Settling the Complexity of Arrow-Debreu Markets under Leontief and PLC Utilities, using the Classes FIXP and \Exists-R
LLeontief Exchange Markets Can Solve Multivariate PolynomialEquations, Yielding FIXP and ETR Hardness ∗ Jugal Garg † Ruta Mehta ‡ Vijay V. Vazirani § Sadra Yazdanbod ¶ Abstract
We show FIXP-hardness of computing equilibria in Arrow-Debreu exchange markets underLeontief utility functions, and Arrow-Debreu markets under linear utility functions and Leontiefproduction sets, thereby settling these open questions [33]. As corollaries, we obtain FIXP-hardness for piecewise-linear concave (PLC) utilities and for Arrow-Debreu markets under linearutility functions and polyhedral production sets. In all cases, as required under FIXP, the setof instances mapped onto will admit equilibria, i.e., will be “yes” instances. If all instances areunder consideration, then in all cases we prove that the problem of deciding if a given instanceadmits an equilibrium is ETR-complete, where ETR is the class Existential Theory of Reals.As a consequence of the results stated above, and the fact that membership in FIXP hasbeen established for PLC utilities [17], the entire computational difficulty of Arrow-Debreumarkets under PLC utility functions lies in the Leontief utility subcase. This is perhaps themost unexpected aspect of our result, since Leontief utilities are meant for the case that goodsare perfect complements, whereas PLC utilities are very general, capturing not only the caseswhen goods are complements and substitutes, but also arbitrary combinations of these and muchmore.The main technical part of our result is the following reduction: Given a set S of simultaneousmultivariate polynomial equations in which the variables are constrained to be in a closedbounded region in the positive orthant, we construct a Leontief exchange market M which hasone good corresponding to each variable in S . We prove that the equilibria of M , when projectedonto prices of these latter goods, are in one-to-one correspondence with the set of solutions ofthe polynomials. This reduction is related to a classic result of Sonnenschein [32, 31]. ∗ Supported by NSF Grants CCF-0914732 and CCF-1216019. † Max-Planck-Institut f¨ur Informatik, Saarbr¨ucken, Germany. [email protected] ‡ College of Computing, Georgia Institute of Technology, Atlanta. [email protected] § College of Computing, Georgia Institute of Technology, Atlanta. [email protected] ¶ College of Computing, Georgia Institute of Technology, Atlanta. [email protected] a r X i v : . [ c s . CC ] N ov Introduction
In economics, it is customary to assume that utility functions are non-separable concave, notonly because of their generality but also their nice properties, e.g., decreasing marginal utilitiesand convexity (of optimal bundles ). Since computer science assumes a finite precision model ofcomputation, we restrict attention to piecewise-linear concave (PLC) utility functions . Extensivestudy of special cases of PLC utility functions has led to a deep understanding of computabilityof market equilibria, ever since the commencement of this study twelve years ago; see details inSection 1.1. However, determining the exact complexity of computing equilibria for Arrow-Debreumarkets under PLC utility functions has remained open. A subcase of the latter, which has beenwidely used in economic modeling [25], is Leontief utility functions, and its exact complexity hasalso remained open, e.g., see [33].In this paper, we resolve both these problems, by showing them FIXP-hard. Very recently, Yan-nakakis [36] and Garg et. al. [17] gave proofs of membership in FIXP for Leontief and PLC utilityfunctions, respectively. However, following the work of Etessami and Yannakakis [16], definingFIXP and proving FIXP-completeness of Arrow-Debreu markets whose excess demand functionsare algebraic, there has been no progress on giving proofs of FIXP-hardness for other market equilib-rium problems. Note that the latter result does not establish FIXP-completeness of Arrow-Debreumarkets under any specific class of utility functions In this paper, we prove FIXP-hardness for Arrow-Debreu markets under Leontief utility func-tions. We also show FIXP-hardness for Arrow-Debreu markets under linear utility functions andLeontief production sets. As corollaries, we obtain FIXP-hardness for PLC utilities and for Arrow-Debreu markets under linear utility functions and polyhedral production sets (membership in FIXPfor production was also shown in [17]). In all cases, as required under FIXP, the set of instancesmapped onto will admit equilibria, i.e., will be “yes” instances. If all instances are under consid-eration, then we prove that the problem of deciding if a given instance admits an equilibrium isETR-complete, where ETR is the class Existential Theory of Reals.As a consequence of the results stated above, the entire computational difficulty of Arrow-Debreu markets under PLC utility functions lies in the Leontief utility subcase. This is perhapsthe most unexpected aspect of our result, since Leontief utilities are meant only for the case thatgoods are perfect complements, whereas PLC utilities are very general, capturing not only the caseswhen goods are complements and substitutes, but also arbitrary combinations of these and muchmore.Perhaps the most elementary way of stating the main technical part of our result is the followingreduction, which we will denote by R : Given a set S of simultaneous multivariate polynomialequations in which the variables are constrained to be in a closed bounded region in the positiveorthant, we construct an Arrow-Debreu market with Leontief utilities, say M , which has one goodcorresponding to each variable in S . We prove that the equilibria of M , when projected onto pricesof these latter goods, are in one-to-one correspondence with the set of solutions of the polynomials.This reduction, together with the fact that the 3-player Nash equilibrium problem (3-Nash) isFIXP-complete [16] and that 3-Nash can be reduced to such a system S , yield FIXP-hardness forthe Leontief case.Reduction R is related to a classic result of Sonnenschein [32, 31] which states that a setof arbitrary multivariate polynomials can be generated as excess demand functions of an Arrow- which is used crucially in fixed point theorems for proving existence of equilibria. Clearly, by making the pieces fine enough, we can obtain a good approximation to the original utility functions. In the economics literature, there are two parallel streams of results on market equilibria, one assumes beinggiven an excess demand function and the other a specific class of utility functions.
The first utility functions to be studied were linear. Once polynomial time algorithms were foundfor markets under such functions [11, 12, 22, 19, 21, 37, 27, 35, 13] and certain other cases [7, 23, 10,34, 20], the next question was settling the complexity of Arrow-Debreu markets under separable,piecewise-linear concave (SPLC) utility functions. This problem was shown to be complete [5, 33]for Papadimitriou’s class PPAD [28]. Also, when all instances are under consideration, the problemof deciding if a given SPLC market admits an equilibrium was shown to be NP-complete [33].The notion of SPLC production sets was defined in [18] and Arrow-Debreu markets under suchproduction sets and linear utility functions were shown to be PPAD-complete.Previous computability results for Leontief utility functions were the following: In contrast toour result, Fisher markets under Leontief utilities admit a convex program [15] and hence theirequilibria can be approximated to any required degree in polynomial time [4, 2]. Arrow-Debreumarkets under Leontief utilities were shown to be PPAD-hard [8]; however, since in this caseequilibria are not rational numbers [14], its complexity is not characterize by PPAD (problemsin PPAD have rational solutions). Leontief utilities are a limiting case of constant elasticity ofsubstitution (CES) utilities [25]. Finding an approximate equilibrium under the latter was alsoshown to be PPAD-complete [6].
We now describe the difficulties encountered in obtaining reduction R and the ideas needed toovercome them; this should also help explain why FIXP-hardness of Leontief (and PLC) marketswas a long-standing open problem. For this purpose, it will be instructive to draw a comparisonbetween reduction R and the reduction from 2-Nash to SPLC markets given in [5]. At the outset,observe the latter is only dealing with linear functions of variables and hence is much easier thanthe former.Both reductions create one market with numerous agents and goods, and the amount of eachgood desired by an agent gets determined only after the prices are set. Yet, at the desired prices,corresponding to solutions to the problem reduced from, the supply of each good need to be exactlyequal to its demand. In the latter reduction, the relatively constrained utility functions give a lotmore “control” on the optimal bundles of agents. Indeed, it is possible to create one large marketwith many agents and many goods and still argue how much of each good is consumed by eachagent at equilibrium.We do not see a way of carrying out similar arguments when all agents have Leontief utilityfunctions. The key idea that led to our reduction was to create several modular units within thelarge market and ensure that each unit would have a very simple and precise interaction with therest of the market. Leontief utilities, which seemed hard to manage, in fact enabled this in a verynatural manner as described below. Interestingly enough, in retrospect, we do not see how to createthese units using only SPLC utility functions. Since the payoff of the row player from a given strategy is a linear function of the variables denoting the proba-bilities played by the column player. losed submarket: A closed submarket is a set S of agents satisfying the following: At everyequilibrium of the complete market, the union of initial endowments of all agents in S exactlyequals the union of optimal bundles of all these agents.Observe that the agents in S will not be sequestered in any way — they are free to choose theiroptimal bundles from all the goods available. Yet, we will show that at equilibrium prices, theywill only be exchanging goods among themselves.These closed submarkets enable us to ensure that variables denoting prices of goods satisfyspecified arithmetic relations. The latter are equality, linear function and product; we show thatthese three arithmetic relations suffice to encode any polynomial equation. Under equality, we wanttwo prices p a and p b to be equal, and under linear functions, we want that p a = Bp b + Cp c + D ,where B, C and D are constants.Under product, we want that p a = p b · p c . Designing this closed submarket, say M , requiresseveral ideas, which we now describe. M has an agent i whose initial endowment is one unit ofgood a and she desires only good c . We will ensure that the amount of good c leftover, after allother agents in the submarket consume what they want, is exactly p b , i.e., the price of good b . Atequilibrium, i must consume all the leftover good c , whose total cost is p b · p c . Therefore the priceof her initial endowment, i.e., one unit of good a , must be p b · p c , hence establishing the requiredproduct relation. The tricky part is ensuring that exactly p b amount of good c is leftover, withoutknowing what p b will be at equilibrium. This is non-trivial, and this submarket needs to haveseveral goods and agents in addition to the ones mentioned above.Once reduction R is established, FIXP-hardness follows from the straightforward observationthat a 3-Nash instance can be encoded via polynomials, where each variable, which represents theprobability of playing a certain strategy, is constrained in the interval [0 , l ∞ -norm is ETR-hard; this entails constraining the variables to bein the interval [0 , / l ∞ -norm. Membership in ETR follows by essentially showing a reduction in thereverse direction: given a Leontief market, we obtain a set of simultaneous multivariate polynomialequations whose roots capture its equilibria.Next we briefly describe the classical Arrow-Debreu market model, the problem of 3-Nash andits relation with the complexity classes FIXP and ETR. Following are a few notations that we willuse in the rest of the paper. Notations.
We mostly follow: capital letters denote matrices of constants, like W ; bold lower caseletters denote vector of variables, like x , y ; and calligraphic capital letters denote sets like A , G .We use [ n ] To denote the set { , . . . , n } . Given an n -dimensional vector x and a number r ∈ R , by x ≤ r , we mean ∀ i ∈ [ n ] , x i ≤ r . The Arrow-Debreu (AD) market [1] consists of a set G of divisible goods, a set A of agents and aset F of firms. Let g denote the number of goods in the market.The production capabilities of a firm is defined by a convex set of production schedules andeach firm wants to use a (optimal) production schedule that maximizes its profit − money earnedfrom the production minus the money spent on the raw materials. Firms are owned by agents: Θ if is the profit share of agent i in firm f such that ∀ f ∈ F , (cid:80) i ∈ A Θ if = 1. Each agent i has a utility3unction U i : R g + → R + over bundles, and comes with an initial endowment of goods; W ij is amountof good j with agent i . Each agent wants to buy a (optimal) bundle of goods that maximizes herutility to the extent allowed by her earned money – from initial endowment and profit shares inthe firms.Given prices of goods, if there is an assignment of optimal production schedule to each firm andoptimal affordable bundle to each agent so that there is neither deficiency nor surplus of any good,then such prices are called market clearing or market equilibrium prices; we note that a zero-pricedgood is allowed to be in surplus. The market equilibrium problem is to find such prices when theyexist. In a celebrated result, Arrow and Debreu [1] proved that market equilibrium always existsunder some mild conditions, however the proof is non-constructive and uses heavy machinery ofKakutani fixed point theorem. We note that an arbitrary market may not admit an equilibrium.A well studied restriction of Arrow-Debreu model is exchange economy , i.e., markets withoutproduction firms. To work under finite precision it is customary to assume that utility functionsare piecewise-linear concave (PLC) and production sets are polyhedral. As stated earlier, agent i ’s utility function is U i : R g + → R + over bundle of goods. These functionsare said to be piecewise-linear concave (PLC) if at bundle x i = ( x i , . . . , x ig ) it is given by: U i ( x i ) = min k { (cid:88) j U kij x ij + T ki } , where U kij ’s and T ki ’s are given non-negative rational numbers. Since the agent gets zero utilitywhen she gets nothing, we have U i ( ) = 0, and therefore at least one T ki is zero. The Leontief utility function is a special subclass of PLC, where each good is required in a fixedproportion. Formally, it is given by: U i ( x i ) = min j ∈ G { x ij A ij } , where A ij ’s are non-negative numbers. In other words the agent wants good j in A ij proportion.Clearly, the agent has to spend (cid:80) j A ij p j amount of money to get one unit of utility. Thus, optimalbundle satisfies the following condition. ∀ j ∈ G , x ij = β i A ij , where β i = (cid:80) j ∈ G W ij p j (cid:80) j ∈ G A ij p j (1) A firm can produce a set of goods using another set of goods as raw materials, and these two setsare assumed to be disjoint. Let P f ∈ R g be the set of production schedules for firm f , then if itcan produce a bundle x s using a bundle x r then x s − x r ∈ P f . The set is assumed to be downwardclosed, contains the origin , no vector is strictly positive ( no production out of nothing ) and ispolyhedral. We call these PLC production sets .Let S f denote the set of goods that can be produced by firm f and R f be the set of goods itcan use as raw material such that S f ∩ R f = ∅ . A PLC production set of firm f can be describedas follows, where x sfj and x rfj denote the amount of good j produced and used respectively.4 f = ( x s − x r ) ∈ R g | (cid:88) j ∈ S f D kfj x sfj ≤ (cid:88) j ∈ R f C kfj x rfj + T kf , ∀ k ; x s ≥ x sfj = 0 , ∀ j / ∈ S f ; x r ≥ x rfj = 0 , ∀ j / ∈ R f where D kfj ’s, C kfj ’s and T kf ’s are given non-negative rational numbers. Since there is no produc-tion if no raw material is consumed, it should be the case that for some k , T kf = 0. The Leontief production is a special subclass of PLC production sets, where a firm f produces asingle good a using a subset of the rest of the goods as raw materials. To produce one unit of a , itrequires D fj units of good j , i.e., x sfa = min j (cid:54) = a (cid:26) x rfj D fj (cid:27) . -player Nash Equilibrium ( -Nash) In this section we describe 3-player finite games and characterize Nash equilibrium. Given a 3-playerfinite game, let the set of strategies of player p ∈ { , , } be denoted by S p . Let S = S × S × S .W.l.o.g. we assume that |S | = |S | = |S | = n s . Such a game can be represented by n s × n s × n s -dimensional tensors A , A and A representing payoffs of first, second and third player respectively.If players play s = ( s , s , s ) ∈ S , then the payoffs are A ( s ), A ( s ) and A ( s ) respectively.Since players may randomize among their strategies, let ∆ p denote the probability distributionover set S p , ∀ p ∈ { , , } (the set of mixed-strategies for player p ), and let ∆ = ∆ × ∆ × ∆ .Given a mixed-strategy profile z = ( z , z , z ) ∈ ∆, let z ps denote the probability with whichplayer p plays strategy s ∈ S p , and let z − p be the strategy profile of all the players at z except p .For player p ∈ { , , } the total payoff and payoff from strategy s ∈ S p at z are respectively, π p ( z ) = (cid:88) s ∈ S A p ( s ) z s z s z s and π p ( s, z − p ) = (cid:88) t ∈ S − p A p ( s, t )Π q (cid:54) = p z qt q Definition 3.1 (Nash (1951) [26])
A mixed-strategy profile z ∈ ∆ is a Nash equilibrium (NE)if no player gains by deviating unilaterally. Formally, ∀ p = 1 , , π p ( z ) ≥ π p ( z (cid:48) , z − p ) , ∀ z (cid:48) ∈ ∆ p . In 1951, John Nash [26] proved existence of an equilibrium in a finite game using Brouwer fixed-point theorem, which is highly non-constructive. Despite many efforts over the years, no efficientmethods are obtained to compute a NE of finite games. Next we give a characterization of NEthrough multivariate polynomials and discuss its complexity.
NE Characterization.
It is easy to see that in order to maximize the expected payoff, only bestmoves should be played with a non-zero probability; by best moves we mean the moves fetchingmaximum payoff. Formally, ∀ p ∈ { , , } , ∀ s ∈ S p , z ps > ⇒ π p ( s, z − p ) = δ p , where δ p = max s (cid:48) ∈ S p π p ( s (cid:48) , z − p ) (2) Assumption.
Since scaling all the co-ordinates of A p ’s with a positive number or adding a constantto them does not change the set of Nash equilibria of the game A = ( A , A , A ), w.l.o.g. we5ssume that all the co-ordinates of each of A p are in the interval [0 , z ps ’s capture strategies, δ p capturespayoff of player p , and β ps are slack variables: F NE ( A ) : ∀ p ∈ { , , } , (cid:80) s ∈ S p z ps = 1 ∀ p ∈ { , , } , s ∈ S p , π p ( s, z − p ) + β ps = δ p and z ps β ps = 0 ∀ p ∈ { , , } , s ∈ S p , ≤ z ps ≤ , ≤ β ps ≤ , ≤ δ p ≤ Lemma 3.2
Nash equilibria of A are exactly the solutions of system F NE ( A ) , projected onto z . Proof :
It is easy to check that NE of A gives a solution of F NE ( A ) using (2); the upper boundson variables of F NE ( A ) holds because all the entries in A are in the interval [0 , z , β , δ ) of F NE ( A ), the first condition ensures that z ∈ ∆. The two parts of the secondcondition imply that z satisfies (2) and therefore is a NE of game A . (cid:50) Let 3-Nash denote the problem of computing Nash equilibrium of a 3-player game. Next wedescribe complexity classes FIXP and ETR, and their relation with 3-Nash.
The class FIXP to capture complexity of the exact fixed point problems with algebraic solutions [16].An instance I of FIXP consists of an algebraic circuit C I defining a function F I : [0 , d → [0 , d ,and the problem is to compute a fixed-point of F I . The circuit is a finite representation of function F I (like a formula), consisting of { max , + , ∗} operations, rational constants, and d inputs andoutputs.The circuit C I is a sequence of gates g , . . . , g m , where for i ∈ [ d ], g i := l i is an input variable.For d < i ≤ d + r , g i := c i ∈ Q is a rational constant, with numerator and denominator encoded inbinary. For i > d + r we have g i = g j ◦ g k , where j, k < i and the binary operator ◦ ∈ { max , + , ∗} .The last d gates are the output gates. Note that the circuit forms a directed acyclic graph (DAG),when gates are considered as nodes, and there is an edge from g j and g k to g i if g i = g j ◦ g k . Sincecircuit C I represents function F I it has to be the case that if we input λ ∈ [0 , d to C I then allthe gates are well defined and the circuit outputs C I ( λ ) = F I ( λ ) in [0 , d . We note that a circuitrepresenting a problem in FIXP operates on real numbers. Reduction requirements:
A reduction from problem A to problem B consists of two polynomial-time computable functions: a function f that maps an instance I of A to an instance f ( I ) of B ,and another function g that maps a solution y of f ( I ) to a solution x of I . If x i = g i ( y ), then g i ( y ) = a i y j + b i , for some j , where a i and b i are polynomial-size rational numbers; every coordinateof x is a linear function of one coordinate of y .In order to remain faithful to Turing machine computation, Etessami and Yannakakis [16]defined following three discrete problems on FIXP. Partial computation
F IXP pc : Given an instance I and a positive integer k in unary, computethe binary representation of some solution, up to the first k bits after the decimal point. Decision
F IXP d : Given an instance I and a rational r return ‘Yes’ if x ≥ r for all solutions,‘No’ if x < r for all solutions, and otherwise either answer is fine. (Strong) Approximation F IXP a : Given an instance I and a rational (cid:15) > x that is within (additive) (cid:15) distance from some solution, i.e., ∃ x ∗ ∈ Sol ( I ) such that | x ∗ − x | ∞ ≤ (cid:15) . 6hereas FIXP is a class of, in general, real-valued search problems, whose complexity can bestudied in a real computation model, e.g., [3], note that F IXP pc , F IXP d and F IXP a are classesof discrete search problems, hence their complexity can be studied in the standard Turing machinemodel. This is precisely the reason to define these three classes. The following result was shown in[16]. Theorem 3.3 [16] Given a -player game A = ( A , A , A ) , computing its NE is FIXP-complete.In particular, the corresponding Decision, (Strong) Approximation, and Partial Computation prob-lems are complete respectively for F IXP d , F IXP a and F IXP pc . The class ETR was defined to capture the decision problems arising in existential theory of reals [30]. An instance I of class ETR consists of a sentence of the form,( ∃ x , . . . , x n ) φ ( x , . . . , x n ) , where φ is a quantifier-free ( ∧ , ∨ , ¬ )-Boolean formula over the predicates (sentences) defined bysignature { , , − , + , ∗ , <, ≤ , = } over variables that take real values. The question is whether thesentence is true. Following is an example of such an instance. ∃ ( x , x ) , ( x + x x − x + 1 = 0 ∧ x x ≥ ∨ ( x − x < < does not change the class ETR. The size of theproblem is n + size ( φ ), where n is the number of variables and size ( φ ) is the minimum number ofsignatures needed to represent φ (we refer the readers to [30] for detailed description of ETR, andits relation with other classes like PSPACE). Schaefer and ˇStefankoviˇc showed the following result;the first result on the complexity of a decision version of 3-Nash. Definition 3.4 (Decision -Nash) Decision -Nash is the problem of checking if a given -playergame A admits a Nash equilibrium z such that z ≤ . , where z is the mixed-strategy profile played. Theorem 3.5 [30] Decision -Nash is ETR-complete. Note that changing the upper bound on all z ps ’s from 1 to 0 . F NE ( A ) (3), exactly capturesthe NE with z ≤ .
5. Thus
Decision -Nash can be reduced to checking if such a system ofpolynomial equalities admits a solution. Next we show a construction of Leontief exchange marketsto exactly capture the solutions of a system of multivariate polynomials, similar to that of F NE ( A ),at its equilibria. Consider the following system of m multivariate polynomials on n variables z = ( z , . . . , z n ), F : { f i ( z ) = 0 , ∀ i ∈ [ m ]; L j ≤ z j ≤ U j , ∀ j ∈ [ n ] } , where L j , U j ≥ f i ’s, and the upper and lower bounds U j ’s and L j ’s are assumed to be rationalnumbers. In this section we show that solutions of F can be captured as equilibrium prices of anexchange market with Leontief utility functions (see Section 2.1.1 for definition). The problems of3-Nash and Decision 3-Nash (see Definition 3.4) can be characterized by a set similar to (4) (see73) in Section 3), in turn we obtain FIXP and ETR hardness results for Leontief markets, and inturn PLC markets, from the corresponding hardness of 3-Nash [16, 30].Polynomial f i is represented as sum of monomials, and a monomial αz d . . . z d n n is representedby tuple ( α, d , ..., d n ); here coefficient α is a rational number. Let M f i denote the set of monomialsof f i , and size [ f i ] = (cid:80) ( α, d ) ∈ M fi size ( α, d ), where size ( r ) for a rational number r is the minimumnumber of bits needed to represent its numerator and denominator. Degree of f i is deg ( f i ) =max ( α, d ) ∈ M fi (cid:80) j d j . The size of F , denoted by size [ F ], is m + n + (cid:80) j ( size ( U j ) + size ( L j )) + (cid:80) i ( deg ( f i )+ size [ f i ]). Given a system F , next we construct an exchange market in time polynomialin size [ F ], whose equilibria correspond to solutions of F . First we transform system F into a polynomial sized equivalent system that uses only the followingthree basic operations on non-negative variables.( EQ. ) z a = z b ( LIN. ) z a = Bz b + Cz c + D, where B, C, D ≥ QD. ) z a = z b ∗ z c (5) Remark 4.1
We note that even though (EQ.) is a special case of (LIN.), we consider it separatelyin order to convey the main ideas.
Next we illustrate how to capture f i ’s using these basic operations through an example. Considera polynomial 4 z z + 3 z z − z − . First, move all monomials with negative coefficients to the right hand side of the equality, sothat all the coefficients become positive.4 z z + 3 z z = z + 2Second, capture every monomials, with degree more than one, using basic operations: z a = z z ≡ z a = z ∗ z , z a = z a ∗ z z b = z z ≡ z b = z ∗ z Third, capture the equality 4 z a + 3 z b = z + 2 using ( LIN. ) and (
EQ. ):4 z a + 3 z b = z + 2 ≡ z e = 4 z a + 3 z b , z f = z + 2 , z e = z f Finally, combine all of the above to represent f i as follows:4 z z + 3 z z − z − ≡ z a = z ∗ z , z a = z a ∗ z z b = z ∗ z z e = 4 z a + 3 z b , z f = z + 2 , z e = z f (6)Since inequalities have to be captured through equalities with non-negative variables, the in-equalities of (4) have to be transformed as follows: ∀ j ∈ [ n ] , z j = s lj + L j , z j + s uj = U j , z j , s lj , s uj ≥ R ( F ) be a reformulation of F , using transformation similar to (6) for each f i , and that of(7) for each inequality. All the variables in R ( F ) are constrained to be non-negative.In order to construct R ( F ) from F , we need to introduce many auxiliary variables (as was donein (6)). Let the number of variables in R ( F ) be N , and out of these let z , . . . , z n be the originalset of variables of F (4). Given a system R ( F ) of equalities, we will construct an exchange market M , such that value of each variable z j , j ∈ [ N ] is captured as price p j of good G j in M . Further,we make sure that these prices satisfy all the relations in R ( F ) at every equilibrium of M . Since equilibrium prices of an exchange market are scale invariant, the relations that these pricessatisfy have to be scale invariant too. However note that in (5) (
LIN. ) and (
QD. ) are not scaleinvariant. To handle this we introduce a special good G s , such that when its price p s is set to 1 weget back the original system.( EQ. ) p a = p b ( LIN. ) p a = Bp b + Cp c + Dp s , where B, C, D ≥ QD. ) p a = p b ∗ p c p s (8)Let R (cid:48) ( F ) be a system of equalities after applying the transformation of (8) to R ( F ). Note that, R (cid:48) ( F ) has exactly one extra variable than R ( F ), namely p s , and solutions of R (cid:48) ( F ) with p s = 1 areexactly the solutions of R ( F ).Let the size of R (cid:48) ( F ) be ( R (cid:48) ( F ) + size ( B, C, D ) in each of (LIN.)-type relations). Recall that M f i denotes the set of monomials in polynomial f i . To bound thevalues at a solution of R (cid:48) ( F ), define H = M max U dmax + 1 , where d = max f i deg ( f i ) , M max = max i |M f i | ,U max = max { max j U j , max f i , ( α, d ) ∈ M f i | α |} . Lemma 4.2 size [ R (cid:48) ( F )] = poly ( size [ F ]) . Vector p is a non-negative solution of R (cid:48) ( F ) with p s = 1 iff z j = p j , ∀ j ∈ [ n ] is a solution of F . Further, p j ≤ H, ∀ j ∈ [ N ] . Proof :
For the first part, it is enough to bound size [ R ( F )]. Note that the number of auxiliaryvariables added to R ( F ) to construct a monomial of f i is at most its degree. Further, to constructthe expression of f i from these, the extra variables needed is at most the number of monomials.Further, the coefficients of ( LIN. ) type relations are coefficients of the monomials of f i ’s. Thus,we get that size [ R (cid:48) ( F )] = O ( (cid:80) i ∈ [ m ] deg ( f i ) size [ f i ] + (cid:80) j ∈ [ n ] size ( U j ) + size ( L j )) = poly ( size [ F ]).The second part follows by construction. For the third part note that ( p , . . . , p N ) is a solutionof R ( F ). Since variables of the original system F is upper bounded by U j s, it is easy to see thatthe maximum value of any variable in a non-negative solution of R ( F ) is at most H . (cid:50) Next we construct a market whose equilibria satisfy all the relations of R (cid:48) ( F ), and has p s > In this section we construct market M consisting of goods G , . . . , G N and G s , such that the prices p , . . . , p N and p s , satisfy all the relations of R (cid:48) ( F ) at equilibrium.First we want price of G s to be always non-zero at equilibrium. To ensure this we add thefollowing agent to market M . Recall that W ij is the amount of good G j agent A i brings to the9arket, U i : R g + → R + is the utility function of A i , and x i denotes the bundle of goods consumedby her. A s : W ss = 1 , W sj = 0 , ∀ j ∈ [ N ]; U s ( x s ) = x ss (9) Lemma 4.3
At every equilibrium of market M , we have p s > , and x ss = W ss . Proof :
At an equilibrium if p s = 0, then agent A s will demand infinite amount of good s , acontradiction. The second part follows using the fact that at any given prices, A s wants to buyonly good s , and has exactly W ss p s amount of money to spend. (cid:50) Since a price p j may be used in multiple relations of R (cid:48) ( F ), the corresponding good has to beused in many different gadgets. When we combine all these gadgets to form market M , the biggestchallenge is to analyze the flow of goods among these gadgets at equilibrium. We overcome this alltogether by forming closed submarket for each gadget. Definition 4.4 (submarket)
A submarket (cid:102) M of a market M consists of a subset of agents andgoods such that the endowment and utility functions of agents in (cid:102) M and the production functionsof firms in (cid:102) M are defined over goods only in (cid:102) M . Definition 4.5 (closed submarket)
A submarket (cid:102) M of a market M is said to be closed if atevery equilibrium of the entire market M , the submarket (cid:102) M is locally at equilibrium, i.e., its totaldemand equals its total supply. The total demand of (cid:102) M is the sum of demands of agents in (cid:102) M andits total supply is the sum of initial endowments of agents in (cid:102) M . In other words, (cid:102) M does not interfere with the rest of market in terms of supply and demand,even if some goods in (cid:102) M are used outside as well. Note that the market of (9) is a closed submarket(due to Lemma 4.3) with only one agent and one good, namely A s and G s respectively. We willsee that the submarket (cid:102) M establishing a relation of type ( EQ. ), (
LIN. ) and (
QD. ) has a setof exclusive goods used only in (cid:102) M , in order to achieve the closed property. Before describingconstruction of closed submarkets for more involved relations, we first describe it for a simpleand important equality relation. Furthermore, we will use equality to construct closed markets for( QD. ).Let there be K relations in R (cid:48) ( F ), numbered from 1 to K , and let M r denote the closedsubmarket establishing relation r ∈ [ K ]. ( EQ. ) p a = p b The gadget for (
EQ. ) consists of two agents with Leontief utility functions, as given in Table 1,where good G r is exclusive to this submarket. M EQ : 2 Agents ( A , A ) and 3 Goods ( G a , G b , G r ) // G r : an exclusive good A : W = (0 , ,
1) and U ( x ) = min { x a , x r } A : W = (1 , ,
1) and U ( x ) = min { x b , x r } Table 1:
Closed submarket M r for r th relation p a = p b In M r , the endowment vector W i ’s should be interpreted as (amount of G a , amount of G b ,amount of G r ), i.e., in the same order of goods as listed on the first line of the table; we use similarrepresentation in the subsequent constructions. 10 emma 4.6 Consider the market M r of Table 1. • M r is a closed submarket. • At equilibrium, M r enforces p a = p b . • Every non-negative solution of p a = p b gives an equilibrium of M r . Proof :
Let α and β denote the utility obtained by A and A at equilibrium respectively. Thenusing (1) which characterizes optimal bundles for Leontief functions, the market clearing conditionsof the two agents give: p b + p r = α ( p a + p r ) p a + p r = β ( p b + p r )Clearly the above conditions imply that αβ = 1 ⇒ β = / α . Note that A and A consume α and β amounts of good G r respectively. And since this good is exclusive to M r , no other agent willconsume it. Further, there are exactly two units of G r available in the entire market M . Hence weget, α + β ≤ β = α in the above condition gives ( α − ≤ ⇒ α = β = 1. Therefore, we get thatevery equilibrium of M r enforces p a + p r = p b + p r ⇒ p a = p b . Further, M r is a closed submarketbecause at equilibrium, demand of every good in M r is equal to its supply in M r even thoughevery good except G r might participate in the rest of the market as well.For the last part, if p a = p b ≥
0, then choosing p r = 1, and x a = x r = x b = x r = 1 gives amarket equilibrium of M r . (cid:50) ( LIN. ) p a = Bp b + Cp c + Dp s The gadget for (
LIN. ) is an extension of (
EQ. ) having two agents with Leontief utility functions,as given in Table 2, where
B, C, D ≥ Remark 4.7
For simplicity, we denote agents of each submarket by A , A , · · · , and sometimesexclusive goods by G , G , · · · , however they are different across submarkets. M r : 2 Agents ( A , A ) and 5 Goods ( G a , G b , G c , G s , G r ) // G r : an exclusive good A : W = (1 , , , ,
1) and U ( x ) = min { x b B , x c C , x s D , x r } A : W = (0 , B, C, D,
1) and U ( x ) = min { x a , x r } Table 2: M r : Closed market for r th relation p a = Bp b + Cp c + Dp s , B, C, D ≥ Lemma 4.8
Consider the market M r of Table 2 with B, C, D ≥ . • M r is a closed submarket. • At equilibrium, M r enforces p a = Bp b + Cp c + Dp s . • Every non-negative solution of p a = Bp b + Cp c + Dp s gives an equilibrium of M r . roof : The proof is similar to that of Lemma 4.6. Let α and β denote the utility obtained by A and A at equilibrium respectively. Then using (1) together with market clearing conditions ofthe two agents, we get: p a + p r = α ( Bp b + Cp c + Dp s + p r ) Bp b + Cp c + Dp s + p r = β ( p a + p r )Clearly the above conditions imply αβ = 1 ⇒ β = / α . Since G r is exclusive to this market,using similar argument as the proof of Lemma 4.6 we get that α + β ≤
2. This together with β = / α gives α = β = 1. Thus every equilibrium of M r enforces p a + p r = Bp b + Cp c + Dp s + p r ⇒ p a = Bp b + Cp c + Dp s . Hence, M r is a closed submarket.For the last part, if p a = Bp B + Cp c + Dp s ≥
0, then setting p r = 1, x r = x r = x a = 1, and x b = B, x ,c = C, x s = D gives a market equilibrium of M r . (cid:50) Using Lemma 4.8, we easily get the following.
Corollary 4.9
There is a simple closed submarket to establish any linear relation of form p a = E p b + · · · + E n p b n + E p s for any n ≥ , where E , E , . . . , E n are non-negative rational constants. ( QD. ) p a = p b p c p s In this section, we derive a closed submarket for establishing the (
QD. ) relation. In order to simplifythe market construction, which is quite involved, we first make the following two assumptions, whichare removed later. First, that p s = 1 and second, that p b (cid:54) = 0. The first assumption violates thescale invariance of prices, see Section 4.2, but simplifies the relation needed to p a = p b p c . Thesecond assumption ensures that no agent can demand an infinite amount of a good of price p b (Note that in the reduction, since the price of a good corresponds to the probability of playing acertain strategy, eventually we do need to allow for p b = 0.).The main idea for enforcing the simpler relation, p a = p b p c , is to ensure that there is an agent A whose initial endowment is one unit of Good 1 priced at p a , and she desires to consume onlyGood 2 priced at p b . The left over amount of Good 2 after everyone, except agent A , consume isexactly p c . Since p b >
0, agent A has to buy all of this left over amount which requires her to spend p b p c . On the other hand her earning from the sell of Good 1 is p a , implying p a = p b p c . Figure1 illustrates the idea. The difficulty in implementing this idea lies in the fact that p b and p c arevariables; if they were both constants, the construction of the submarket would have been easy. A ( p b , p c ) (1 , p b p c )2 1 w = 1 p a = p Figure 1:
The main idea for enforcing the relation p a = p b p c . Wires are numbered in circle, and wire i carries good G i . The tuple on each wire represents (amount, price). In order to present the submarket in a modular manner, we will first define some devices.Each of these devices will be implemented via a set of agents with Leontief utility functions. Eachdevice ensures a certain relationship between the net endowment left over by these agents and thenet consumption of these agents; for convenience, we will call these the net endowment and net onsumption of the device . Clearly, at equilibrium prices, for each device, the total worth of its netendowment and net consumption must be equal. Submarkets for the devices
In this section, we show implementation of three devices to be used in the submarket for (QD.) relation.
Converter (Conv( q )): The net consumption of this device is one unit of good G , whose priceis p , and the net endowment is p/q units of good G , whose price is q . Table 3 and Figure 2 illustratethe implementation. In the figure tuple on edges represent (amount, price) of the correspondinggood shown in circle. Table 3 has two parts: Part 1 describes the market and Part 2 enforces linearrelations among prices using the submarkets described in Sections 4.3.1 and 4.3.2. Conv ( q )(1 , p )( pq , q ) A A (1 , p )( pq , q )(1 , Hq − p )( H − pq , q )1 : 1 3221 w = H Figure 2:
Flow of goods in Part 1 of Table 3 for Conv( q ). Wires are numbered in circle, and wire i carriesgood G i . The tuple on each wire represents (amount, price). Part 1: Input: G at price p Output: p/q units of G at price q A , A ), 3 goods ( G , G , G ) A : W = H and U ( x ) = min { x , x } A : W = 1 and U ( x ) = x Part 2:
Closed submarkets for the following linear relations p = qp = Hq − p Table 3:
A closed submarket for Conv( q ) There are two agents A and A , and three goods G , G and G . The endowment of A is H units of G , whose price is set to q . Recall that H is a constant defined in Section 4.2. A likes toconsume G and G in the ratio of 1:1. The net consumption of this device, i.e., one unit of G atprice p , is consumed by A . Agent A ’s endowment is one unit of G , whose price is set to Hq − p . A wants to consume G , whose price is q . Hence, it consumes H − p/q units of G (observe thatthere is no need to perform the division involved in p/q explicitly). The remaining p/q units of G form the net endowment of the device, as required.13 ombiner (Comb( l, p a , p b )): The net consumption of this device is l units each of goods G and G , whose prices are p a and p b , respectively. The net endowment is l units of a good G , whoseprice is p a + p b . Table 4 and Figure 3 illustrate the implementation. (1 , l ( p a + p b ))( l, p b )( l, p a )
21 4 A A C omb ( l, p a , p b ) w = H (1 , Hp a + Hp b − p )( H − l, p a + p b ) ( l, p a + p b ) A C omb ( l, p a , p b ) ( l, p a + p b ) ( l, p a ) ( l, p b ) Figure 3:
Flow of goods in Part 1 of Table 4 for Comb( l, p a , p b ). Wires are numbered, and wire i carriesgood G i . The tuple on each wire represents (amount, price). Part 1: Input: l units of G and G at price p a and p b respectively Output: l units of G at price p a + p b A , A , A ), 5 goods ( G , G , G , G , G ) A : W = 1 and U ( x ) = min { x , x } A : W = H and U ( x ) = min { x , x } A : W = 1 and U ( x ) = x Part 2:
Closed submarkets for the following linear relations p = p a + p b p = Hp a + Hp b − p Table 4:
A closed submarket for Comb( l, p a , p b ) Agent A wants G and G in the ratio 1:1, and no other agent wants these goods. Therefore, A will consume all of the available G and G and hence the price of her endowment, i.e., one unitof G , will be l ( p a + p b ) (observe that there is a multiplication involved in this price; however, it isnot performed explicitly).Agent A wants G and G in the ratio 1:1. The price of A ’s endowment, i.e., one unit of G is set to H ( p a + p b ) − p . Hence the endowment of A , i.e., H units of G , has a price of ( p a + p b ).Of this, A must consume ( H − l ), leaving l amount of G as the net endowment of this device.14 plitter (Spl( l, p a , p b )): The net endowment of this device is l units each of two goods G and G , whose prices are p a and p b , respectively. The net consumption is l units of Good 1, whoseprice is p a + p b . Table 5 and Figure 5 illustrate the implementation. ( H − l, p a )( H − l, p b )(1 , Hp a + Hp b − p )(1 , l ( p a + p b )) ( l, p a + p b )( l, p b )( l, p a )
32 532 14 1 : 1 A A A Spl ( l, p a , p b ) w = w = H Spl ( l, p a , p b ) ( l, p b ) ( l, p a ) ( l, p a + p b ) Figure 4:
Flow of goods in Part 1 of Table 5 for Spl( l, p a , p b ). Wires are numbered, and wire i carries good G i . The tuple on each wire represents (amount, price). Good G is desired only by Agent A . Hence, the price of her initial endowment, i.e., one unit of G , is forced to be l ( p a + p b ) (observe that the multiplication involved is not done explicitly). Agent A wants goods G and G in the ratio 1:1. The price of G is set explicitly to H ( p a + p b ) − p .The endowment of A is H units each of G and G , whose prices have been set to p a and p b ,respectively. Agent A wants these two goods in the ratio 1:1, and because of the setting of theprice of her initial endowment, she must consume ( H − l ) units of each of these two goods. Theremaining amounts, i.e., l each, form the net endowment of the device, as required. Part 1: Input: l units of G at price p a + p b Output: l units of G and G at price p a and p b respectively3 Agents ( A , A , A ), 5 goods ( G , G , G , G , G ) A : W = 1 and U ( x ) = x A : W = W = H and U ( x ) = min { x , x } A : W = 1 and U ( x ) = min { x , x } Part 2:
Closed submarkets for the following linear relation p = p a p = p b p = Hp a + Hp b − p Table 5:
A closed submarket for Spl( l, p a , p b ) ubmarket construction for p a = p b p c Now we are ready to describe a closed submarket that enforces p a = p b p c . Consider the sub-market given in Table 6. In this market, the 7 goods, G , . . . G are exclusive to the submarket;the price of good G j is p j . The prices of G , G , G , G , G , G are set to appropriate using ( EQ. )and (
LIN. ) relations and prices p a , p b and p c , as specified in the second part of the table. Thesubmarket uses two Converters, one Combiner and one Splitter. Each of these devices is specifiedby giving its (net endowment, net consumption). Besides the agents needed to implement thesedevices, the submarket requires two additional agents, A and A . Part 1: A , A ), 2 Converters ( Conv , Conv ), 1 Combiner ( Comb ),1 Splitter (
Spl ), and 7 Goods ( G , . . . , G ) A : W = 1 and U ( x ) = x A : W = 1 and U ( x ) = x Conv = Conv (1): ( G , G ) Conv = Conv ( p b ): ( G , G ) Comb ( p c , p b , G , G ) , G ) Spl ( p c , p b , G , ( G , G )) Part 2:
Closed submarkets for the following linear relations p = p c p = 1 p = 1 p = p b p = p a p = p b Table 6:
A closed submarket M (cid:48) r that enforces p a = p b p c Lemma 4.10
The submarket given in Table 6 (and illustrated in Figure 5) enforces p a = p b p c andis closed at equilibrium under the assumption p b (cid:54) = 0 . Proof :
Let p be an equilibrium price vector of the entire market, where p b >
0. It is easy to seethat prices p , p , p and p are strictly greater than zero, so these have to be consumed completely.Further, since the ( EQ. ) and (
LIN. ) submarkets implementing the second part of Table 6 areclosed (Lemmas 4.6 and 4.8), the net endowment of goods G , . . . , G for agents in this submarket,including those in devices, is exactly what they bring.We have set p = p c and the endowment of A is one unit of G . Good G is desired only bythe first agent in Conv ; hence its net consumption costs p c . Furthermore, since the price of G isset to 1 and the parameter of Conv is 1, the net endowment of Conv will be p c units of G .The first agent in Comb wants G and G in the ratio 1:1 both of whose prices are positive.Moreover, net endowment p c of Good G is desired only by this agent, and therefore using (1) atequilibrium the agent has to consume p c units of both the goods. Since prices of G and G are 1and p b respectively, the price of the net endowment of Comb will be p b + 1, and the amount willbe p c . 16 p c , , p c ) ( p c ,
1) ( p c , p b )(1 , p b p c )( p c , p b + 1) ( p c , p b ) A Conv (1) A Spl ( p c , p b , Comb ( p c , p b , Conv ( p b ) Figure 5:
Flow of goods in Part 1 of Table 6. Wires are numbered, and wire i carries good r i . The tupleon each wire represents (amount, price). Thus, output of
Comb is p c amount of good G priced at p b + 1, which has to be consumed by Spl . The prices p and p are set to 1 and p b , respectively, thereby ensuring that the total worthof the net endowment and net consumption of Spl are equal. Finally, agent A gets p c amountof G whose price is p b and has one unit of G as her endowment. Hence the price of G mustbe p = p a = p b p c , as required. Good G is only desired by the first agent in Conv . The netendowment of this device is p c units of G whose price is set to p b , and this good is fully consumedby the first agent of Comb .Note that it may be possible that p c is zero which may force prices of some of G , . . . , G goodsto be zero, like G . However, whoever consumes this good also want to consume another good withnon-zero price, in the same proportion. And therefore demand of no good will exceed the supply.Since the total supply and demand of each of the seven goods are equal, the submarket is closedat these (equilibrium) prices. (cid:50) Now we will modify the construction of Table 6 in order to remove the assumption p b > p s = 1. Consider the implementation of Table 7. Lemma 4.11
Consider the submarket M r of Table 7, • M r is a closed submarket. • At equilibrium, M r enforces p a = p b p c p s , and p c / p s ≤ H . • Every non-negative solution of p a = p b p c , where p s > and p c ≤ H , gives an equilibrium of M r with p s = 1 . Proof :
The only difference between the market of Tables 7 and 6 are that p b is replaced with p b + p s , and p c with p c / p s . As p s > p b + p s >
0, and p c p s is well-defined.17 art 1: A , A ), 2 Converters ( Conv , Conv ), 1 Combiner ( Comb ),1 Splitter (
Spl ), and 7 Goods ( G , . . . , G ) A : W = 1 and U ( x ) = x A : W = 1 and U ( x ) = x Conv = Conv ( p s ): ( G , G ) Conv = Conv ( p b + p s ): ( G , G ) Comb ( p c / p s , p b + p s , p s ): (( G , G ) , G ) Spl ( p c / p s , p b + p s , p s ): ( G , ( G , G )) Part 2:
Closed submarkets for the following linear relations p = p c p = p s p = p s p = p b + p s p = p b + p s p a + p c = p Table 7:
A closed submarket M r that enforces p a = p b p c p s Further, the prices to be set in Part 2 of devices are still linear, even when l = p c / p s in Comb and
Spl . Thus using Lemma 4.10 it follows that the market is closed and p = ( p b + p s ) p c p s , Then usingthe last linear relation enforced in Part 2 of Table 7 we get p a = p b p c p s .For the last part, consider a non-negative p a , p b and p c such that p a = p b p c . Set p s = 1, theprices of goods G , G , G , G , G as per Part 2 of Table 7, and p = p b + 2 p s and p = ( p b + p s ) p c p s .For goods within devices, set their prices as per Part 2 of Tables 3, 4 and 5 respectively. For good G in Comb ( p c p s ) and good G in Spl ( p c p s ), set their prices to to ( p b + 2 p s ) ∗ p c p s . It is easy to verifythat this gives an equilibrium for market M r of Table 7. (cid:50) In this section we prove the main results using the claims established in Section 4.3. Given a system F of multivariate polynomials as in (4), construct an equivalent set of relations R (cid:48) ( F ) consistingof only three types of basic relations given in (8). Let K be the number of relations in R (cid:48) ( F ). Foreach relation r ∈ [ K ], depending on its type, we construct a market M r as described in Tables 1,2 and 7. Further, for each r of type ( QD. ) replace the corresponding devices with agents of Tables3, 4 and 5 respectively. Combine all the M r ’s to form one market M . Also add the agent of (9) in M .Since equilibrium prices of an Arrow-Debreu market are scale invariant, i.e., if p = ( p , . . . , p g )is an equilibrium price vector then so is α p , ∀ α >
0, it is without loss of generality (wlog) toassume some kind of normalization. For example, (cid:80) j p j = 1, or choose a good to be num`eraire , i.e., fix its price to 1. For the latter case, we require price of the num`eraire good to be non-zero We note that, the reduction in [16] from fixed-point to Arrow-Debreu market with algebraic excess demandfunction, it is assumed that (cid:80) j p j = 1 at equilibrium.
18t equilibrium, and any good, for which an agent is non-satiated , qualifies.Given an equilibrium price vector p of M , we know that p s > p of M w.l.o.g. we mean equilibrium prices with p s = 1. In the next twolemmas we establish that the equilibria of market M exactly capture the solutions of system F . Lemma 4.12 If p is an equilibrium price vector of M , then z j = p j , ∀ j ∈ [ n ] is a solution of F . Proof :
Due to Lemma 4.2, it is enough to show that p is a solution of R (cid:48) ( F ). The submarket M r , constructed for relation r of R (cid:48) ( F ), is closed and enforces r at p (first two statements ofLemmas 4.6, 4.8, and 4.11). Since M is a union of M r ’s, p has to satisfy each of the relation of R (cid:48) ( F ). (cid:50) Next we map solutions of F to equilibria of market M . Lemma 4.13 If z is a solution of F , then there exists equilibrium prices p of market M , where p s = 1 and p j = z j , ∀ j ∈ [ n ] . Proof :
Using Lemma 4.2, we can construct a non-negative solution p (cid:48) of R (cid:48) ( F ) using z suchthat p (cid:48) s = 1, p (cid:48) j = z j , ∀ j ∈ [ n ], and p (cid:48) j ≤ H, ∀ j ∈ [ N ].Construct prices p of market M , where set p j = p (cid:48) j , ∀ j ∈ [ N ] and p s = 1. Set x ss = 1 foragent A s of (9). Last statement of Lemmas 4.6, 4.8, and 4.11, imply that in each M r , p can beextended to yield an equilibrium. Since equilibrium in M consists of equilibrium in each M r withsame prices for common goods, combining these gives an equilibrium of M . (cid:50) Thus establishing the strong relation between solutions of F and equilibria of market M , nextwe prove the main theorem of the paper which will give all the desired hardness results as corollaries. Theorem 4.14
Equilibrium prices of market M , projected onto ( p , . . . , p n ) , are in one-to-onecorrespondence with the solutions of F . Further M can be expressed using polynomially many bitsin the size [ F ] , i.e., size [ M ] = poly ( size [ F ]) . Proof :
First part follows using Lemmas 4.12 and 4.13. For the second part, it is enough toshow that size [ M ] = poly ( size [ R (cid:48) ( F )]), due to Lemma 4.2. Let L be the size of R (cid:48) ( F ).For each of the relation r ∈ [ K ] of system R (cid:48) ( F ), we add O (1) agents and goods in M r , asdescribed in Tables 1, 2, and 7 together with Tables 3, 4 and 5. Let n g and n a be the goods and agents in M , then we have n g = O ( K ) and n a = O ( K ).Clearly, each agent of M r brings O (1) goods and has utility for exactly those many goods, andthese markets are closed submarkets (see Definition 4.5). The utility functions of the agents areLeontief which can be written as (good id, coefficient). The endowments can be captured similarly.Thus the encoding of endowments and utility functions of agents in M r requires: (i) O(log K )if r is of type ( EQ. ), ( ii ) O(log K + size ( B, C, D )) if r is of type ( LIN. ), and ( iii ) O(log K + L ) if r is of type ( QD. ) as size [ H ] = O ( L ), where H is a constant defined in Section 4.2.Since M is a union of M r , ∀ r ∈ [ K ], and the agent of (9), the size of M is at most O ( KL ). (cid:50) Theorem 4.14 shows that finding solutions of F can be reduced to finding equilibria of anexchange market with Leontief utility functions. An agent is said to be non-satiated for good j if at any given bundle she can obtain more utility by consumingadditional amount of good j .
19s discussed in Section 3, the problem of computing a Nash equilibrium of a 3-player game A can be formulated as finding a solution of system F NE ( A ) (3) of multivariate polynomials inwhich variables are bounded between [0 ,
1] (Lemma 3.2). Note that size [ F NE ( A )] = O ( size ( A )).Further, since taking projection on a set of coordinates is a linear function, the next theorem followsusing the formulation of (3), together with Lemma 3.2, and Theorems 3.3 and 4.14 (see Section 3.1for the reduction requirements for class FIXP). Theorem 4.15
Computing an equilibrium of an exchange market with Leontief utility functionsis FIXP-hard. In particular, the corresponding Decision, (Strong) Approximation, and PartialComputation problems are hard for
F IXP d , F IXP a and F IXP pc , respectively. Further since the class of Leontief utility functions is a special subclass of piecewise-linearconcave (PLC) utility functions, under which goods need not be only complementary (like Leontief)and substitute (like separable PLC), but can be arbitrary combination of these and much more,the next theorem follows.
Theorem 4.16
Computing equilibrium of an exchange market with piecewise-linear concave utilityfunctions is FIXP-hard. In particular, the corresponding Decision, (Strong) Approximation, andPartial Computation problems are hard for
F IXP d , F IXP a and F IXP pc , respectively. Note that in Theorems 4.15 and 4.16 the resulting market is guaranteed to have an equilibrium,since it was constructed from an instance of 3-Nash which always has a NE (Nash’s theorem [26]).However in general an AD market may not have an equilibrium. Checking if an arbitrary exchangemarket with SPLC utility function has an equilibrium is known to be NP-complete [33]. We studythe analogous question for Leontief (and in turn PLC) markets. It turns out that the complexityof these questions is captured by the class ETR.Theorem 3.5 shows that checking if a 3-player game A has NE within 0 . l ∞ norm is ETR-complete. Clearly, this problem can be reduced to finding a solution of F NE ( A ) of(3) with upper-bound on z ps ’s changed from 1 to 0 . M , then M will have an equilibrium if and only if game A has a NE within 0 . Theorem 4.17
Checking existence of an equilibrium in an exchange market with Leontief utilityfunctions, and in market with PLC utility functions is ETR-hard. [18] gave a reduction from an exchange market M with arbitrary concave utility functions toan equivalent Arrow-Debreu market M (cid:48) with firms, where utility functions of all the agents arelinear. It turns out that M (cid:48) has all the goods of M , in addition to others, and equilibrium pricesof M are in one-to-one correspondence with the equilibrium prices of M (cid:48) projected onto the pricesof common goods. Further the production functions of M (cid:48) are precisely the utility functions in M ,hence representation of M (cid:48) is in the order of the representation of M . Therefore, this reductiontogether with Theorems 4.15 and 4.17, gives the next two results. Corollary 4.18
Computing equilibrium of an Arrow-Debreu market with linear utility functionsand Leontief production, and in turn PLC (polyhedral) production sets, is FIXP-hard. In particular,the corresponding Decision, (Strong) Approximation, and Partial Computation problems are hardrespectively for
F IXP d , F IXP a and F IXP pc . Corollary 4.19
Checking existence of an equilibrium in an Arrow-Debreu market with linear utilityfunctions and Leontief production, and in turn PLC (polyhedral) production sets, is ETR-hard.
Next we show that checking existence of an equilibrium in markets with PLC utility functionsand PLC production sets is in ETR. 20
Existence of Equilibrium in ETR
Using the nonlinear complementarity problem (NCP) formulation of [17] to capture equilibria ofPLC markets, in this section we show that checking for existence of equilibrium in PLC markets isin ETR, and therefore ETR-complete using Corollary 4.19. For the sake of completeness next wepresent the NCP formulation derived in [17].Recall the PLC utility functions and PLC production sets defined in Sections 2.1 and 2.2respectively. Using the optimal bundle and optimal production plan conditions at equilibrium forsuch a market, [17] derived the nonlinear complementarity problem (NCP) formulation AD-NCPfor market equilibrium as shown in Table 8, and showed the Lemma 5.1. All the variables in theNCP of Table 8 are non-negative, and we omit this condition for the sake of brevity.
Table 8:
AD-NCP ∀ ( f, k ) : (cid:88) j D kfj x sfj ≤ (cid:88) j C kfj x rfj + T kf and δ kf ( (cid:88) j D kfj x sfj − (cid:88) j C kfj x rfj − T kf ) = 0 ∀ ( f, j ) : p j ≤ (cid:88) k D kfj δ kf and x sfj ( p j − (cid:88) k D kfj δ kf ) = 0 ∀ ( f, j ) : (cid:88) k C kfj δ kf ≤ p j and x rfj ( (cid:88) k C kfj δ kf − p j ) = 0 ∀ ( i, j ) : (cid:88) k U kij γ ki ≤ λ i p j and x ij ( (cid:88) l U kij γ ki − λ i p j ) = 0 ∀ ( i, k ) : u i ≤ (cid:88) j U kij x ij + T ki and γ ki ( u i − (cid:88) j U kij x ij − T ki ) = 0 ∀ i : (cid:88) j x ij p j ≤ (cid:88) j W ij p j + (cid:88) f Θ if φ f and λ i ( (cid:88) j x ij p j − (cid:88) j W ij p j − (cid:88) f Θ if φ f ) = 0 ∀ j : (cid:88) i x ij + (cid:88) f x rfj ≤ (cid:88) f x sfj and p j ( (cid:88) i x ij + (cid:88) f x rfj − − (cid:88) f x sfj ) = 0 ∀ i : (cid:88) k γ ki = 1 and u i = λ i ( (cid:88) j W ij p j + (cid:88) f Θ if φ f ) + (cid:88) k γ ki T ki ∀ f : φ f = (cid:88) k δ kf T kf and (cid:88) j p j = 1 Lemma 5.1 [17] If ( p , x , x s , x r , λ , γ , δ ) is a solution of AD-NCP, then ( p , x , x s , x r ) is a marketequilibrium. Further, if ( p , x , x s , x r ) is a market equilibrium, then ∃ ( λ , γ , δ ) such that ( p , x , x s , x r , λ , γ , δ ) is a solution of AD-NCP. Due to Lemma 5.1, checking if the market has an equilibrium is equivalent to checking if AD-NCP admits a solution. Since all the inequalities and equalities in AD-NCP are polynomial, andall the coefficients in these polynomials are rational numbers, AD-NCP can be represented usingsignature { , , − , + , ∗ , <, ≤ , = } . The denominators of the coefficients can be removed by takingleast common multiple (LCM) while keeping the size of coefficients polynomial in the original size.Therefore, checking if AD-NCP has a solution can be formulated in ETR (see Section 3.2 fordefinition), and we get the following result using Theorem 4.17.21 heorem 5.2 Checking existence of an equilibrium in an exchange market with piecewise linearconcave utility functions is ETR-complete.
The next result follows using Corollary 4.19.
Theorem 5.3
Checking existence of an equilibrium in an Arrow-Debreu market with PLC utilityfunctions and PLC (polyhedral) production sets is ETR-complete.
Is computing an equilibrium for a Fisher market under PLC utilities FIXP-hard? Clearly theproblem is in FIXP since Fisher markets are a subcase of Arrow-Debreu markets. We believethat existing techniques, for example of [33] establishing hardness for Fisher markets under SPLCutilities via reduction from Arrow-Debreu markets, will not work and new ideas are needed. Asstated in Section 1.1, finding an approximate equilibrium under CES utilities was also shown to bePPAD-complete [6]. Is computing an exact equilibrium FIXP-complete?In economics, uniqueness of equilibria plays an important role. In this vein, we ask what is thecomplexity of deciding if a PLC or Leontief market has more than one equilibria. We note that thereduction given in this paper blows up the number of equilibria and hence it will not answer thisquestion in a straightforward manner.
Acknowledgement:
We wish to thank Mihalis Yannakakis for valuable discussions.
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