CConstrained Trading Networks
Can Kizilkale & Rakesh Vohra June 22, 2020
Abstract
Trades based on bilateral (indivisible) contracts can be represented by a network.Vertices correspond to agents while arcs represent the non-price elements of a bilateralcontract. Given prices for each arc, agents choose the incident arcs that maximizetheir utility. We enlarge the model to allow for polymatroidal constraints on the setof contracts that may be traded which can be interpreted as modeling limited one-for-one substitution. We show that for two-sided markets there exists a competitiveequilibrium however for multi-sided markets this may not be possible.
Trades based on bilateral (indivisible) contracts are represented by a network. The verticescorrespond to agents while the arcs represent the non-price elements of a bilateral contract.The arc’s orientation identifies which agent is the “buyer” and which the “seller”. Themodel is rich enough to allow an agent to be a buyer in some trades and a seller in others.It subsumes the classic assignment model (Shapley and Shubik 1971). Given prices for eacharc/trade, an agent chooses the subset of incident arcs that maximize her utility. A centralquestion is whether a competitive equilibrium in this economy exists.In general it does not. Assuming quasi-linearity and a full substitutability conditionon agents’ preferences, (John William Hatfield et al. 2013) have shown that a competitiveequilibrium exists. Full substitutability is a generalization of the gross-substitutes prop-erty introduced by Kelso-Crawford and is equivalent to valuations over contracts being M � concave (Fujishige and Yang 2003, Murota and Tamura 2003). Competitive equilibria oftrading networks are also stable outcomes in that they cannot be blocked by any coalition ofagents and trades. A blocking set is a set of contracts and corresponding prices such that allagents party to these contracts (strictly) prefer them (while possibly declining some of theirequilibrium contracts)(John W Hatfield et al. 2013). Conversely, in any stable outcome it is Research supported in part by DARPA grant HR001118S0045. any subset of contracts to which she is a party to. In otherwords, a vertex is free to choose any subset of its incident arcs. This is unsatisfying if thecontracts under consideration involve the exchange of resources in limited supply. Certaincombinations of contracts could be infeasible. This can be modeled by defining an agentsvalue for infeasible subsets of contracts to be −∞ , which may result in a violation of M � concavity.In this paper we enlarge the trading networks model to allow for explicit constraints on theset of contracts that may be traded. It is obvious that for arbitrary constraints, it is unlikelythat a competitive equilibrium will exist. Therefore, we restrict attention to constraints thatcan be represented by polymatroids. Polymatroids are polyhedrons associated with integervalued submodular functions. They play an important role in combinatorial optimizationbut also make an appearance in some market design settings. In the context of tradingnetworks they can be interpreted as modeling limited one-for-one substitution. Informally,the marginal rate of technical substitution between two products evaluated at any bundle iseither zero or one. This interpretation appears in (Milgrom 2009).The first contribution of this paper is establishing the existence of a competitive equilib-rium for two sided markets, ones in which agents are either sellers or buyers but not both. Closest comparable results we are aware of are (Kojima, Sun, and Yu 2018) and (Gul, Pe-sendorfer, and Zhang 2019). Both consider one-sided markets, the first without transfersand the second with. Theorem 3 of the second paper is a special case of our result. Oursecond result shows that existence of a competitive equilibrium fails in multi-sided settings.The next section introduces the notation used in this paper. The subsequent sectionsummarizes result needed from the literature on discrete convexity. The remaining sectionsdescribe our main results.
A trading network is represented by a directed multigraph G = ( N, E ) where N is the setof vertices and E the set of arcs. Each vertex i ∈ N corresponds to an agent and each arc e ∈ E corresponds to the non-price elements of a trade that can take place between theincident pair of vertices. For each e ∈ E , the source vertex e + corresponds to the seller andthe sink vertex e − corresponds to the buyer in the trade. Let δ + ( i ) and δ − ( i ) be the outgoingand incoming arcs incident to vertex i ∈ N , and δ ( i ) = δ + ( i ) ∪ δ − ( i ). So as to avoid havingto distinguishing between a trade involving one unit of a good and two units of the same This was already observed in (John W Hatfield et al. 2013). This is equivalent to a multi-sided setting where the utility of the corresponding agents is separable overthe contracts it buys and sells though. outcome is any vector x ∈ Z E + where x e denotes the number of copies of trade e that were executed. We define aprice vector p ∈ R E , where p e is the price associated with the trade that corresponds to thearc e . Denote by p X the price vector restricted to the arcs in X .It is more convenient to represent the volume of trade associated with arc e betweenagents i and j using two variables rather than one: y ie and y je . If e ∈ δ + ( i ) then, y ie ≥ y ie ≤
0. We will call the y variables net-flows .Every x ∈ Z E + that is an outcome, corresponds to a new flow. For any arc e betweenagents i and j we have | y ie | = | y je | = x e and y ie + y je = 0. A net-flow y = ( y , . . . , y | N | ) is calledfeasible if it corresponds to an outcome, i.e., for every arc e we have y ie + y je = 0 whenever e ∈ δ + ( i ) ∩ δ − ( j ). If y i is the incidence vector of trades in which i is a part of, the valuefunction of agent i will be written as w i : y i → R ∪ {−∞} . Agent i ’s surplus or utility forthe vector of trades y i will be u i ( y i , p ) = w i ( y i ) + � e ∈ δ + ( i ) y ie p e + � e ∈ δ − ( i ) y ie p e . Given a price vector p ∈ R δ ( i ) , agent i ∈ N ’s demand correspondence is D i ( p ) = arg max { u i ( y i , p ) : y i ∈ Z δ + ( i )+ × Z δ − ( i ) − } . Definition 2.1
A feasible net-flow y = ( y , . . . , y | N | ) along with a price vector p ∈ R E is a competitive equilibrium ( y, p ) if, for all i ∈ N , y i ∈ D i ( p ) . Definition 2.2 An efficient outcome is one that solves the following problem: max � i ∈ N w i ( y i ) s.t. y ie + y je = 0 ∀ i = e + , j = e − ∀ e ∈ Ey ie ∈ Z + ∀ e ∈ δ + ( i ) ∀ iy ie ∈ Z − ∀ e ∈ δ − ( i ) ∀ i Under quasi-linearity, every competitive equilibrium outcome y is efficient and conversely.Further, if y is efficient, there exists a price vector p such that the pair ( y, p ) is a competitiveequilibrium. In this section we review definitions and properties of M -convex sets/functions. For a moreextensive discussion see (Murota 2003b). For all y ∈ Z n we define supp + ( y ) = { i | y i > } and supp − ( y ) = { i | y i < } . The notion of M − convexity is based on the exchanges betweenthe sets supp + and supp − . Let’s start by defining the ”M-convex” set. Having −∞ in the range of the value function allows for incorporating trading constraints as describedin John W Hatfield et al. 2013, e.g., if a trader cannot sell goods without procuring its inputs first, this canbe incorporated by specifying −∞ for bundles of trades where this happens. efinition 2.3 A set B ⊆ Z E is called M-convex if for all x, y ∈ B and for all u ∈ supp + ( x − y ) there exists a v ∈ supp − ( x − y ) such that x − χ u + χ v and y + χ u − χ v are bothin the set B . In definition 2.3, χ u refers to the basis vector where the u th element is one and all others arezero. M-convex functions are defined similarly in (2.4). Definition 2.4
A function f : Z E → R for an M-convex domain B is called M-convex iffor all x, y ∈ B , ∀ u ∈ supp + ( x − y ) there exists a v ∈ supp − ( x − y ) such that f ( x ) + f ( y ) ≥ f ( x − χ u + χ v ) + f ( y + χ u − χ v ) . Definition 2.5
A function f, is called M-concave if − f is M-convex. M-convexity has strong ties with submodularity. A real valued submodular function f definedon E is one that satisfies f ( S ) + f ( T ) ≥ f ( S ∪ T ) + f ( S ∩ T ) for all S, T ⊆ E . Theorem 2.1
For any M-convex set B , the function f ( S ) = max { � j ∈ S x j | x ∈ B } forevery S ⊆ E is a submodular function. Theorem 2.1 gives a connection between M-convex sets and integral submodular setfunctions. The converse is also true.
Corollary 2.1 If f is an integer valued submodular function on E , then there exists anM-convex set B ⊆ Z E such that f ( S ) = max { � j ∈ S x j | x ∈ B } . If f is a submodular function on E , then, the polymatroid associated with f is thepolyhedron P f = { x ∈ R E : � e ∈ S x e ≤ f ( S ) ∀ S ⊆ E } . It is called an integral polymatroidif f is integer valued. In fact, the M-convex set B in Corollary 2.1 is precisely P f .The definition of M-convexity relies on the exchange between two different elements andit can be weakened to cover single element exchanges. We start with the set definition first. Definition 2.6
A set B ⊆ Z E is called M � -convex if for all x, y ∈ B and for all u ∈ supp + ( x − y ) we have one of the following: x − χ u and y + χ u are both in B .There exists a v ∈ supp − ( x − y ) such that x − χ u + χ v and y + χ u − χ v are both in theset B . M � − convex functions are defined as follows. Definition 2.7
A function f : Z E → R is called M � -convex if for all x, y ∈ B , ∀ u ∈ supp + ( x − y ) we have one of the following. f ( x ) + f ( y ) ≥ f ( x − χ u ) + f ( y + χ u ) .There exists a v ∈ supp − ( x − y ) such that f ( x ) + f ( y ) ≥ f ( x − χ u + χ v ) + f ( y + χ u − χ v ) . f : Z E → R is called M � -concave if − f is M � -convex. The domain of a function f : Z E → R is defined to be domf = { x ∈ Z n | f ( x ) < ∞} . A consequence of a functionbeing M � -convex is the following: Lemma 2.1
The domain of an M � -convex function is an M � -convex set. Definition 2.8
The convex-extension of a function is as follows (the intersection of thesupporting hyperplanes of the epigraph): ˆ f ( x ) = min { � z λ z f ( z ) | � z λ z z = x, ≤ λ z ≤ , � z λ z = 1 } We can define concave-extension similarly.
Definition 2.9 ˆ f ( x ) is the concave-extension of f ( x ) if − ˆ f ( x ) is the convex-extensionof − f ( x ) . Suppose each agent’s valuation function, w i is M � -concave and let f ( y ) = � i w i ( y i ). Note, f ( y ) is also M � concave because the argument of each w i are disjoint. The problem of findingan efficient outcome in the trading network setting can be formulated as:maximize : f ( y ) s.t. y ∈ B. Here B represents the set of feasible trades: y ie + y je = 0 ∀ i = e + , j = e − ∀ e ∈ Ey ie ∈ Z + ∀ e ∈ δ + ( i ) ∀ iy ie ∈ Z − ∀ e ∈ δ − ( i ) ∀ i A convex relaxation of this problem is:maximize : ˆ f ( y ) s.t. y ∈ B ∗ (1)where ˆ f ( y ) is the concave extension (definition 2.8) of the objective function f ( y ) and B ∗ isthe set of solutions to y ie + y je = 0 ∀ i = e + , j = e − ∀ e ∈ Ey ie ≥ ∀ e ∈ δ + ( i ) ∀ iy ie ≤ ∀ e ∈ δ − ( i ) ∀ i It is known that problem (1) has an integral solution (Murota 2003a), the Lagrange multi-pliers associated with the inequality representation of B will be supporting prices and thisestablishes the existence of a competitive equilibrium.5he set B ∗ in problem (1) had a particular structure one might wonder if integrality ofthe convex relaxation might hold more generally. Here we show that it does if the set ofnet-flows is constrained to lie in an integral polymatroid, denoted P . Consider:maximize ˆ f ( y ) s.t. y ∈ P (2)We show problem 2 has an integer optimal. This result, is, as far as we know new, howeverspecial cases of it do appear, for example (Gul, Pesendorfer, and Zhang 2019).Let us recall the following theorem. Theorem 2.2 (Thm 4.22 Murota 2003a) For M-convex sets B and B the convex closureof their intersection is equal to the intersection of their convex closures, that is ¯ B ∩ ¯ B = ( B ∪ B ) . Definition 2.10 B f ( x ) = { z ∈ N ( x ) �∃ λ, � z λ z = 1 , � z λ z z = x, � z λ z f ( z ) = ˆ f ( x ) } where N ( x ) is the unit hypercube containing x and ˆ f ( x ) is the convex extension of function f . B f ( x ) will have an important role in our analysis. The convex extension of function f ( y ) isa piece-wise linear function and B f ( x ) represent the maximal set that contains x such thatconvex extension is linear. This can be viewed as the facets of the epigraph. Lemma 2.2
For every M � concave function f and x in the convex hull of the domain of f , B f ( x ) is an M-convex set. Proof.
Let x , x ∈ B f ( x ). Since f ( x ) is an M � concave function there exists u ∈ supp + ( x − x ) and v ∈ supp − ( x − x ) such that, f ( x − χ u + χ v ) + f ( x + χ u − χ v ) ≥ f ( x ) + f ( x ) . Now take convex coefficients λ , λ such that � z ∈ N ( x ) λ z z = � z ∈ N ( x ) λ z z = x and � z ∈ N ( x ) λ i f ( z ) = ˆ f ( x ) (by definition such a λ i exists). Then for λ = λ + λ we have � z λ z z = x , � z ∈ N ( x ) f ( z ) = ˆ f ( x ), λ x > λ x >
0. If we set λ δ = min { λ x , λ x } weobtain the following inequalityˆ f ( x ) ≥ [ � z ∈ N ( x ) −{ x ,x } λ z f ( z )]+ λ δ ( f ( x − χ u + χ v )+ f ( x + χ u − χ v ))+( λ x − λ δ ) f ( x )+( λ x − λ δ ) f ( x )6 � z ∈ N ( x ) f ( z ) = ˆ f ( x ) . But then, by definition ( x − χ u + χ v ) , ( x + χ u − χ v ) are both in B f ( x ) hence B f ( x ) is anM-convex set. Corollary 2.2
For all probability vectors λ such that � z ∈ B f ( x ) λ z z = x we have ˆ f ( x ) = � z λ z f ( z ) . Proof follows trivially from the definition of B f ( x ). Theorem 2.3 If f is M -concave function and P is an integral polymatroid, then, thereexists an integer optimizer for problem 2. Proof.
Let x be an optimizer for problem 2. If N ( x ) ⊆ B then we are done since at least oneextreme point of N ( x ) will also be an optimizer. Else let F x be the face of B that contains x . F x is an M-convex set (as we have seen earlier M-convex sets are essentially polymatroids).From lemma 2.2 we know that B f ( x ) is an M-convex set and from theorem 2.2 we have x ∈ F x ∩ B f ( x ) = ( F x ∪ B f ( x )) . But then there exists convex coefficients λ such that � z ∈ ( F x ∪ B f ( x )) λ z z = x. From corollary 2.2 we have � z ∈ ( F x ∪ B f ( x )) λ z f ( z ) = ˆ f ( x ) . But then there exists a y ∗ ∈ ( F x ∪ B f ( x )) such that f ( y ∗ ) ≤ ˆ f ( x ) and since x is an optimizer, y ∗ has to be an optimizer too. Since y ∗ is integral this concludes the proof. Consider the case of one buyer (agent 1) and one seller (agent 2). Thus, the vertex cor-responding to agent 1 has no outgoing arcs only incoming ones. The vertex correspondingto agent 2 has no incoming arcs only outgoing arcs. Hence, y = − y = x where x is anoutcome. Assume that x is constrained to a polymatroid P . w ( y ) is M -concave but w ( y ) is linear. Denote by the ˆ w i the convex extension of w i . Our objective is to maximizeˆ w ( y ) + ˆ w ( y ) = ˆ w ( x ) + ˆ w ( − x )over x ∈ P where ˆ w i is the concave extension of w i .7 emma 3.1 The maximizer(s) of ˆ w ( y ) + ˆ w ( y ) = ˆ w ( x ) + ˆ w ( − x ) over x ∈ P are integral. Proof.
Let x be a maximizer of the objective. Observe that w is linear over the set B w ( x ).Then w ( x ) + w ( − x ) is linear over the set B w ∩ P (both w and w are linear on this set).But then, this is a linear program and there exists an extreme point of B w ∩ P that achievesthe optimum, since the intersection of two integral M-convex sets is an integral polyhedraall of whose extreme points are integral. The proof follows.We generalize Lemma 3.1 to trade networks where each agents utility w i ( y i ) is separableas follows. w i ( y i ) = w + i ( y i + ) + w − i ( y i − )where w + i is M � -concave, w − i is linear and y i + , y i − represent the net-flows that correspondto the outgoing/incoming edges respectively. The problem of finding an efficient allocation,denoted problem EA , can be written as follows.max � i ( w + i ( y i + ) + w − i ( y i − )) (3)s.t. x ∈ P (4) y ie = x e = − y je , ∀ e ∈ δ + ( i ) ∩ δ − ( j ) (5)where P is an integral polymatroid. Theorem 3.1
Problem EA has an integral optimum. Proof.
For i � = j we have δ + ( i ) ∩ δ + ( j ) = ∅ and supp ( y i − ) ∩ supp ( y j − ) = ∅ . Then, � i ( w + i ( y ( δ + ( i )))(represented by w + ( x )) is an M -concave function while � i w − i ( y ( δ − ( i )))(representedby w − ( − x )) is linear. Substituting w + and w − into the objective function allows us to invokeLemma (3.1) to conclude the proof.The separability requirement clearly holds in two sided markets where an agent is either abuyer or a seller but not both. Hence, buyers can have M � -utilities over the edges they ‘con-sume’ while sellers have linear costs over the edges they ‘sell’. The separability requirementcan be relaxed. For all i let the utility function be w i ( y i ) = w i ( y i + , y i − ) such that when y i + iskept constant the function is linear over y i − and when y i − is kept constant it is M � -concaveover y i + . In other words, when the incoming/outgoing links are kept constant, the functionis M � -concave/linear on the outgoing/incoming edges.8 Impossibility
In this section we show that if we relax separability and allow the market to be multi-sided,then, a competitive equilibrium need not exist.Let the outcome vector x ∈ Z E is constrained to lie in a polymatroid. This means thattrade between two distinct pairs of agents can be linked to each other. For example, thevolume of trade that a pair a, b can conduct could be limited by the volume of trade thatanother pair, c, d , conduct. This can happen if the execution of trades require the use ofshared infrastructure. For instance, a could be selling electricity to b and similarly c is sellingelectricity to d . If they use a common grid electricity to distribute power, then, the tradesof one pair constrain the trades that the other pair can engage in.We append to problem (1) the requirement that the outcome vector, x ∈ Z E , lie in apolymatroid. arbitrary polymatroidal constraints on the links ignoring their orientation.We examine whether this augmented problem has an integral solution. The example belowshows this is not the case and demonstrates that a competitive equilibrium in this settingdoes not exist. Example 1
Consider the network given in figure 1 with two agents, N , N . Each agenthas one incoming and one outgoing link denoted e and g . The value functions of each agentare displayed below. They are M � -concave. w ( y e , y g ) = (6) − . y e = 0 , y g = 0 (7) − y e = 1 , y g = − − y e = 1 , y g = 0 (9) − y e = 0 , y g = − − ∞ elsewhere. (11) w ( y e , y g ) = (12) − . y e = 0 , y = 0 (13) − y g = 1 , y e = − − y g = 1 , y e = 0 (15) − y g = 0 , y e = − − ∞ elsewhere. (17) Since the arguments of the function w and w are disjoint, the function f ( y ) = w ( y e , y g ) + w ( y e , y g ) is also M � -Concave. Consider the polymatroid on x ∈ Z E defined below: P = { x � x e ≤ , x g ≤ , x e + x g ≤ } . The net-flows corresponding to x are y e = − y e = x e and − y g = y g = x g . Given an outcome x , denote by y ( x ) the corresponding net-flows. Then, we have f ( y (0 , − , f ( y (1 , , f ( y (1 , f ( y (0 , − . However, max x ∈ P ˆ f ( y ( x )) ≥ / f ( y (0 , / f ( y (1 , − . and this is larger than each integral feasible flow which will lead to a larger objective value.This shows that we cannot have an integral efficient outcome. In trading networks even relatively simple constraints on the set of feasible trades are anobstacle to the existence of a competitive equilibrium. Under polymatroidal constraintsseparability of the utility function is crucial for the existence of competitive equilibrium.
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