Specifying a Structural Matching Game of Trading Networks with Transferable Utility†

American Economic Review: Papers & Proceedings 2017, 107(5): 256–260 https://doi.org/10.1257/aer.p20171114

By Jeremy T. Fox*

Matching games model relationship formation b( ) I and s( ) I. In some models, trades in a market setting. In transferable utility match- indexω other∈ features,ω ∈ such as the hours of weeklyω ing games, matched agents exchange monetary work in a labor market. Each trade has an transfers or prices. There is increasing interest equilibrium price p . ω ∈ Ω in the structural estimation of matching games An agent i I whoω buys trades and sells ∈ Φ with transferable utility in many fields of eco- trades at the prices p Ω has a profit of nomics including development, family, finance, Ψ industrial organization, labor, marketing, and v i , p p , strategy e.g., Ahlin 2016; Akkus, Cookson, and (Φ Ψ) − ∑ ω + ∑ ω Hortaçsu( 2016; Baccara et al. 2012; Chiappori ω∈Φ ω∈Ψ i and Salanié 2016; Chiappori, Salanié, and Weiss where v ( , ) is the valuation of agent i for 2016; Chen and Song 2013; Choo 2015; Choo the buyer ΦandΨ seller trades. The fact that the and Siow 2006; Fox 2010, 2016; Fox, Yang, and prices enter additively is why this model has i Hsu forthcoming; Galichon and Salanié 2015; transferable utility; v ( , ) is expressed in Graham 2011; Mindruta, Moeen, and Agarwal monetary units before anyΦ Ψscale normalization 2016; Siow 2015; Yang, Shi, and Goldfarb in estimation. The valuation from making no 2009 . This paper explores matching games of trades is normalized to zero. ) trading networks where agents can make more A market outcome is sets of trades ( i, i) i I than one match and where agents are not neces- for all agents and a vector of the pricesΦ Ψof all∈ sarily a priori divided into two sides, as in men trades ( p ) . In the structural estimation of a and women. matchingω game,ω∈Ω the dependent variable is some This article lays out many modeling decisions aspect of the market outcome and the indepen- that need to be made when structurally estimat- dent variables are some aspects of the spaces ing matching games. These are issues that are of agents I and of trades . The estimation germane to prior studies but are not discussed object is some aspect of the valuationsΩ of agents i there in detail for the sake of conciseness. I do v ( , ). not discuss particular estimators or inference ThereΦ Ψ are many types of matching games procedures. nested within the structure of matching with trading networks. In one-to-one matching, I. Matching with Trading Networks agents make only a single match; this can be encompassed by setting valuations to if the − ∞ Models of matching with trading networks use number of elements of i i is more than 1. fairly flexible notation Hatfield et al. 2013 . Let In two-sided matching, Φan ∪agentΨ is specified to ( ) i a matching market be indexed by agents i I. be a buyer or seller ex ante; v ( i , i) if Let there be a space of trades . A trade ∈ a buyer agent sells trades or a ΦsellerΨ agent= − buys∞ includes the identity of the buyerΩ and ωseller∈ Ω, trades. The notation also encompasses many- to-many matching, where an agent can buy and sell multiple trades as part of the same market * Department of Economics MS-22, Rice University, PO outcome. Box 1892, Houston, TX 77251 e-mail: jeremyfox@gmail. An econometric model must specify which com . ( † )Go to https://doi.org/10.1257/aer.p20171114 to visit the variables are observed by the agents in the article page for additional materials and author disclosure matching game but unobserved by the econo- statement. metrician and what variables are observed by the 256 VOL. 107 NO. 5 Structural Many-to-Many Matching Games 257 agents and also measured in the data. Note that allows externalities from say post-merger com- most of the papers cited above impose symmet- petition between non-merging firms and the ric information: there are no variables observed modeling of asymmetric information. Basing by one agent in the game but not another agent. structural estimation off of any solution concept Decompose agent i’s valuation for trades will lead to inconsistency if the solution concept ( , ) as is misspecified. However, casual intuition sug- Φ Ψ gests that noncooperative solution concepts are i i v ( , ) ( xi , , ) , , often overly specific for particular decentralized Φ Ψ = π Φ Ψ + ϵΦ Ψ markets and so are particularly likely to be mis- where ( xi , , ) is a function of observables specified. One researcher might model mergers to the πresearcherΦ Ψ , including a vector of i’s as a first-price auction of complete information, i characteristics xi , and , is unobserved het- another a first-price auction of incomplete infor- ϵΦ Ψ erogeneity in i’s valuation. If ( xi , , ) is mation, another a second-price auction, another known up to a finite vector ofπ parametersΦ Ψ , as some sort of bargaining model, and so forth. as in x , , X x , , for a vecθ- Meanwhile, it is possible that several noncoop- π( i Φ Ψ) = ( i Φ Ψ)′θ tor of observables X( xi , , ) , then this func- erative games have equilibria that are consistent tion is parametrically specified;Φ Ψ otherwise it is with the same cooperative solution concept. For nonparametric. Likewise, if the distribution example, Fox and Bajari 2013 estimate a FCC i i I ( ) of the vector ( , ) ∈ , is known to the spectrum auction matching model using pair- researcher afterϵ ΦimposingΨ Φ⊆Ω Ψ⊆Ω location and scale wise stability while arguing that certain nonco- normalizations, as in Choo and Siow 2006 , or operative dynamic games of private information is known up to a finite vector of parameters,( ) as in such ascending auctions have implicitly in Galichon and Salanié 2015 , this distribution collusive Nash equilibria that satisfy pairwise is parametric. ( ) stability.

II. Specifying the Structural Matching Game B. Matching Pairwise Stability versus Networks Pairwise Stability A. Cooperative versus Noncooperative Games The definition of pairwise stability used in the A well specified matching game needs a solu- matching literature including matching with tion concept. Matching games can in principle trading networks is stronger( than the most com- be analyzed under both cooperative game theo- mon definition in) the related networks literature retic solution concepts and noncooperative solu- e.g., Jackson and Wolinsky 1996 . The match- tion concepts. One cooperative solution concept ing( definition of pairwise stability) says there is pairwise stability: a market outcome is pair- exists no vector of prices for trades (p ) wise stable if there exists no vector of prices such that two agents would prefer to makeω ω∈Ω a for trades (p ) such that two agents would trade instead of or in addition to their current prefer to makeω ω∈Ωa trade instead of or in addi- tradesω and no agents would prefer to drop trades tion to their trades in theω market outcome and while the networks or “Jackson” definition of typically no agent would prefer to drop trades. pairwise stability says only that no two agents One( noncooperative) solution concept would be would prefer to add or drop trades. There is no to specify an auction of complete or incomplete swapping of trades considered under the net- information, for example where sellers submit works definition. sealed bids to buyers for each buying opportu- Any pairwise stable market outcome under nity and buyers pick the lowest bidder for each the matching definition is pairwise stable under buying opportunity if that bid is lower than a the weaker networks definition. The networks reserve price. definition, while weaker and hence more robust, The empirical papers cited previously use can thus lead to additional pairwise stable mar- cooperative solution concepts while some struc- ket outcomes and hence a wider identified set tural models of say mergers in industrial orga- for the parameters of interest if the researcher nization use noncooperative concepts e.g., is agnostic about equilibrium selection e.g., Gowrisankaran 1999; Jeziorski 2014; Perez-( Ciliberto and Tamer 2009 . Nonparametric (iden- Saiz 2015 . Noncooperative game theory easily tification of valuation functions) has been studied ) 258 AEA PAPERS AND PROCEEDINGS MAY 2017 for the matching definition but not the weaker E. Continuum versus Truly Finite Markets networks definition Fox 2010; Graham 2011 . ( ) A truly finite market is when the set of agents C. Pairwise Stability versus Competitive in the data is equal to the set of agents in the Equilibrium structural matching game for a market, the set I. A continuum market is when the set of agents A market outcome that is pairwise stable is in the data is a finite subset of a continuum of robust to deviations by two agents at a time agents in the matching game, I. In either case, a while a market outcome that is groupwise sta- researcher might model one or multiple markets ble is robust to deviations by groups of arbitrary and explore various asymptotic arguments for size. One can prove that a groupwise stable mar- frequentist estimation and inference. ket outcome is also a competitive equilibrium Researchers working with marriage datasets and that competitive equilibria are efficient in or data from one large market have typically the sense of maximizing economywide out- assumed that the true market is a continuum put Hatfield et al. 2013 .1 This distinction is e.g., Choo and Siow 2006; Fox and Bajari important( in Fox and Bajari) 2013 , who impose 2013( . Assumptions on a continuum mar- pairwise stability but not groupwise( ) stability, ket are) typically made such that the aggregate and hence argue that they can use their structural market outcome is a function of the distribu- i estimates of valuations to measure inefficiency tion of ( , ) , across agents i but not in a spectrum auction. The trades in a compet- particularϵ ΦrealizationsΨ Φ⊆Ω Ψ⊆Ω of the unobservables. itive equilibrium are unique if the set of trades Therefore, the market outcome is econometri- that maximizes economywide output is unique.2 cally deterministic in the aggregate. By focusing on the equilibrium played in the data, there is D. Solution Concept Existence no need to specify an equilibrium selection rule. Researchers in other settings view each mar- Some researchers have estimated matching ket as truly finite and allow the market outcome games without a proof of the existence of the in each market to be a function of the reali- i i I imposed solution concept. For example, con- zations of ( , ) ∈ , . In the truly finite sider a matching game with a finite number of market, multipleϵΦ Ψ equilibriaΦ⊆Ω Ψ⊆Ω typically lead to set agents where there exists only one agent who identification if the researcher is completely can make multiple trades as a seller and that agnostic about the selection rule over multiple agent has complementarities across the multi- values of a solution concept e.g., Ciliberto and ple trades. Then there exist preferences for the Tamer 2009 . Fox 2010 discusses( a “medium” other agents where no competitive equilibrium assumption )on the( equilibrium) selection rule exists Hatfield et al. 2013 . On the other hand, that maintains point identification. an equilibrium( exists if there) is instead a continuum of agents in the market or agents can F. Data on Prices of Trades make divisible investments in consummated trades Azevedo and Hatfield 2015; Hatfield In all papers on structurally estimating match- and Kominers( 2015 . Imposing solution con- ing games, a researcher has data on aspects of ) cepts that might not exist is dangerous ground the trades ( i , i) i I. A few matching papers but working with a continuum of agents is one have exploredΦ estimationΨ ∈ also using data on the way of precisely stating an approximation that is prices of trades p e.g., Akkus, Cookson, ( ω)ω∈Ω ( needed to ensure existence. and Hortaçsu 2016; Fox and Bajari 2013; Uetake and Watanabe 2016 . Assumptions ) can be imposed on the distribution of i i I 1 , ∈ , such that the prices of A competitive equilibrium is a price vector p Ω such that (ϵΦ Ψ)Φ⊆Ω Ψ⊆Ω all agents pick trades to maximize profits and the market trades p are not functions of the reali- clears: supply equals demand on each trade. ( ω)ω∈Ω 2 i i I If preferences satisfy a substitutes condition, all pair- zations of , ∈ , . This simplifies both wise stable markets outcomes are also groupwise stable and (ϵΦ Ψ)Φ⊆Ω Ψ⊆Ω competitive equilibria Hatfield et al. 2013 . Many empirical identification arguments and the computation applications focus on complementarities.( ) of estimators. The related hedonics literature VOL. 107 NO. 5 Structural Many-to-Many Matching Games 259 focuses more on data on the prices of trades Baccara, Mariagiovanna, Ayse Imrohoroglu, e.g., Heckman, Matzkin, and Nesheim 2010 . Alistair J. Wilson, and Leeat Yariv. 2012. “A ( An econometric selection problem may arise) Field Study on Matching with Network Exter- if the prices of trades are jointly distributed nalities.” American Economic Review 102 5 : i i I ( ) with , ∈ , and the prices of trades 1773–804. (ϵΦ Ψ)Φ⊆Ω Ψ⊆Ω are observed only for consummated trades. Chen, Jiawei, and Kejun Song. 2013. “Two-Sided Typically, the so-called generalized Roy selec- Matching in the Loan Market.” International Journal of Industrial Organization 31 2 : tion model leads to issues of partial identifi- ( ) cation and slow rates of convergence. Similar 145–52. selection issues may arise if the researcher has Chiappori, Pierre-Andre, and Bernard Salanié. data on say the profits or valuations of consum- 2016. “The Econometrics of Matching Mod- mated trades only. els.” Journal of Economic Literature 54 3 : 832–61. ( ) Chiappori, Pierre-Andre, Bernard Salanié, and G. Quotas Yoram Weiss. 2016. “Partner Choice and the Marital College Premium: Analyzing Marital Define a quota qi to be the maximum number Patterns over Several Decades.” Unpublished. of trades an agent i can undertake; valuations are Choo, Eugene. 2015. “Dynamic Marriage Match- if the agent makes more trades. In marriage, ing: An Empirical Framework.” Econometrica the− ∞ quota of each agent is known to be 1 Choo 83 4 : 1373–423. and Siow 2006 . In the matching of suppliers( of Choo,( Eugene,) and Aloysius Siow. 2006. “Who car parts to the) assemblers of cars, the quota of Marries Whom and Why.” Journal of Political each assembler is the number of different car Economy 114 1 : 175–201. parts needed from an engineering perspective Ciliberto, Federico,( ) and Elie Tamer. 2009. “Mar- but the quota of each supplier is unknown Fox ket Structure and Multiple Equilibria in Airline forthcoming . To compute market outcomes( to a Markets.” Econometrica 77 6 : 1791–828. matching game) within an estimator, one needs to Fox, Jeremy T. 2010. “Identification( ) in Matching specify quotas or integrate them out over some Games.” Quantitative Economics 1 2 : 203–54. distribution. The matching maximum score esti- Fox, Jeremy T. Forthcoming. “Estimating( ) Match- mator of Fox forthcoming bases estimation off ing Games with Transfers.” Quantitative of inequalities( that may remain) valid in the pres- Economics. ence of missing data on quotas. 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