j : [0, e()] R + as follows: g j(x) = 0 x fj (y) dy. p j Ths functon gves the utlty derved by on spendng x dollars on good j at prce p j. Observe that by scalng the functons f j approprately we can assume w.l.o.g. that b j = 1 for each good j. Each buyer also has utlty for the part of her money that she does not spend. For B, let f 0 : [0, e()] R + specfy the rate at whch derves utlty per dollar as a functon of the amount she does not spend. If returns wth x dollars, the utlty derved from ths unspent money s gven by g 0(x) = 0 x f 0(y)dy. If fj s contnuous and monotoncally decreasng, gj wll be strctly concave and dfferentable. It s easy to see that for such functons, at any prces of the goods, there s a unque allocaton that maxmzes s utlty. The Arrow-Debreu model s the classcal exchange model. The market conssts of agents, each havng an ntal endowment of goods, and havng specfed utlty functons for the goods. The problem s to fnd prces for goods such that after each agent sells her ntal endowment at these prces and buys her optmal basket of goods, the market clears. Ths model nvolves feedback the money possessed by an agent depends on the prces of goods. Hence, any algorthm that teratvely modfes prces has to keep track of the changng amounts of money possessed by buyers, makng ths model computatonally more dffcult. The endowment of each agent s a bundle of goods e [0, 1] A (nstead of money, as before). e s satsfy: j A, B ej = 1. For ths model, the spendng constrant utlty functon specfes the rate at whch an agent derves utlty for a good as a functon of the fracton of money spent, fj : [0, 1] R. The total money earned by the buyer s e() = j A ejpj. Va an approprate scalng, the rate functon can be treated the same way as n the Fsher settng. 3. EXISTENCE AND UNIQUENESS RESULTS 3.1 Characterzng the optmal bundle n the Fsher settng Throughout the paper, we wll use the notaton x to denote a vector, and x j to denote the j th component of x. When the utltes are lnear,.e., f j s a constant functon, buyer buys only those goods that maxmze hs bang per buck, whch s f j/p j. The correspondng characterzaton for the spendng constrant model s that the buyer buys only those goods that maxmze the rate at whch he derves hs bang per buck for that good. More precsely, suppose that spends M j amount of money on good j when he buys hs optmum bundle. (Note that j A M j = e().) Let α := max j A fj (Mj)/p j be the maxmum rate at whch derves hs bang per buck. Then Mj > 0 fj (Mj)/p j = α ; Mj = 0 fj (0)/p j α. Suppose that f : [a, b] R + s contnuous and strctly decreasng. Then f s nvertble n [f(b), f(a)]. Note that f : [f(b), f(a)] [a, b] s also contnuous and strctly decreasng. If f s dentcally zero, then defne f to be dentcally zero as well. Suppose that each fj s ether contnuous and strctly decreasng n [0, e()], or zero. Further, f for each there s at least one j such that fj s non-zero, then say that the rate functons are nce for ths nstance. Let fj be the nverse of f j. Gven a target rate of bang per buck value α for buyer, the money that he should spend 1 on good j s gven by Mj = fj (αp j). Ths suggests a way to compute α. Recall that j A M j = e(). Therefore α s the soluton to the equaton (n the unknown α): fj (αpj) = e(). (1) j A Note that α p j could be greater than fj (0), n whch case the nverse does not exst. We can fx ths by defnng fj (x) = 0 for all x f j (0). Ths preserves the contnuty of fj snce fj s zero at f j (0). Smlarly, we extend f j to x f j (e()) by defnng t to be e() at all these ponts. As long as the rate functons are nce, (1) has a unque soluton. Snce we defned the nverse of a zero functon to be zero, ths ensures that a buyer never spends any money on goods for whch he has no utlty. We next establsh the behavor of α wth the change n prces. Suppose that for some j, p j s ncreased nfntesmally. Then fj (αpj) ether does not change, or decreases nfntesmally. Therefore α, the soluton to (1), ether does not change, or decreases nfntesmally. We state ths observaton for future reference: Lemma 1. α s a contnuous and non-ncreasng functon of p j for all j. 3.2 Exstence and unqueness theorems Let ξ() : R n + R n + be the demand functon n terms of money,.e., ξ j(p) denotes the total amount of money the buyers are wllng to spend on good j at the gven prce vector p. In terms of the above notaton, we get that ξ j(p) = B M j. Therefore, ξ j(p) = f B j (αp j). (2) Snce we already establshed that f j s contnuous, and that α s a contnuous functon of p (Lemma 1), ξ(p) s a contnuous functon of p. Snce we assumed that there s a unt amount of each good, the market equlbrum condton can be restated as ξ(p) = p. We now recall Brouwer s Fxed Pont Theorem that we use n establshng the exstence of equlbrum. Theorem 2 (Brouwer). Let g : S S be a contnuous functon from a non-empty, compact, convex set S R n nto tself, then there s an x S such that g(x ) = x. The goal s to defne a contnuous functon g from S = {p : p j 0, j A pj = B e()}, the scaled smplex of prce vectors, nto tself so that the fxed pont corresponds to the equlbrum. ξ almost works for ths purpose, except that t s not defned when some p j = 0. 1 Another way to thnk about f j (αpj) s that t s the length of the lne segment y = α between the y-axs and the curve y = f j (x)/p j.

4 Consder any j A. Let B be such that f j s non-zero. Let e() l j > 0 be such that f j (l j) l j f j (e()/n) e()/n, j j. There exsts such an l j snce f j (l j ) l j tends to nfnty as l j tends to zero. It can be shown that f an equlbrum prce p exsts, then p j l j. We dvde S nto two parts, Sn := {p S : p j l j, j A} and Sout := S \ Sn. When x Sn we let g(x) = ξ(x); when x Sout, we let g(x) = ξ( x) where x s the pont n Sn that s closest to x. It s easy to check that g s ndeed contnuous on S and that the fxed pont of g s an equlbrum. Theorem 3. For all markets n the Fsher settng wth spendng constrant utltes, there exsts an equlbrum prce vector f the rate functons are nce. For the unqueness of equlbrum, we prove a more general result. Defnton 4. A demand functon f satsfes Weak Gross Substtutablty f f j(p) does not decrease on ncreasng the prce of any good j other than j: f j p j 0, j j A. Lemma 5. The demand functon ξ satsfes Weak Gross Substtutablty. Proof. From (2), t s enough to prove that f j (αp j) s a non-decreasng functon of p j. But the only dependence on p j comes va α. Snce f j s non-ncreasng, t s enough to prove that α s a non-ncreasng functon of p j. But ths has already been establshed n Lemma 1. Defnton 6. A demand functon f satsfes Scale Invarance f f does not change when all the prces are multpled by the same non-zero scalar: f(p) = f(θp), θ > 0. Lemma 7. The demand functon ξ satsfes Scale Invarance. Proof. Note that f α s the soluton to (1) at the prce vector p, then α /θ s the soluton at the prce vector θp. Therefore f j (αp j) s Scale Invarant, and n turn, ξ s Scale Invarant. Lemma 8. If the demand functon of a market satsfes Weak Gross Substtutablty, and Scale Invarance then the equlbrum prces are unque. Proof. Suppose that there are two prce vectors, p and q at whch the market clears. Consder θ := max ( pj ). j A q j W.l.o.g θ 1. It s enough to prove that θ 1. Note that for all j, θq j p j,.e., θq s component-wse bgger than or equal to p. However, at least one component s exactly equal; let that component be j,.e., θq j = p j. Now consder the process that starts wth p and rases the prce of each good untl t s θq. Snce we only ncrease the prces of goods other than j, ξ j does not decrease durng ths process (by Weak Gross Substtutablty of ξ),.e., ξ j(θq) ξ j(p). Snce p s market clearng, ξ j(p) = p j. Further, p j = θq j by choce of j. On the other hand, ξ j(θq) = ξ j(q) (by Scale Invarance of ξ), whch n turn s equal to q j snce q s market clearng. So we have q j = ξ j(q) = ξ j(θq) ξ j(p) = p j = θq j. Hence θ 1 and we are done. From Lemmas 5, 7 and 8, the followng theorem follows. Theorem 9. The equlbrum prce vector for any market n the Fsher settng, wth spendng constrant utltes s unque f the rate functons are nce. 3.3 The Arrow-Debreu settng We follow the exposton of the Arrow-Debreu Theorem n [1] and prove an analogous theorem for the Arrow-Debreu settng wth spendng constrant utltes usng Kakutan s Fxed Pont Theorem. We show that the excess demand n our case has essentally all the propertes that the classc model has. We only state those propertes here. For the complete proof, see [6]. Let ζ be the excess demand functon,.e., ζ j(p) = ξ j(p)/p j 1. The market clearng condton translates to ζ(p) = 0. Let S = {p R m + : p 1 + p p m = 1, p > 0 A}. If ζ satsfes the followng propertes, 1. ζ s contnuous and bounded from below. 2. ζ satsfes the Walras law,.e, p.ζ(p) = 0 holds for all p S. 3. If a sequence {p n} of strctly postve prces p n = (p n 1, p n 2,..., p n l ) p = (p 1, p 2,..., p l ) and p k > 0 holds for some k, then the sequence {ζ k (p n)} of the k th components of {ζ k (p n)} s bounded. 4. p n p S, wth {p n} S mply lm n ζ(p n) =. then there exsts at least one vector p S such that ζ(p) = 0. In the Arrow-Debreu settng, prces are not unque. Consder two agents, wth endowments (1, 0) and (0, 1) respectvely. Suppose that for each agent, the utlty for hs good far outweghs the utlty for the other good. Then the market clears for many dfferent prces, n whch each agent buys only hs own good. 4. THE BASIC ALGORITHM Consder the case where f j s are decreasng step functons. If so, g j wll be a pecewse-lnear and concave functon. We wll call each step of f j a segment. The set of segments defned n functon f j wll be denoted seg(f j ). Suppose one of these segments, s, has range [a, b] [0, e()], and f j(x) = c, for x [a, b]. Then, we wll defne value(s) = b a, rate(s) = c, and good(s) = j; we wll assume that good 0 represents money. Let segments() denote the set of all segments of buyer,.e., segments() = n j=0 seg(f j). Let us assume that the gven problem nstance satsfes the followng (mld) condtons:

5 For each good, there s a potental buyer,.e., j A B s seg(f j ) : rate(s) > 0. Each buyer has a desre to use all her money (to buy goods or to keep some unspent),.e., B : s segments(), rate(s)>0 value(s) e(). Theorem 10. Under the condtons stated above, there exst unque market clearng prces. W.l.o.g. we may assume that each e() and the value of each segment s ntegral. Gven non-zero prces = (p 1,..., p n), we characterze optmal baskets for each buyer relatve to. Defne the bang per buck relatve to prces for segment s seg(fj ), j 0, to be rate(s)/p j. The bang per buck of segment s seg(f0) s smply rate(s). Sort all segments s segments() by decreasng bang per buck, and partton by equalty nto classes: Q 1, Q 2,.... For a class Q l, defne value(q l ) to be the sum of the values of segments n t. At prces, goods correspondng to any segment n Q l make equally happy, and those n Q l make strctly happer than those n Q l+1. Fnd k such that value(q l ) < e() value(q l ). 1 l k 1 l k By the condtons of Theorem 10, segments n Q k have nonzero rate. At prces, s optmal allocaton must contan goods correspondng to all segments n Q 1,..., Q k, and a bundle of goods worth e() ( 1 l k value(q l)) correspondng to segments n Q k. We wll say that for buyer, at prces, Q 1,..., Q k are her forced parttons, Q k s her flexble partton, and Q k+1,... are her undesrable parttons. At any ntermedate pont n the algorthm, not all forced parttons w.r.t. the current prces wll be allocated. However, the forced parttons that are allocated must form an ntal set from the sorted lst. More precsely: Invarant 1: For each buyer, there s an nteger t such that the forced allocatons to correspond exactly to all segments n parttons Q 1,..., Q t, where Q 1, Q 2,... s the sorted lst of parttons of relatve to current prces. We wll say that Q t s the current partton for buyer, and we wll denote t by Q (). The exact value of t depends on the order n whch events happen n the algorthm; however, when the algorthm termnates, t = k as defned n above. 4.1 Ensurng monotoncty of prces The algorthm teratvely rases prces untl equlbrum prces are reached. Clearly, a prme consderaton s to ensure that the equlbrum prce s not exceeded for any good. We descrbe below a condton that ensures ths. At any ntermedate pont n the algorthm, certan segments are already allocated. By allocatng segment s, s seg(f j ), j 0, we mean allocatng value(s) worth of good j to buyer. The exact quantty of good j allocated wll only be determned at termnaton, when prces are fnalzed. In addton, at an ntermedate pont n the algorthm, some money would be returned to buyer. Let returned(s), s seg(f0), denote the amount of money returned to, correspondng to segment s, where returned(s) value(s). If returned(s) > 0, then all segments s seg(f0) havng hgher rate must be fully returned,.e., there s at most one partally returned segment for each buyer. Let allocated(j) denote the total value of good j, j 0 already allocated and let spent() denote the sum of the amount spent by buyer on allocated segments and the amount of money already returned to her. Thus, when segment s s allocated, value(s) s added to allocated(j) and to spent(), and when returned(s) money s returned to, correspondng to segment s seg(f0), returned(s) s added to spent(). Also, defne the money left over wth buyer, m() = e() spent(). Defne the current bang per buck of buyer, α(), to be the bang per buck of partton Q (). Ths s the rate at whch derves utlty, per dollar spent, for allocatons from Q () at current prces. Next, we defne the equalty subgraph G = (A, B, E) on bpartton A, B and contanng edges E. Correspondng to each buyer and each segment s Q (), E contans the edge (, j), where good(s) = j. The capacty of ths edge, c j = value(s). Denote by, and the current allocatons, amounts spent and left over money,.e., (allocated(j), j A), (spent(), B) and (m(), B), respectvely. We wll carry over all these defntons to sets, e.g. for a set S A, (S) wll denote j S m(j). We next defne network N(,, ), whch s a functon of the current prces, allocatons and amounts spent. Drect all edges of the equalty subgraph, G, from A to B. Add a source vertex s, and drected edges (s, j), for each j A and havng capacty p j allocated(j). Add a snk vertex t, and drected edges (, t), for each B and havng capacty m(). Throughout the algorthm, we wll mantan the followng: Invarant 2: (s, A B t) s a mn-cut n network N(,, ). For S A, defne ts neghborhood n the equalty subgraph to be Γ(S) = { B j S wth(, j) G}. For A A and B B, defne c(a ; B ) to be the sum of capactes of all edges from A to B n N(,, ). For S A, defne best(s) = mn { (T ) + c(s; Γ(S) T )}, T Γ(S) and defne bestt(s) to be a maxmal subset of Γ(S) that optmzes the above expresson. Observe that best(s) s the capacty of the mn-cut separatng t from S n N(,, ). Also observe that f T 1 and T 2 optmze the above expresson, then T 1 T 2 must satsfy m() = c(s; ). Hence bestt(s) s unque. We can now gve a characterzaton of Invarant 2 n terms of cuts n the network. Lemma 11. Network N(,, ) satsfes Invarant 2 ff S A : (S) (S) best(s). A set S A that satsfes the nequalty n Lemma 11 wth equalty wll be called a tght set. By the followng lemma, f Invarant 2 holds, there s a unque maxmal tght set. Lemma 12. Assume that Invarant 2 holds. If S 1 A and S 2 A are two tght sets, then S 1 S 2 s also a tght set.

6 Corollary 13. If Invarant 2 holds, the maxmal tght set s unque. 4.2 Algorthm 1 Observe that f S s a tght set, market clearng prces have been acheved for these goods unless the equalty subgraph undergoes change snce the prces of goods n S are just rght to exactly exhaust the money of all buyers nterested n these goods. Hence, the algorthm wll stop rasng prces of these goods (ndeed, rasng them wll volate Invarant 2). The algorthm parttons the equalty subgraph G = (A, B, E) nto two: frozen and actve, consstng of bparttons (A 1, B 1) and (A 2, B 2), respectvely (throughout ths paper, A 1, A 2 wll be subsets of A and B 1, B 2 wll be subsets of B). A 1 s the maxmal tght set of G, B 1 = Γ(A 1), and the frozen subgraph satsfes (A 1) (A 1) = (B 1). The actve subgraph satsfes S A 2 : (S) (S) < best(s) and so prces of goods n A 2 can be rased wthout volatng Invarant 2. The crucal job of parttonng the equalty subgraph s performed by subroutne freeze (see Secton 4.3), whch also performs other related functons. As argued n Secton 4.3, the frozen and actve subgraphs are dsconnected and hence decoupled. A buyer s sad to have a partally returned segment s seg(f 0) f 0 < returned(s) < value(s). Ths happens f moves to the frozen subgraph before value(s) money correspondng to segment s could be fully returned. If so, when returns to the actve subgraph, the algorthm attempts to return the rest of value(s) to. In order to ensure Invarant 2 at the start of the algorthm, the followng steps are executed: Fx all prces at 1/n. Snce all goods together cost one dollar and all e() s are ntegral, the ntal prces are low enough that each buyer can afford all the goods. Clearly, each buyer s current partton wll be her frst partton. Next, we have to ensure that each good j has an nterested buyer,.e., has an edge ncdent at t n the equalty subgraph. Compute α for each buyer at the prces fxed n the prevous step and compute the equalty subgraph. If good j has no edge ncdent, reduce ts prce to p j = max B max s seg(f j ) rate(s) α. Next, partton the equalty subgraph nto frozen and actve by callng subroutne freeze (see Secton 4.3). Market clearng prces have not been reached for goods n the actve subgraph and ther prces need to be ncreased. We want to do ths n such a way that the equalty subgraph remans unchanged. Observe that f buyer has equalty edges to goods j and j then u j p j = uj,.e., p j p j p j = uj. u j Ths suggests ncreasng prces n such a way that the rato of prces of any two goods s not affected, whch n turn s accomplshed as follows: Multply the current prce, p j, of each good j A 2 by x. Intalze x = 1, and start rasng x contnuously. As x s rased, one of four events could take place (these are executed n the order below, for reasons explaned n the proof): Event 1: Ths event happens f the actve subgraph contans buyer wth rate(s) = α() for s seg(f0), where returned(s) < value(s). (Ths event happens n one of two ways: Frst, buyer wth a partally returned segment just moved to the actve subgraph. Second, as prces ncrease, the bang per buck of decreased to the pont where she s equally happy leavng wth money correspondng to segment s unspent. In ether case, the algorthm must return money correspondng to s before t can rase prces of goods.) The algorthm starts rasng returned(s) contnuously untl one of two events happens: Event 1(a): Observe that for a set S A 2 such that bestt(s), best(s) s decreasng as returned(s) s rased (snce m() s decreasng). As a result such a set may go tght. When ths happens, subroutne freeze s called to compute the frozen subgraph; n the process, wll be frozen. Event 1(b): returned(s) = value(s). In ths case, money correspondng to segment s has been fully returned. Event 2: As prces ncrease, a subset of A 2 may go tght. If so, subroutne freeze s called to recompute the frozen and actve subgraphs. Event 3: For buyers n B 2, goods n A 1 are becomng more and more desrable (snce ther prces are not changng, whereas prces of goods n A 2 are ncreasng). As a result, a segment s seg(f j), B 2, j A 1 may enter nto the current partton of buyer, Q (). When ths happens, edge (, j) s added to the equalty subgraph. As a result, A 1 s not tght anymore, and therefore subroutne freeze s called to recompute the frozen and actve subgraphs. Event 4: Suppose B 1 has a segment s seg(f j ) allocated to t, where j A 2. Because the prce of j s ncreasng, at some pont the bang per buck of ths segment may equal α,.e., segment s enters s current partton. When ths happens, we wll deallocate segment s,.e., subtract value(s) from allocated(j) and from spent() and add edge (, j) to the actve subgraph. Snce m(j) ncreases, A 1 s not tght anymore, and therefore subroutne freeze s called to recompute the frozen and actve subgraphs. Algorthm 1 s summarzed below. Ths algorthm rases varables contnuously at two places; we frst need to replace these wth dscrete procedures. Compute the mnmum value of x at whch each of the four events takes place, and mnmum of these s the event that happens frst. For Events 1, 3 and 4, the computaton s straghtforward. Let x be the value of x at whch Event 2 happens. We gve a procedure for computng x n Secton 4.4. Also, let y be the value of returned(s) at whch Event 1(a) occurs. In Secton 4.4 we gve a procedure for computng y as well.

7 Intalzaton: j A, p j 1/n; B, α rate(s)/good(s), s Q () ; Compute equalty subgraph G; j A f deg G(j) = 0 then rate(s) p j max B max s seg(f j ) α Recompute G; (A 1, B 1) (, ) (The frozen subgraph); (A 2, B 2) (A, B) (The actve subgraph); whle A 2 do x 1; Defne j A 2, prce of j to be p jx; Rase x contnuously untl one of four events happens: f rate(s) = α(), for s seg(f0) wth returned(s) < value(s) then Start rasng returned(s) contnuously untl returned(s) = value(s) or the followng events happens: f S A 2 goes tght then Call freeze; f S A 2 goes tght then Call freeze; f segment, s, correspondng to B 2, j A 1 enters Q (), then Add (, j) to G wth c j = value(s); Call freeze; f allocated segment, s, correspondng to B 1, j A 2 enters Q (), then Deallocate s; Add (, j) to G wth c j = value(s); Call freeze; Algorthm 1: 4.3 Subroutne freeze Subroutne freeze operates as follows: Va max-flow, fnd a mn-cut n N(,, ) that maxmzes the number of vertces n the s sde (there s a unque such maxmal mn-cut). Let t be (s A 1 B 1, A 2 B 2 t) 2. Clearly, B 1 = bestt(a 1). Correspondng to each edge connectng A 1 to Γ(A 1) B 1, allocate goods. As a result of ths allocaton, there may be buyers n B 2 that do not have any equalty edges ncdent at them (however, by the maxmalty of the cut found, they must have money left over). For each such buyer, compute her parttons relatve to current prces and nclude edges correspondng to the frst unallocated partton. If none of these edges s ncdent at a good n A 1, subroutne freeze halts. Otherwse, t starts all over agan to fnd a mn-cut n the modfed network. Before returnng, subroutne freeze parttons the equalty subgraph nto two: frozen and actve. The frozen subgraph conssts of the bpartton (A 1, B 1) and the actve subgraph conssts of (A 2, B 2). Observe that buyers n B 1 may desre goods n A 2. By 2 Throughout ths paper, we wll assume that A 1, A 2 A and B 1, B 2 B. Lemma 14, the prces of these goods can be rased wthout volatng Invarant 2. As soon as ths happens, buyers n B 1 who have equalty edges to goods n A 2 wll not be nterested n these goods anymore, and such edges can be dropped. The frozen and actve graphs are hence decoupled. After these changes, the followng holds: Lemma 14. The actve and frozen subgraphs satsfy Invarant 2. Furthermore, the actve subgraph satsfes: S A 2 : (S) (S) < best(s), and the frozen subgraph satsfes: (A 1) (A 1) = (B 1). 4.4 Computng x and y : mn-cuts n parametrc networks For smplcty of notaton, assume that the actve subgraph s (A, B). Throughout ths secton, wll denote prces at the begnnng of the current phase,.e., at x = 1. We frst show how to compute x, the value of x at whch Event 2 occurs,.e., a new set goes tght. Let S A denote the tght set. In N(,, ), replace the capactes of edges (s, j), j A, by p j x allocated(j) to obtan the parametrc network N (,, ). By Invarant 2, at x = 1, (s, A B t) s a mn-cut n N (,, ). Lemma 15. The smallest value of x at whch a new mncut appears n N (,, ) s gven by x = mn S A best(s) + (S), (S) and the unque maxmal set mnmzng the above expresson s S. Lemma 16. The followng hold: If x x, then (s, A B t) s a mn-cut n N (,, ). If x > x, then for any mn-cut (s A 1 B 1, A 2 B 2 t) n N (,, ), S A 1. For B, denote the sum of capactes of edges ncdent at n N(,, ) by c(). Defne m () = mn{m(), c()}, and to be the vector consstng of m (), B. Observe that replacng by n N(,, ) does not change the mn-cut or ts capacty. Defne N (,, ) to be the network obtaned by replacng by n N (,, ). The reason for workng wth s that the cut (s A B 1, B 2 t) has the same capacty as the cut (s A B, t). Ths property wll be used crtcally n the next lemma. Lemma 17. Set x = ( (B) + (A))/ (A) and fnd the mnmal mn-cut n N (,, ) (.e., the unque mn-cut mnmzng the s sde). Let t be (s A 1 B 1, A 2 B 2 t). If A 1 = B 1 = then x = x and S = A. Otherwse, x > x and A 1 s a proper subset of A. Lemma 18. x and S can be found usng n max-flow computatons. Next, we show n case of Event 1 how to determne whether Event 1(a) or Event 1(b) occurs, and n the former case, how to compute y. Compute prces of all goods for the current value of x, and let them be denoted by. Let denote all

8 forced allocatons made so far. Compute the money returned to buyers; for, assume that segment s s fully returned. Let denote the vector of money spent. Construct network N(,, ) and fnd a maxmal mn-cut n t. If (s, A B t) s the only mn-cut n t, then Event 1(b) occurs,.e., the entre money correspondng to segment s can be returned to wthout a set gong tght. Next assume that the maxmal mn-cut n the network s (s A 1 B 1, A 2 B 2 t), wth A 1. If so, Event 1(a) occurs. Clearly, the procedure stated above uses one max-flow computaton. Lemma 19. If Event 1(b) occurs, y = value(s) ( (A 1) (A 1) ( (B 1)+c(A 1; Γ(A 1) B 1)). 4.5 Termnaton wth market clearng prces Observe that despte the return polcy, the algorthm monotoncally keeps rasng prces of goods, and ths provdes us wth a natural measure of progress the dfference between total money possessed by buyers (after takng nto consderaton money returned) and the sum of the prces of all goods. When ths dfference becomes zero, all goods must be frozen, and the algorthm termnates. If Invarants 1 and 2 hold, termnatng prces are market clearng. Let M denote the total amount of money possessed by the buyers, U denote the largest rate of a segment, and Z denote the total number of segments n all specfed utlty functons. Let = nu n. Let us partton the runnng of the algorthm nto phases each phase ends when a new set goes tght,.e., Event 1(a) or Event 2 occurs. Partton each phase nto teratons each teraton ends when Event 3 or Event 4 happens. Observe that f Event 1(b) occurs whle returnng money correspondng to segment s, then ths segment wll never be consdered agan. Hence the number of occurrences of Event 1(b) s bounded by the number of segments n functons f0, for all, whch n turn s bounded by Z. Each teraton requres computaton of x, Lemma 17, whch requres n max-flow computatons. The total number of max-flow computatons executed by subroutne freeze n a phase s bounded by the total number of forced allocatons and s Z. Hence we get: Theorem 20. Algorthm 1 termnates wth market clearng prces and executes at most O(M 2 (Z + n 2 )) max-flow computatons. 4.6 Establshng polynomal runnng tme We next present Algorthm 2 whch s a polynomal tme mplementaton of Algorthm 1. The only dfference between the two algorthms s the way a actve set s defned at the start of a phase and the manner n whch t s updated n each teraton. Algorthm 2 ncludes n the actve subgraph only those buyers that have a large surplus so that substantal progress can be guaranteed n a phase. We wll need the noton of a balanced flow from [8] n order to specfy the actve subgraph precsely. Gven flow f n the network N(,, ) let R(,,, f) denote the resdual graph w.r.t. f. Defne the surplus of buyer, γ (, f), to be the resdual capacty of the edge (, t) wth respect to f,.e., m mnus the flow sent through the edge (, t). The surplus vector s defned to be (, f) := (γ 1(, f), γ 2(, f),..., γ n(, f)). Let v denote the l 2 norm of vector v. A balanced flow n network N(,, ) s a maxmum flow that mnmzes (, f). Property 1 If there s a path from node B to node j B n R(,,, f) then (, f) j(, f). Property 1 helps characterze balanced flows as follows; the proof s dentcal to that n Theorem 17 n [7]: A maxmum flow f n network N(,, ) s balanced ff t satsfes Property 1. Moreover, any two balanced flows have the same surplus vector. We let ths surplus vector be ( ). Lemma 21. A balanced flow can be computed n network N(,, ) usng n max-flow computatons. Proof. Let S = ( ) denote the total surplus, and let s av = S/n denote the average surplus. Subtract s av money from all buyers to obtan network N from network N(,, ). Compute a maxmal mn-cut n N, say (s A 1 B 1, A 2 B 2 t). In case A 1 = A, the surplus vector s unque. Otherwse, for each edge (, j), Γ(A 1) B 2, j A 1, make the correspondng forced allocaton, and then partton the equalty subgraph nto two: (A 1, B 1) and (A 2, B 2). Compute balanced flows n each separately. (In case one of these graphs s empty, we need to compute a balanced flow n only one graph; however, below we wll deal wth the general case.) We wll show that the surplus of buyers n B 1 s s av and the surplus of buyers n B 2 s > s av. We wll prove the second fact only; the proof of the frst fact s analogous. The proof s by contradcton. Let C B 2 be the set of buyers havng smallest surplus and let D A be Γ(C). By Property 1, there are no resdual paths from B 2 C to C. Hence all flow from D A 2 goes through C; say ths flow s α. Furthermore, observe that edges gong from D A 1 to C must form part of the forced allocaton made above and hence are fully saturated. Let ths flow be β. Therefore, the flow gong from C to t s α + β. Now the capacty of edges from C to t n the prevous stuaton,.e., on subtractng s av money from each buyer, s < α + β. Therefore, n ths stuaton, movng C and D to B 1 and A 1, respectvely, wll result n a smaller cut, contradctng the mnmalty of cut found. Ths proves that each buyer n B 2 has surplus > s av. Now consder the unon of the two balanced flows. We wll show that t s a balanced flow n N(,, ). For ths t suffces to show that there are no resdual paths from B 1 to B 2 n the resdual graph of the unon flow, snce there are no resdual paths from low surplus to hgh surplus buyers wthn B 1 or wthn B 2 by Property 1. Two observatons prove ths: All edges from A 1 to B 2 are nvolved n forced allocatons made above and so are fully saturated and therefore cannot provde a path from B 1 to B 2. Edges from A 2 to B 1 carry no flow, and hence are all drected from A 2 to B 1 n the resdual graph. Therefore, these edges also cannot provde a path from B 1 to B 2. The argument gven above naturally suggests a dvde and conquer method for computng a balanced flow n N(,, ): subtract s av money from each buyer and compute mnmal mn-cut to determne the two parttons. Make forced allocatons and then recursvely compute balanced flows n each partton. We next defne subroutne freeze2 whch plays a role analogous to that of subroutne freeze. Subroutne freeze2 uses

9 the dvde and conquer procedure outlned n Lemma 21 to compute a balanced flow n network N(,, ); t makes the requred forced allocatons n the process. As a result of forced allocatons, there may be buyers that do not have equalty edges ncdent at them. If each buyer has equalty subgraph edges ncdent at t, subroutne freeze2 halts. Otherwse, for each buyer not havng such edges, t computes the current partton of ths buyer and adds edges correspondng to t. freeze2 goes back to recomputng a balanced flow n the resultng network. The four events executed by Algorthm 1 are also executed by Algorthm 2, and the condtons that trgger them are also dentcal. The only dfference s n the defnton of actve subgraph at the begnnng of a phase and the manner n whch t s modfed wth each event. We gve these detals below. At the begnnng of a phase, Algorthm 2 computes a balanced flow n network N(,, ). Let δ be the maxmum surplus of a buyer n ths flow and let B 2 B be the set of buyers havng ths surplus. Let A 2 A be the set of goods that are adjacent to these buyers n the equalty subgraph. Then, the startng actve subgraph s (A 2, B 2). The algorthm starts rasng prces of goods n A 2 as n Algorthm 1. We next gve the actons to be taken n each event. Event 1(a): The current phase comes to an end. Event 1(b): The money correspondng to segment s s fully returned. Event 2: The current phase comes to an end. Event 3: Add edge (, j) (drected from j to and havng capacty s)) to the equalty subgraph and call subroutne freeze2. The new actve subgraph conssts of all buyers and goods that have a resdual path n R(,,, f) {s, t} to the current actve subgraph (and contans the current actve subgraph). Event 4: Same as Event 3, except that edge (, j) s drected from to j (snce s) amount of flow has already been sent from j to ). Let s analyze ts runnng tme. Event 3 can happen at most n tmes, because each tme a new good enters the equalty subgraph. Event 4 can happen at most Z tmes, snce any segment can be deallocated at most once n a phase. The total number of executons of subroutne freeze2 n a phase s O(Z + n); each executon requrng n max-flow computatons. The followng lemma follows almost mmedately from [8]. Lemma 22. ([8]) If 0 s the prce vector at the begnnng of the algorthm and the prce vector after t phases, then t = O n 2 log (p 0 ) (p ). Hence the total number of phases s bounded by O n 2 log ( 2 M) = O n 2 (log n + n log U + log M). Hence we get Theorem 23. Algorthm 2 fnds market clearng prces for the spendng constrant model usng O n 3 (n + Z)(log n + n log U + log M) max-flow computatons. 5. EXTENSIONS TO CONTINUOUS FUNCTIONS 5.1 The Fsher settng We defne an approxmate market equlbrum by relaxng the Market Clearng condton. Defnton 24 (Approxmate Market Equlbrum). A prce vector p s an ɛ-approxmate market equlbrum f nether defcency nor surplus of goods s too hgh n value: j A ξ j(p) p j ɛ j A p j. We now gve an algorthm for the Fsher settng when the rate functons are nce. Assume that the algorthm s gven oracle access to the fj s. The algorthm s smple: approxmate the gven functons wth step functons where all the segments are of length ɛ. More precsely, Fj (x) := fj ( x ɛ) ɛ s the requred approxmaton. Now run Algorthm 2 wth Fj s as nput, and return the prce vector thus obtaned, say p. Let Mj be the money that buyer spends on good j when he buys the optmal bundle at prces p (w.r.t the functons fj s), and M j be the money that spends on j accordng to Algorthm 2. (Note that the Fj s are step functons, so there need not be a unque optmal bundle. Hence we consder the allocaton gven by Algorthm 2.) We show that M j s n fact a good approxmaton to Mj as n the followng lemma: Lemma 25. Let M j and M j be as defned above. Let n = A. Then, B, j A, M j nɛ M j M j + ɛ. (3) Proof. Let n j := Mj /ɛ,.e., n jɛ Mj < (n j + 1)ɛ. If we show that n jɛ M j, then we get that Mj M j + ɛ. We may assume that 0 < n j, snce otherwse the nequalty trvally follows. Note that we chose our approxmaton Fj of fj such that Fj (x) fj (x) for all x. Therefore max j A Fj (M j)/p j s α = fj (Mj )/p j snce 0 < n j. Therefore Fj (M j) fj (Mj) fj (n jɛ) = Fj (n jɛ), by the defnton of Fj. Snce Fj s non-ncreasng, we get that n jɛ M j. Recall that j A M j = j A M j = e(). Therefore, for any j, M j Mj = j A,j j (M j M j ) nɛ. Note that B M j = ξ j(p) (by defnton) and B M j = p j (snce p s market clearng for the F j s). Now summng (3) over all B, we get that j A, p j n nɛ ξ j(p) p j + n ɛ, where n = B. Therefore, ξ j(p) p j n nɛ. (4) Summng over all j A, we get that p s ndeed an n n 2 ɛ- approxmate market equlbrum, snce the prces are all at least 1. We have actually proved a stronger verson of approxmaton,.e., ξ s component-wse close to p, and that the error s absolute (addtve). The defnton only needed that the respectve sums be close, and the error be relatve (multplcatve). In fact, more s true: that the allocaton returned by the algorthm (.e., spends M j on j,) s almost optmal w.r.t the f j s. We leave the detals of ths to the full verson [6].

10 5.2 The Arrow-Debreu Settng We frst extend Algorthm 2 to the Arrow-Debreu settng wth step functons. Note that the man dfference n the Arrow-Debreu settng s that the ncome of a buyer, e(), s not constant. It s a functon of p. Ths dffculty s overcome n Algorthm 3, whch s as follows: Start wth prces p j = 1 j, and compute all the e() s. Now run Algorthm 2 wth these e() s fxed, untl we fnd a p such that γ(p) s 3 at most nɛ. Let one such run be called an epoch. At the end of an epoch, recompute the e() s and repeat, unless p s an ɛ-approxmate equlbrum. We show that the number of epochs needed so that one s guaranteed an ɛ-approxmate equlbrum s at most 1/ɛ 2. Let P denote the sum of the prces of all the goods, and M denote the sum of the ncomes of all the buyers at any pont of tme n the algorthm. Let f (by abuse of notaton) be allocated(a)+ the value of a max-flow n the network N(,, ). Lemma 26. A prce p s 2ɛ-approxmate market clearng f w.r.t. p, P f P ɛ. Lemma 27. The value of P f at the end of each subsequent epoch does not ncrease. Proof. Note that for each good j, the value of P f contrbuted by j s the resdual capacty of the edge (s, j),.e., p j f j, where f j s allocated(j)+ the flow n the edge (s, j). We wll argue that at the end of each epoch, p j f j never ncreases. Suppose that p j ncreased n that epoch. Snce Algorthm 2 only ncreases the prces of those goods j whose (s, j) edge s saturated, p j f j s zero. So we may assume that p j remaned the same n that epoch. Snce no prces are decreased, and the ncomes have ncreased, all the segments of j that were (partally or fully) flled n the earler epoch wll stll be flled to at least the same extent n the new epoch as well. Hence p j f j does not ncrease. Lemma 28. If at the end of an epoch, ether P M nɛ or P n, then p s a 4ɛ-approxmate equlbrum. ɛ Proof. Note that (p) s actually equal to M f. Hence, at the end of each epoch, M f nɛ. Snce, we start wth prces all 1, P f n to begn wth. Snce from Lemma 27, P f does not ncrease at the end of each subsequent epoch, P f n at the end of each epoch. Now suppose that P M nɛ, then P f = (P M) + (M f) 2nɛ 2P ɛ. On the other hand, f P n, then ɛ n P ɛ. So P f n P ɛ. The result now follows from Lemma 26. If n each epoch P M > nɛ, then after 1 epochs P ɛ n 2, whch means that we are guaranteed an ɛ-approxmate ɛ equlbrum wthn O( 1 ) epochs. ɛ We now show that the 2 number of phases n each epoch s polynomal. Snce at the begnnng of each epoch (p) P n/ɛ and the epoch ends f (p) nɛ, Lemma 22 says that there are O(n 2 log n ) phases n each epoch. Moreover, each phase needs O(n(n + Z)) max-flow computatons. ɛ Hence we get the followng theorem: 3 v denotes the l 1-norm of the vector v. Theorem 29. Algorthm 3 fnds an ɛ-approxmate equlbrum usng O n 3 (n+z) ɛ 2 log n ɛ max-flow computatons. We can extend ths algorthm to the contnuous case as n the Fsher settng, the only dfference beng that we use Algorthm 3, whch only gves an ɛ approxmate equlbrum. It can be shown that the composton of the two algorthms s stll ɛ approxmate. For detals, see [6]. 6. REFERENCES [1] Alprants, Brown, and Burknshaw. Exstence and Optmalty of Compettve Equlbra, Sprnger-Verlag, [2] K. K. Arrow, and G. Debreu, Exstence of an Equlbrum for a Compettve Economy, Econometrca, Vol. 22, pp , [3] G. Debreu. Economes wth a fnte set of equlbra. Econometrca, 38:387 92, [4] N. R. Devanur, V.V. Vazran. Improved Approxmaton Scheme for Computng Arrow Debreu Prces n The Lnear Case. In Proc. FSTTCS, [5] N. R. Devanur, V.V. Vazran. Algorthmc, Exstence and Unqueness Results for the Spendng Constrant Model. Manuscrpt 4, [6] N.R. Devanur, C.H. Papadmtrou, A. Saber, V.V. Vazran. Market Equlbrum va a Prmal-Dual-Type Algorthm. In Proc. FOCS, [7] N.R. Devanur, C.H. Papadmtrou, A. Saber, V.V. Vazran. Market Equlbrum va a Prmal-Dual-Type Algorthm. Full verson. [8] X. Deng, C. H. Papadmtrou, and S. Safra, On the Complexty of Equlbra. In Proc. STOC, [9] E. Esenberg and D. Gale. Consensus of subjectve probabltes: The par-mutuel method. Annals Of Mathematcal Statstcs, 30: , [10] R. Garg and S. Kapoor Aucton Algorthms for Market Equlbrum. In Proc. STOC, [11] K. Jan. A polynomal tme algorthm for computng the Arrow-Debreu market equlbrum for lnear utltes. Manuscrpt, [12] K. Jan, M. Mahdan, and A. Saber. Approxmatng Market Equlbrum. In Proc. APPROX, [13] C. Papadmtrou. Algorthms, games, and the nternet. In Proceedngs of the 33rd Annual ACM Symposum on the Theory of Computng, pages , [14] H. Scarf. The Computaton of Economc Equlbra (wth collaboraton of T. Hansen), Cowles Foundaton Monograph No [15] V.V. Vazran. Market Equlbrum When Buyers Have Spendng Constrants. Submtted, [16] L. Walras. Elements d econome poltque pure; ou, Theore de la rchesse socale (Elements of Pure Economcs; Or the Theory of Socal Wealth). Lausanne, Pars, 1874 (1954, Engl. transl.). 4 The papers [6], [8] and [16] are avalable at and