While the story has been different in each case, fundamentally, we ve maintained:

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1 Econ 805 Advanced Micro Theory I Dan Quint Fall 2009 Lecture 22 November What the Hatfield and Milgrom paper really served to emphasize: everything we ve done so far in matching has really, fundamentally, been about one model While the story has been different in each case, fundamentally, we ve maintained: Two sides of the market Agents on one side are just looking for one match Preferences of agents on the other side satisfy some sort of substitutes restriction either trivially (because they want to match to just one partner), or explicitly And we found that this gave us a series of nice results: a natural equivalence between a stability notion and the core existence of stable matchings, and the fact that they form a lattice a constructive procedure to find the two extreme points in this lattice, each of which is one side s unanimously most-preferred and the other side s least-preferred stable matching some simple comparative statics about how these extreme points respond to changes in the number of agents Today, we consider a model that is actually fundamentally different, Michael Ostrovsky s model of supply chain networks There are more than two sides to the market there are suppliers of basic goods, consumers of final goods, and potentially several layers of firms that turn intermediate goods into other intermediate goods The many-to-one assumption is abandoned every agent in the market could potentially be interested in multiple contracts with multiple partners Once we re in a many-to-many world, the easy equivalence between stability and the core breaks down In fact, even in a two-sided many-to-many model, it s much less obvious what is the appropriate notion of stability Ostrovsky spends some time discussing this, and defending his choice for this model and since it s a very recent paper, the references list is very up-to-date, so if you re interested in many-to-many models, check it out 1

2 So, let s start with Ostrovsky s model We begin with a finite set of firms A (he calls them nodes, since he ll talk in terms of chains and networks), and a partial order on A Basically, for two firms, a b means firm a is upstream of b, which means firm a sells something firm b might need (although it could be indirect a could supply raw materials to another firm, who processes them into an input that firm b uses) DRAW IT, including the fact that they don t have to be neatly organized in layers Relationships between firms are contracts; a contract specifies the buyer, the seller, the price, and an exact description of the good for sale Ostrovsky assumes trade takes place in discrete increments for example, all contracts for iron ore are for one ton of iron ore so if a steel producer wants to buy five tons from a particular supplier, they sign five different contracts In fact, he assumes that each contract is for a particular ton of iron ore, not a generic ton of iron ore each contract lists the serial number of the unique good being traded. (If there are lots of equivalent tons of iron ore for sale by the same seller, the buyer would be indifferent among them, but would have a particular tie-breaking rule for resolving indifferences.) So each contract c = (s, b, l, p), with the universe of potential contracts C finite and exogenously given (He points out that C could just be all possible contracts, or we could remove all the contracts between particular pairs of firms to account for, say, the trade embargo with Cuba, by not allowing U.S. and Cuban firms to trade with each other) And he shows how special cases of this model incorporate the marriage model, the college admissions model, and the Kelso and Crawford firms-and-workers model 2

3 Each firm (node) can be involved in lots of contracts simultaneously, but it can t have two contracts with the same partner and the same serial number each unique good can only be traded once Subject to this constraint, each agent has preferences over collections of contracts. He gives the example of quasilinear preferences, which would give the payoff to node a to a set of contracts X as V a (X) = W a ({(s c, b c, l c ) c X}) + p c p c c D c U where D is the set of contracts with downstream firms (that is, contracts where a is the seller, and therefore earns the price), U is the set of contracts with upstream firms (so a is the buyer and pays the price), and W incorporates the benefit from the products purchased and the cost of generating the products sold For a given set of contracts X, let D a (X) be the subset that have a as the seller (so they are contracts between a and a downstream partner), and U a (X) the subset of X that have a as the buyer (contracts between a and an upstream partner) And let Ch a (X) be firm a s demanded (chosen) set of contracts out of the set X Ostrovsky introduces two conditions, which extend the usual substitutes assumption to this setting Contracts are same-side substitutes if a firm views contracts with all downstream partners as substitutes, and all contracts with upstream partners as substitutes, by the usual Hatfield- Milgrom definition That is, preferences satisfy the same-side substitutes condition for firm a if for any two sets of contracts X and Y, if D a (X) = D a (Y ) and U a (X) U a (Y ), U a (X) U a (Ch a (X)) U a (Y ) U a (Ch a (Y )) (leave the downstream contracts the same, add an upstream contract for firm a, and like before, the set of rejected upstream contracts grows any upstream contract that firm a was already rejecting, he still rejects) and likewise if you switch U a and D a So that s same-side substitutes. if there were no intermediate firms only basic goods suppliers and final goods consumers we d be in a two-sided market, and this would exactly be the usual assumption of substitutes preferences However, if there are middle firms, we also need a condition about how they view upstream and downstream contracts; Ostrovsky assumes that contracts are cross-side complements Firm a s preferences satisfy the cross-side complements condition if, when you add a downstream contract, the set of upstream contracts accepted grows; and vice versa 3

4 Formally, again take two sets of contracts X and Y. Cross-side complements says that if D a (X) = D a (Y ) and U a (X) U a (Y ), then D a (Ch a (X)) D a (Ch a (Y )) So by adding new possible upstream contracts, firm a now demands all the downstream contracts he was before, plus maybe some new ones And likewise when D a and U a are switched Ostrovsky points out these conditions are restrictive For example, they rule out economies of scale in production If a firm could turn one ton of iron ore into one ton of steel at a cost of $4,000, or two tons into two tons at $6,000, there would be prices at which two contracts for a ton of iron ore would violate the same-side substitutes condition Similarly, suppose a firm could turn one ton of iron into one ton of steel, or one ton of wood into one ton of furniture, but not both due to capacity constraints or whatever In that case, cross-side complements would be violated: a drop in the price of iron ore would lead to the firm demanding iron and steel contracts rather than the wood and the furniture contracts; so iron and furniture would be substitutes, not complements However, he does point out that there are lots of production technologies that would satisfy these conditions. He again links them to the endowed assignment preferences, and gives an example Next is the notion of stability In marriage markets, we only worried about individuals and pairs blocking In many-to-one markets, we also worried about a single hospital and a group of doctors blocking Here, we worry about a buyer-seller chain blocking DRAW IT simple three-layer, two-firms-in-each-layer example Ostrovsky calls a collection of contracts a network And he worries about: an individual blocking a network because it s not individually rational (one firm could drop one or more of its contracts, keep some of the others, and be better off) 4

5 a buyer-seller pair blocking a network, by introducing a new contract (maybe dropping some of their existing contracts, maybe keeping some of them) a buyer-seller-seller chain blocking a network, by introducing one new contract between each (Show it on example) Formally, for a given network µ, a chain block is a chain of contracts (c 1,..., c n ) such that: none of these contracts are in the existing network µ the buyer in one contract is the seller in the next that is, b c1 = s c2, b c2 = s c3, and so on everyone in the chain would demand the new contracts, if their only options were the new contracts plus the contracts in µ Ostrovsky also points out firms are non-strategic they treat each contract independently, and don t worry how they interconnect That is, a firm in the chain could demand the new contract but plan to drop an existing contract with another member of the chain, and so on Ostrovsky points out that the model may therefore be inappropriate for settings where a few large players dominate and manipulate a market; but more appropriate in competitive markets with small players He also points out from the beginning that nodes don t have to be firms they could be countries, or regions 5

6 So that s the notion of stability It will help to see some examples Consider a simple market with six agents: two top-level suppliers a 1 and a 2, two intermediary firms b 1 and b 2, and two consumers of the final outputs c 1 and c 2 Suppose for simplicity that there are no prices or quantities: each firm can form just one link to the adjacent layer So there are eight possible contracts: (a i, b j ) and (b j, c k ) for i, j, k {1, 2} For preferences: Supplier a i prefers trade with b i to trade with b j to no trade; so a 1 prefers the contract (a 1, b 1 ) to the contract (a 1, b 2 ) to no contract End consumer c i prefers trade with b j to b i to nothing, so c 1 prefers (b 2, c 1 ) to (b 1, c 1 ) to no contract For intermediary b i, trading with only one partner is worse than no contract; he prefers to trade with supplier a j, and consumer c i. So firm b 1 prefers the set of contracts {(a 2, b 1 ), (b 1, c 1 )} to either pair {(a 1, b 1 ), (b 1, c 1 )} or pair {(a 2, b 1 ), (b 1, c 2 )} to pair {(a 1, b 1 ), (b 1, c 2 )} to nothing (prefs over the middle two cases don t matter) Ostrovsky (p 905) shows four sample networks (collections of contracts). A contract is represented by a solid line connecting two nodes. SHOW EXAMPLES. The first network is not stable because there is a chain of two contracts that blocks it. The second is not stable because there is a single contract (chain of length 1) that blocks it. The third is not individually rational. The fourth is stable. Since we re in a many-to-many world, this concept of stability is not equivalent to the core That is, there could potentially be a bigger coalition of firms that could block, even though there is no blocking chains Ostrovsky argues that chain stability is still the appropriate measure to use, because bigger coalitions would be difficult to assemble, while a chain involves only pairs of firms that are used to interacting with each other (He also points out that if a deviating coalition involved multiple firms in the same layer, it might be illegal due to antitrust laws) He does a pretty good job of defending the choice take a look It also turns out that, if each firm a is limited to just one upstream contract and one downstream contract and the preference restrictions hold, then the set of chain-stable networks is equivalent to the weak core (just like in many-to-one) 6

7 So that s the basic setup for the Ostrovsky paper. Rather than two sides, there are multiple sides (not completely isolated from each other), and a partial ordering of who can trade with who in which role. Each firm sees contracts with its suppliers, or with its customers, as substitutes, but it sees a supplier contract and a customer contract as complements. And stability is defined as a network which does not allow any blocking chains. What he finds is that most of the results we ve seen for many-to-one markets go through: Chain-stable networks exist, and form a lattice, and there is an algorithm for computing the maximal and minimal element There is one chain-stable network which is unanimously most-preferred by all top-level suppliers, and unanimously least-preferred by all final-level consumers, although the preferences of intermediate firms are not clear......and one chain-stable network which is unanimously most-preferred by all bottom-level consumers, and unanimously least-preferred by all top-level suppliers, but the preferences of intermediate firms are not clear And these best and worst chain-stable networks respond in the way you d expect when top- or bottom-level players are added: when a top-level supplier is added, the other top-level suppliers end up worse off, and the final-level consumers end up better off, and vice versa Like with all the other matching papers we ve seen, the essence of the proof is an iterative procedure for finding the maximal and minimal chain-stable networks The algorithm is confusing, and it will help to work with a simple example (the one we ve already defined) Ostrovsky claims the algorithm is a generalization of the one in Hatfield and Milgrom (which in turn is a generalization of Kelso and Crawford, which was a generalization of Gale and Shapley) 7

8 Now, a network has edges which are bi-directional a contract between a buyer b and a seller s is drawn as an edge connecting the two The algorithm Ostrovsky uses requires us to define a pre-network, which has the same nodes but has directional edges (arrows) A given arrow r is a vector (o r, d r, c r ), where o r is the origin of the arrow, d r the destination of the arrow, and c r the contract represented by that arrow (o r and d r bear no relation to which node is the buyer and which the seller in contract c r ; they re more about which node wants to add that contract) For a given pre-network ν, ν(a) will be the set of contracts attached to arrows point to a, that is, ν(a) = {c : r = (o r, a, c) ν} In a pre-network, there can be multiple arrows from the same node to the same node, but they must list different contracts; if there are arrows in both directions between two nodes, they could list the same contract or different ones So there are two potential arrows corresponding to each contract c: one from b c to s c, one from s c to b c Like in Hatfield and Milgrom, the algorithm for finding the chain-stable networks is just the iteration of a monotone function, this time a mapping from a pre-network to another pre-network The fixed points of this mapping will correspond to the chain-stable networks The mapping T from the set of pre-networks to itself is defined by T (ν) = {r R : c r Ch or (ν(o r ) c r )} That is, an arrow r tied to a contract c r, pointing from o r (origin) to o r s trading partner under c r, is in the pre-network T (ν) if o r would choose c r, when given the choice set of c r plus all the contracts with arrows pointing towards o r in the pre-network ν 8

9 Let s work through the example in Ostrovsky We start with the same six firms and preferences as before And we start with the pre-network that has all possible arrows pointing upstream, and no arrows pointing downstream, which we call ν min Consider the first iteration Firm a 1 has two arrows pointing to it, from b 1 and from b 2 So it adds an arrow to b i if it would choose to trade with b i, given a choice between b i, b 1, and b 2 Since a 1 prefers to trade with b 1, then, T (ν min ) contains the arrow from a 1 to b 1, and no other arrows from a 1 Similarly, T (ν min ) contains an arrow from a 2 to b 2, and no other arrows from a 2 Next, consider b 1 At ν min, b 1 has arrows from c 1 and c 2 So b 1 adds an arrow to an agent x if it would choose the contract with agent x, when choosing between x, c 1, and c 2 If x = a 1 or x = a 2, then b 1 would demand x and c 1 ; so we add arrows from b 1 to both a 1 and a 2 But if x = c 1 or x = c 2, then b 1 would demand nothing, since he only wants to trade if he can both buy and sell; so there are no other arrows from b 1 We similarly add arrows from b 2 to a 1 and a 2 Next, c 1 c 1 has no arrows pointing toward him under ν min, so we add arrows from c 1 if he would choose them given a choice of that contract or nothing Since c 1 would prefer to trade with either b 1 or b 2 to nothing, we put arrows from c 1 to b 1, and to b 2 And similarly, arrows from c 2 to b 1 and b 2 And that defines T (ν min ) 9

10 Ostrovsky shows one more iteration of the algorithm, and then doesn t show the fact that T 2 (ν min ) turns out to be a fixed point of T the algorithm has already converged Finally, once we have a pre-network ν which is a fixed point of T, we define the network F (ν), which has all the contracts c r which are represented by arrows in both directions under ν In the case of the example, this contains four contracts between a 1 and b 1, b 1 and c 1, a 2 and b 2, and b 2 and c 2 To get the convergence result, we have to first put a partial order on the set of pre-networks We do this by defining ν 1 ν 2 if ν 1 fewer downstream arrows and more upstream arrows (by set inclusion) This is why we called our starting point before ν min since it had no downstream arrows and every possible upstream arrow, it was by definition less than or equal to any other pre-network Ostrovsky then shows that the mapping T is isotone that if ν 1 ν 2, T (ν 1 ) T (ν 2 ) And so T is an isotone mapping from a finite lattice to itself, so it converges monotonically to a fixed point I won t take you through the proof that once we have a fixed point ν, that F (ν) is chain-stable but basically, since a one-directional arrow means one party wants to add a contract, if a contract is missing from F (ν), that means one of the two parties wouldn t want to add it. It s a little more complicated, since a firm could add an upstream and a downstream contract simultaneously, but it ends up working So, the fixed points of T correspond to the chain-stable networks; and we can find the highest and lowest fixed points, by iterating T from the maximal and minimal pre-network. (We did the minimal one; for the maximal, start with the pre-network with no upstream arrows and all possible downstream arrows.) 10

11 In this particular example, Ostrovsky shows there are four chain-stable networks (p 910) If we let ν min and ν max be the highest and lowest fixed points of T, he shows that F (ν min ) is unanimously preferred to all other chain-stable networks by all top-level suppliers, and unanimously least-preferred by all final-level consumers; and vice versa for F (ν max) And just like in the last two models, he shows that if you start off at one of these two extreme chain-stable networks, add a top-level supplier, and restart the algorithm, you end up at the corresponding new extreme chain-stable network, and all the top-level suppliers are weakly worse off and all the final-level consumers are weakly better off As I said, Ostrovsky spends some time discussing different stability concepts and defending his choice of chain-stability He also discusses a few applications One that I like: he shows that we can use the supply-chain model to capture two-sided markets with complementarities Suppose there are firms that views workers and machines as complements, but views workers as substitutes among themselves, and machines as substitutes among themselves Let machines be the top-level supplier, firms be the intermediate nodes, and workers the end-level consumers Then same-side substitutes and cross-side complements would be satisfied, so all these results would hold (Of course, we actually care a lot more about firm s preferences than machines preferences, and the theory says very little about the preferences of the intermediate layers) Tuesday, a couple applications of matching probably school choice and kidney exchange Then after Thanksgiving, you guys are up! COURSE REVIEWS 11