Dichotomies in Equilibrium Computation and Membership of Plc Markets in Fixp

Piecewise-linear, concave (PLC) utility functions play an important role in work done at the intersection of economics and algorithms. We prove that the problem of computing an equilibrium in Arrow-Debreu markets with PLC utilities and PLC production sets is in the class FIXP. Recently it was shown that these problems are also FIXP-hard (Garg et al., arXiv:1411.5060), hence settling the long-standing question of the complexity of this problem. Central to our proof is capturing equilibria of these markets as fixed points of a continuous function via a nonlinear complementarity problem (NCP) formulation. Next, we provide dichotomies for equilibrium computation problems, both Nash and market. There is a striking resemblance in the dichotomies for these two problems, hence providing a unifying view. We note that in the past, dichotomies have played a key role in bringing clarity to the complexity of decision and counting problems.


Introduction
Piecewise-linear, concave (PLC) utility functions play a central role in work done at the intersection of economics and algorithms for the following reasons.In economics, it is customary to assume that utility functions are concave, and production sets are convex -this leads to several nice properties such as decreasing marginal utilities and convexity in optimal bundles; the latter is indispensable for applying fixed point theorems for proving existence of equilibria (and for obtaining algorithms for computing them).Since in computer science we assume a finite precision model of computation, differentiable utility functions, which assume infinite precision arithmetic, are not appropriate, thereby making PLC utility functions important.
Additionally, we will assume that production sets are polyhedral.We call this PLC production since the boundary of polyhedral production set can be defined by a PLC correspondence.Clearly by making the pieces fine enough, the approximation to the original utilities and production sets can be made as good as needed.
Over the last decade and a half, considerable insight has been gained on the complexity of computing equilibria in markets with utility functions being subclasses of PLC functions [14,4,37,7,22,19].However, the question of pinning down the complexity of PLC markets has remained open.In this paper, we take a step towards resolving it by showing that the problem of computing an equilibrium in Arrow-Debreu markets with PLC utilities and PLC production sets is in the class FIXP of Etessami and Yannakakis [16]; recently it was show that these problems are also FIXP-hard [21], hence settling the complexity of this long-standing open problem.Etessami and Yannakakis have also shown that the class FIXP captures the complexity of computing an equilibrium for k-player Nash, henceforth denoted k-Nash, for k ≥ 3 [16].
Central to our proof is capturing equilibria of these markets as fixed points of a continuous function via a nonlinear complementarity problem (NCP) formulation.We note that at present very few problems have been shown to be in FIXP and we believe this technique, using an NCP formulation, will find use in the future.
Impressive results obtained over the last decade and a half on equilibrium computation have resulted in deep insights; however, these results may seem disparate and there is a need to find a unifying picture to gain even better understanding.In this paper, we attempt this by organizing some of the most important results into dichotomies.We start by observing that the results already known on Nash equilibrium lead to a dichotomy that respects three different criteria, computational complexity being one of them, see Table 1.In a nutshell, this dichotomy establishes a qualitative difference between 2-Nash and k-Nash for k ≥ 3. The two results stated above, together with other results, lead to analogous dichotomies for market equilibrium.
In the endeavor within TCS, to classify natural computational problems by their complexity, dichotomies have played a key role in bringing much clarity; these dichotomies characterize how the complexity of a certain problem changes as a certain parameter is changed.Perhaps the most well known of these dichotomies is Schaefer's theorem, which gives a complete characterization of when a restriction of SAT, defined via relations over the Boolean domain, is in P and when it is NP-complete.Following this result, a lot of work was done on dichotomies for decision problems, e.g., see [2,10], and for counting problems, see the extensive survey [3]; in the latter case the dichotomy is between P and #P-complete.
An early result of Megiddo and Papadimitriou [28] indicated that the computability of equilibrium problems should have its own unique character, different from, say, the computability of decision or counting problems.They showed that Nash Equilibrium cannot be NP-hard unless NP = co-NP, and hence new classes were needed to establish evidence of intractability for this problem.The classes PPAD [31] and FIXP [16] have been invaluable in this respect.The dichotomies identified in this paper, which are quite different from those of decision and counting problems, also help highlight the unique character of equilibrium computation.

The classes PPAD and FIXP
The two complexity classes PPAD, defined by Papadimitriou [31], and FIXP, defined by Etessami and Yannakakis [16], have played an important role in the theory of equilibrium computation.e.g., they capture the complexity of 2-Nash and k-Nash, for k ≥ 3, respectively.These classes appear to be quite disparatewhereas solutions to problems in the former are rational numbers, those to the latter are algebraic numbers.And whereas the former is contained in function classes NP ∩ co-NP, the latter lies somewhere between P and PSPACE, and is likely to be closer to the harder end of PSPACE [38].
Informally, PPAD is the class of problems that allow for "path-following algorithms" and for this reason (see Section 1.2 for a brief discussion on such algorithms), PPAD has an intimate connection with complementary pivot algorithms: obtaining such an algorithm for a problem gives, together with Todd's result [36], membership of the problem in PPAD.Furthermore, the Lemke-Howson result [26], which gave a complementary pivot algorithms for 2-Nash, provided a key motivation for the definition of this class.On the other hand, a problem is in FIXP if its solutions are in one-to-one correspondence with the fixed points of a function which is defined using an arithmetic circuit with operations of +, * , /, max, and an arbitrary number of rational constants. 1nother important distinction between the two classes is the nature of approximation allowed: whether the algorithm finds a point x that is almost a fixed point in the sense that |x − f (x)| is small, or the point x is itself close to a fixed point, x * , i.e. |x − x * | is small, where by "small" we mean inverse-exponential.Assuming f is Lipschitz continuous, as is the case for all problems considered in this paper, a point satisfying the latter condition also satisfies the former condition; however, a point x satisfying the former condition may be exponentially far away from any fixed point, so the two notions are quite different.Although exact 3-Nash is not in PPAD, if the first kind of approximation is allowed, the problem turns out to be in PPAD; this is a classic result of Daskalakis et.al. [12].In this case, the strategies computed are within inverse-exponential of being best responses.Yet, this solution may be exponentially far from any exact Nash equilibrium.However, if the second kind of approximation is required, the problem is FIXP-complete and hence unlikely to be in PPAD.
The only results showing membership in FIXP or proving FIXP-hardness for market equilibrium questions we are aware of are the following: [16] prove that the problem of computing an equilibrium in an Arrow-Debreu market is FIXP-complete provided the excess demand is an algebraic function of the prices and this model is a simplified version of the standard model in that individual utility functions are not given, only the aggregate excess demand function is given.[6] show that an Arrow-Debreu market under CES utility functions is in FIXP provided the elasticity parameter for each agent is a rational number ρ i < 1 and is given in unary.[38] show that an Arrow-Debreu market under Leontief utility functions is in FIXP; observe that in the latter cases as well, excess demand is an algebraic function of the prices.No markets with production have been shown to be in FIXP.Further, containment in FIXP for exchange markets with general PLC utilities is also not known.On the other hand exchange markets with even Leontief utilities functions, a simple subclass of PLC utilities, and markets with linear utilities and Leontief production both are shown to be FIXP-hard [21].
For markets under PLC utility functions, considered in this paper, optimal bundles of buyers are not unique.Therefore, excess demand will not be a function, it will be a correspondence -this is one of the major difficulties we need to overcome.For markets with production, the amount of each good available is not a constant, which leads to another difficulty to be overcome.
As stated above, PLC markets may not have rational equilibria and so don't admit a linear complementarity problem (LCP) formulation.Instead, we give a nonlinear complementarity problem (NCP) formulation whose solutions are in one-to-one correspondence with market equilibria.We then design a continuous function F over a convex, compact domain which is computable by a FIXP circuit, and we show that the fixed points of F are in one-to-one correspondence with the solutions of the NCP, and hence market equilibria.We believe this technique for proving membership in FIXP using an NCP formulation will find use in the future.

Complementary pivot algorithms
An algorithm that walks on the one-skeleton of a polyhedron to find a solution, which is necessarily at a vertex of the polyhedron, is called a pivoting-based algorithm or a path-following algorithm.The classic example of such an algorithm is the simplex algorithm of Dantzig [11] for linear programming.This algorithm is known to take exponential time in the worst case [24], yet it is the most used algorithm for solving linear programs in practice; moreover, this is despite the fact that linear programming is in P. It turns out that the polynomial time algorithms are too slow in practice.
A pivoting-based algorithm which additionally attempts to satisfy certain complementarity conditions is called a complementary pivot algorithm, classic examples being the Lemke-Howson algorithm [26] for 2-Nash and Eaves' algorithm [15], which is based on Lemke's algorithm [25], for the linear case of the Arrow-Debreu market model.
The common feature of these three algorithms is that, similar to the simplex algorithm, they run fast on randomly chosen examples (established in [35,18], respectively) even though they take exponential time in the worst case (established in [24] and [32] for the first two algorithms and left as an open problem in [18] for the third); the worst case examples are artificially contrived to make the algorithm perform poorly.These algorithms also tend to yield deep structural properties of the underlying problem, e.g., strong duality; index, degree and stability for 2-Nash equilibria [34]; and oddness of number of equilibria [18], respectively.Additionally, complementary pivot algorithms are path-following algorithms and therefore they yield membership of the problem in PPAD.
A FIXP-complete problem is unlikely to have a complementary pivot algorithm since such problems are unlikely to be in PPAD.On the other hand, a PPAD-complete problem may still allow for a fast computability in practice via a complementary pivot algorithm, Hence, the classes PPAD and FIXP are quite different from the viewpoint of computability and it is useful to study the dichotomy between these two classes of problems.We do this in the next section.

The Arrow-Debreu Market Model
The Arrow-Debreu market model [1] consists of a set G of divisible goods, a set A of agents and a set F of firms.Let n denote the number of goods in the market.
The production capabilities of a firm is defined by a set of production schedules.If a firm can produce a bundle x p of goods using bundle x r as raw material, then such a production schedule defines a production possibility vector (PPV) (x p − x r ).The set of PPVs of a firm determines its production capabilities.Let S f ∈ R n denote the PPV set of firm f .Following are the standard and natural assumptions on S f (see [1]).
1.The set S f is closed and convex, and contains the origin.

The set of produced goods and raw goods of a firm are
to be the set of raw goods and 3. Downward close -Adding to raw material does not decrease the production, i.e., if v ∈ S f and w ≤ v then w ∈ S f .

5.
No vacuous production -Condition S ∩ −S = 0 of [1] implies that no subset of firms can do nontrivial production such that their sum total is 0. Formally, if for The goal of a firm is to produce as per a profit maximizing (optimal) schedule.At prices p ∈ R n + , profit maximizing schedules of firm f are arg max x∈S f x • p. Firms are owned by agents: Each agent i comes with an initial endowment of goods; W i j is the amount of good j with agent i.The preference of an agent i over bundles of goods is captured by a non-negative, non-decreasing and concave utility function U i : R n + → R + .Non-decreasingness is due to free disposal property, and concavity captures the law of diminishing marginal returns.Each agent wants to buy a (optimal) bundle of goods that maximizes her utility to the extent allowed by her earned money -from initial endowment and profit shares in the firms; at prices p if φ f is the profit of firm f then money earned by agent i is ∑ j W i j p j + ∑ f Θ i f φ f .Without loss of generality, we assume that the total initial endowment of every good is 1, i.e, ∑ i∈A W i j = 1, ∀ j ∈ G3 .Given prices per unit of goods, if there is an assignment of optimal production schedule to each firm and optimal affordable bundle to each agent so that there is no deficiency of any good and goods with surplus amount have zero prices, then such prices are called market clearing or market equilibrium prices.The market equilibrium problem is to find such prices when they exist.In a celebrated result, Arrow and Debreu [1] proved that a market equilibrium always exists under some mild conditions, however the proof is non-constructive and makes heavy use of Kakutani's fixed point theorem.
A well studied restriction of Arrow-Debreu model is exchange economy, i.e., markets without production firms.

Dichotomies in Equilibrium Computation
We will assume throughout this paper that all numbers given in an instance are rational.Table 1 gives the dichotomy for Nash equilibrium computation.The rationality of 2-Nash was first established as a corollary of the Lemke-Howson algorithm [26], and the first 3-Nash game having only irrational equilibria was given by Nash [30].The complexity of finding a Nash equilibrium was shown to be PPAD-complete for two-player games [31,12,5], while for three or more it turned out to be FIXP-complete [16]; see Section 1.1 for detailed description of both these classes.However checking if a specific type of equilibrium exists, such as equilibrium with payoff at least h, equilibrium where i th strategy is played with non-zero probability, or simply two or more equilibria, turns out to be NP-complete [23,8] and ETR-complete [20], respectively.
A well-studied subclass of games is symmetric games, in which all the players have the same set of strategies.Nash [30] showed that such a game always has a symmetric equilibrium, i.e., one in which each player plays the same strategy.The problem of finding such a Nash equilibrium in a symmetric k player game is called symmetric k-Nash.Results analogous to the general case give a similar dichotomy between symmetric 2-Nash and symmetric k-Nash for k ≥ 3, as shown in Table 2.
Recent results have yielded analogous dichotomies for market equilibrium computation as well, and are presented in Tables 3 and 4, for consumption and production, respectively.These results include the complexity results of [4,37], establishing PPAD-completeness of computing equilibria for Arrow-Debreu  [33,20] Table 1: Nature of solution Rational [26] Algebraic; irrational example [30] Complexity PPAD-complete [31,12,5] In FIXP [16]; FIXP-hard [20] Practical algorithms Lemke-Howson [26] ?Decision NP-complete [8] ETR-complete [20] Table 2: markets under Separable PLC (SPLC) utilities, the new complementary pivot algorithms [18] and [22], and a proof of membership of PLC markets in FIXP, which is established in the current paper.Note that in the tables, results of the current paper have been indicated as CP.
The existence of a market equilibrium is established for a class of markets satisfying certain sufficiency conditions [1,27]; however, in general a market may or may not have an equilibrium.Similar to the case of Nash equilibrium, the decision problem of checking if a given market has an equilibrium is NP-complete and ETR-complete, respectively, for SPLC and PLC markets [37,4,22,21].
For Table 4, which gives dichotomy for production, we also need to specify the class of utility functions of agents.For this, we have used the following convention.For "negative" results, such as PPAD-hardness, NP-hardness, ETR-hardness, or irrational example, we assume the most restricted utilities, i.e., linear.For "positive" results, such as containment in PPAD or NP, or rationality of equilibria, we assume the most general utilities, i.e., SPLC.

Membership in FIXP
In this section, we show that the equilibrium computation problem in markets with PLC utility functions and PLC production functions is in FIXP [16].
We first obtain a characterization of market equilibrium in terms of the solutions of a nonlinear complementarity problem 4 (NCP) formulation and then design a continuous function F over a convex and compact

Nature of solution
Rational [22] Algebraic: CP (Theorem 4.9) irrational example [22] Complexity PPAD-complete [22] In FIXP: CP (Theorem 4.17); FIXP-hard [21] Practical algorithms GV [22] ?(based on Lemke [25]) Decision NP-complete [22] ETR-complete [21] Table 4: domain, computable by a FIXP circuit, i.e., algebraic circuit with {max, min, +, −, * , /} operators and rational constants.Further we show that assuming the weakest known sufficiency conditions for the existence of market equilibrium given by Arrow and Debreu [1] 5 , fixed points of F are in one-to-one correspondence with the solutions of NCP, and hence are related to market equilibria.Etessami and Yannakakis [16] showed membership in FIXP for exchange markets (markets without production) with explicit algebraic demand function, however this approach does not work for markets with PLC utilities.A major difficulty is that the demand of an agent (or firm) is not an explicit algebraic function of given prices; it is not even unique.The same difficulty was experienced by [37] in proving membership of exchange markets with Separable PLC (SPLC) utilities in PPAD, and they resort to the characterization of PPAD (given in [16]) as a class of exact fixed-point computation problems for polynomial time computable piecewise-linear Brouwer functions.No such characterization for FIXP is known.Further we also consider markets with production firms, which has its own difficulties, like handling market clearing conditions becomes non-trivial due to indefinite quantities of goods in the market.
We develop a novel technique for proving membership in FIXP (PPAD) from NCP (LCP), which may be of independent interest.The primary intuition for this technique is to use complementarity conditions in the FIXP circuit to push the mapping of any point that is not an equilibrium away from the point itself.To keep things simple, first we show our result for the exchange markets with PLC utilities, and then extend it to also include PLC production.

Exchange economy
The piecewise-linear concave (PLC) utility function, of agent i, u i : R + n → R + can be described as where U i jk 's and T i k 's are given non-negative rational numbers, and ) is a bundle of goods [13].Given prices per unit p, agent i's optimal bundle is a solution to the following linear program (LP): Let γ i k and λ i be the non-negative dual variables of constraints in the above LP.From the optimality conditions, we get the following linear constraints and complementarity conditions.Note that the constraints are linear assuming prices are given.All variables introduced will have a non-negativity constraint; for the sake of brevity, we will not write them explicitly.
From strong duality, (4.1) and (4.2) are equivalent.Further by taking the sums of complementarity conditions, these conditions also give Hence u i is a redundant variable and can be eliminated using the above expression, however for clarity we keep it as a placeholder variable for the above expression.We get the above constraints for each agent i and all together, they capture the optimal bundle and budget constraints of every agent.At market equilibrium, we also need market clearing of each good, which is essentially, ∑ i x i j ≤ 1, ∀ j.By putting these together and now treating prices p as variables, we get the nonlinear complementarity problem (NCP) formulation as shown in Table 5.Since equilibrium prices are scale invariant in Arrow-Debreu market, we have put ∑ j p j = 1 as well.Sufficiency Conditions.Market equilibrium may not exist, and it is NP-complete to decide whether there exists an equilibrium even in markets with SPLC utility functions [37].Arrow-Debreu [1] showed that a market equilibrium exists under the following sufficiency conditions: endowment matrix W > 0 and each agent is non-satiated.In case of PLC utilities, non-satiation condition implies that for every k, there exists a j such that U i jk > 0.
Next we define a continuous function F : D → D, where D is convex and compact and show that the fixed points of F are in one-to-one correspondence with the solutions of E-NCP, and hence are related to market equilibria using Lemma 4.1.Since F is continuous on a convex and compact D, there exists a fixed point.
Clearly, for such a theorem, we need to assume sufficiency conditions.
To define D, first we obtain upper bounds on all variables at equilibrium.Let , and λ max def = 2n (U max +T max ) /W min .Note that W min > 0 under sufficiency conditions.Since the total quantity of every good is 1, 0 ≤ x i j < x max at equilibrium.Using (4.3), we get where N is the total number of variables, and let (p, x, γ, λ ) def = F(p, x, γ, λ ) as given in Table 6.

Table 6: FIXP Circuit for Exchange Economy
The following claim is straightforward using Lemma 4.1 and we omit its proof.
Claim 4.2.Every market equilibrium gives a fixed point of F.
Next assuming sufficiency conditions for the existence of market equilibrium, we show that every fixed point of F gives a market equilibrium.Table 7 gives all the conditions that might lead to a fixed point of F based on the update rules in Table 6.
Next we show that none of the conditions in shaded rows, namely (1.2), (2.2), (3.3), and (4.3), are satisfied at fixed points of F, which implies that each fixed point of F gives a solution of E-NCP, and hence market equilibrium. 2) 2) 2) 3) 2) Table 7: Conditions for a Fixed Point Based on the Update Rules in Table 6 Claim 4.3.At every fixed point of F, 0 < λ i < λ max , ∀i.
Proof.First suppose that λ i = λ max for some i at a fixed point.It implies that for every good j such that p j ≥ W min 2n , we have x i j = 0 (from (3.1)).Hence, ∑ j x i j p j < W min , which contradicts ∑ j x i j p j ≥ ∑ j W i j p j (from (4.3)).Hence λ i < λ max , ∀i at a fixed point.
Next suppose that λ i = 0 for some i at a fixed point.It implies that for every γ i k > 0 and U i jk > 0, we have x i j = x max (from (3.3)).Note that here we use the sufficiency condition that for every k there exists a j such that U i jk > 0. Further p j > 0 for such goods and ∑ i x i j > 1 for all goods whose p j > 0 (from (1.2)).By this, we get ∑ i, j x i j p j > 1 = ∑ i, j W i j p j .This further implies that ∃i such that ∑ j x i j p j > ∑ j W i j p j and λ i = λ max , which is a contradiction.Claim 4.4.At every fixed point of F, ∑ i x i j ≤ 1, ∀ j.
Proof.Suppose ∃ j such that ∑ i x i j > 1.It implies that p j > 0 (from (1.2)).This further implies that whenever p j > 0, we have ∑ i x i j > 1.Hence, we have ∑ i j x i j p j > 1 = ∑ i, j W i j p j .By this, we get that ∃i such that ∑ j x i j p j > W i j p j and hence λ i = λ max (from (4.3)), contradicting Claim 4.3.
Note that Claim 4.4 implies that x i j < x max , ∀(i, j) at every fixed point of F.
From Claims 4.4 and 4.3, we have ∑ k U i jk γ i k x i j = λ i p j x i j , ∀(i, j) and ∑ j x i j p j λ i = ∑ j W i j p j λ i , ∀i.Putting these together, we get that u i > ∑ j W i j p j λ i + ∑ k T i k γ i k , which is a contradiction.Claims 4.3, 4.4, 4.5 imply that none of the conditions (1.2), (2.2), (3.3), (4.3) are satisfied at fixed points of F. Therefore, we get the following theorem.Theorem 4.6.Assuming sufficient conditions of the existence of market equilibrium, every fixed point of F gives a solution of E-NCP and hence a market equilibrium.Furthermore, F can be computed by a FIXP-circuit and hence market equilibrium computation problem for PLC utilities is in FIXP.
Remark 4.7.This technique can be used to obtain a Linear-FIXP (equivalent to PPAD) circuit for markets with SPLC utilities using the linear complementary problem (LCP) formulation given in [18], thereby giving an alternate proof of membership in PPAD for such markets.However, the same approach for proving membership in Linear-FIXP does not seem to work for 2-Nash using its LCP formulation.

Markets with production
Recall from Section 2 that each firm has a production technology to produce a set of goods from a set of raw goods.The PLC production technology of firm f can be described as where D f jk 's, C f jk 's and T f k 's are given non-negative rational numbers, and x f ,p j and x f ,r j denote the amount of good j produced and used respectively [17].In the above expression, the first summation is on goods j which can be produced by firm f , and the second summation is on goods j which can be used as a raw material.These two sets of goods are disjoint as described in Section 2, however for simplicity we do not introduce more symbols and taking summation over all goods.
Given prices p, firm f 's profit maximizing plan is a solution of the following linear program (LP): Let δ f k be the non-negative dual variables of constraints in the above LP.From the optimality conditions, we get the following linear constraints and complementarity conditions.Note that the constraints are linear assuming prices are given.All variables introduced will have a non-negativity constraint; for the sake of brevity, we will not write them explicitly.
From strong duality, (4.4) and (4.5) are equivalent.Let φ f captures the profit of firm f , i.e., φ f = ∑ j p j x f ,p j − ∑ j p j x f ,r j .Further by taking the sums of complementarity conditions, these conditions also give We get the above constraints for each firm k and all together, they capture the optimal production plan of every firm.Next we need to add constraints capturing optimality of the bundles of agents and market clearing.For this, we only need to modify the market clearing constraints in Table 5 appropriately and we get the nonlinear complementarity problem (NCP) formulation AD-NCP for market equilibrium as shown in Table 8.
The next lemma and theorem follow from the construction.
Theorem 4.9.Assuming sufficient conditions of the existence of equilibrium, a market with PLC utilities and PLC production has at least one equilibrium with algebraic prices.
Sufficiency Conditions.For markets with production, Arrow-Debreu [1] gave the following sufficiency conditions for the existence of equilibrium: endowment matrix W > 0, each agent is non-satiated, no production out of nothing and no vacuous production (defined in Section 2).In case of PLC production, the last two conditions mean that the following linear constraints define a bounded polyhedron.6 where first is production constraint and second is supply constraint.Let x * be the maximum possible value of a variable over these constraints.Note that the bit length of x * is polynomial in the size of input and can be computed in polynomial time.
Next we define a continuous function F : D → D, where D is convex and compact and show that the fixed points of F are in one-to-one correspondence with the market equilibria.Since F is continuous on a convex and compact D, there exists a fixed point.Clearly, for such a theorem, we need to assume sufficiency conditions.
To define D, first we obtain upper bounds on all variables at equilibrium.Let x p max def = x * + 1 and x max def = 2lx p max + 2, where l is the total number of firms and x * is as discussed above in the sufficiency conditions.Next define min and max of every input C, D, T,U,W as we get an upper bound on δ f k at equilibrium as: where N is the total number of variables, and let (p, x, x p , x r , γ, δ , λ ) def = F(p, x, x p , x r , γ, δ , λ ) as given in Table 9.
The following claim is straightforward using Lemma 4.8 and we omit its proof.
Claim 4.10.Every market equilibrium is a fixed point of F.
Next assuming sufficiency conditions for the existence of market equilibrium, we show that every fixed point of F is a market equilibrium.Table 10 gives all the conditions that might lead to a fixed point of F based on the update rules in Table 9.We show that none of the conditions in shaded rows, namely (1.2), (2.2), (3.3), (4.3), (5.3), (6.3) and (7.3), are satisfied at fixed points of F, which implies that each fixed point of F gives a solution of AD-NCP in Table (8) and hence a market equilibrium.
Proof.Suppose ∃( f , j) such that ∑ k C f jk δ f k > p j , then we have x f ,r j = x r max (from (5.3)).It implies that whenever C f jk > 0, we have δ f k = 0 (from (7.1)), which contradicts the starting assumption.
Claim 4.13.If ∑ i, j x i j p j + ∑ f , j x f ,r j p j > ∑ i, j W i j p j + ∑ f , j x f ,p j p j , then ∃i such that ∑ j x i j p j > ∑ j W i j p j + ∑ f ,k Θ i f T f k δ f k .
Proof.This proof is by contradiction.Suppose we have ∑ j x i j p j ≤ ∑ j W i j p j + ∑ f ,k Θ i f T f k δ f k , ∀i, then summing it over all i and using ∑ i Θ i f = 1, we get x i j p j > ∑ i, j W i j p j + ∑ f , j x f ,p j p j − ∑ f , j x f ,r j p j (4.8) Claims 4.11 and 4.12 imply that x f ,r j (∑ k C f jk δ f k − p j ) = 0, ∀( f , j) and δ f k (∑ j D f jk x f ,p j − ∑ j C f jk x f ,r j − T f k ) = 0, ∀( f , k), and it further implies that ∑ f ,k T f k δ f k = ∑ f , j,k δ f k D f jk x f ,p j − ∑ f , j x f ,r j p j .Using this and (4.8) we get ∑ f , j,k δ f k D f jk x f ,p j > ∑ f , j x f ,p j p j , which is a contradiction because (6.1), (6.2) and (6.3) imply that ∑ f , j,k δ f k D f jk x f ,p j ≤ ∑ f , j x f ,p j p j .
Claim 4.14.At every fixed point of F, • 0 < λ i < λ max , ∀i • ∑ i x i j + ∑ f x f ,r j ≤ 1 + ∑ f x f ,p j , ∀ j • x i j < x max , ∀(i, j) p = p case 1: ∑ i x i j + ∑ f x f ,r j ≤ 1 + ∑ f x f ,p j , ∀ j (1.1) p j = 0 and ∑ i x i j + ∑ f x f ,r j ≤ 1 + ∑ f x f ,p j case 2: p j > 0 and ∑ i x i j + ∑ f x f ,r j > 1 + ∑ f x f ,p j (1.2) γ i k = 0 and u i ≤ ∑ j U i jk x i j + T i k case 2: γ i k > 0 and u i > ∑ j U i jk x i j + T i k (2.2) x i j = x max ∑ k U i jk γ i k ≥ λ i p j (3.3) 2) x f ,r j = x r max x f ,p j = x f ,p 2) Table 10: Conditions for a Fixed Point Based on the Update Rules in Table 9 Proof.First suppose that λ i = λ max for some i at a fixed point.It implies that for every good j such that p j ≥ W min/2nx max , we have x i j = 0 (from (3.1)).Hence, ∑ j x i j p j < W min , which contradicts (4.3).Hence 0 ≤ λ i < λ max , ∀i at a fixed point.
Next suppose that λ i = 0 for some i at a fixed point.It implies that for every γ i k > 0 and U i jk > 0, we have x i j = x max (from (3.3)).Note that here we use the sufficiency condition that for every k there exists a j such that U i jk > 0. Since x max is much larger than 1 + ∑ f x f ,p j , we have p j > 0 for such goods and ∑ i x i j + ∑ f x f ,r j > 1 + ∑ f x f ,p j for all goods whose p j > 0 (from (1.2)).By this, we get ∑ i, j x i j p j + ∑ f , j x f ,r j p j >

Claim 4 . 11 .
At every fixed point of F

Table 9 :
FIXP Circuit for Markets with ProductionClaim 4.12.At every fixed point of F, ∑ k