A Rational Convex Program for Linear Arrow-Debreu Markets

A Rational Convex Program for Linear Arrow-Debreu Markets

Nikhil R. Devanur Microsoft Research, Redmond, USA. (nikdev@microsoft.com)    Jugal Garg College of Computing, Georgia Institute of Technology, Atlanta, USA. (jgarg@cc.gatech.edu)    László A. Végh Department of Management, London School of Economics, London, UK. (l.vegh@lse.ac.uk)
Abstract

We give a new, flow-type convex program describing equilibrium solutions to linear Arrow-Debreu markets. Whereas convex formulations were previously known ([17, 15, 6]), our program exhibits several new features. It gives a simple necessary and sufficient condition and a concise proof of the existence and rationality of equilibria, settling an open question raised by Vazirani [21]. As a consequence we also obtain a simple new proof of Mertens’s [16] result that the equilibrium prices form a convex polyhedral set.

1 Introduction

The exchange market model is a classical model of a market along with a notion of equilibrium, introduced by Walras in 1874 [24]. In this model, agents arrive at the market with an initial endowment of divisible goods, and a utility function for consuming goods. A market equilibrium assigns prices to the goods such that when every agent uses the revenue from selling her initial endowment for purchasing a bundle of goods that maximizes her utility, the market clears, i.e, the total demand for every good is equal to its supply. The celebrated theorem by Arrow and Debreu [1] proves the existence of a market equilibrium under mild necessary conditions on the utility functions - therefore it is commonly known as the Arrow-Debreu market model. Since then, understanding equilibrium behavior and computing equilibrium prices has been extensively studied in mathematical economics and more recently in theoretical computer science.

In this paper we study the linear Arrow-Debreu model, where the utility functions of agents are linear. Let us first mention results pertaining to a well-studied further special case, the linear Fisher model, that was formulated by Fisher in 1891, who also studied the computability of equilibrium, via a hydraulic machine no less! (See Brainard and Scarf [4] for a fascinating account.) In this model, the agents are separated into two types, buyers and sellers; buyers arrive to the market with a certain amount of money they wish to spend on goods offered by the sellers. This model turned out to be substantially easier from a computational perspective than the linear Arrow-Debreu model. A convex programming formulation was given by Eisenberg and Gale in 1959 [11]. The problem of equilibrium computation was introduced to the theoretical computer science community by Devanur et al. [8], who gave a polynomial time combinatorial primal-dual algorithm. This initiated an intensive line of research, most notable among which is a strongly polynomial time algorithm by Orlin [19]; for a survey, see [18, Chapter 5] or [21]. Also, Shmyrev [20] gave a new type of convex program (which was discovered independently by Birnbaum et al. [2]) capturing the equilibria.

Let us now turn to the linear Arrow-Debreu model. The first important algorithmic result was a finite Lemke-type path following algorithm for finding an equilibrium solution by Eaves [10]. A remarkable consequence of this algorithm is that when the utilities are given by rational numbers, there also exists an equilibrium among rational numbers.

The history of convex programming formulations for the linear Arrow-Debreu model is somewhat convoluted. Jain [15] formulated a convex program whose solutions correspond to market equilibria; this can be used to obtain a polynomial time algorithm via the Ellipsoid algorithm. It turned out later that the same convex program was already formulated by Nenakov and Primak in 1983 [17]. Interestingly, the computer science community so far seems to have been unaware of the 1989 paper by Cornet in [6] giving a similar, yet better convex program. This is not mentioned even in the survey paper by Codenotti et al. [5] exploring the background of the problem. (These convex programs will be exhibited in Section 3.)

An unsatisfactory aspect of the program in [15, 17] is that it fails to show the existence of an equilibrium; it only shows that if there exists an equilibrium, then any feasible solution to the convex program is one. In contrast, Cornet’s program provides a proof of existence assuming that a stronger sufficient condition given by Gale in 1957 [13] holds, however fails to show it for the weaker necessary and sufficient condition given by Gale in 1976 [14]. An efficient interior point algorithm to compute an equilibrium was given by Ye [25] based on the convex program in [15, 17]. An important recent result is a combinatorial primal-dual algorithm by Duan and Mehlhorn [9]; this does not rely on the convex programming formulation but adapts techniques from the algorithm by Devanur et al. [8] for linear Fisher markets.

The convex programs [17, 15] and [6] for the linear Arrow-Debreu model is of substantially different nature from those [11, 20, 2] for the linear Fisher model. The latter ones have linear constraints only, with separable convex objectives, in contrast to the nonlinear constraints in [17, 15]. Whereas [6] is formulated with only very simple linear constraints, the max-min type objective in fact hides similar nonlinear constraints.

The feasible region for both formulations for the linear Fisher model are indeed classical polyhedra, [11] a generalized flow polyhedron and [20, 2] a circulation polyhedron. This also explains why for the linear Fisher model, classical flow techniques are applicable (see [22, 23]) and strongly polynomial time algorithms exist. Also interestingly, the convex programs of [11] and [20, 2] fall into the class of rational convex programs, defined by Vazirani [21]: for a rational input, there exists a rational optimal solution with bitsize bounded polynomially in the input size. For the previous programs [6, 17, 15] the proof of the existence of a rational optimal solution requires further nontrivial arguments (e.g. [10, 9]). An open problem in [21] asks for the existence of a rational convex program for the linear Arrow-Debreu model with a simple, direct proof of rationality.

In this paper, we exhibit a rational convex program for the linear Arrow-Debreu model, that also guarantees the existence of an equilibrium, thus settling the open questions of [21]. Our convex program draws from the convex programs in [11, 20, 2]; more precisely, it is a combination of the convex program of [20, 2] and the dual of the convex program of [11] (see Devanur [7]). The objective function has terms from both the convex programs and there are two sets of constraints, one describing a circulation polyhedron as in [20, 2], and another that is similar to the dual of the Eisenberg-Gale [11] convex program. The main technical contribution is to show the existence of an equilibrium based on the Karush-Kuhn-Tucker (KKT) conditions for this convex program. Our program is feasible if and only if Gale’s [14] necessary and sufficient conditions on the existence of equilibria hold. The existence of a rational optimal solution for rational input follows by showing that there exists an optimal solution that is an extreme point of the feasible region.

Now we give a formal description of the model and give our convex program. We are given set of agents, and assume that there is a one-to-one mapping between agents and goods, every agent arrives with one divisible unit of good of type . This is without loss of generality: the general case with an arbitrary set of goods and arbitrary initial endowments can be easily reduced to this setting; see Section 3. The utility of agent for the good of agent is . The directed graph contains an arc for every pair with ; it may also contain loops expressing that some agents are interested in their own goods. We make the standard assumption that for each agent , contains at least one incoming and one outgoing arc incident to .

By a market equilibrium, we mean a set of prices and allocations satisfying the following conditions.

  • Market clearing: Demand equals supply.

    • , for every , i.e., every good is fully sold.

    • for every , i.e., the money spent by agent equals his income .

  • Optimal bundle: Every agent is allocated a utility maximizing bundle subject to its budget constraint. That reduces to

    • For every , if then is the maximal value over .

    • for every ;

It is easy to see that the following condition is necessary for the existence of an equilibrium:

For every strongly connected component of the digraph ,
if then there is a loop incident to the node in .

Indeed, assume is a singleton strongly connected component without a loop. Let denote the set of nodes different from that can be reached on a directed path in from . In an equilibrium allocation, the agents in spend all their money on the goods of the agents in ; this implies , contrary to our assumption.

We formulate the following convex program, with variables representing the prices, the ’s the inverse best bang-per-bucks, and the money paid by agent to agent .

Theorem 1.

Consider an instance of the linear Arrow-Debreu market given by the graph and the utilities . The convex program (1) is feasible if and only if (1) holds, and in this case the optimum value is 0, and the prices in an optimal solution give a market equilibrium with allocations . Further, if all utilities are rational numbers, then there exists a market equilibrium with all prices and allocations also rational, of bitsize polynomially bounded in the input size.

Here, the bitsize of the rational number is defined as . The rational optimum property follows by observing that there exists an optimal extremal point solution. The following results easily follow from the above theorem:

Corollary 2.

The following hold for linear Arrow-Debreu markets.

  1. For every agent the utility is the same at every equilibrium.

  2. The vectors at equilibrium form a convex set. In particular, the set of price vectors at equilibrium is convex.

Property (i) was already proved by Gale [14] in 1976 and also follows from Cornet [6]. Whereas the convexity of equilibrium prices was proved by Mertens [16] and by Florig [12], both these proofs are quite involved, whereas it is a straightforward consequence of Theorem 1. We are not aware of previous proofs on the convexity of . In contrast, Cornet [6] proved that is convex at equilibria; here is the amount of good allocated to agent .

The Lagrangian dual of (1) is similar to Cornet’s program [6] (see (CP-C) in Section 3) but is different from it. Also, analyzing the optimal Lagrangian multipliers for (1) we can derive the feasibility convex program [17, 15]; these correspondences will be explained in Section 3. Our program exhibits some new and advantageous features as compared to Cornet’s:

  • The program (1) provides necessary and sufficient condition of the existence of equilibria. In contrast, Cornet’s program provides only a stronger sufficient condition given by Gale [13].

  • The program (1) is feasible if and only if there exists an equilibrium. In contrast, Cornet’s program can be feasible also if there exists no equilibrium; in this case the objective is unbounded.

  • The program (1) also demonstrates the existence of a rational equilibrium, which is not the case with Cornet’s program.

  • All constraints in (1) are linear.

  • Our program establishes links to known convex programs for the Fisher model.

We think that the discovery of this convex program will pave the way for more efficient (and in particular, strongly polynomial time) algorithms for this model.

The rest of the paper is structured as follows. Section 2 is dedicated to the proof of Theorem 1. This is based on the KKT conditions, however, the argument is not straightforward, in contrast to similar arguments for the convex programs of [11, 20, 2]. Section 3 shows the equivalence of our existence condition (1) to previous results by Gale [13, 14], exhibits the previous convex programs [6, 17, 15], and explains the correspondence between our formulation (1) and these programs.

2 Proof of Theorem 1

Let us first verify that (1) is actually a convex program. The feasible region is defined by linear constraints, so we only have to check that the objective is convex. The terms corresponding to the ’s are linear. The term is the relative entropy of and and is well-known to be convex in the nonnegative variables and .111Let us give a simple proof. We need to verify that for every and , we have This can be derived using the convexity of for , with the linear combination . Let us now verify the feasibility claim.

Claim 3.

The convex program (1) is feasible if and only if (1) holds.

Proof.

Assume that (1) is violated, that is, there is a strongly connected component consisting of a single node , and there is no loop in incident to (that is, .) For a contradiction, assume (1) admits a feasible solution . Then gives a feasible circulation on the graph such that there is a positive amount of flow entering (and leaving) every node. The circulation can be decomposed to a weighted sum of directed cycles: , where for each , is the 0-1 incidence vector of a directed cycle , and . Clearly every cycle must be contained inside a strongly connected component. Hence no cycle may be incident to , that is, the flow entering this node is 0, a contradiction.

Assume now that (1) is satisfied. Consequently, there is a directed cycle in incident to every node . Set , and let denote the amount of entering the node . This gives a feasible solution to (1) with . ∎

Claim 4.

The objective value in 1) is non-negative, and it is 0 if and only if the prices and the allocations form a market equilibrium. Conversely, for every market equilibrium , we get an optimal solution to 1) by setting , , , where .

Proof.

By the third inequality, . Hence the second term in the objective is at least

This implies that the objective value is . Moreover, the lower bound is tight if and only if whenever . This is equivalent to all transactions being best bang-per-buck purchases. It is easy to verify that the solution represents a market equilibrium. The second part also follows easily. ∎

The proof of the assertion in Theorem 1 that optimal solutions to (1) correspond to market equilibria is complete by the following lemma.

Lemma 5.

Whenever (1) is feasible, the optimum value is 0.

Let us now formulate the Karush-Kuhn-Tucker conditions on optimality. Since all constraints in (1) are linear, these are sufficient and feasible for optimality. Consider an optimal solution , and let us associate Lagrangian multipliers , , and to the inequalities in the order as described in (1). We obtain the following conditions.

(1)
(2)
(3)

Also, (1) must be tight for all , and (3) must be tight for all . Further, implies , and implies . Note that in an optimal solution every , and hence (3) always holds with equality. We can therefore derive the following from (3):

(4)

The following remark can be interpreted as a “self-duality” property: a market equilibrium does not only provide a primal optimal solution to (1) but also optimal Lagrangian multipliers.

Remark 6.

Assume there exists a market equilibrium ; by re-scaling, we may assume for all . As in Claim 4, , and give an optimal solution to (1). It is straightforward to check that , , and give optimal Lagrangian multipliers.

The next claim expresses the optimum objective value of (1) in terms of the Lagrangian multipliers.

Claim 7.

Let be a primal optimal solution, and let be optimal Lagrangian multipliers. Then

Proof.

By complementary slackness, (1) is tight whenever . Taking the combination of these equalities multiplied by , we get

In the second equality, we used the degree constraints in (1). Next, let us add the equalities (2) multiplied by . We obtain

Here we used (4) for the second term, and that whenever for the third term. Adding this to the previous inequality proves the claim. ∎

Using the previous claim, Lemma 5 follows from the next lemma.

Lemma 8.

For the optimal Lagrange multipliers , it follows that

Proof.

The proof is by induction on the number of agents . We assume that for all markets with agents, the assertion holds. Let us introduce and . These are quantities playing a similar role to and : the conditions (1) can be rewritten as

and furthermore by complementary slackness it follows that if then equality must hold. The ’s are therefore the inverse best bang-per-buck values for the prices . Let denote the set of arcs with and the set of arcs with . By complementary slackness, and . Let us define

Claim 9.

We have for every . Further, if , and , then holds.

Proof.

The first claim is equivalent to if . This follows since

For the second part, assume for a contradiction that for some best bang-per-back arc with . This would imply that the inequality above is strict, giving a contradiction. ∎

Together with (2), this gives

(5)

with equality only if . Let

denote the sets of agents having all their best bang-per-buck goods in with respect to prices . Recall that . By the definition of , we get from (4) that

(6)

Combining this with the straightforward , for all , we obtain

Rearranging the sums gives

(7)

The next step requires the following observation.

Claim 10.

For every arc with , it follows that .

Proof.

For a contradiction, assume , that is, there exists a good with . Then

a contradiction. ∎

Recall that , and therefore if and , then must hold by the above Claim. Hence if , then . Combining this with (5) and (7), we get

(8)

We must have equality throughout, and therefore for all it follows that and ; the latter was a necessary condition for equality in (5). Now the second part of Claim 9 guarantees that .

Using (6), we have . On the other hand, the above equalities guarantee . We can therefore conclude . Moreover, the following holds.

Claim 11.

No arc in enters or leaves the set .

Proof.

Recall that and . Since , the definition of implies that no arc leaves ; recall that . The second inequality in (8) must hold with equality, implying that , whenever , . Claim 10 implies that no arc enters , and . The first to equalities in (1) imply that . Hence no arc with may leave . ∎

If , then the proof of Lemma 8 is complete. If , then consider the restrictions of and to , and to the arcs inside . The first gives a feasible solution to (1) on the restricted graph, whereas the second give optimal Lagrange multipliers, since the primal-dual slackness conditions are satisfied. According to our assumption on being a minimal counterexample, it follows that for all , completing the proof. ∎

To complete the proof of Theorem 1, it is left to verify the claim on the existence of a rational optimal solution. This will follow from the next structural observation; note that the feasible region is a polyhedron.

Claim 12.

There exists an optimal solution to (1) that is an extremal point of the feasible region.

Proof.

Consider an optimal solution to (1); by the above, we know that it corresponds to a market equilibrium. As every point in the feasible region, can be written as the sum of extremal rays and a convex combination of extremal points. Pick an arbitrary extremal point from the combination. We claim that this is also an optimal solution to (1). By Claim 4, it suffices to show that it corresponds to a market equilibrium, which is equivalent to whenever . For a contradiction, assume and holds for an . Since is included in the convex combination giving , every strict inequality for must also be strict for ; this would contradict the optimality of . ∎

Since every extremal point of a rational polyhedron is rational with polynomially bounded size, the proof of Theorem 1 is complete. Next we derive the bound on the values of equilibrium prices and allocation. For this, we assume that all ’s are integers, since scaling them by a constant does not change the equilibrium.

Lemma 13.

Assume all utilities are integers and we let . Then there exists equilibrium prices that are quotients of two integers , along with allocations that are quotients of two integers .

Proof.

From Claim 12, an optimal solution to (1) is achieved at an extremal point, say , of the associated polyhedron. Let denote the number of non-zero ’s at . We claim that . Indeed, consider the bipartite graph , where , . If this graph contains a cycle, then the ’s can be modified such that every binding constraint remains binding and we get one more pair with , in a contradiction with being a vertex.

Let denote a subset of binding constraints for in the linear system defining the feasible region of (1), after removing the columns corresponding to the variables. The number of columns is . Note that the equalities corresponding to the nodes are linearly dependent, and therefore the rank of the matrix is at most .

By Cramer’s rule, every , and is quotient of two integers bounded by the maximum sub-determinant of . Using Hadamard’s bound, this is at most the product of the largest column norms of . Note that , as the only constraints containing nonzero constants are the inequalities. The norm of each of the columns corresponding to the variables is as each is contained in two constraints with coefficient 1. Similarly, the norm of each of the columns corresponding to the ’s is at most , and the norm of each of columns corresponding to the ’s is at most . We need the largest columns and therefore may remove one of those of norm 2. From this, we can conclude that every and is quotient of two integers bounded by . Since the allocation , we get that every is quotient of two integers bounded by . ∎

Remark 14.

The above bound can be further strengthened to .

3 Relation to previous work

3.1 Existence results

The Arrow-Debreu market is traditionally formulated in a more general setting. Besides the set of agents , there is a set of goods , and each agent arrives to the market with an initial endowment of good . A market is given as . Our setting corresponds to the special case when , and if and 0 otherwise. We shall refer to our special case as bijective markets.

Again, a market equilibrium consists of prices and allocations of goods , where represents the amount of good sold by agent to agent such that:

  • , , i.e., every good of every agent is fully sold.

  • For every , whenever for some and , then is the maximal value over .

  • , , that is, the money spent by agent equals his income.

  • for every .

The general case can be easily reduced to bijective markets (see e.g. Jain [15]). First if a good is included in the initial endowment of multiple agents, we give a different name for each such occurrence. If an agent has goods in the endowment, we split the agent into copies with the same utility function, each owning one of the goods.

Consider now a market in the general form . We say that a subset of agents is self-sufficient whenever , for some implies that . That is, agents in are not interested in the goods owned by agents not in . We say that a market is irreducible if there exists no self-sufficient proper subset of the agents. The following sufficient condition was given by Gale in 1957:

Theorem 15 ([13]).

If the market is irreducible then there exists an equilibrium.

The above condition is sufficient but not necessary. Later, in 1976 Gale [14] gave a strengthening of the above theorem. We say that a subset of agents is super self-sufficient if in addition to above, such that and . That is, an agent in owns a good for which no agent in is interested.

Theorem 16.

[Existence Theorem [14]] There exists an equilibrium in the market if and only if no subset of is super self-sufficient.

We show that in our special case of bijective markets (i.e. , and if and 0 otherwise), the existence condition in Theorem 1 is equivalent to that in Theorem 16.

Lemma 17.

A bijective market is irreducible if and only if the directed graph is strongly connected. Further, (1) holds if and only if no subset of is super self-sufficient.

Proof.

The first part follows since in a bijective market a subset of agents is self-sufficient if and only if no arc enters in the directed graph . For the second part, assume first that (1) is violated for node , and let denote the set of nodes different from that can be reached on a directed path in from . Now let . It is easy to check that is super self-sufficient, since and .

Conversely, assume there exists a super self-sufficient set . According to the condition, there exist , such that and . Clearly is a singleton component with no self-loop in the strongly connected components of graph , verifying (1). ∎

3.2 Previous convex programs

Let us first exhibit Cornet’s convex program [6]. It was originally given for the general case of arbitrary endowments, but we present it here for bijective markets. Also, it was originally formulated with a max-min objective over the feasible region for all , ; we unfold the max-min objective here in the natural way. The variable corresponds to the amount of good purchased by agent , whereas corresponds to the logarithm of the price of good .

(CP-C)
Theorem 18 ([6]).

If (CP-C) is bounded then , and is an optimal solution if and only if corresponds to a market equilibrium where for all . Further, if the market is irreducible then (CP-C) is bounded.

The proof uses a nontrivial argument on Lagrangian duality. Note that the existence of equilibrium follows on under Gale’s sufficient condition from 1957 (Theorem 15), as opposed to (1), where it follows under the necessary and sufficient condition in Theorem 16.

According to Theorem 18 and Lemma 17, if the market is irreducible then , and must hold for every . By taking logarithms we get that the following convex program has a feasible solution:

(CP-J)

This is precisely the convex program by Nenakov and Primak [17], and by Jain [15].

We can write the Lagrangian dual of our program (1), see Boyd and Vandenberghe [3]. This gives

(CP-D)

Note that the variables in an optimal solution correspond to optimal Lagrangian multipliers satisfying the KKT-conditions (1)-(3). Theorem 1 implies that strong duality holds: if (1) is feasible then there exists a market equilibrium, that easily provides a solution (CP-D).

Despite certain similarities, this formulation appears to be different from (CP-C), namely, it has a larger feasible region. Indeed, for every feasible solution of (CP-C), , , gives a feasible solution to (CP-D). Nevertheless, the converse is not true since may not hold for feasible solutions of (CP-D).

We further note that following the argument of Section 2, we can derive the feasibility of (CP-C). It follows that in an optimal solution we must have and for all . Using these, we can substitute , . This yields a feasible solution to (CP-C).

References

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