Settling the Complexity of Arrow-Debreu Equilibria in Markets with Additively Separable Utilities

Settling the Complexit y of Arro w-Debreu Equilibria in Mark ets with Ad ditiv ely Separable Uti lities Xi Chen ∗ Dec heng Da i † Y e Du ‡ Shang-Hua T eng § Abstract W e prov e that the problem of computing a n Arrow-Debreu market equilibrium is PP AD-complete even when all traders use additively separ able, piecewise- line a r a nd conc ave utilit y functions. In fa ct, our pro of shows that this ma rket-equilibrium problem do e s not hav e a fully p olyno mial-time appro - ximation scheme unless e very problem in PP AD is solv able in p olynomia l time. ∗ Department o f Computer Science, Princeton Universit y . † Department of Computer Science, Tsinghua Universit y . W ork done while visiting Princeton U n iversi ty . This wo rk was supp orted in part by the National Natural S cience F oundation of China Grant 60553001 , and the National Basic Research Program of China Gran t 2007C B80790 0, 2007CB 807901 . ‡ Department o f Electrical Engineering and Computer Science, Universit y of Mic higan. § Microsof t Research New England. Affiliatio n starting from F all 2009: D epartment of Computer Science, U niversit y of Southern California. 1 In tro duction One of the central d ev elopments in mathematical economics is the general equilibrium theory , wh ic h pro vides th e foun dation for comp etitiv e pricing [1, 35]. When sp ecialized to exc hange economies, it con- siders an exchange market in whic h there are m traders and n d ivisib le goo d s, where trader i has an initial endowment of w i,j ≥ 0 of go o d j and a utility function u i : R n + → R . The individual go al of trad er i is to obtain a new b undle of go o ds that maximizes her utilit y . This new bundle can b e sp ecified by a column v ector x i ∈ R n + , where the j th en try x i,j is the amount of go o d j that trader i is able to obtain after the exchange. Naturally , the exc h ange should satisfy P i x i,j ≤ P i w i,j , f or all go o d j . The pioneering equilibriu m th eorem of Arrow and Debreu [1] s tates that if all the utilit y functions u 1 , ..., u m are quasi-conca v e, then under some m ild cond itions, the market h as an e qu ilibrium pric e p = ( p 1 , ..., p m ): A t this price, in dep end en tly , eac h trader can sell her endo wmen t vir tually to th e market to obtain a budget and then buys a bund le of go o ds with this budget from the marke t — which con tains the union of all go o ds — that maximizes h er u tilit y . The equilibriu m condition guaran tees that the supply equals the d emand and h ence the marke t clears: Ev ery go o d is sold and ev ery trader’s budget is completely sp en t. In the case when the utilit y fun ctions are strictly conca v e, there is a uniqu e optimal bund le of go o ds for eac h tr ader at an y given price p . Nev ertheless, th e th eorem extends to quasi-conca ve utilit y functions su ch as linear or piecewise linear utility functions [29, 21], ev en though they are n ot strictly qu asi-conca v e, and there could b e m u ltiple optimal bun dles of go o d s for eac h trader at a giv en price. The existence pro of of Arrow and Debreu [1], based on Kakutani’s fixed p oin t th eorem [28], is non- constructiv e in th e view of p olynomial-time computabilit y . Despite the p rogress b oth on algorithms for and on the complexity- theoretic und erstanding of market equilibria, sev eral f undamental questions con- cerning mark et equilibria, including some seemingly simp le ones, remain un settled. Vija y V azirani [31] wrote: “Conc ave u tility functions, even if they ar e additively sep ar able over the go o ds, ar e not e asy to de al with algorithmic al ly. In fact, obtaining a p olynomial time algorithm for such functions is a pr emier op en question to day.” A fu nction u ( x 1 , ..., x n ) from R n + to R is an additively sep ar able and c onc ave function if there exist real- v alued conca v e fu nctions f 1 , ..., f n suc h that u ( x 1 , ..., x n ) = n X j =1 f j ( x j ) . Noting th at ev ery conca v e f u nction can b e appro ximated b y a piecewise linear and conca ve (PLC) func- tion, V azirani [31] further ask ed wh ether o ne can compute a mark et equilibr ium with additiv ely separable PLC utilities in p olynomial time; or wh ether the pr oblem is PP AD-hard. This op en question has b een ec ho ed in sev eral w ork sin ce then [13, 24, 20, 37]. 1 1.1 Our Con tribution In th is pap er, we settle the complexit y of fin ding an Ar r o w-Debreu equ ilibr ium in an exc hange market with additiv ely separable PLC utilities. W e sh o w that this equilibriu m p roblem is PP AD-complete. F or an intege r t > 0, a r eal-v alued fun ction f ( · ) is t -segmen t piecewise linear o ve r R + = [0 , + ∞ ) if f is contin uous and R + can b e divided in to t su b-int erv als such that f is a linear function ov er ev ery sub-inte rv al. If eac h trader’s utilit y is an add itiv ely separable t -segment PLC fu nction, then w e r efer to the mark et as a t -line ar market . Clearly , a mark et with linear utilities is a 1-linear mark et. In con trast to the f act that an Arro w-Debreu m ark et equ ilibr ium of a 1-linear mark et can b e found in p olynomial time [18, 30, 12, 14, 25], we sho w that ev en computing an Arro w -Debreu equilibr ium in a 2-linear market is PP AD-complete, via a r eduction fr om Sp arse Bima trix [6]: the problem of find ing an approximate Nash equilibrium in a sparse t wo-pla y er game (see Section 2.1 f or the definition). Our construction of the PP AD-complete markets h as sev eral nice tec hn ical elements. First we intro- duce a sequen ce of sim p le line ar mark ets {M n } with n go o d s, whic h we refer to as the pric e-r e gulating mark ets. M n has the follo wing nice pric e-r e gulation pr op erty : If p is a normalize d 1 appro ximate equi- librium pr ice vec tor of M n , then p k ∈ [1 , 2] for all k ∈ [ n ]. Th is price-regulation prop ert y allo ws u s to enco de n free v ariables x 1 , ..., x n b et w een 0 and 1 u sing the n en tr ies of p b y s etting x k = p k − 1. As a k ey step in our analysis, we sho w that the p rice-regulation p r op erty is stable with resp ect to “ smal l p erturb ations ” to M n : When new tr ad er s are added to M n (without in tro ducing new go o ds), this prop erty r emains h old as long as the amoun t of go o ds these traders carry with them is small compared to those of the traders in M n . W e then show ho w to set the initial endowmen ts and utilit y functions of new traders so that we can con trol the flo w s of go o d s in the mark et and set new r equ iremen ts that ev ery appro ximate equilibrium price ve ctor p has to satisfy . Using the stabilit y of the price-regulating m ark et, w e giv e a reduction fr om a t w o-pla y er game to an exc hange mark et M : Given an n × n tw o-pla yer game ( A , B ), w e construct an add itively separable P LC mark et by addin g n ew traders — whose initial endo w men ts are relativ ely sm all — to M 2 n +2 , the p rice- regulating m arket with 2 n + 2 goo d s. W e u se the first 2 n entries of p to en co de a pair of p robabilit y v ectors ( x , y ): x k = p k − 1 and y k = p n + k − 1, k ∈ [ n ]. W e then devel op a no vel wa y to enforce the Nash equilibrium constrain ts o ver A , B , x and y b y carefully sp ecifying the b eha viors of th e new tr ad er s . In doing so, we constr u ct a marke t M with the pr op ert y th at from ev ery app ro xim ate mark et equilibrium p of M , the pair ( x , y ) obtained ab ov e (after normalization) is an appro ximate Nash equilibriu m of ( A , B ). Moreo v er, if ( A , B ) is a sparse t w o-pla y er game, then the relation of whic h trader s are inte rested in w hic h go o ds in M is also sparse (see Section 2.3 for details). In the construction of M , the price-regulation prop ert y plays a critical role. I t enab les u s to design the utilit y functions of new traders so that we kno w exactly their preferences ov er the go o d s with resp ect to an y approximat e equ ilibrium price p , ev en though w e h a ve no idea in adv ance ab out the entries of p when constructing M . W e ant icipate that our reduction tec hniqu es will help to resolv e more complexity-t heoretic questions concerning other families of exc hange mark ets suc h as the general CES mark ets an d the hybrid linear- Leon tief mark ets [7]. 1 W e say a p rice vector p is n ormalized if th e smallest nonzero entry of p is equal to 1. 2 1.2 Related W ork The computation of a market equilibrium p rice in an exc hange market has b een a chall enging problem in m athematical economics [34, 31]. The matter is more complex b ecause s ome market s only h a ve irr a- tional equilibria, making th e computation of exact equilibria with a fin ite-precision algorithm imp ossible. One alternativ e appr oac h to handle irr ationalit y is to express equilibria in some simple algebraic form. Ho wev er, it turns out that fi nding an exact m ark et equilibrium in general is not computable [33]. T o circumv en t the irr ationalit y , one usu ally uses some notion of appr o ximate market equilibria. Th ere are v arious notions of approximat e equ ilibria: Some require that the approxima tion solution is within a small geometric distance fr om an exact equilibrium , while others only r equire that the s upply-demand condition and/or the individual optimalit y cond ition are appr o ximately satisfied. In this pap er, f ollo wing Scarf [34], w e consider the latter n otion of approxima te mark et equilibria. 1.2.1 Algorithms for Mark et Equilibria Scarf pioneered th e algorithmic d ev elopment for compu ting general comp etitiv e equilibria [34]. His ap- proac h com bined n umerical app ro x im ation with com binatorial insights used in Sp erner’s lemma [36] for fixed p oint s and in Lemk e and Howson’s algorithm for t wo-pla y er games. Although his algorithm may not alw ays run in p olynomial time, Scarf ’s work has pr ofoun d imp act to computational economics. Building on the su ccess of con v ex pr ogramming [18], p olynomial-time algorithms hav e b een d ev elop ed for sp ecial markets whose sets of equilibria enjoy some degree of conv exit y . F or Arrow-De breu mark ets with lin ear utilit y functions, Nenak o v and Primak ga ve a p olynomial-time algorithm [30], and there are no w s ev eral p olynomial-time algorithms f or compu ting or app r o ximating market equilibria with linear utilit y functions [12, 14, 25, 19, 26, 15, 39]. Other p olynomial-time algorithms for sp ecial marke ts include Ea ves’s algorithm f or Cobb-Douglas markets [17] and Dev an ur and V azirani’s algorithm for mark ets with sp end ing constrain t utilities [16] (also s ee [37 ]). The conv ex programming based approac h for appr o x- imating equilibria has b een extended to all mark ets whose utilities satisfy wea k gross sub stitutabilit y (W GS) by C o denotti, Pemmara j u , and V aradara jan [10]. In [9], Co d enotti, McCun e, and V aradara jan sho wed that for m arkets that satisfy WGS, th ere is a p rice-adjustment mec hanism called t ˆ a onnement that con verge s to an appro x im ate equilibr ium efficien tly . A closely related market mo del is Fisher’s mo del [2]. In th is mo del, th ere are t w o differen t t yp es of traders in the mark et: pr o duc ers and c onsumers . Eac h consumer comes to the mark et w ith a budget and a utilit y function. Each p ro du cer comes to the mark et with an endowmen t of go o ds, and will sell them to the consum ers for money . A market equilibrium is then a pr ice ve ctor p for goo ds so that if eac h consumer sp ends all her budget to maximize h er utilit y , then th e mark et clears. An (appr o ximate) mark et equilibrium in a Fisher’s marke t with C ES 2 utilit y functions [18, 39, 38, 14, 27] or w ith piecewise 2 CES ( standing for constant elasticit y of substitution) is a p opular family of utility functions. Let s > 0 b e a parameter called the elasticity of substitution , t h en a CES function with elasticit y of substitution s has the follo wing form: u ( x 1 , ..., x m ) = m X j =1 d j x r j ! 1 /r , where r = s − 1 s . 3 linear utilit y fun ctions [38] can b e found in p olynomial time. In [3], Chen, Deng, S un, and Y ao ga ve an algorithm for market s with logarithmic utilit y fu nctions. I ts r unnin g time is p olynomial wh en either the n um b er of sellers or the num b er of buye rs is b ound ed b y a constan t. Ho wev er, p rogress on Arro w-Debreu mark ets wh ose sets of equilibr ia do not enjoy con vexit y has b een relativ ely slow. Th ere are only a few algorithms in this category . Dev an ur and Kannan [13] ga ve a p oly- nomial-time algorithm for exc hange mark ets with PLC u tilit y fun ctions and a constan t num b er of go o ds . Co denotti, McCun e, Pen umatc h a, and V aradara jan ga v e a p olynomial-time algorithm for CES mark ets when the elasticit y of substitution s ≥ 1 / 2 [8]. 1.2.2 The Complexit y of Equilibrium Problems P apadim itriou initiated the complexit y-theoretic s tudy of fixed-p oint computations [32]. He in tro duced the complexit y class PP AD, and pro v ed that the problem of fi nding a Nash equilibrium in a tw o-pla yer game, the computational ve rsion of Sp ern er’s Lemma, and the problem of computing an approximat e fixed p oint are mem b ers of PP AD. Recen tly , there was a series of deve lopmen ts that c haracterized the computational complexit y of se- v eral equilibrium pr ob lems in game theory . Dask alakis, Goldb erg, and P apad im itriou [22] pr o ved that the p roblem of compu tin g an exp onential ly-precise Nash equilibrium of a f our-pla yer game is PP AD- complete. Chen and Deng [4] th en prov ed that the pr oblem of computing a t w o-play er Nash equilibrium is also PP AD-complete. Ch en and Deng’s resu lt, together with an earlier redu ction of [11], implies that computing a mark et equilibrium in an Arro w-Debreu m arket with L eon tief utilities 3 is PP AD-hard . On the approximat ion fr ont, Ch en, Deng and T eng [5] pro ved that it is PP AD-complete to fi nd a p olyno- mially-precise app ro ximate Nash equilibr ium in t wo- pla yer or multi-pla y er games. Huang and T eng [24] then extended th is approxi mation resu lt to Leon tief market equilibria. Their appr oximati on result also implies that th e mark et equilibrium p roblem with CES utilit y fun ctions is PP AD-hard, if the elasticit y of substitution s is sufficient ly small. 2 Preliminaries 2.1 Complexit y of Nash Equilibria in Sparse Two-Pla y er Games A t wo-pla y er game is d efined by the pa yoff matrices ( A , B ) of its t w o play ers. In this pap er, we assume that b oth play ers h a ve n c hoices of actions and hence b oth A and B are square matrices with n ro w s and columns. W e use ∆ n ⊂ R n to denote the set of probabilit y distributions of n dim en sions. A pair of pr obabilit y vec tors ( x , y ) (i.e., x ∈ ∆ n and y ∈ ∆ n ) is a Nash equilibrium of ( A , B ) if for all i and j in [ n ] = { 1 , 2 , ..., n } , A i y T < A j y T = ⇒ x i = 0 and xB i < xB j = ⇒ y i = 0 , where we use A i and B i to denote the i th row v ector of A and th e i th column vec tor of B , resp ectiv ely . W e will use the follo w ing notion of appro xim ate Nash equilibria. 3 Leonti ef functions are sp ecial cases of CES functions with s approaching 0. A Leonti ef funct ion h as th e follo wing form: u ( x 1 , ..., x m ) = min j ∈ S x j /d j , where S ⊆ [ m ] is a subset of go od s and d j > 0 for all j ∈ S . 4 Definition 1 (W ell-Supp orted Nash Equilibria) . F or ǫ > 0 , ( x , y ) is an ǫ - wel l-supp orte d Nash e quilibrium of ( A , B ) , if x , y ∈ ∆ n and for al l i, j ∈ [ n ] , A i y T + ǫ < A j y T = ⇒ x i = 0 , and (1) xB i + ǫ < xB j = ⇒ y i = 0 . (2) Definition 2 (Sparse Normalized T w o-Pla y er Games) . A two-player game ( A , B ) is normalized if every entry of A and B is b etwe en − 1 and 1 . We say a two-player g ame ( A , B ) i s sparse if every r ow and every c olumn of A and B have at most 10 nonzer o entries. Let S p arse Bima trix d en ote the problem of find ing an n − 6 -w ell-supp orted Nash equilibrium in an n × n sparse normalized tw o-pla yer game, then we h a ve Theorem 1 (Ch en-Deng-T eng [6]) . Sp a rse Bima trix is PP AD -c omplete. 2.2 Mark ets with Additiv ely Separable PLC Ut ilities Let G = { G 1 , ..., G n } denote a set of n divisible go o d s, and T = { T 1 , ..., T m } denote a set of m traders. F or eac h trader T i ∈ T , w e use w i ∈ R n + to denote her initial endowmen t, u i : R n + → R + to denote her utilit y function, and x i ∈ R N + to denote her allo cation vect or. In this pap er, w e will fo cus on m ark ets with additiv ely separable piecewise linear and conca ve utilities. Definition 3. A f unction r ( · ) : R + → R + is said to b e t -segment piecewise linear and conca v e (PLC) if 1. r (0) = 0 and r ( · ) i s c ontinuous over R + ; 2. ther e exists a tuple of length 2 t + 1 ,  θ 0 > θ 1 > ... > θ t ; a 1 < a 2 < ... < a t  ∈ R 2 t +1 + , such that (a) for any i ∈ [0 : t − 1] , the r estriction of f over [ a i , a i +1 ] ( a 0 = 0) is a se gment of slop e θ i ; (b) the r estriction of f over [ a t , + ∞ ) is a r ay of slop e θ t . The 2 t + 1 -tuple [ θ 0 , θ 1 , ..., θ t ; a 1 , a 2 , ..., a t ] is also c al le d the represent ation of r ( · ) . Mor e over, we say r ( · ) is s tr ictly monotone if θ t > 0 , and is α - b ounded for some α ≥ 1 if α ≥ θ 0 > θ 1 > ... > θ t ≥ 1 . Definition 4. A utility function u ( · ) : R n + → R + is said to b e an additiv ely separable PLC function if ther e exist a set of n PLC functions r 1 ( · ) , ..., r n ( · ) : R + → R + such that u ( x ) = X j ∈ [ n ] r j ( x j ) , for al l x ∈ R n + . (3) In su c h a m arket, we use, for eac h trader T i ∈ T , r i,j ( · ) : R + → R + to denote her PLC f u nction with resp ect to go o d G j ∈ G . In another wo rd, w e ha ve u i ( x ) = X j ∈ [ n ] r i,j ( x j ) , for all x ∈ R n + . 5 W e use p ∈ R n + to denote a price vec tor, wh ere p 6 = 0 and p j is the p r ice of G j , j ∈ [ n ]. W e alw ays assume that p is normalize d , that is, the smallest n onzero entry of p is equal to 1. Giv en p , w e use OPT ( i, p ) ⊂ R n + to denote the set of allocations that maximize the utilit y of T i : OPT ( i, p ) = argmax x ∈ R n + , x · p ≤ w i · p u i ( x ) . W e use X = { x i ∈ R n + : i ∈ [ m ] } to denote an allo cation of the mark et: F or eac h trader T i ∈ T , x i ∈ R n + is th e amoun t of go o ds that T i receiv es. I n particular, the amount of G j that T i receiv es in X is x i,j . Definition 5 (Arr o w-Debreu [1]) . A market e quilibrium i s a (normalized) pric e ve ctor p ∈ R n + such that ther e exists an al lo c atio n X which has the fol lowing pr op erties: 1. The market cle ars: F or eve ry go o d G j ∈ G , X i ∈ [ m ] x i,j ≤ X i ∈ [ m ] w i,j . (4) In p articular, if p j > 0 , then X i ∈ [ m ] x i,j = X i ∈ [ m ] w i,j . (5) 2. Every tr ader gets an optimal bu nd le: F or ev ery T i ∈ T , we have x i ∈ OPT ( i, p ) . In general, not ev ery marke t has such an equilibr ium price v ector. F or the add itiv ely separable PLC mark ets considered here, the follo wing condition guaran tees the existence of an equilibrium. Definition 6 (Econom y Graph s) . Given an additively sep ar able PL C market, we build a dir e cte d gr aph G = ( T , E ) as fol lows. The vertex set of G is exactly T , the set of tr aders in the market. F or e very two tr aders T i 6 = T j ∈ T , we have an e dge fr om T i to T j if ther e exists an inte ger k ∈ [ n ] such that w i,k > 0 and r j,k ( · ) is strictly monoto ne. In another wor d, T i p ossesses a go o d which T j wants. G is c al le d the econom y graph of the market [29, 8 ] . W e say the market is strongly connected if G is str ongly c onne cte d. The follo wing theorem is a corollary of Maxfield [29], and the pro of can b e found in App endix A. Theorem 2. L et M b e a market with additively sep ar able PLC utilities. If it is str ongly c onne cte d, then a market e quilibrium p exi sts. Mor e over, i f al l the p ar ameters of M ar e r ational nu mb ers, then it has a r ational market e quilibrium p . The numb er of bits we ne e d to describ e p is p olynomial in the input size of M ( that is, the numb er of bits we ne e d to describ e the market M ) . 2.3 Definition of t he Sparse Marke t Equilibrium Problem By Theorem 2, the f ollo wing searc h problem Market is w ell defined: The input of the problem is an additiv ely separable PLC mark et M that is b oth rational and strongly connected; and the output is a rational mark et equilibrium p of M . 6 In the rest of the section, w e define a muc h more r estricted v ers ion of Mark et : Sp arse Mar ket . Th e main result of the pap er is that Sp arse Mark et is PP AD-complete. First of all, the inpu t of Sp arse Marke t is an additivel y separable PLC m ark et whic h not only is strongly connected, but also satisfies the follo w in g three conditions: Definition 7 ( α -Bounded Mark ets) . We say an additively sep ar able PLC market M is α -b oun ded , for some α ≥ 1 , if for al l T i and G j , r i,j ( · ) is e ither the zer o f unction ( r i,j ( x ) = 0 for al l x ) or α -b ounde d. Definition 8 (2-Linear Mark ets) . W e c al l an additively sep ar able PL C market M a 2-linear market if for al l T i ∈ T and G j ∈ G , r i,j ( · ) has at most two se gments. Definition 9 ( t -Sp arse Mark ets) . We say an additively sep ar able PLC market M is t - sp arse for some inte ger t > 0 if 1) F or every T i ∈ T , we have | supp( w i ) | ≤ t ; and 2) F or every T i ∈ T , the nu mb er of j ∈ [ n ] such that r i,j ( · ) is not the zer o function is at most t . In another wor d, every tr ader owns at most t go o ds at the b e ginning and is inter este d in at most t go o ds. W e use the follo wing definition of appr oximate marke t equilibria: Definition 10 ( ǫ -Appr o ximate Mark et Eq u ilibrium) . Given an additively sep ar able PLC market M , we say p is an ǫ - appr oximate market e quilib rium of M , for some ǫ ≥ 0 , if ther e is an al lo c ation X = { x i ∈ R n + : i ∈ [ m ] } such that every tr ader gets an optimal bund le with r esp e ct to p : x i ∈ OPT ( i, p ) f or al l i ∈ [ m ] ; and the market cle ars appr oximately: F or ev ery G j ∈ G ,       X i ∈ [ m ] x i,j − X i ∈ [ m ] w i,j       ≤ ǫ · X i ∈ [ m ] w i,j . (6) W e r emark that there are v arious notions of appro ximate market equilibria. T he r eason we adopted the on e ab o ve is to sim p lify the analysis. The construction in Section 4 actually works for some other notions of approximate equilibria, e.g., the one that also allo ws the allo cation to b e ju s t appro ximately optimal for eac h trader. Finally , we let Sp arse Mar ket d enote the f ollo wing searc h problem: The input of the problem is a 2-linear mark et M that is strongly connected, 27-b ound ed, and 23-sparse; and the output is an n − 13 -appro x im ate market equilibrium of M , where n is the n um b er of go o ds in the marke t. It is tedious but not v er y hard to sho w that S p arse Market is a problem in PP AD 4 . 4 In [21], the auth or show ed how t o construct a contin uous map from any market with qu asi-conca ve utilities such that the set of fix ed p oints of the map is precisely the set of equilibria of th e market. Wh en th e market is add itively separable PLC, one can show that th e contin uous map is indeed Lipsc hitz contin u ou s. A s a result, one can reduce the problem of finding an approximate market equilibrium to the p roblem of finding an approximate fix ed p oint in a Lipschitz contin uous map. This implies a redu ction from S p arse Market to the d iscrete fixed p oin t problem stud ied in [23] (also see [5] for the high-dimensional versi on) which is in PP A D, and thus, t he former is also in PP AD. 7 One can in fact r eplace th e constan t 27 h ere by any constan t larger than 1 and our main result, Theorem 3, b elo w still holds. The constan t 23, ho wev er, is related to the constan t 10 in Definition 2. The main result of the pap er is the follo wing theorem: Theorem 3 (Main) . Sp arse Market is PP AD -c omplete. 3 A Price-Regulating Mark et W e now construct th e family of pr ice-regulating mark et {M n } . F or eac h p ositiv e inte ger n ≥ 2, M n has n go o ds and satisfies the follo wing strong price regulation pr op erty . Prop ert y 1 (Price Regulation) . A pric e v e ctor p is a normalize d n − 1 -appr oximate e qu ilibrium of M n if and only if 1 ≤ p k ≤ 2 , for al l k ∈ [ n ] . W e start with some notation. The go o ds in M n are G = { G 1 , ..., G n } , and the tr ad er s in M n are T = n T s : s ∈ S o , w here S = n s = ( i, j ) : 1 ≤ i 6 = j ≤ n o . F or eve ry tr ad er T s ∈ T , we u se w s ∈ R n + to denote h er initial end o wm ent, u s : R n + → R + to denote her utilit y fu nction, r s ,k ( · ) to denote her PL C fu nction w ith resp ect to G k , and OPT ( s , p ) to denote the set of bund les that maximize her utilit y with resp ect to p . Mark et M n is a linear market in wh ic h for all s ∈ S and k ∈ [ n ], r s ,k ( · ) is a ra y starting at (0 , 0). In the construction b elo w , we let r s ,k ( · ) ⇐ [ θ ] den ote the action of setting r s ,k ( · ) to b e the linear fu nction of slop e θ ≥ 0. Construction of M n : First, we set the initial endo wmen t vecto rs w s : F or ev ery s = ( i, j ) ∈ S , we set w s ,k = 1 /n if k = i ; and w s ,k = 0 otherwise. Second, w e set the PLC functions r s ,k ( · ): F or all s = ( i, j ) ∈ S and k ∈ [ n ], we set r s ,k ( · ) ⇐ [ θ ] and θ = 0 if k 6 = i, j ; θ = 1 if k = j ; and θ = 2 if k = i . It is easy to c heck th at M n constructed ab o ve is strongly connected, 2-b ounded, and 2-sparse. Pr o of of Pr op erty 1. The first direction is trivial. If 1 ≤ p k ≤ 2 for all k ∈ [ n ], then one can verify th at X =  x s = w s : s ∈ S  is a m ark et clearing allocation that provides an optimal bu n dle of go o ds for eac h trader at pr ice p . The second direction is less trivial. L et p b e a normalized (1 /n )-appro ximate market equilibrium of M n , and X b e an optimal allocation that clears the mark et. First, it is easy to c hec k that p k m u st b e p ositiv e for all k ∈ [ n ] since otherwise, w e h av e x s ,k = + ∞ for all s = ( i, j ) such that k = i or j , whic h con tradicts the assumption that p is an appro ximate equilibrium. Since p is normalized, w e ha v e p k ≥ 1 for all k ∈ [ n ]. No w assume f or con tradiction that Prop erty 1 is not tru e, then without loss of generalit y , we may assume that p 1 = max k p k > 2 and p 2 = min k p k = 1. 8 T o reac h a con tradiction, w e fo cus on the amount of G 1 eac h trader gets in the allocation X . First, if 1 / ∈ { i, j } where s = ( i, j ), then w e ha ve x s , 1 = 0; Second, if i = 1 and j = 2, then x s , 1 = 0 sin ce 2 p 1 < 1 p 2 and T s lik es G 2 b etter than G 1 with resp ect to the price ve ctor p ; Third, if j = 1, then x s , 1 = 0 sin ce 1 p 1 < 2 p i and T s lik es G i b etter than G 1 ; Finally , for all s = ( i, j ) suc h th at i = 1 and j 6 = 2, w e h a ve x s , 1 ≤ 1 /n since the budget of T s is exactly (1 /n ) · p 1 . As a result, we h a ve X s ∈ S x s , 1 ≤ n − 2 n , wh ile X s ∈ S w s , 1 = n − 1 n , whic h con tradicts the assumption that p is a (1 /n )-appro ximate equilibrium since     n − 2 n − n − 1 n     > 1 n · n − 1 n . The price-regulation prop ert y then follo ws. Let x k = p k − 1 for k ∈ [ n ], then M n pro vides u s a w a y to enco de n f r ee v ariables x 1 , ..., x n b et w een 0 and 1. In the next section, w e will use M 2 n +2 and the fi rst 2 n en tries of p : x k = p k − 1 and y k = p n + k − 1 , for k ∈ [ n ], to enco de a pair of distributions ( x , y ). Starting from an n × n sparse t w o-play er game ( A , B ), we will sho w ho w to add m ore trad er s to “p erturb” the p rice-regulating market M 2 n +2 so th at an y app ro xim ate equilibrium p of the new mark et yields an appro ximate Nash equilibrium ( x , y ) of ( A , B ). 4 Reduction from Sp arse Bima trix to Sp arse Market In th is section, w e give a p olynomial-time redu ction fr om S p arse Bima trix to Sp arse Market . Giv en an n × n sparse tw o-pla ye r game ( A , B ), w here A , B ∈ [ − 1 , 1] n × n , we construct an ad d itiv ely separable PLC mark et M by add ing more traders to the price-regulating mark et M 2 n +2 . There are 2 n + 2 go o d s G = { G 1 , ..., G 2 n , G 2 n +1 , G 2 n +2 } in M , and th e traders T in M are T = n T s , T u , T v , T i : s ∈ S, u ∈ U, v ∈ V , i ∈ [2 n ] o , where S =  s = ( i, j ) : 1 ≤ i 6 = j ≤ 2 n + 2  , U =  u = ( i, j, 1) : 1 ≤ i 6 = j ≤ n  and V =  v = ( i, j, 2) : 1 ≤ i 6 = j ≤ n  . 9 Note that |T | = O ( n 2 ). The traders T s , where s ∈ S , h a ve almost th e same initial endowmen ts w s and PLC functions r s ,k ( · ) as in M 2 n +2 ; w e will only s lightly mo dify these parameters to ease th e analysis in the next section. F or eac h agen t T ∈ T , we will set h er PL C f unction r ( · ) with resp ect to G k , k ∈ [2 n + 2], to one of the follo wing f unctions: 1. r ( · ) is th e zero function: r ( x ) = 0 for all x ≥ 0 (denoted by r ( · ) ⇐ [0]); or 2. r ( · ) is a ra y: r ( x ) = θ · x for all x ≥ 0 (denoted by r ( · ) ⇐ [ θ ]); or 3. r ( · ) is a 2-segmen t PLC f unction with representa tion [ θ 0 , θ 1 ; a 1 ] (d en oted b y r ( · ) ⇐ [ θ 0 , θ 1 ; a 1 ]). 4.1 Setting up the Mark et F or eac h trader T ∈ T , we set her initial endowmen t and PLC utilit y fun ctions as follo w ing: 4.1.1 T raders T s , where s ∈ S F or eac h trader T s ∈ T , where s = ( i, j ) ∈ S , w e set her initial end owmen ts w s and her PLC functions r s ,k ( · ) almost the same as hers in M 2 n +2 . The initial endowmen t w s is set as: w s ,k = 1 /n if k = i ; and w s ,k = 0 otherwise, w here k ∈ [2 n + 2]. The PLC fun ctions r s ,k ( · ) is set as: r s ,k ( · ) ⇐ [ θ ] and θ = 0 if k / ∈ { i, j } ; θ = 1 if k = j ; and θ = 2 if k = i , wh ere k ∈ [2 n + 2]. 4.1.2 T raders T u , where u ∈ U Let u = ( i, j, 1) b e a triple in U with 1 ≤ i 6 = j ≤ n . W e u se A i and A j to d enote th e i th and j th ro w v ectors of A , resp ectiv ely . W e defi ne C and D to b e the f ollo wing n -dimensional v ectors: F or k ∈ [ n ], ( C k , D k ) = ( A i,k − A j,k , 0) if A i,k − A j,k ≥ 0; and ( C k , D k ) = (0 , A j,k − A i,k ) otherwise . By d efinition, w e h a ve C − D = A i − A j while b oth vect ors C and D are nonnegativ e. Moreo v er, b ecause A is a sparse matrix, the num b er of nonzero en tries in either C or D is at most 20 and ev ery entry is b et w een 0 and 2. W e also let E and F b e th e follo wing t wo nonnegativ e n um b ers: ( E , F ) =   X k ∈ [ n ] D k − X k ∈ [ n ] C k , 0   if X k ∈ [ n ] D k ≥ X k ∈ [ n ] C k ; ( E , F ) =   0 , X k ∈ [ n ] C k − X k ∈ [ n ] D k   otherwise . Accordingly , we hav e E , F ≥ 0 and E + X k ∈ [ n ] C k = F + X k ∈ [ n ] D k . 10 Moreo v er, since C and D are sp arse, w e also ha ve 0 ≤ E , F ≤ max   X k ∈ [ n ] C k , X k ∈ [ n ] D k   ≤ 40 . W e set the initial endowmen t ve ctor w u = ( w u , 1 , ..., w u , 2 n +1 , w u , 2 n +2 ) of T u as follo ws: 1. w u ,i = 1 /n 4 ; w u ,k = w u , 2 n +2 = 0 for all other k ∈ [ n ]; 2. w u ,n + k = C k /n 5 for all k ∈ [ n ]; and 3. w u , 2 n +1 = E /n 5 . It is easy to v erify that the num b er of nonzero en tries in w u is at most 22. W e set the PLC utilit y fun ctions r u ,k ( · ), where k ∈ [2 n + 2], of T u as follo ws: 1. r u ,i ( · ) ⇐ [9 , 1; 1 /n 4 ]; and r u ,k ( · ) ⇐ [0] for all other k ∈ [ n ]; 2. r u , 2 n +2 ( · ) ⇐ [3]; 3. r u ,n + k ( · ) ⇐ [0] for all k ∈ [ n ] such that D k = 0; 4. r u ,n + k ( · ) ⇐ [27 , 1; D k /n 5 ] f or all k ∈ [ n ] such that D k > 0; and 5. r u , 2 n +1 ( · ) ⇐ [0] if F = 0; and r u , 2 n +1 ( · ) ⇐ [27 , 1; F /n 5 ] if F > 0. Note that the n u m b er of k ∈ [2 n + 2] su c h that r u ,k ( · ) is not the zero function is at most 23. The constants 1, 3, 9 and 27 in the constru ction may lo ok str an ge at first sight . The motiv atio n is that, if the pr ice-regulatio n pr op erty still holds for the n ew market M (whic h turns out to b e true), then w e kno w exactly the p reference of T u o ver the go o ds since 3 > 2. See the pro of of Lemma 4 for more details. 4.1.3 T raders T v , where v ∈ V The b ehavio r of T v , v ∈ V , is v ery sim ilar to that of T u except that it w ork s on the second matrix B . Let v = ( i, j, 2) b e a triple in V w ith 1 ≤ i 6 = j ≤ n . W e u s e B i and B j to den ote th e i th and j th column v ectors of B , r esp ectiv ely . Similarly , we defi n e the follo wing n -d imensional v ectors C and D : ( C k , D k ) = ( B k ,i − B k ,j , 0) if B k ,i − B k ,j ≥ 0; and ( C k , D k ) = (0 , B k ,j − B k ,i ) otherwise . As a result, we ha ve C − D = B i − B j while b oth C and D are nonnegativ e. W e also d efine E , F ≥ 0 in a similar wa y so that E + X k ∈ [ n ] C k = F + X k ∈ [ n ] D k and 0 ≤ E , F ≤ 40 . W e set the initial endowmen t ve ctor w v = ( w v , 1 , ..., w v , 2 n +1 , w v , 2 n +2 ) of T v to b e 11 1. w v ,n + i = 1 /n 4 ; w v ,n + k = w v , 2 n +2 = 0 for all other k ∈ [ n ]; 2. w v ,k = C k /n 5 for all k ∈ [ n ]; and 3. w v , 2 n +1 = E /n 5 . W e set the PLC utilit y functions r v ,k ( · ), where k ∈ [2 n + 2], of T v as follo ws: 1. r v ,n + i ( · ) ⇐ [9 , 1; 1 /n 4 ]; and r v ,n + k ( · ) ⇐ [0] for all other k ∈ [ n ]; 2. r v , 2 n +2 ( · ) ⇐ [3]; 3. r v ,k ( · ) ⇐ [0] for all k ∈ [ n ] suc h th at D k = 0; 4. r v ,k ( · ) ⇐ [27 , 1; D k /n 5 ] f or all k ∈ [ n ] such that D k > 0; and 5. r v , 2 n +1 ( · ) ⇐ [0] if F = 0; and r v , 2 n +1 ( · ) ⇐ [27 , 1; F /n 5 ] if F > 0. Again, the n um b er of nonzero en tries in w v is at most 22, and the num b er of indices k suc h that r v ,k ( · ) is n ot the zero function is at most 23. 4.1.4 T raders T i , where i ∈ [2 n ] Finally , for eac h i ∈ [2 n ], we set the in itial end o wm en t v ector w i = ( w i, 1 , ..., w i, 2 n +2 ) of T i as follo ws: w i, 2 n +1 = 1 /n 12 and w i,k = 0 , for all other k ∈ [2 n + 2]. W e set the PLC utilit y functions r i,k ( · ), where k ∈ [2 n + 2], of T i as follo ws: r i,i ( · ) ⇐ [1] and r i,k ( · ) ⇐ [0] , for all other k ∈ [2 n + 2]. 4.2 F rom Approxima te Mark et Equilibria to Appro ximate Nash Equilibria By definition, M is a 2-linear additive ly sep arab le PLC mark et which is strongly connected, 27-b ounded and 23-sparse. Let N = 2 n + 2, the n u m b er of go o ds in M . T o pro v e Theorem 3, w e only need to show that from an y N − 13 -appro x im ate market equilibrium p of M , one can construct an n − 6 -w ell-supp orted Nash equilibrium ( x , y ) of ( A , B ) in p olynomial time. T o this end, let ( x ′ , y ′ ) denote th e follo wing t wo n -dimensional v ectors: x ′ k = p k − 1 and y ′ k = p n + k − 1 , for all k ∈ [ n ]. (7) Then, w e n orm alize ( x ′ , y ′ ) to get a p air of distributions ( x , y ) (we will s ho w later that x ′ , y ′ 6 = 0 ): x k = x ′ k P i ∈ [ n ] x ′ i and y k = y ′ k P i ∈ [ n ] y ′ i , for all k ∈ [ n ]. (8) Theorem 3 follo ws d irectly from Theorem 4 wh ic h w e w ill prov e in the next section. Note that if p is a N − 13 -appro x im ate equilibr ium, then it is also an n − 13 -appro x im ate equilibrium by definition. 12 Theorem 4. If p is an n − 13 -appr oximate market e quilibrium of M , then ( x , y ) c onstructe d ab ove is an n − 6 -wel l-supp orte d Nash e quilibrium of ( A , B ) . 5 Correctness of the Reduction In this section, we pro v e Theorem 4. Let p = ( p 1 , ..., p 2 n +2 ) b e an norm alized n − 13 -appro x im ate mark et equilibrium of M . By the s ame argumen t used earlier, we can pro ve that p k > 0 for all k ∈ [2 n + 2]. Therefore, w e ha ve p k ≥ 1 for all k and min k p k = 1. Let X b e an optimal allo cation with r esp ect to p that clears the mark et appro x im ately: X = n a s , a u , a v , a i ∈ R 2 n +2 + : s ∈ S, u ∈ U, v ∈ V , i ∈ [2 n ] o . W e start with the follo wing notation. Let T ′ ⊆ T be a sub set of traders, and k ∈ [2 n + 2]. W e us e w k [ T ′ ] to denote th e amount of go o d G k that traders in T ′ p ossess at the b eginn in g an d a k [ T ′ ] to denote the amoun t of goo d G k that T ′ receiv es in the fi nal allo cation X . According to our constru ction, w k [ T ] ∈ [2 , 3] for ev ery k ∈ [2 n + 2]. Because X clears the mark et appro ximately , w e ha ve   w k [ T ] − a k [ T ]   ≤ w k [ T ] /n 13 ≤ 3 /n 13 , for all k ∈ [2 n + 2]. (9) W e fur ther divid e the traders into tw o group s: T 1 = { T s : s ∈ S } and T 2 = T − T 1 . Then (9) imp lies   w k [ T 1 ] − a k [ T 1 ] + w k [ T 2 ] − a k [ T 2 ]   ≤ 3 /n 13 , for all k ∈ [2 n + 2]. (10) 5.1 The Price-Regulation Prop erty First, we sh o w that, the p r ice v ector p m u s t still satisfy the price-regulation p rop erty as in the price- regulating marke t M 2 n +2 . W e will use the fact that traders in T 1 p ossess almost all the go o ds in M . Lemma 1 (Pr ice Regulation) . F or al l k ∈ [2 n + 2] , 1 ≤ p k ≤ 2 . Pr o of. Assume for con tradiction that p do es not satisfies the price-regulation prop erty . Then without loss of generalit y , w e assume that p 1 = max k p k > 2 and p 2 = 1. By the same argumen t used in the p ro of of Prop erty 1, w e ha ve w 1 [ T 1 ] = (2 n + 1) · 1 n , a 1 [ T 1 ] ≤ 2 n · 1 n , and thus, w 1 [ T 1 ] − a 1 [ T 1 ] ≥ 1 n . By (10), w e ha ve w 1 [ T 2 ] − a 1 [ T 2 ] ≤ − 1 n + 3 n 13 = ⇒ a 1 [ T 2 ] ≥ w 1 [ T 2 ] + 1 n − 3 n 13 ≥ 1 n − 3 n 13 (11) b ecause w 1 [ T 2 ] ≥ 0. Ho w ever, this cannot b e true since th e amount of goo ds the traders in T 2 p ossess at 13 the b eginnin g is muc h smaller compared to 1 /n . Even if they sp end all the money on G 1 , w e still ha ve a 1 [ T 2 ] ≤ P k ∈ [2 n +2] p k · w k [ T 2 ] p 1 ≤ X k ∈ [2 n +2] w k [ T 2 ] = O ( n − 2 ) ≪ 1 n , since w e assumed th at p 1 = max k p k . This con tradicts with (11). 5.2 Relations b et w een p k and w k [ T 2 ] − a k [ T 2 ] Next, w e p ro ve t wo very u seful relations b et w een p k and w k [ T 2 ] − a k [ T 2 ]. Lemma 2. L et p b e a normalize d n − 13 -appr oximate market e qu ilibrium and X b e an optimal al lo c ation that cle ars the market appr oximately. If w k [ T 2 ] − a k [ T 2 ] > 3 /n 13 for some k ∈ [2 n + 2] , then p k = 1 . Pr o of. Without loss of generalit y , we prov e the lemma for the case when k = 1. By (10), w e ha ve w 1 [ T 1 ] − a 1 [ T 1 ] < 0 . This means that, in th e mark et participated by traders T s , the amount of G 1 whic h they would like to buy is strictly m ore than the amount of G 1 they p ossess at th e b eginning. Int uitiv ely this implies that the price p 1 of G 1 is lo wer th an what it should b e, and indeed we sho w b elo w that p 1 = min k p k = 1. On one hand, b y th e construction, only the follo wing traders T s are interested in G 1 : S 1 = { s = (1 , j ) : j 6 = 1 } and S 2 = { s = ( i, 1) : i 6 = 1 } . On the other hand, w e ha ve a 1 [ T s , s ∈ S 1 ] ≤ w 1 [ T s , s ∈ S 1 ] = w 1 [ T 1 ] due to th e b udget limitation. As a result, there must exist an s = ( i, 1) ∈ S 2 suc h that a s , 1 > 0 . Since a s is an optimal bund le for T s with resp ect to p , w e ha ve 1 p 1 ≥ 2 p i = ⇒ p 1 ≤ p i 2 . By Lemma 1, th e price-regulation prop ert y , we conclude th at p 1 = 1 and th e lemma is p ro ved. Lemma 3. L et p b e a normalize d n − 13 -appr oximate market e qu ilibrium and X b e an optimal al lo c ation that cle ars the market appr oximately. If w k [ T 2 ] − a k [ T 2 ] < − 3 /n 13 for some k ∈ [2 n + 2] , then p k = 2 . Pr o of. Without loss of generalit y , we prov e the lemma for the case when k = 1. By (10), w e ha ve w 1 [ T 1 ] − a 1 [ T 1 ] > 0 . This means that, in th e mark et participated by traders T s , the amount of G 1 whic h they would like to buy is strictly less than the amount of G 1 they p ossess at the b eginning. Intuitiv ely , this imp lies that the price p 1 of G 1 is higher than what it should b e, and indeed we show b elow that p 1 = 2 = max k p k . 14 Since a 1 [ T 1 ] < w 1 [ T 1 ], there must exist a j ∈ [2 n + 2] w ith j 6 = 1 su c h th at s = (1 , j ) and a s , 1 < w s , 1 . (Otherwise a 1 [ T 1 ] ≥ w 1 [ T 1 ]). This means that T s sp end s some of its money to buy G j and thus, 1 p j ≥ 2 p 1 = ⇒ p 1 ≥ 2 p j . By Lemma 1, th e price-regulation prop ert y , we conclude th at p 1 = 2 and th e lemma is p ro ved. W e also need the follo wing t wo lemmas. W e only prov e the first one. The second one can b e p ro ved symmetrically . Lemma 4. L et u = ( i, j, 1) b e a triple in U and u ′ = ( j, i, 1) ∈ U . Then for any k ∈ [2 n + 1] , we have w u ,k + w u ′ ,k ≥ a u ,k + a u ′ ,k . (12) Lemma 5. L et v = ( i, j, 2) b e a triple in V and v ′ = ( j, i, 2) ∈ V . Th en for any k ∈ [2 n + 1] , we have w v ,k + w v ′ ,k ≥ a v ,k + a v ′ ,k . Pr o of of L emma 4. Without loss of generalit y , w e only n eed to prov e Lemma 4 for the case wh en u = (1 , 2 , 1) and u ′ = (2 , 1 , 1). L et C an d D d en ote th e f ollo wing tw o n -d imensional vecto rs: F or k ∈ [ n ], ( C k , D k ) = ( A 1 ,k − A 2 ,k , 0) if A 1 ,k − A 2 ,k ≥ 0; and ( C k , D k ) = (0 , A 2 ,k − A 1 ,k ) otherwise . (13) W e also define E and F to b e the follo wing t w o n onnegativ e num b ers: ( E , F ) =   X k ∈ [ n ] D k − X k ∈ [ n ] C k , 0   if X k ∈ [ n ] D k ≥ X k ∈ [ n ] C k ; ( E , F ) =   0 , X k ∈ [ n ] C k − X k ∈ [ n ] D k   otherwise . (14) Then b y the constru ction, w e ha ve w u ,n + k = C k /n 5 and w u ′ ,n + k = D k /n 5 for all k ∈ [ n ], w u , 1 = w u ′ , 2 = 1 /n 4 , w u , 2 n +1 = E /n 5 , w u ′ , 2 n +1 = F /n 5 , and all other en tries of w u and w u ′ are 0. W e now fo cus on the preferen ce of T u . After selling its initial endo wmen t, the budget of T u is p 1 · 1 n 4 + X k ∈ [ n ] p n + k · C k n 5 + p 2 n +1 · E n 5 = Ω  1 n 4  b y Lemma 1. The PLC u tilit y functions r u ,k ( · ) of T u are designed carefully , so that ev en though w e d o not know w hat exactly p is, we kno w the b eha vior of T u due to the p rice-regulation p rop erty: T u first 15 buys the follo wing bund le of go o ds from the market n D k n 5 amoun t of G n + k and F n 5 amoun t of G 2 n +1 : k ∈ [ n ] o . (15) As D has at most 20 n onzero en tries and ev ery en try is b et w een 0 and 2, the cost of this bu ndle is X k ∈ [ n ] p n + k · D k n 5 + p 2 n +1 · F n 5 = O  1 n 5  ≪ 1 n 4 . T u then buys as m uc h G 1 as it can u p to 1 /n 4 , and s p ends all the money left, if an y , on G 2 n +2 . The b ehavio r of T u ′ is similar. It first buys the follo wing bund le of go o ds fr om th e market: n C k n 5 amoun t of G n + k and E n 5 amoun t of G 2 n +1 : k ∈ [ n ] o . (16) It th en buys as m u c h G 2 as it can u p to 1 /n 4 , and sp ends all the money left, if any , on G 2 n +2 . No w we are ready to p ro ve the lemma. T he case when k ∈ [ n ] b ut k 6 = 1 , 2 is trivial sin ce w u ,k = w u ′ ,k = a u ,k = a u ′ ,k = 0 . When k = 1, we ha ve w u , 1 + w u ′ , 1 = 1 /n 4 , a u ′ , 1 = 0, a u , 1 ≤ 1 /n 4 and thus, (12) follo ws. T he case wh en k = 2 can b e pro v ed similarly . F or th e case of n + k where k ∈ [ n ], w e hav e w u ,n + k = C k n 5 , w u ′ ,n + k = D k n 5 , a u ,n + k = D k n 5 , and a u ′ ,n + k = C k n 5 , and (12) follo ws. When k = 2 n + 1, we ha v e w u , 2 n +1 = E n 5 , w u ′ , 2 n +1 = F n 5 , a u , 2 n +1 = F n 5 , and a u ′ , 2 n +1 = E n 5 , and (12) follo ws. This fin ishes th e pro of of the lemma. By Lemma 4, L emma 5 and Lemma 2, w e immediately get the f ollo wing corollary concerning p 2 n +1 . Corollary 1. p 2 n +1 = 1 . Pr o of. First, by Lemma 4 and Lemma 5, w e h av e w 2 n +1 [ T u , T v : u ∈ U, v ∈ V ] − a 2 n +1 [ T u , T v : u ∈ U, v ∈ V ] ≥ 0 . Ho wev er, the constru ction implies that w 2 n +1  T i : i ∈ [2 n ]  = 2 n · 1 n 12 = 2 n 11 and a 2 n +1  T i : i ∈ [2 n ]  = 0 . As a r esult, w 2 n +1 [ T 2 ] − a 2 n +1 [ T 2 ] ≥ 2 /n 11 ≫ 3 /n 13 . It then follo ws fr om L emma 2 th at p 2 n +1 = 1. 16 5.3 Pro of of Theorem 4 No w w e let x ′ and y ′ denote the tw o vec tors obtained in (7). By Lemma 1 w e ha ve x ′ k , y ′ k ∈ [0 , 1] for all k ∈ [ n ]. W e will pro ve the f ollo wing t wo prop erties of ( x ′ , y ′ ) and use them to pro v e Th eorem 4. Prop ert y 2. F or al l 1 ≤ i 6 = j ≤ n , we have ( A i − A j ) y ′ T < − ǫ = ⇒ x ′ i = 0; and (17) x ′ ( B i − B j ) < − ǫ = ⇒ y ′ i = 0 , (18) wher e ǫ = n − 6 , A i denotes the i th r ow ve ctor of A , and B i denotes the i th c olumn ve ctor of B . Prop ert y 3. Ther e exist i and j ∈ [ n ] such that x ′ i = 1 and y ′ j = 1 . No w assu me that x ′ and y ′ satisfy b oth pr op erties. In p articular, Prop er ty 3 implies that x ′ , y ′ 6 = 0 . As a r esu lt, we can n orm alize them to get tw o pr ob ab ility distribution x and y using (8). Before pro vin g these t wo prop erties, we sho w that ( x , y ) m u st b e an ǫ -w ell-sup p orted Nash equilibrium of ( A , B ). Pr o of of The or em 4. Since b oth x and y are p robabilit y distrib u tions, we only need to sho w that ( x , y ) satisfies (1) and (2) for all i, j : 1 ≤ i 6 = j ≤ n . W e on ly pr o ve (1) h ere. Assume A i y T + ǫ < A j y T , then w e ha ve ( A i − A j ) y ′ T = ( A i − A j ) y T ·   X k ∈ [ n ] y ′ k   < − ǫ since P k ∈ [ n ] y ′ k ≥ 1 by Prop ert y 3. As a result, by Prop ert y 2 we h a ve x ′ i = 0 and thus, x i = 0. Finally , we prov e Prop erty 2 and Prop ert y 3. Pr o of of Pr op erty 2. W e only prov e (17) for the case when i = 1, j = 2. (18) can b e p ro v ed similarly . Let u = (1 , 2 , 1) and u ′ = (2 , 1 , 1). Let C and D b e the t wo nonnegativ e vecto rs d efined in (13), and E and F b e the t wo n onnegativ e n um b ers defined in (14). W e ha v e C − D = A 1 − A 2 and E + X k ∈ [ n ] C k = F + X k ∈ [ n ] D k . (19) Assume ( A 1 − A 2 ) y ′ T < − ǫ . Th en the mon ey of T u left after purchasing the bund le in (15) is p 1 · 1 n 4 + X k ∈ [ n ] p n + k · C k n 5 + p 2 n +1 · E n 5 − X k ∈ [ n ] p n + k · D k n 5 − p 2 n +1 · F n 5 . By Corollary 1, w e ha ve p 2 n +1 = 1. Using (19), we can simplify the equation to b e the f ollo wing: p 1 · 1 n 4 + 1 n 5 X k ∈ [ n ] y ′ k · ( C k − D k ) = p 1 · 1 n 4 + 1 n 5 ( A 1 − A 2 ) y ′ T < p 1 · 1 n 4 − ǫ n 5 . (20) 17 This implies that the amount a u , 1 of G 1 that T u buys is sm aller than 1 n 4 − ǫ p 1 n 5 ≤ 1 n 4 − 1 2 n 11 since ǫ = n − 6 . How ev er, we hav e w u , 1 = 1 /n 4 and th us, w u , 1 − a u , 1 > 1 / (2 n 11 ) . (21) On the other hand, it is easy to c h eck that w u ′ , 1 = 0 and a u ′ , 1 = 0. By Lemma 4 and 5, we hav e w 1 [ T u , T v : u ∈ U, v ∈ V ] − a 1 [ T u , T v : u ∈ U, v ∈ V ] > 1 2 n 11 . (22) Next w e b ound w 1  T i : i ∈ [2 n ]  − a 1  T i : i ∈ [2 n ]  . By th e construction, a 1  T i : i ∈ [2 n ] , i 6 = 1  = 0 and a 1 , 1 = p 2 n +1 · 1 n 12 p 1 ≤ 1 n 12 , since p 2 n +1 = 1. Th erefore, w 1  T i : i ∈ [2 n ]  − a 1  T i : i ∈ [2 n ]  ≥ − 1 /n 12 . Com bining (22), w e h a ve w 1 [ T 2 ] − a 1 [ T 2 ] > 1 2 n 11 − 1 n 12 ≫ 3 n 13 . It th en follo ws from Lemma 2 that p 1 = 1 and thus, x ′ 1 = 0. Pr o of of Pr op erty 3. Let ℓ ∈ [ n ] b e one of the ind ices that maximizes A ℓ y ′ T , then we s h o w that x ′ ℓ = 1. Without loss of generalit y , we ma y assume that ℓ = 1. First, w e consider v = ( i, j, 2) and v ′ = ( j, i, 2) in V . In the p ro of of Lemma 4, w e sho wed that w u ,n + k + w u ′ ,n + k = a u ,n + k + a u ′ ,n + k , for all p airs u = ( i, j, 1) and u ′ = ( j, i, 1), and all k ∈ [ n ]. Similarly , we can pr o ve that w v , 1 + w v ′ , 1 = a v , 1 + a v ′ , 1 . (23) Second, for ev er y u = ( i, j, 1) ∈ U , we alw ays ha v e w u , 1 = a u , 1 . Th is is b ecause 1. If i 6 = 1, then w u , 1 = a u , 1 = 0; and 2. If i = 1, then by (20), the money of T u left after purchasing the bundle of go o d s in (15) is at least p 1 /n 4 , so w u , 1 = a u , 1 = 1 /n 4 . As a r esult, we h a ve w 1 [ T u , T v : u ∈ U, v ∈ V ] = a 1 [ T u , T v : u ∈ U, v ∈ V ]. 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M athematic al Pr o gr amming , 111:31 5–34 8, 2008. 21 A Pro of of Theorem 2 In this section, we pro v e Theorem 2. T o this end, we fi rst sho w that un der the conditions of Th eorem 2, M has at least one quasi-e qu ilibrium (see the definition b elo w). T hen we sh o w that any quasi-equilibrium of M is in d eed a mark et equ ilibr ium. Definition 11. A quasi-e quilibrium of M is a (normalized) pric e ve ctor p ∈ R n + such that ther e exists an al lo c ation X = { x i ∈ R n + : i ∈ [ m ] } which has the fol lowing pr op erties: 1. The market cle ars: F or eve ry go o d G j ∈ G , X i ∈ [ m ] x i,j ≤ X i ∈ [ m ] w i,j ; In p articular, if p j > 0 , then X i ∈ [ m ] x i,j = X i ∈ [ m ] w i,j ; 2. F or every tr ader T i ∈ T , at le ast one of the f ol lowing is true: (a) x i ∈ OPT ( i, p ) ; (b) p · x i = p · w i = 0 ( zer o inc ome ) . The difference b etw een market equilibr ia and quasi-equilibria is that in the latter, w e do not requ ire the optimalit y of allo cations for traders with a zero income: If a tr ad er has a zero income, then w e can assign her an y bund le of zero cost. How ev er, if p is a quasi-equilibriu m and the income of every trader is p ositiv e with resp ect to p , then by defin ition p must b e a market equ ilibr ium. In [29 ] Maxfield ga v e a s et of cond itions that are sufficient for the existence of a qu asi-equilibriu m in an exc hange m ark et. W e u se th e follo wing simplified v ers ion [29]: Theorem 5 ([29 ]) . An exchange market M has a quasi-e qu ilibrium p if 1. F or e ach tr ader T i ∈ T , its utility function u i : R n + → R is b oth c ontinuous and quasi-c onc ave; and 2. F or e ach tr ader T i ∈ T , u i is non-satiable, i.e ., f or any x ∈ R n + , ther e exists a ve ctor y ∈ R n + such that u i ( y ) > u i ( x ) . No w we u se Theorem 5 to p ro v e Th eorem 2. Pr o of of The or em 2. First, it is easy to c h eck that if M is an additiv ely s eparable P L C marke t that is strongly conn ected, th en it satisfies b oth conditions in Theorem 5. In particular, u i is n on-satiable s ince the econom y graph of M is strongly connected and thus, there exists a j ∈ [ n ] suc h that r i,j ( · ) is strictly monotone. As a result, M must ha ve a quasi-equilibriu m p . W e u se X = { x i ∈ R n + : i ∈ [ m ] } to d enote an allo cation that clears the m ark et. Since p 6 = 0 , there is at least one trader in T , sa y T 1 ∈ T , has a p ositiv e income. 22 Second, we sh o w that for ev er y trader, its income is p ositiv e and thus, p is indeed an equ ilibrium of M . Sup p ose this is not true, then there is at least one trader T 2 whose income is zero. Since the econom y graph is strongly connected, ther e is a directed p ath from T 2 to T 1 . As a result, th ere must b e a directed edge T 3 T 4 on the p ath such that the income of T 3 is zero and the income of T 4 is p ositiv e. By definition, there exists a j ∈ [ n ] suc h that the amount of G j that T 3 o wn s at the b eginning is p ositiv e and the PLC utilit y fun ction of T 4 with resp ect to G j is strictly monotone. Ho wev er, s in ce the income of T 3 is zero, w e h a ve p j = 0 and thus, the amount of G j that T 4 w ants to b uy is + ∞ , contradicting the assu m ption that p is a quasi-equilibrium of M (since the income of T 4 is p ositive but the bu ndle she receiv es is not optimal). No w w e ha ve pro v ed the existence of a market equilibriu m p . The second part of Theorem 2 f ollo ws from the work of Dev an u r and Kannan [13]. In [13], the authors pr op osed an algorithm for computing a mark et equilibr ium in an additiv ely sep arab le PLC m arket 5 . They divide the whole searc h s pace R n + of p in to “cells” C ⊂ R n + using h y p erplanes. Then for eac h cell C , there is a rational linear program LP C that c haracterizes the set of mark et equilibria in C : p ∈ C is an equilibrium of M if and only if it is a feasible solution to LP C (In particular, if LP C has no feasible solution then there is no equilibrium in C ). Moreo v er, the size of L P C , f or an y cell C , is p olynomial in the size of M . No w let p b e a market equilibrium of M , whic h is n ot necessarily rational. W e let C ∗ denote the cell that p lies in, then p must b e a feasible solution to LP C ∗ . Since LP C ∗ is rational, it m u st hav e a rational solution p ∗ and the num b er of bits one n eed to describ e p ∗ is p olynomial in the size of LP C ∗ and th u s, is p olynomial in the size of M . Theorem 2 then follo ws since p ∗ is also an equilibrium of M . 5 When t he num b er of goo ds is constant, the algorithm is p olynomial-time. 23