AA Theory of Market Efficiency
Anup RaoUniversity of Washington [email protected]
February 7, 2017
Abstract
We introduce a mathematical theory called market connectivity that gives concrete ways toboth measure the efficiency of markets and find inefficiencies in large markets. The theory leadsto new methods for testing the famous efficient markets hypothesis that do not suffer from the joint-hypothesis problem that has plagued past work. Our theory suggests metrics that can beused to compare the efficiency of one market with another, to find inefficiencies that may beprofitable to exploit, and to evaluate the impact of policy and regulations on market efficiency.A market’s efficiency is tied to its ability to communicate information relevant to marketparticipants. Market connectivity calculates the speed and reliability with which this communi-cation is carried out via trade in the market. We model the market by a network called the tradenetwork , which can be computed by recording transactions in the market over a fixed interval oftime. The nodes of the network correspond to participants in the market. Every pair of nodesthat trades in the market is connected by an edge that is weighted by the rate of trade, andassociated with a vector that represents the type of item that is bought or sold.We evaluate the ability of the market to communicate by considering how it deals with shocks . A shock is a change in the beliefs of market participants about the value of the productsthat they trade. We compute the effect of every potential significant shock on trade in themarket. We give mathematical definitions for a few concepts: • The tension and energy of the network are related concepts that measure the strengthof the connections between sets of participants that trade similar items. They measurethe amount of trade that is affected by significant shocks. They are high when there aremany paths of high rate of trade that connect those with differing beliefs about the valueof items. They are low when information from some large set of participants must take along time to reach some other large set via trade. • A bottleneck in the network is a small set of nodes that monopolizes an unusually largeshare of the trade in the network. The nodes in the bottleneck have an incentive to setprices incorrectly and interfere with the fair transmission of information in the market.We give explicit mathematical definitions that capture these concepts and allow for quantitativemeasurements of market inefficiency. a r X i v : . [ q -f i n . E C ] F e b ontents Introduction
Market-based economies have come to be the dominant system for the production and distributionof goods and services. The power of markets stems from their decentralized nature [Hay45]. Amarket is able to channel the effort of individuals in fruitful directions, even though no individualis aware of all the information necessary to decide how their effort ought to be expended. Complexdecisions about how to react to scarcities, surpluses, and spikes in demand are handled without theintervention of a central entity. Over time, markets have proven themselves to be adept at takingadvantage of economies of scale, and forming reliable and efficient distribution networks.The forces of self-interest, competition, and the balancing of supply and demand all exertstabilizing influences on prices in the market. They suggest the concept of an efficient market ,defined by Fama [Fam70, Fam91] as follows:
Definition 1. An efficient market is one in which prices always fully reflect available information. It has long been theorized that financial markets are efficient, or close to being efficient. This isthe well-known efficient markets hypothesis [Fam70, Fam91]. The efficient markets hypothesis is aseductive idea, an idea that is simultaneously deep and accessible. The hypothesis is supported bycompelling intuition—if prices did not accurately reflect all information, then traders would havean opportunity to exploit the discrepancy to make a profit, which would lead to prices becomingmore accurate. So, prices should not drift far from being correct. As Malkiel notes, “[. . .] [W]heninformation arises, the news spreads very quickly and is incorporated into the prices of securitieswithout delay.” [Mal03, p. 59]However, much is left open to interpretation in this reasoning and in Definition 1. What doesit really mean that prices fully reflect information? What is the mechanism by which informationspreads? If a market is not efficient, how can one quantify its inefficiency? Exactly what informationis available to the market and what information is not available? These questions are not addressedby Definition 1 or the reasoning associated with the efficient markets hypothesis. Even if we knew all information, and we knew what information is available to all market participants, we mightfind it difficult to agree on what prices ought to be. If a tree falls in a forest and no one is aroundto hear it, should the price of wood change? What if a woodcutter hears the tree falling, how muchshould the price of wood change then? What if a carpenter hears the tree falling, how much shouldthe price change then?Still, the efficient markets hypothesis is extremely useful, and an extensive theory has beendeveloped to test it. The definitional problems discussed above lead to a sticking point in thisestablished theory, called the joint-hypothesis problem . As Fama explains it:[. . .] [M]arket efficiency per se is not testable. It must be tested jointly with some modelof equilibrium, an asset-pricing model. [. . .] [W]e can only test whether information isproperly reflected in prices in the context of a pricing model that defines the meaning of“properly.” As a result, when we find anomalous evidence on the behavior of returns, theway it should be split between market inefficiency or a bad model of market equilibriumis ambiguous [Fam91, p. 1575].The aim of this work is to develop a rigorous mathematical theory that can concretely quantifymarket inefficiencies and so replace Definition 1 with concepts that are testable. We wish toeliminate the joint-hypothesis problem and minimize the guesswork involved in deciding whether3r not prices properly take into account information that they should. We seek a model that canbe calibrated with data from the real world, and analyzed with computers to find and quantifyinefficiencies.Such a model could have far reaching consequences. Catastrophic recessions or dramatic swingsin prices are less likely to occur in a market that is more efficient. An efficient market should expe-rience a large change in prices only if market participants suddenly become aware of an unexpected external change in the world. A useful mathematical model could provide an important parameterthat indicates the health of the market. Regulators and policy makers could use it to objectivelyevaluate their decisions. Firms could use it to identify inefficiencies that can be exploited for profit,helping to eliminate the inefficiencies.
Figure 1: A trade network with 8 nodes,represented as circles. Edges connectnodes that trade. Edges are weightedby the rate of trade, and colored accord-ing to the type of item being traded.We introduce a new mathematical theory of efficiencythat we call market connectivity . Market efficiency isclosely tied to a market’s ability to communicate eco-nomic information via trade. So, market connectivitymeasures the speed and reliability with which the mar-ket is able to carry out this communication. Market par-ticipants do learn information from sources external tothe market , but the theory focuses on the ability of themarket itself to communicate information by trade; theadvantage is that this can be measured with data. In ourtheory, a market that is able to disseminate informationquickly and reliably is deemed efficient, and conversely, amarket that cannot disseminate information quickly andreliably is deemed inefficient. Our concepts maintain thespirit of Definition 1, but avoid the joint-hypothesis prob-lem, because we measure the ability of the market to com-municate, rather than the accuracy of prices.We represent the market by a network called the tradenetwork . Every participant in the market corresponds toa node of the network—every firm, individual, or otherentity that buys and sells in the market is represented bya node. Every pair of nodes is connected by an edge weighted by the rate of trade between thecorresponding participants, and associated with a vector that represents the type of the item beingtraded. The rate of trade can be estimated by computing the ratio of the total monetary value ofthe trade between the two participants during an interval of time (say a year), divided by the lengthof the interval. The vectors that represent the type of item can be computed from the distributionof trade in the item (more on this in Section 3). These type vectors encode the similarity of differentitems traded in the network. The trade network itself evolves with time as traders discover new As we discuss at the end of this section, our theory is likely to account for all significant external channels ofinformation as well, because such channels will lead to parallel channels of trade. The evolution of the trade network over time does encode some information about the efficiency of the market. Itcaptures how quickly the market is able to form new trade connections in response to inefficiencies. We do not studythis further here, but this seems to be a direction for future work that is worth pursuing. More on this in Section 5. a) Two trade networks with the same total volume of trade. Every node in bothnetworks trades with 3 other nodes, and every edge has the same rate of trade, yet thenetwork on the left is better connected. Information will take a long time to propagateacross the gray divider in the network on the right. A B CA B C (b) Two trade networks where A , B , C provide a service to a set of customers. A , B , C have the same share of the market in both networks, but the network on the left is moreefficient, because most of A ’s customers also trade with B and C . In the market on theright, if B and C reduce their prices, A has no incentive to reduce its price, since itscustomers do not trade with B and C . Figure 2: The trade network can be used to find market inefficiencies.trading partners and react to changes in prices. However, we restrict our attention to a fixednetwork that corresponds to a snapshot of the market during a particular interval of time.The trade network of a market is likely to be sparse—most nodes will trade with a relativelysmall number of nodes. Individuals tend to trade with local businesses more than they do withbusinesses located far away. Firms tend to trade with other firms with whom they have priorrelationships, and large contracts are not often renegotiated. So, firms will tend to be sparselyconnected in the market. Many trading relationships—like subscription to cell phone service orrental agreements—are maintained with contracts that deter participants from switching tradingpartners. Even markets that may seem well connected at first, like the stock market, may not turnout to be so, on closer inspection. Traders in the stock market trade only with other participantsthat either hold the same stocks, or are interested in the same stocks. If two publicly traded firms A , B have businesses that operate in the same space, yet traders that own stocks of A are not wellconnected to traders that own stocks of B in the trade network, the market may have an inefficiency.This is the kind of inefficiency that our model finds.It is worth noting that one network can be better connected than another, even though bothhave the same volume of trade. For example, Figure 2a shows two trade networks, each involving 12participants. Every edge has the same rate of trade. In both cases, every participant is connectedto 3 others, but the network on the left is better connected. There are only two edges crossing thegray divider on the right, so information that is known to the nodes on one side of the gray dividercan take a long time to propagate to the other side. This illustrates the fact that measuring theconnectivity of trade networks is not as straightforward as counting the total volume of trade, orestimating the number of participants involved in a lot of trade. The structure of the connectionsbetween the participants is important. The structure also reveals nodes that have undue control ofprices because of their position. Figure 2a illustrates this using two networks where the nodes A , B , C A , B and C have the same share of the trade in both examples, so market share alone cannot discerna difference between the two networks. However, in the market on the right, few customers tradewith more than one provider. So, if B , C reduce their prices, A has very little incentive to followsuit. In the market on the left, customers do often trade with multiple providers, leading to a moreefficient market. Observing the trade network can reveal such inefficiencies even though measuringmarket share does not reveal them.A key difference between our approach and past work is that we do not attempt to describe theevolution of prices and do not attempt to estimate what prices ought to be. Instead, we measurethe connectivity of the market by computing the effect of potential shocks . A shock is a change inthe beliefs of market participants about the value of the products that they trade. For example, ifmany corn farmers experience a loss of their crop due to infestation, that event generates a shockin the market—some subset of market participants would then believe that corn is relatively morevaluable. A significant shock is one that substantially changes the beliefs of many participants inthe market. In an efficient market, a large volume of trade should be affected by any significantshock, leading to pressure on the participants to readjust their beliefs to a new norm. We give 3definitions to quantify market connectivity: Tension and Energy
Intuitively, the tension of the network is the total normalizing force thatmarket participants experience after any significant shock, and the energy of the network isthe minimum amount of work that needs to be done to overcome market forces and bringthe market to the state of any significant shock. Suppose two participants trade an item atrate r , and after the shock, their beliefs for the amount of the item that is worth a unit ofcurrency are x and y . Then the force generated by the shock is proportional to r · | x − y | .The energy stored in the edge is equal to the amount of work that needs to be done to bringthe market to this state: (1 / · r · ( x − y ) . There is no force or energy if both nodes have thesame beliefs. The force and energy are maximized when the difference in their beliefs is largeand the rate of trade is high. The tension of the shock is the magnitude of the net force feltby all nodes. Up to normalization factors, the tension/energy of the network is the minimumover all significant shocks, of the tension/energy experienced by the nodes after the shock.The tension and energy of a shock do have several qualitative differences. The tension is thetotal net force felt by all market participants, which corresponds to quantifying how quicklysmall changes in the beliefs of nodes helps to bring the nodes closer to having the same beliefs.The energy corresponds to quantifying the total potential energy stored in the edges when thenodes experience the shock. The energy is large when most nodes trade with partners thathave conflicting beliefs from themselves. The tension is large when most nodes have beliefsthat differ from the average of the beliefs of their trading partners.Two large countries that do not trade correspond to a trade network with low tension/energy.Prices in one country can easily stray far from prices in the other country. These conceptscan be used to distinguish the two examples shown in Figure 2a. Bottlenecks A bottleneck for tension/energy is a small set of nodes that is responsible for a largefraction of the tension/energy of the network after some significant shock. Market participantsthat correspond to the bottleneck have an incentive to engage in anticompetitive practices,blocking the flow of information and interrupting the transmission of shocks.6 firm that has a monopoly on the production of an item that cannot be easily substitutedwith items available from competitors is a particular kind of bottleneck, but not the only one.Prices can be held at artificial levels by such a bottleneck. The concept of bottlenecks can beused to distinguish the two examples shown in Figure 2b.A market with high connectivity is a market that has high energy, high tension, and no bottlenecks.The advantage of our theory over prior work is that it does not use an asset pricing model, sostatements analogous to the efficient markets hypothesis are falsifiable in our theory. One can useour definitions to explicitly find and quantify inefficiencies if they exist. An inefficiency in ourtheory corresponds to a significant shock with low tension/energy, or a significant shock and asmall set of nodes that form a bottleneck for the shock. Such an inefficiency identifies an obstaclefor the flow of information that may be exploited for profit, reducing the inefficiency, or addressedby a change in policy or regulations.In the restricted case where all trade in the market involves only one type of item, the formulaswe use for tension and energy have long been used to model many physical systems, and are closelyrelated to concepts studied in spectral graph theory [Spi, Chu96]. The same equations describe thebehavior of electrical networks (where beliefs correspond to voltages, tension corresponds to thecurrent flowing into the network, and energy corresponds to the power of the network), networksof springs, as well as the flow of heat in conducting materials. The fact that these equations haveproven their worth in so many diverse settings is evidence that they capture something fundamentalabout the trade network. Viewed as a spring system, the rate of trade along an edge correspondsto the spring constant of the spring that it represents as in Hooke’s Law. Each node’s beliefcorresponds to a location on the number line. The placement of the nodes at these locations by theshock induces forces at the nodes, and the tension is the total magnitude of all these forces. Theenergy is the total potential energy stored in the system.When multiple items are traded, the formulas we develop are natural multidimensional analogsof their single item counterparts—they loosely correspond to what happens when nodes are placedin a multidimensional space and the directions of the forces are twisted according to the type ofitem being traded. We associate each item with a vector that we call the type vector of the item.Similar items have nearly identical type vectors, and items that are different from each other havenearly orthogonal type vectors. We show how to use data from the trade network to generate thetype vectors. Each edge applies a force to a node only in the direction of the type vector. Thesechoices allow us to model how trade in one item can carry information about a shock in anotheritem. To the best of our knowledge, the mathematics we have developed for this general setting isnew.Our intuition is that the trade network actually encodes all significant channels of information,even those that are external to the market. Let us make a few points to justify this belief. First,when we refer to the “market”, we mean the whole market. Prices on two different stock exchangescan be correlated even if stocks in one exchange cannot be traded on the other, because participantsthat trade on these exchanges are connected in the market via their trade in other items. Second,even if there are significant channels for the transfer of information that are truly external to themarket, traders that act on this information will create a channel of trade that parallels the flow ofinformation in the external channel, leaving a trace of that flow visible in the trade. For example,if traders in one country use the price of iron ore in another to make decisions about whether tosell or buy iron ore in their local market, then discrepancies in the price of iron ore between thetwo countries will give traders incentives to trade iron ore or products related to iron ore between7he countries. Conversely, if trade between the two countries is impossible, then information aboutthe price of iron ore in the other country is not actually a very useful source of information for thelocal trader. So, the extent to which trade connects different parts of the market is a good proxyfor the extent to which those parts are able to communicate information relevant to prices. If alarge number of people make trading decisions using information they obtain from the same publicsource, then there will be correlations in their trading decisions, which will lead to many of thembeing connected to each other via short routes of trade in the market. Consequently, this source ofinformation is reflected in the trade of the market, and will be accounted for by the mathematics ofthe theory. Finally, the theory we develop is intended to quantify the macroeconomic health of themarket, and the mathematics is tuned to finding large scale inefficiencies. A channel of informationis important to our theory only if it affects decisions that lead to a significant volume of trade—allsuch channels are likely to be reflected in the trade network.After briefly discussing related work, in Section 2 and Section 3, we explain the intuitions behindthe choice of the model and develop the theory. We show how several ideas in economics can bebetter understood using the concepts we develop in our work. In Section 5, we discuss some openquestions, and possibilities for future work that we find worthwhile. The efficient markets hypothesis is of fundamental importance, and there is a vast literature ofprior work that has sought to understand it. This literature has proven to be illuminating and im-portant, both theoretically and practically, and has had a major impact on the views and practicesof professional investors. It is responsible for bringing data driven methods to bear on understand-ing finance. We refer the reader to the reviews of Fama [Fam70, Fam91] for details. Empiricaleconomists have used different types of tests to try and understand if the market is efficient, whichhave been categorized in [Fam91] as follows. In tests for return predictability , historical data areused along with an asset pricing model to compute estimates for the correct returns on investments,and these are compared to actual returns realized by the market. In event studies , the speed atwhich prices adjust to significant new events is measured. In tests for private information , re-searchers attempt to understand whether or not market participants have private information thatis not reflected in prices. To conclude that the market is not efficient using such tests, one mustbe sure that the information gathered by the test is at least as comprehensive as the informationknown to the market, and that the asset pricing model used is more trustworthy than the marketitself. Perhaps not surprisingly, most of these tests have concluded that the financial market isefficient. That said, these tests do provide valuable insights into the inner workings of financialmarkets.The theory we develop here is based on intuitions from this past work, especially the notionsof event studies and tests for private information. However, the mathematics and data used inour theory are quite different from past work. We do not attempt to make any quantitativestatements about the evolution of prices, nor do we attempt to understand what information themarket should take into account. Accurately modeling such phenomena is extremely difficult, andpast work suffers from the joint-hypothesis problem because of this difficulty. Instead, we evaluatemarkets as systems for communication. Unlike past work, we give explicit ways to quantify marketinefficiencies and find them if they exist, so market efficiency as we define it can be unambiguouslydetermined. If markets do not have high connectivity, this can be conclusively proven to be thecase within our theory, so our theory does not suffer from the joint-hypothesis problem that plagues8raditional concepts of efficiency. The primary motivation for our work is to strengthen the conceptof market efficiency by giving it these features.A different line of past work has tested a consequence of the efficient markets hypothesis:no trader should expect to make unusually large returns from trading in the financial market.Again, this is a vast area that we cannot hope to summarize here, so we refer the reader to thesurvey [Mal03]. This work is also extremely useful in practice, but the fact that traders cannotexpect positive returns is not conclusive evidence of market efficiency. Traders may have difficultygenerating positive returns even in an inefficient market due to high transaction costs, the difficultyof finding inefficiencies that they are capable of exploiting before they disappear, the lack of capitalto compete with an entrenched system, or compliance with regulations.Perhaps the biggest challenge to the efficient markets hypothesis has come from the field ofbehavioral economics [PS15]. Behavioral economics argues that the efficient markets hypothesiscannot hold because market participants are not rational, and are subject to human psychology. Ourown model is built on the idea that market efficiency is limited by the bandwidth of informationavailable to individual participants, so our work is inspired by similar considerations. However,unlike behavioral economics, our theory provides an alternative to the established theory of marketefficiency that we believe has greater explanatory and predictive power.There is a significant history of using networks to model social and economic phenomena. Thebook [Jac10] is a good reference. Several subfields here have developed very similar mathematicsto the mathematics in our paper, but to the best of our knowledge, none are well suited to un-derstanding market efficiency. One of the most related concepts from this past work is the
Bass model of diffusion [Jac10, Chapter 7], which was invented to understand the spread of contagion ina social network. Contagion has been used to model the spread of financial crises [KRV03]. Herethe emphasis is on understanding how a crisis spreads over financial relationships, rather than howuseful information spreads, so the role of trade is quite different from in our model. Another bodyof work with similar mathematics arises in the study of learning networks [Jac10, Chapter 8]. Inthe DeGroot model [DeG74], every node of the network holds an opinion, and each node updatesits opinion according to the opinions of its neighbors in the network. All of these past models aremathematically and conceptually related to our own model for the single item case, but do notappear to take into account the subtleties that arise from trade in multiple types of items. Indeed,the equations we use for the single item case have a much longer history, going back to the study oftopics like electrical networks, networks of springs and the diffusion of heat. DeGroot also developsa notion of social influence that seems on the surface similar to our notion of bottlenecks, but ismathematically quite different. Social influence of a node in an undirected network is determinedsolely by the fraction of the total weight of edges that touch the node—the location of the node isnot relevant. The concepts of efficient networks , and graphical games [Jac10, Chapter 6] are lessrelated to the topic of this paper than they may seem from their names. They are concerned withunderstanding the space of strategies for playing games on networks.
We use the notion of a trade network to understand the limits and power of markets to disseminateinformation via trade. The ideas we develop are simplest when there is only one item being producedand consumed in the market. We start by discussing this restricted case, because it is easier tounderstand. Later, we show how to model markets where an arbitrary number of items are traded,9
7 2 6
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Figure 3: On the left: a single item trade network with 4 participants. The numbers on the edgesdenote the rate of trade. On the right: the network is shown with a mapping to the number line.The locations of the nodes correspond to a shock: a change to beliefs about the amount of the itemthat is worh a unit of currency. The arrows show the forces induced on each node by trade.which is much more applicable to real markets.Recall that we represent every market participant by a distinct node, and connect two nodesby an edge if the corresponding market participants directly trade with each other. We use V todenote the set of nodes and E to denote the set of edges. Every edge e = { u, v } connects two nodes, u and v of the network. The edge is assigned a non-negative number called the weight of the edge,denoted w ( e ), or w ( { u, v } ). The weight is the rate of the trade between the two participants. If w ( e ) = 0, that is equivalent to there being no trade along the edge e . To estimate the weight ofeach edge in real world markets, one can use past transactions to compute the monetary value ofthe trade between every pair of participants over a fixed time interval and then divide this numberby the length of the interval.Since single item markets correspond to spring systems, we start by giving some intuition aboutthe correspondence between the two. It is helpful to think of placing each node at the locationon the number line that corresponds to its belief for the value of the single item being traded inthe market. In equilibrium, every node has the same belief for the value of the item, and thereis no tension in the network. A shock is a change in the beliefs for the amount of the item thatis worth a unit of currency. It can be represented by a n dimensional vector of numbers x ∈ R n ,where x v denotes the node v ’s belief about the change. The shock induces forces in the tradenetwork that helps to reduce the shock. The force on a node u along the edge { u, v } is equal to w ( { u, v } ) · ( x u − x v )—its magnitude is proportional to the product of the weight and the differencebetween the new beliefs of u, v about the value of the item. The larger the net forces felt by thenodes, the faster the shock dissipates in the market. The shock will be resolved quickly if manyparticipants with differing beliefs about the value of the item trade with each other at a high rate. This is just one possible interpretation for the model. The model can be interpreted in a number of differentways, each with its own merits.
2 3 5
B DA F J IC E HGD F JIC E HG-7 -6 -5 0 5 6 7shockBA
Figure 4: A mapping of a network to the number line. Every node is colored according to itslocation on the line, which is its belief about the amount of the item that a unit of currency isworth. The gray arrows show the forces experienced by nodes A , B and J . All other nodes experienceno net force. The tension of the network for this shock is the sum of the magnitudes of the net forceon each node, which is T( x ) = 3 + 2 + 5 = 10. By Fact 2, the energy for this shock is E( x ) = 35.We are most concerned with the dissipation of shocks that are felt by a large part of the market. Next we review the basic laws of spring networks (the book [Bol98] is a good reference). Throughoutthe discussion we work with springs whose relaxed length is 0: namely the springs always pull nodescloser together and never push them apart. Hooke’s law says the node u experiences a force of w ( { u, v } ) · ( x v − x u ) , because of the edge { u, v } . The energy stored in the edge { u, v } is equal to w ( { u, v } ) · ( x u − x v ) . The energy is the amount of work that needs to be done to pull the two nodes of the spring apart.Given a shock x , we define the tension induced by x to be the total magnitude of all the net forcesfelt by the nodes: T( x ) = X u ∈ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X v ∈ V w ( { u, v } ) · ( x u − x v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . The total energy of x is the sum of the energies of each edge:E( x ) = (1 / · X { u,v }∈ E w ( { u, v } ) · ( x v − x u ) .
1 3 1 1
A B DCA shock CB D
1 1 B shock
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Tension = 8, Energy = 3 Tension = 6, Energy = 3 Tension = 6, Energy = 1.5 (a) Tension and energy are incomparable.
1 1 1 1 4 1 1
A B DCA shock CB D
4 1 1
A B DCA shock CB D EE Tension = 10 Energy = 9 Tension = 8 Energy = 10 (b) The tension may decrease by trade with new nodes, but the energy does not.
Figure 5: Examples illustrating the difference between the tension and energy of a shock.One can check that the gradient of the energy is the vector of forces that need to be applied tothe nodes to hold them in place:( ∇ E( x )) u = X { u,v }∈ E w ( { u, v } ) · ( x u − x v ) , (1)so −∇ E( x ) is the vector of net forces experienced by the nodes, and the tension is the L norm ofthe gradient vector: T( x ) = |∇ E( x ) | = X u ∈ V | ( ∇ E( x )) u | . The energy of the network is equal to the amount of work that needs to be done to bring thenetwork to its current position from a 0 energy state. The following fact is useful to keep in mindto understand the examples we discuss next:
Fact 2.
If some subset of the nodes are held at location a and some other subset are held at b , andthe rest are left to settle at equilibrium, the energy is always equal to | ( a − b ) t/ | , where t is thetension. The gradient is the vector of partial derivatives. B cannot help to reduce the energy. This is what leads to thesecond example having lower tension than the first. In the final example, the energy is much lowerbecause the participants are much closer to having the same beliefs than in the other two examples.Loosely speaking, the tension corresponds to the ability of most participants to reduce the distanceto equilibrium, while the energy corresponds to the total work that needs to be done to return themarket to equilibrium. In Figure 5b, we see that the presence of a new node can actually decreasethe tension of the network, though it can never decrease the energy. For simplicity of explanation,these subtleties are ignored in the examples we discuss next. The shocks in all of these exampleshave been chosen so that they do not distinguish between tension and energy. The model we define in this paper can be used to explain diverse phenomena that pertain tomarket efficiency. Here we consider several examples of well known ideas in economics that canbe explained using our model. We stress here that a proper treatment of actual markets shouldtake into account that the items traded in the market are different from each other. We discussthe theory that handles this more realistic setting in Section 3. To compute the connectivity ofthe market, we find significant shocks that have low energy, low tension or reveal the presence ofbottlenecks.Suppose the market under consideration consists of n farmers that produce and consume corn.Every farmer buys and sells corn by trading in a common market place, and every farmer tradeswith every other farmer at a rate of 1. Then every pair of nodes in the trade network is connectedby an edge of weight 1. If a single farmer produces less corn than expected, the effects of thisshock are quickly felt by the whole market. The farmer will seek more corn from others, whichincreases the price of corn and incentivizes the others to produce and sell more corn to compensate.If a large fraction of the farmers experience a drop in production, the effect on the network iseven more pronounced. Viewed as spring network, where every edge has weight 1, we see that thetension between any two disjoint sets S, T that have different values in the shock is quite large. Forexample, suppose that
S, T are each of size at least n/
3, every node in S has a value of +1, andevery node of T has a value of −
1. Then there are at least n / S to T ,so the tension of any such shock is at least 4 n /
9, and the energy is at least 2 n /
9. So every shockgives high energy and tension, and the market has high connectivity.It is a commonly held view that free trade increases market efficiency, a view that can bequantified with our model. Suppose that the market consists of farmers living in two adjacenttowns, each with n/ −
1, andevery node in the other is placed at 1. For this shock, we have T( x ) = E( x ) = 0. This matches theintuition that there is no way for information to flow between the two towns via trade. It will notbe long before traders emerge between the two towns. Consider now the trade network depicted inFigure 6b, where there are three traders trading corn at a rate of 1 along a road connecting the twotowns. This market is more efficient than the disconnected market that we had before, since trade13 (a) T( x ) = E( x ) = 0 when all the white nodes in one part are placed at −
1, and all the gray nodesfrom the other part are placed at 1. The shock identifies the disconnection in the network. -1 10shock (b) Only the nodes at − x ) = 1 , E( x ) = 0 . Figure 6: All edges have unit weight in this sequence of examples showing that tension and energyincrease with connectivity.does communicate information between the two towns. In line with our intuition, if every node inone town is placed at −
1, and every node in the other is placed at 1, and the rest of the nodesare allowed to relax at equilibrium, a force of 0 . .
5. Now suppose that there was just 1 trader betweenthe two towns as in Figure 7a. Then the market would be better at communicating informationfrom one town to the other, since only one trader is involved in communicating the shock from onetown to the other. This is reflected in the fact that the tension of the shock has increased to 2.The market is most efficient when there are multiple independent traders trading between the twotowns, as in Figure 7b. The tension of the shock has increased to 6.Trade networks can be used to explain the success of many modern innovations in finance.Consider the role of exchange traded funds in the stock market. These are funds that hold assetslike stocks or bonds, and allow investors to trade financial instruments that correspond to diversifiedportfolios. In recent years, the volume of trade in exchange traded funds has risen dramatically.The total annual trade in these funds exceeds the U.S. GDP, and the funds have a turnover ratethat is more than 4 times larger than average stocks [Bal15]. Our model suggests that this highlevel of trade contributes tremendously to increasing market connectivity. It is quite hard for a14
1 1 -1 10shock (a) A short trade route generates more tension than the trade route in Figure 6b. T( x ) = 2 , E( x ) =1. The node at 0 experiences no net force. -0.5 0.5
3 3 -1 10shock (b) Many parallel trade routes generate more tension than in the network of Figure 7a. T( x ) =6 , E( x ) = 3. Only the nodes at − Figure 7: All edges have unit weight in this sequence of examples showing that tension and energyincrease with connectivity.typical investor to monitor and trade a truly diversified portfolio of stocks, so if these (or similar)funds were not available, investors would be trading only with others that are interested in thesame stocks. Since the set of participants interested in trading a highly diversified exchange tradedfund is much larger, trade in these funds makes the trade network much more connected thanbefore. Investors that were formerly focused on disjoint portfolios that are nevertheless associatedwith the same exchange trade fund will now begin to trade with each other. This makes the tradenetwork much more likely to have high energy and tension for a given shock . Exchange tradedfunds offer a channel of communication that reaches a much broader set of participants than tradein the constituent stocks can provide.A popular view in economics is that the availability of liquid assets is vital to the functioningof a healthy economy. For example, Chordia, Roll and Subrahmanyam write “Liquidity facilitatesefficiency, in the sense that the market’s capacity to accommodate order flow is larger during periodswhen the market is more liquid.” [CRS08] This view is supported by our model as well. A market If two publicly traded companies operate in the same space, but most shareholders of one company do not holdshares of the other, our model suggests that the introduction of an exchange traded fund giving exposure to bothcompanies can improve market connectivity.
2 2 10 10 -1 10shock
Figure 8: Two nodes that are responsible for most of the tension/energy form a bottleneck.with high liquidity is one in which nodes are more likely to be connected to many trading partnersvia trade. Such a network is much more likely to have high energy and tension than a sparselyconnected market.A second kind of market inefficiency that we study corresponds to monopolies in the market.Suppose a small set of traders comes to control a very large fraction of the trade between the twotowns, as in Figure 8. We have T( x ) = 24 , E( x ) = 12, but the tension and would drop dramaticallyto T( x ) = 4 , E( x ) = 2 if the highlighted nodes are removed. This is what we call a bottleneck inthe trade network. It is a small set of nodes that contributes most of the tension/energy.In some settings, the set of nodes involved in the bottleneck have an incentive to commit fraud,or engage in monopolistic practices, because they have a large market share. Even if these nodesbehave in good faith, information flow in the market relies too heavily on the bottleneck, whichcan lead to the market being inefficient if these nodes make errors in their adjustments to prices. s h o c k Figure 9: In a centralized economy, every nodetrades only with the central authority, which is abottleneck.An extreme example of a bottleneck can beobserved in the trade network of a centralizedeconomy, shown in Figure 9. This network iswell connected. In fact, from the perspectiveof the tension and energy, it is nearly as goodfas the fully-connected network, because there isa short path of length 2 connecting every twonodes in the network. However, every individ-ual trades only with the central authority, de-picted as the node in the middle. Even thoughthe tension and energy of the network are quitehigh, the network clearly has a bottleneck. Theburden for recognizing and transmitting shocksfalls entirely onto a single entity, making it un-likely for information to be reliably communi-cated.It is widely recognized that the financial crisis of 2007-2009 began because of errors in accuratelypricing mortgage backed securities (MBS). A crash in the value of these securities initiated a globalrecession. Data gathered about the organization of the market prior to the crisis suggest that abottleneck was to blame. As Fligstein and Goldstein report:16nother commonly voiced myth about the MBS market is that it was highly dispersed,with too many players to control any facet of the market. On the contrary, we show thatover time all of the main markets connected to MBS, the originators, the packagers, thewholesalers, the servicers, and the rating companies became not only larger, but moreconcentrated. By the end, in every facet of the industry 5 firms controlled at least 40% ofthe market (and in some cases closer to 90%). Separate market niches also increasinglycondensed around the same dominant firms. As a result, the mortgage field was notan anonymous market scattered across the country, but instead consisted of a few largefirms. This concentration meant that firms very much collaborated and competed inthese various markets. Firms would join together in MBS packages and assume differentroles with each other. This meant that they had a great deal of knowledge of the marketand what the others’ moves were. [FG10, p. 33]One of the major goals of our work is to develop tools that can be used to detect structural problemslike bottlenecks before a crisis occurs. Economists and regulators could use the theory to identifybottlenecks in the trade network and investigate methods to tackle them before they lead to a crisis.Market connectivity can also be used to assess financial solutions that mitigate the effects ofbottlenecks. For instance, consider the role of futures in the commodities market. Futures contractsguarantee delivery of a commodity at a date in the future. Perhaps surprisingly, it is often the casethat only a small proportion of contracts that are traded are settled by actual delivery. Our modelsuggests an explanation for this counterintuitive fact: futures contracts serve as a mechanism tobreak bottlenecks in the trade network, and they are most effective when the trade in the futuresexceeds trade in the actual item. We illustrate this using the following anecdote by Hieronymus[Hei77] about the origins of futures contracts in corn:A country merchant rides into town in January and proceeds to the place of businessof the terminal merchant to whom he regularly sells. On offering 20,000 bushels forJune delivery at the current price he hears, “I would like to bid that much but withthe large stocks in Chicago and a large crop coming in I can only pay 15 cents belowtoday’s price.” Our friend mentions the high price that he has already paid farmers,comments on the ancestry of terminal grain merchants in general, and takes himself tothe nearest saloon to find solace. There he comments to all and sundry regarding thegreed and cowardice of grain merchants in general and one in particular. On hearingthis one stalwart soul says, “I know nothing of corn, being a builder of houses myself,but it occurs to me that the price of corn will be quite as high in June as it is now.”Being true to his occupation, just as we know country merchants today, our man asks,“Is that a firm offer?” “I shouldn’t want to go quite that far but I will bid five centsbelow today’s price. That will take you off of the hook and out of your cups and leaveroom for a bit of a profit for me.” “Done,” replies the country merchant and they signa contract. Some weeks later the builder of houses, who has now become interested inthese matters, notes that the price of corn for June delivery is five cents above the pricethat he has paid. He figures that $1000 in his pocket is better than 20,000 bushels ofcorn in his lap in June so he peddles his contract to the nearest terminal merchant andwonders why he had not discovered this easy road to riches sooner. [Hei77, p. 74]The transformation to the trade network described in this anecdote is shown in Figure 10a. Thecountry merchants are represented by the nodes A and B , and the terminal merchants by the nodes17 C D C D s h o c k s h o c k F FFF FF C D C DFF FFF F A B BAA B A B (a) A futures market can help to break a bottleneck. On the left, the merchants A and B can onlytrade with merchants C and D . C and D are a bottleneck for the depicted shock. The introductionof futures traders labeled F substantially changes the network to the figure on the right. A , B cannow sell their contracts to the futures traders, which reduces the role of C , D in generating thetension in the network. C and D must now compete with the traders F for the contracts and cannotfix prices. A B CAC AB CBA B C s h o c k s h o c k (b) High switching costs can give rise to trade networks with bottlenecks. In the network on the left, A , B , C provide service to a set of customers that correspond to the nodes on the bottom. Switchingcosts are high, so most customers trade with a single provider. A is a bottleneck for the depictedshock. If switching costs are low, we might see the network shown on the right, where A is no longera strong bottleneck. Figure 10: Bottlenecks explain the impact of high switching costs and the benefits of a futuresmarket. 18 and D . If futures contracts were not available, the network would look like the one on the left,where C and D clearly form a bottleneck for the depicted shock. They are able to charge unfairprices from A and B because A and B cannot sell their corn to anyone else. The introduction of thefutures market dramatically alters the trade network, as shown on the right. There are two reasonswhy C and D become less of a bottleneck according to our definitions. The first is more obvious:the tension and energy that they are responsible for has been reduced. For the given shock, C and D are responsible for only a small part of the tension, since a sizable fraction of the trade nowinvolves the futures traders instead of C and D . Consequently their ability to extort higher priceshas been greatly diminished. A second reason is that the significance of the depicted price shockhas been reduced, because a larger fraction of the traders have the same beliefs in the depictedshock. A shock is now significant only if the futures traders have significant disagreement aboutthe values in the shock. C and D cannot manipulate prices very easily because the high volume oftrade in futures contracts will reveal accurate prices for corn.Next we consider the effects of switching costs on market efficiency. Markets where participantsare locked-in to trade with particular nodes will tend to be less efficient than markets whereparticipants can freely switch trading partners. As Farrell and Kemperer observe,Switching costs [. . .] bind customers to vendors if products are incompatible, lockingcustomers or even markets in to early choices. Lock-in hinders customers from changingsuppliers in response to (predictable or unpredictable) changes in efficiency, and givesvendors lucrative ex post market power [. . .] [FK07, p. 1970]The effects of two types of switching costs can be seen in Figure 10b. The figure shows three serviceproviders, A , B , C that serve a group of customers, represented as the nodes on the bottom. Ifswitching costs are prohibitively high, one might observe the network shown on the left. The typicalcustomer trades with only one of the service providers. If a shock occurs where the customers of A have a different belief from the rest of the market, A becomes a bottleneck for the shock. Thenetwork on the right shows what the network might look like if switching costs are lower. Nowmost customers trade with multiple providers, and we see that A is less of a bottleneck for the sameshock, because customers also experience tension and energy because of the edges that connectthem to B and C . A can no longer fix prices because the competition with B and C is much moreof a threat. Note that this example shows that trade networks are more effective at identifyingmonopolies than counts of market share. In this last example, A has only a 1 / vertical integration . Vertical integration refers to an arrangement where a firm owns severalparts of the supply chain of a product. Does market efficiency improve with vertical integration?On the one hand, vertical integration does increase the energy and tension of most shocks; onthe other hand, it has been observed that vertical integration is associated with increased rates offraud. Fligstein and Roehrkasse [FR16] make the case that vertical integration was to blame forthe elevated levels of fraud that contributed to the financial crisis of 2007-2009. They write,[. . .] [T]he motivation toward fraud increases with vertical integration because the diffu-sion of fraud is a precondition of continuing normal operations. [. . .] [T]he transactionpoints that vertical integration eliminates are not only sites of potential fraud. Theyalso represent key opportunities for quality control and due diligence. [FR16, p. 624]19 .5-0.50 G H s h o c k A B CD E FI
Supply Chain123 -11 0.5-0.50 s h o c k -11 G HA B CD E FI HB CV E FI HB CV E FI Figure 11: Vertical integration can lead to bottlenecks. The network on the left shows a supplychain involving the firms A , B , C , D , E , F , G , H , I . Each firm trades at an equal rate with theadjacent levels of the supply chain. The network on the right shows the same shock applied afterfirms A , D , G have been combined into the firm V . The tension and energy have increased, but V now contributes 1 / / / / V have beenstretched to be twice as long as the tension generating edges that V replaced. The real power of markets emerges when one considers trade networks involving many differentkinds of items. It is a mistake to view such a market as a collection of distinct markets tradingeach item, because shocks in one item can be dissipated via trade in another item. This makesmulti-item networks more efficient than any of the subnetworks obtained by restricting attentionto individual items.Consider the market shown in Figure 12, which is a market of farmers that grow both corn andwheat. Viewing the market as a network for trading corn and a separate network for trading wheat20uggests that the corn network has 0 tension and energy for a significant shock, since no corn istraded between the left and the right. But this is misleading. If the farmers on the left experiencea sharp drop in the production of corn, the price of corn will rise. This will increase the demand forwheat on the left, making the price of wheat rise on the right, and eventually leading to a rise inthe price of corn on the right—the trade in wheat transmits shocks in the corn market. However,if the trade between the left and right is in a commodity, say iron ore, that is not closely linked tocorn, then the market is much less effective at communicating shocks in the corn network. Theseconsiderations motivate the choices we make next. wheat corn
Figure 12: Trade in wheat can carry informationabout corn, so one should not view the trade net-work as two different trade networks for each item.We associate every item traded in the mar-ket with a unit vector τ ∈ R d , for some number d . We refer to τ as the type vector of the item.The type vectors are chosen in such a way thatthe inner product h τ , η i captures the correla-tion between the items represented by the vec-tors τ , η . So two items that are completely dif-ferent from each other are represented by nearlyorthogonal vectors— h τ , η i ≈ h τ , η i ≈
1. Every edge e ofthe trade network corresponds to an item, andwe let τ ( e ) denote the corresponding type vec-tor . To allow for the fact that two participantsmay trade multiple types of items, we allow theset of edges E to contain multiple edges between two vertices u, v . The mathematics we develophas the feature that the values of the inner products between all pairs of items is the only relevantfeature of the type vectors to the model. For example, rotating all the type vectors by the samedegree will preserve all the quantities we define, because doing so does not change any of the innerproducts.One can treat the vectors that represent items in the market as parameters of the model, oruse the trade network itself to generate them. If the market has n participants, and E denotesthe subset of the edges that trade item z , let τ be the d = n dimensional vector such that τ v = P v ∈ e ∈ E w ( e ) is equal to the total amount of trade of item z that node v engages in. Thenrepresent item z using the unit vector τ = τ k τ k . Two items that are largely traded by disjoint partsof the market will correspond to nearly orthogonal vectors, and two items that are often tradedby the same nodes will correspond to vectors that have a large inner product. Two items thatare substitutable will likely be sold and traded by the same nodes in the market, and two itemsthat are unrelated to each other will likely be traded by different nodes. This choice correspondssomewhat to our intuitions, but may not always be ideal. One may need to come up with thesevectors using a combination of data driven methods and information about the underlying items.Note that if I types of items are traded in the market, it is always possible to set d ≤ I , sincewe can always project the type vectors down to an I dimensional space while preserving the inner These type vectors can be used to encode the geography of the market even in single item markets. If we wantto take into account the fact that items that are nearby are likely to be more correlated, we can associate every edgewith a unit vector that is proportional to the vector encoding the location of the trade. heatcorn ⌧ ( e ) (a) If the white nodes are placed at the point ( − , , iron orecorn ⌧ ( e ) (b) If the white nodes are placed at the point ( − , , Figure 13: A shock on a muti-item network, showing the effect of choosing different type vectors.products. Moreover, by the Johnson-Lindenstrauss lemma [JL84] one can always project the typevectors to a space whose dimension is proportional to log I , while introducing tiny errors in theinner products. So, one can ensure that d is much smaller than n .A shock in the system is an nd dimensional vector x ∈ R nd , that can be interpreted as n vectorsof dimension d . The coordinates x u ∈ R d correspond to the beliefs of node u about the amounts ofall items that are worth a unit of currency, and the inner product h τ , x u i corresponds to u ’s beliefabout the value of the item whose type vector is τ .Trade along an edge e = { u, v } induces a normalizing force on v of w ( e ) · h τ ( e ) , x u − x v i · τ ( e ) . The energy of the edge is (1 / · w ( e ) · h τ ( e ) , x u − x v i . x u − x v is orthogonal to the type vector, then no force is induced and the energy is 0,since the parties agree on their beliefs for the item they trade. Note that these quantities remainidentical if the type vector is replaced with its negation.Given a shock x , we define the tension of the shock to be the sum of the magnitudes of allforces felt by the nodesT( x ) = X u ∈ V (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X { u,v }∈ e ∈ E w ( e ) · h τ ( e ) , x v − x u i · τ ( e ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , where here k y k = qP dj =1 y j denotes the length of the vector y . Figure 13a shows the tension anetwork experiences when the shock is along a direction in one of the items, and the other item hasa type vector that is not orthogonal. Figure 13b shows a similar situation when the type vectorsare orthogonal. We define the energy of the shock to be the sum of the energies on all edges:E( x ) = (1 / X { u,v } = e ∈ E w ( e ) · h τ ( e ) , x u − x v i . As in the single item case, the gradient of the energy is the amount of force needed at each nodeto hold the network in place:( ∇ (E( x ))) u = X u ∈ e = { u,v } w ( e ) · h τ ( e ) , x u − x v i · τ ( e ) , So adjusting beliefs in the directions of the forces felt at each node is the quickest way to reducethe energy.
In this section, we give mathematical definitions for the tension and energy of trade networksand show how to identify bottlenecks mathematically. We aim to give numbers that capture thebehavior of the network for all significant shocks. To ease comprehension, we explain the definitionfor the single item case first, and then generalize the definitions to handle multiple items.
Key to the definition is what kinds of shocks should be considered significant. A small, completelydisconnected part of the network can give rise to 0 tension/energy. Suppose the network consistsof two disjoint sets A , B that have no edges between and the sum of all weights of edges in A is (cid:15) and the sum of all weights in B is w − (cid:15) . If (cid:15) is very small, and B is well connected, this should beconsidered an efficient market. But placing all the nodes of A at w − (cid:15) and all the nodes of B at − (cid:15) gives E( x ) = T( x ) = 0, even though P u ∈ V w ( u ) · x u = 0. Such shocks focus too much attentionon small anomalies in the network and should not be considered significant.Observe that the tension and energy of a shock remain the same if every entry of the shock isshifted by the same number. Namely, if denotes the all 1’s vector, then E( x ) = E( x + γ ) andT( x ) = T( x + γ ), for every number γ . Keeping this in mind, it will be convenient to shift each23hock so that the average belief of the nodes (weighted by the volume of trade) is 0. For any shock x , define x = x − P u ∈ V w ( u ) · x u w · . x is the shock obtained by shifting x so that the average shock value is 0: X u ∈ V w ( u ) · x u = X u ∈ V w ( u ) · x u − X u ∈ V w ( u ) · x u = 0 . For any node u , x u is a measure of how far node u ’s belief is from the average belief in thenetwork. In order to preclude shocks that put undue weight on small parts of the network, wedefine the set of significant shocks of magnitude α , for 0 ≤ α ≤
1, to be: V α = ( x : for every v ∈ V , X u ∈ V w ( u ) · | x u | w ≥ α · | x v | ) . Significant shocks are those where the magnitude of the shock is well spread among the nodesin the sense that the shock affects at least an α (weighted) fraction of the trade in the market. Ourmeasure of connectivity should be scale invariant. One can always increase or decrease the energyand tension by multiplying the shock by a scalar. To ignore such scaling factors, we measure theratio of the tension and energy to a normalization factor: Definition 3.
The α -tension of the trade network is T α = min x ∈V α T( x )2 P u ∈ V w ( u ) · | x u | . Next we define the energy of the trade network. Significant shocks in this context are givenby: U α = ( x : for every v ∈ V , X u ∈ V w ( u ) · x u w ≥ α · x v ) . Definition 4.
The α -energy of the trade network is E α = min x ∈U α E( x ) P u ∈ V w ( u ) · x u . The differences in the choices of the normalization factors and the choices of significant shocksfor tension and energy can be explained by the properties they ensure. The proofs of these factsare elementary, and can be found in Appendix A.
Fact 5. ≥ T α ≥ and ≥ E α ≥ . Both quantities are relatively large when the network is well connected: The energy of the network is closely related to the concept of conductance in spectral graph theory. The expressionfor the energy is the same as the expression for the conductance of the normalized Laplacian of a graph. act 6. For every α , the fully connected network on n nodes with unit edge weights has T α = 12 + 12( n −
1) = E α . When the number of participants is n = 2, Fact 6 shows that T α = 1 = E α . Since the set ofsignificant shocks only gets bigger as α gets smaller, we have: Fact 7.
For every α ≤ β , T α ≤ T β , E α ≤ E β . Although the set of significant shocks for tension and energy are different, they are closelyrelated:
Lemma 8. U α ⊆ V α ⊆ U α . Finally, we have:
Fact 9. E α = 0 ⇒ T α = 0 ⇒ E α = 0 . Let B be a subset of k nodes. Define T B ( x ) to be the tension of the shock after deleting theedges of the network that do not touch B . Definition 10. A ( k, α, β ) bottleneck for tension is a set of k nodes B such that that there is avector x ∈ V α with T( x ) > and T B ( x )T( x ) ≥ β. Similarly, let E B ( x ) denote the energy due to all the edges that touch the set B . Definition 11. A ( k, α, β ) bottleneck for energy is a set of k nodes B such that there is a vector x ∈ U α with E( x ) > and E B ( x )E( x ) ≥ β. A ( k, α, β ) bottleneck is a serious obstruction to efficiency when k is small and α, β are relativelylarge. Intuitively, when the network has a ( k, α, β ) bottleneck, there is a small set of k nodes B that is responsible for an unusually large share β of the tension/energy of the network when thenetwork experience a large shock of magnitude α . As in the single item case, we define: w ( u ) = X e u w ( e ) , to be the total weight of all edges that touch u , and w = X u ∈ V w ( u )to be sum of all the weights of each node.We need a technical lemma whose proof is deferred to Appendix A:25 emma 12. For every shock x , there is a vector z ∈ R d such that X u ∈ e ∈ E w ( e ) h τ ( e ) , x u i · τ ( e ) = X u ∈ e ∈ E w ( e ) h τ ( e ) , z i · τ ( e ) . Let y be the vector such that for every u ∈ V , y u = z , and define x = x − y . The point of this definition is that the energy and tension of x remain exactly the same as thoseof x , E( x ) = E( x ) , T( x ) = T( x ) , but the weighted average belief of x is 0: X u ∈ e ∈ E w ( e ) · h τ ( e ) , x u i · τ ( e ) = 0 , and x still captures the inefficiencies described in x .Towards defining the α -tension, for each α >
0, we define the set of significant shocks to be V α = ( x : for every v ∈ V , X u ∈ e ∈ E w ( e ) · | h τ ( e ) , x u i | w ≥ α · k x v k ) . Definition 13.
For α > , the α -tension of the trade network is T α = min x ∈V α T( x )2 · P u ∈ e ∈ E w ( e ) · | h τ ( e ) , x u i | . For the energy, we define significant shocks with the set: U α = ( x : for every v ∈ V , X u ∈ e ∈ E w ( e ) · h τ ( e ) , x u i w ≥ α · k x v k ) . Definition 14.
For α > , the α -energy of the trade network is E α = min x ∈U α E( x ) P u ∈ e ∈ E w ( e ) · h τ ( e ) , x u i . In the case that τ ( e ) is the same vector for every edge e , these definitions are identical to thosedefined for the single item case. To understand the definitions, we prove several facts whose proofscan be found in Appendix A. Continuing the analogy with the case of single item markets, we have: Fact 15. ≥ T α ≥ and ≥ E α ≥ . Fact 16.
For any choice of type vectors, if the network has n nodes such that every node tradesevery item with every other node at unit rate, we have T α ≥ α , and E α = + n − . n = 2, Fact 16 shows that E α = 1. For large n , the energy of this completely connected network is close to 1 /
2. To contrast, the tension canactually be small. In Appendix B, we give an example showing that one can have fully connectednetworks where the tension is close to α/ α gets smaller, we have: Fact 17.
For every α ≤ β , T α ≤ T β and E α ≤ E β . Lemma 18. U α ⊆ V α ⊆ U α Finally, we have:
Fact 19. E α = 0 ⇒ T α = 0 ⇒ E α = 0 . Given the changes we have already made, the definitions that capture bottlenecks are identicalto the corresponding definitions in the single item case. Define T B ( x ) to be the tension of theshock after deleting the edges of the network that do not touch B . Definition 20. A ( k, α, β ) bottleneck for tension is a set of k nodes B such that that there is avector x ∈ V α with T( x ) > and T B ( x )T( x ) ≥ β. Similarly, let E B ( x ) denote the energy due to all the edges that touch the set B . Definition 21. A ( k, α, β ) bottleneck for energy is a set of k nodes B such that there is a vector x ∈ U α with E( x ) > and E B ( x )E( x ) ≥ β. A ( k, α, β ) bottleneck is a serious obstruction to efficiency when k is small and α, β are relativelylarge. Intuitively, when the network has a ( k, α, β ) bottleneck, there is a small set of k nodes B that is responsible for an unusually large share β of the tension/energy of the network when thenetwork experience a large shock of magnitude α . The principal contribution of this work is the definition of the trade network and the conceptsof tension, energy and bottlenecks, which allow for a theory of market efficiency that does notsuffer from the joint-hypothesis problem. The trade network provides a bridge from observabledata to concepts like market efficiency, without making any assumptions about the evolution orcorrectness of prices. The trade network encodes data about the information that is visible tomarket participants and the information participants are able to act on. Market efficiency relies onwho knows what, rather than what everyone knows, and the concept of the trade network providesmetrics about who knows what.We believe that the mathematics in our theory is useful to measure efficiency because the singleitem case is similar to equations used to model analogous concepts in physics. Experiments haveconfirmed that these equations accurately describe electrical networks, networks of springs, and thediffusion of heat. That said, one can only know if the choices we have made in the formulas for27ension, energy and bottlenecks are well-founded by experimenting with real data. Experimentscan help to validate the model, or find issues that suggest refinements to the model. Gatheringdata to approximate the trade network would be extremely useful.It would be very useful to find sampling based techniques to estimate the energy/tension ofthe network, and use them to compute the statistics we have defined in this work. Can we samplesmall amounts of data about the trade network and use it to compute the tension and energy, andfind bottlenecks?There are several ways to extend our model that we believe could be fruitful: • One could study the evolution of the trade network over time. An efficient market should havea rapidly evolving trade network, with traders switching trading partners to take advantageof new information and better prices. If nodes do not switch trading partners in response toshocks, that is an inefficiency. One could look for a model that rigorously captures this kindof market efficiency using data. • One could study the relationship between the rates of trade and prices. If nodes do not oftentrade with partners offering them the best price, the market is likely to be inefficient. Again,it remains open to rigorously define this kind of efficiency.
Thanks to Paul Beame, Morgan Dixon, Abe Friesen, Shayan Oveis Gharan, Kevin Miniter, ShayMoran, Alex Jaffe, Anna Karlin, Kayur Patel, Sivaramakrishnan Natarajan Ramamoorthy, DarcyRao, Tim Roughgarden, Brad Wagenaar, Alex White and Amir Yehudayoff for useful comments.Thanks to Kayur Patel for many suggestions that improved the figures in this paper.
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Proof of Fact 5.
The tension is clearly non-negative. We have:T( x ) = T( x ) = X u ∈ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X v ∈ V w ( { u, v } ) · ( x u − x v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X u ∈ V X v ∈ V w ( { u, v } ) · ( | x u | + | x v | )= 2 X u ∈ V w ( u ) · | x u | , proving that T α ≤
1, for x ∈ V α , since such x must have 2 P u ∈ V w ( u ) · | x u | > x ) = E( x ) = (1 / X { u,v }∈ E w ( { u, v } ) · | x u − x v | ≤ X { u,v }∈ E w ( { u, v } ) · ( | x u | + | x v | ) since 2 a + 2 b ≥ ( a + b ) for all a, b .= X u ∈ V w ( u ) · | x u | , proving that E α ≤
1, for x ∈ U α . Proof of Fact 6.
For any shock x ∈ V α , we haveT( x ) = T( x ) = X u ∈ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X v ∈ V ( x u − x v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = X u ∈ V (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n · x u − X v ∈ V x v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = X u ∈ V n · | x u | . by the definition of x Thus we have T α = min x ∈ V α T( x )2 P u ∈ V ( n − | x u | = n n −
1) = 12 + 12( n − . For any x ∈ U α , we haveE( x ) = E( x ) = (1 / X { u,v }∈ E ( x u − x v ) = (1 / X { u,v }∈ E x u + x v − x u x v = (1 / X u ∈ V ( n − x u − X { u,v }∈ E x u x v . Now since P u x u = 0, we have that for every u , P u = v x v = − x u . SoE( x ) = (1 / X u ∈ V ( n − x u − X u ∈ V x u ( − x u )= (1 / X u ∈ V ( n − x u + 2 X u ∈ V x u = (1 / X u ∈ V ( n + 1) x u . Thus E α = min x ∈U α P u ∈ V ( n + 1) x u P u ∈ V ( n − x u = n + 12( n −
1) = 12 + 22( n − . roof of Lemma 8. Suppose x ∈ V α . Then we have α · | x v | ≤ X u ∈ V w ( u ) · | x u | w ≤ sX u ∈ V w ( u ) w · sX u ∈ V w ( u ) w · x u by the Cauchy-Schwartz inequality= sX u ∈ V w ( u ) w · x u , proving that X u ∈ V w ( u ) · x u w ≥ α · x v , and so x ∈ U α . For the other containment, suppose that x ∈ U α , and x v is the coordinate thathas maximum magnitude. Then for every v ∈ V , α · | x v | ≤ α · | x v | = α · | x v | | x v | ≤ | x v | X u ∈ V w ( u ) · x u w since x ∈ U α = | x v | X u ∈ V w ( u ) · x u w · x v ≤ | x v | X u ∈ V w ( u ) · | x u | w · | x v | since | x u || x v | ≤ X u ∈ V w ( u ) · | x u | w , so x ∈ V α . Proof of Fact 9.
Suppose x ∈ U α is a shock such that E( x ) = 0. Then by lemma 8, x ∈ V α .Moreover, for every edge { u, v } in the network, we have x u = x v , or else the energy would bepositive. So we must have T( x ) = 0, proving that T α = 0.Suppose x ∈ V α and T( x ) = 0. Then by lemma 8, x ∈ U α . Let f ( t ) ∈ R n be the vector offorces felt by the nodes during the shock t x . Then we see that f ( t ) is t times the vector of forcesfelt when the shock is x , so f ( t ) = for every t . Since this vector is the gradient of the energy at t x , we must have E(1 · x ) = E(0 · x ) = 0. Proof of Lemma 12.
This proof requires some knowledge about positive semidefinite matrices.View τ ( e ) as a d dimensional column vector, and consider the d × d matrix A = X u ∈ e ∈ E w ( e ) · τ ( e ) τ ( e ) | . We have A z = X u ∈ e ∈ E w ( e ) h τ ( e ) , z i · τ ( e ) , A is positive semidefinite, since for every z , we have z | A z = X u ∈ e ∈ E w ( e ) · h τ ( e ) , z i ≥ . Thus there is an orthonormal basis e , e , . . . , e d for the space R d and non-negative numbers γ , . . . , γ d such that A = d X i =1 γ i · e i e i | . Moreover, every type vector is contained in the span of vectors e i for which γ i >
0, because if,for example there is some edge for which τ ( e ) is not contained in this span, then we can express τ ( e ) = a + b , where a is in the span, and b = is orthogonal to it. Then we have0 = b | d X i =1 γ i · e i e i | ! b = b | A b = X u ∈ e ∈ E w ( e ) (cid:10) τ ( e ) , b (cid:11) ≥ w ( e ) h τ ( e ) , b i = w ( e ) · k b k > , which is a contradiction. So, for q = P u ∈ e ∈ E w ( e ) · h τ ( e ) , x u i · τ ( e ) , we can set z = X i : γ i > (1 /γ i ) h q , e i i · e i , and then we get X u ∈ e ∈ E w ( e ) h τ ( e ) , z i · τ ( e ) = A z = d X i =1 h q , e i i · e i = q , as required. Proof of Fact 15.
The tension is clearly non-negative. We have:T( x ) = T( x ) = X u ∈ V (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X { u,v } = e ∈ E w ( e ) · h τ ( e ) , x v − x u i · τ ( e ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ X u ∈ V X { u,v } = e ∈ E w ( e ) · kh τ ( e ) , x v − x u i · τ ( e ) k≤ X u ∈ V X { u,v } = e ∈ E w ( e ) · ( | h τ ( e ) , x v i | + | h τ ( e ) , x u i | )= 2 X u ∈ e ∈ E w ( e ) · | h τ ( e ) , x u i | , proving that T α ≤
1, for x ∈ V α .Similarly, the energy is clearly non-negative. We haveE( x ) = E( x ) = (1 / X { u,v } = e ∈ E w ( e ) · h τ ( e ) , x u − x v i ≤ X { u,v } = e ∈ E w ( e ) · ( h τ ( e ) , x u i + h τ ( e ) , x v i ) since a + b ≥ ( a + b ) / a, b .= X u ∈ e ∈ E w ( e ) · h τ ( e ) , x u i , proving that E α ≤
1, for x ∈ U α . 32 roof of Fact 16. Let I denote the set of items, and let τ ( i ) denote the type vector of item i . Thenfor any shock x ∈ V α , we haveT( x ) = T( x ) = X u ∈ V (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X v ∈ V,i ∈ I h τ ( i ) , x v − x u i · τ ( i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = X u ∈ V (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( n − X i ∈ I h τ ( i ) , x u i · τ ( i ) − X v ∈ V,i ∈ I h τ ( i ) , x v i · τ ( i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = X u ∈ V (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( n − X i ∈ I h τ ( i ) , x u i · τ ( i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . by the definition of x To bound this expression, we use the Cauchy-Schwartz inequality twice:( n − X u ∈ V (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X i ∈ I h τ ( i ) , x u i · τ ( i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≥ ( n − X u ∈ V (cid:10)P i ∈ I h τ ( i ) , x u i · τ ( i ) , x u (cid:11) k x u k by Cauchy-Schwartz ≥ ( n − X u ∈ V,i ∈ I h τ ( i ) , x u i k x u k . Now since x ∈ V α , we have ≥ ( n − X u ∈ V,i ∈ I h τ ( i ) , x u i · αn | I | P u ∈ V,i ∈ I | h τ ( i ) , x u i | since x ∈ V α ≥ ( n − (cid:16)P u ∈ V,i ∈ I | h τ ( i ) , x u i | (cid:17) n | I | · αn | I | P u ∈ V,i ∈ I | h τ ( i ) , x u i | by Cauchy-Schwartz= α ( n − X u ∈ V,i ∈ I | h τ ( i ) , x u i | . So we get T α = min x ∈ V α T( x )2 P u ∈ V,i ∈ I ( n − | h τ ( i ) , x u i | ≥ α . Similarly, we have that for any shock x ∈ U α ,E( x ) = E( x ) = (1 / X u = v,i ∈ I h τ ( i ) , x u − x v i = (1 / X u = v,i ∈ I h τ ( i ) , x u i + h τ ( i ) , x v i − h τ ( i ) , x u i h τ ( i ) , x v i = (1 / X u,i ∈ I ( n − h τ ( i ) , x u i − X u ∈ u,i ∈ I h τ ( i ) , x u i · * τ ( i ) , X v = u x v + . (2)33ow by the definition of x , = X u ∈ V,i ∈ I h τ ( i ) , x u i τ ( i ) = X i ∈ I * τ ( i ) , X u ∈ V x u + · τ ( i ) , so 0 = *X i ∈ I * τ ( i ) , X u ∈ V x u + · τ ( i ) , X u ∈ V x u + = X i ∈ I * τ ( i ) , X u ∈ V x u + , proving that for every i , (cid:10) τ ( i ) , P u ∈ V x u (cid:11) = 0, and D τ ( i ) , P v = V x v E = − h τ ( i ) , x u i . Returning to(2), we have E( x ) = (1 / X u,i ∈ I ( n − h τ ( i ) , x u i − X u ∈ u,i ∈ I h τ ( i ) , x u i * τ ( i ) , X v = u x v + = (1 / X u,i ∈ I ( n − h τ ( i ) , x u i + X u ∈ u,i ∈ I h τ ( i ) , x u i = (1 / X u,i ∈ I ( n + 1) h τ ( i ) , x u i . This proves thatE α = min x ∈U α E( x ) P u ∈ V,i ∈ I ( n − h τ ( i ) , x u i = n + 12( n −
1) = 12 + 12( n − . Proof of Lemma 18. If x ∈ U α , let v be the node for which k x v k is maximized. Then for every v ∈ V , α · k x v k ≤ α · k x v k k x v k ≤ X u ∈ e ∈ E w ( e ) · h τ ( e ) , x u i w · k x v k = k x v k X u ∈ e ∈ E w ( e ) · h τ ( e ) , x u i w · k x v k ≤ k x v k X u ∈ e ∈ E w ( e ) · | h τ ( e ) , x u i | w · k x v k = X u ∈ e ∈ E w ( e ) · | h τ ( e ) , x u i | w , x ∈ V α . If x ∈ V α , then for every v ∈ V , α · k x v k ≤ X u ∈ e ∈ E w ( e ) · | h τ ( e ) , x u i | w ≤ s X u ∈ e ∈ E w ( e ) w · vuut X u ∈ e ∈ E w ( e ) · h τ ( e ) , x u i w by the Cauchy-Schwartz inequality= vuut X u ∈ e ∈ E w ( e ) · h τ ( e ) , x u i w , proving that x ∈ U α . Proof of Fact 19.
Suppose x ∈ U α is a shock such that E( x ) = 0. Then by Lemma 18, x ∈ V α .Moreover, for every edge e = { u, v } in the network, we have h τ ( e ) , x u − x v i = 0, or else the energywould be positive. So we must have T( x ) = 0, proving that T α = 0.Suppose x ∈ V α and T( x ) = 0. Then by Lemma 18, x ∈ U α . Let f ( t ) ∈ R n be the vector offorces felt by the nodes during the shock t x . Then we see that f ( t ) is t times the vector of forcesfelt when the shock is x , so f ( t ) = for every t . Since this vector is the gradient of the energy at t x , we must have E(1 · x ) = E(0 · x ) = 0. B Example Matching Fact 16 ⌧ ( e )( ↵, p ↵ )( ↵, p ↵ ) Figure 14: Even the fully connected network with 2 participants has T α = α .Here we show that one can have a fully connected network whose tension is proportional to α ,matching the lower bound proved in Fact 16 upto a factor of 2. Similar ideas can be used to givea large network that asymptotically matches the bounds given by Fact 16.Consider the network shown in Figure 14, which has just two participants, 1 ,
2. Suppose the firstedge has type vector τ (1) = ( α, √ − α ), and the second has type vector τ (2) = ( − α, √ − α ),and both have weight 1. Then if we set x = (1 ,
0) and x = ( − , x is a significant35hock with x = x . Indeed, k x k , k x k ≤
1, and X i =1 , X u =1 , | h τ ( i ) , x u i | = 4 α = αw, so x ∈ U α . But the tension of the shock is:T( x ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X i =1 , h τ ( i ) , x − x i τ ( i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X i =1 , h τ ( i ) , x − x i τ ( i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 2 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X i =1 , h τ ( i ) , x − x i τ ( i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 2 (cid:13)(cid:13)(cid:13) α · ( α, p − α ) − α · ( − α, p − α ) (cid:13)(cid:13)(cid:13) = 8 α , so we get T α ≤ T( x )2 P i =1 , P u =1 , |h τ ( i ) , x u i| = α α = αα