{{short description|Field of market economics}} {{More footnotes|date=December 2020}} {{Economics sidebar}} In economics, '''matching markets<ref>{{Cite book |last=Atila Abdulkadiroglu and Tayfun Sonmez |first= |url=https://books.google.com/books?hl=iw&lr=&id=gO1UBQAAQBAJ&oi=fnd&pg=PA3&dq=%22Matching+markets%22&ots=JmImVbob16&sig=GWKU2i_Q3h9Jqji_QFaRW4ERwEc |title=Matching Markets: Theory and Practice. In: Advances in Economics and Econometrics: Volume 1, Economic Theory: Tenth World Congress |last2= |first2= |last3= |first3= |date=2013-05-13 |publisher=Cambridge University Press |year=2013 |isbn=978-1-107-71780-0 |language=en}}</ref>''' are markets in which there are two groups of people and/or objects, where each person from one group wants to be matched to a person/object from the other group. Different people in each group may have different preferences regarding who they want to match in the other group; hence the theory is often called '''matching under preferences'''.<ref name=":0">{{Cite web |last=Manlove |first=David |date=2013 |title=Algorithmics of Matching Under Preferences {{!}} Series on Theoretical Computer Science |url=https://www.worldscientific.com/worldscibooks/10.1142/8591 |access-date=2026-05-13 |website=www.worldscientific.com |language=en |doi=10.1142/8591#t=aboutBook}}</ref><ref>{{Cite book |last=Bettina Klaus, David F. Manlove, Francesca Rossi|title=Matching under preferences|date=2016|publisher=Cambridge university press|year=2016}}</ref>

The theory of matching markets aims to design rules and algorithms for computing matchings that satisfy some desirable normative properties. The most important such property is '''stability'''.

In 2012, the Nobel Memorial Prize in Economic Sciences was awarded to Alvin E. Roth and Lloyd Shapley for their work on matching markets, particularly "the theory of stable allocations and the practice of market design".<ref>{{Cite web|url=https://www.nobelprize.org/prizes/economic-sciences/2012/summary/|title=The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2012|access-date=2020-12-29|website=The Nobel Prize}}</ref>

Matching markets are broadly divided into two main categories: markets with and without money. Markets with money are often said to have *transferrable utility* (TU), as money can be used to transfer utility from one agent to another. Similarly, markets without money are said to have *non-transferrable utility* (NTU).

== Matching markets without money (NTU) == This strand of the literature emerged from the Gale and Shapley (1962) introduction of the Gale–Shapley algorithm. Problems studied in this strand include:<ref name=":0" />

* Stable marriage problem - a one-to-one matching between agents of two disjoint groups, each of whom has strict preferences over agents of the other group, so the preferences form a bipartite graph. The typical example is men and women, as in the campus Marriage Pact. * Hospitals-residends problem - a one-to-many matching between agents of two disjoint groups. The typical example is matching medical school graduates for internships in hospitals, as in the National Resident Matching Program. * Stable matching with indifference - a generalization of the stable marriage problem, where agents on each group may be indifferent between two or more agents of the other group. * Stable roommates problem - a one-to-one matching between agents of a single group. This is a generalization of stable marriage, as the preferences form a general graph. * Pareto optimal matchings; * Popular matching<nowiki/>s; * Rank-maximal matching and generous matching.

== Matching markets with money (TU) == This trand dates back to work on Monge (1781) and Kantorovich (1942) work on optimal transportation theory, in particular following Koopmans and Beckmann (1957) who studies the problem with pricing. Modern TU matching follows work by Shapley and Shubik (1971), who provided a TU equivalent of Gale and Shapley (1962), as well as Becker (1973) who applied TU matching to the marriage market.<ref>{{Cite journal|title=Sorting through Search and Matching Models in Economics|journal=Journal of Economic Literature|url=http://www.janeeckhout.com/wp-content/uploads/20.pdf|last=Chade|first=Hector|volume=55|pages=493–544|last2=Eeckhout|first2=Jane|year=2017|last3=Smith|first3=Lones}}</ref> == Related theories == '''Walrasian markets''' are markets with people and commodities, where people buy and sell commodities, and have different preferences regarding various bundles of commodities, but do not match with other people. In contrast, the theory of matching markets focuses of who matches with whom.

'''Search-and-matching theory''' is a descriptive theory describing the ''decentralized'' formation of matches when there are search frictions (the search is costly). In contrast, matching markets theory usually considers normative (desirable) properties of matchings, and ''centralized'' rules that guarantee such properties. Under certain contexts, the search and matching equilibrium converges to a stable matching when search frictions disappear.<ref>{{Cite journal|title=Stable marriages and search frictions|journal=Journal of Economic Theory|url=https://ideas.repec.org/a/eee/jetheo/v151y2014icp163-195.html|last=Lauermann|first=Stephan|volume=151|pages=163–95|last2=Nöldeke|first2=Georg|year=2014}}</ref>

==Applications== Matching theory typically focuses on two-sided matching, where two types of workers are considered (e.g. men and women in the marriage market, firms and workers in the labor market, and students matching with colleges). A smaller literature considers other types of matching, such as one-sided matching (e.g. the stable roommates problem) and many-sided matching (e.g. man-woman-child matching). Within two sided matching, three types of matches are considered: one-to-one, many-to-one, and many-to-many.<ref>{{cite book |last1=Niederle |first1=Muriel |last2=Roth |first2=Alvin |last3=Sönmez |first3=Tayfun |date=2007 |title=The New Palgrave Dictionary of Economics, 2nd edition |url=https://web.stanford.edu/~niederle/Palgrave%20Matching.Approved.pdf|publisher=Palgrave Macmillan}}</ref>

Matching theory has been applied to study a wide set of applications, including: marriage, housing allocation, kidney exchange, the National Resident Matching Program, school choice,<ref>{{cite web |url=https://piazza.com/class_profile/get_resource/h7c8a0q5bh717w/h7gxc4dztaz7k3 |title=Matching Theory and Its Applications |last=Levin |first=Jonathan |date= 2011}}</ref> and hedging strategies in options financial markets.<ref>{{cite journal|title=Financial option insurance|journal=Risk Management–Journal of Risk Crisis and Disaster|date=February 2017|first1=Q.-W.|last1=Wang|first2=Jian-Jun|last2=Shu|volume=19|issue=1|pages=72–101|doi=10.1057/s41283-016-0013-5|arxiv=1708.02180}}</ref> == References == {{reflist}}

Category:Market (economics) Category:Matching markets