A Brief Introduction to Matching Theory and Its Applications İpek Özkal Sanver Bilgi University Bilgi University Summer School, PACD2019 Universite de Caen Basse Normandie August 29, 2019
Outline • Basic Concepts and Results on Matching Theory; • Characterization of the Core by Consistency and Converse Consistency Characterization of the Core by Consistency and Converse Consistency Axioms; • Applications and Current Research Areas
Matching Problems • Two-sided matching problems • One-sided matching problems Applications: Applications: Roommate problems Marriage problems (one to one) Marriage problems (one to one) Matching of workers to firms Kidney exchange (many to one) Matching of doctors to hospitals (many to many)
Nobel Prize in Economic Sciences (2012) • was awarded jointly to Alvin E. Roth and Lloyd S. Shapley • "for the theory of stable allocations and the practice of market design."
A simple model: Marriage Markets Two finite and disjoint sets of agents The set of women W, and the set of men M. The set of women W, and the set of men M. A society is the union of men and women. The potential mates of an agent is the other set and himself/herself
A matching problem • Each agent has a (strict) preference relation over its potential mates, denoted by P i • P= (P i ) A is a preference profile. • A matching problem is a pair p=(A,P).
Illustration of a Matching .m 1 .w 1 .m 2 2 .w .w 2 .m 3 .w 3 .m 4
P m1 P m2 P m3 P w1 P w2 P w3 w 2 w 3 w 3 m 2 m 2 m 1 w 3 w 1 w 2 m 3 w 2 m 2 w 1 w w w 2 w w 1 m 1 m m m 1 m m 3 m 1 m 2 m 3 w 1 m 3 w 3
A mild axiom: STABILITY • Consider the matching µ 1 where • Consider the matching µ 2 where • µ 1 (m 1 ) = w 2 • µ 2 (m 1 ) = w 3 • µ 1 (m 2 ) = w 3 • µ 2 (m 2 ) = w 2 • µ 1 (m 3 ) = w 1 µ (m ) = w • µ 2 (m 3 ) = w 1 µ (m ) = w NOT INDIVIDUALLY RATIONAL • BLOCKED BY THE PAIR (m 2 ,w 1 )
The Equivalence of the Core and the Set of Stable Matchings • The Core is the set of matchings which are not dominated by any other matching; i.e which are not blocked by any coaliton. • Nothing is lost by ignoring coalitions other than singletons and pairs. • The Existence of the Core in Marriage Problems (Gale and Shapley) • The core is not equal to the set of individually rational and Pareto optimal matchings, e.g. The Serial Dictatorship Rule
Core in Roommate Problems Only one set of agents A Each agent i has preference P i over the other agents. The core = The set of stable matchings However no more existence result…..
Gale and Shapley (1962), and more…. • Existence of a stable matching • Men-(Women-) optimal matching • Deferrred acceptance procedure • Deferrred acceptance procedure • Lattice Theorem (Conway, Knuth) • Decomposition Lemma (Demange, Sotomayor) • Lonely Wolf Lemma • Blocking Lemma (Hwang) • Dubins, Friedman (1981)’s Result on the Limits of Manipulation
COMMON INTEREST AND CONFLICT • The agents on one side of the market have a common interest regarding the set of stable matchings , since they are in agreement on the best stable matching. • However, it turns out that agents on opposite sides of the market • However, it turns out that agents on opposite sides of the market have opposite interests in this regard. • The optimal stable matching for one side of the market is the worst stable matching for the agents on the other side of the market.
An example in which not all agents have strict preferences P m1 P m2 P m3 P w1 P w2 P w3 w 2 , w 3 w 2 w 3 m 1 m 1 m 1 w 1 w 1 w 1 w 1 w 1 w 1 m 2 m 2 m 2 m 2 m 3 m 3 m 3 Consider µ 1 where µ 1 (m 1 ) = w 2 Consider µ 2 where µ 2 (m 1 ) = w 3 µ 1 (m 2 ) = w 1 and µ 1 (m 3 ) = w 3 . µ 2 (m 2 ) = w 2 and µ 2 (m 3 ) = w 1 . • m 3 prefers µ 1 to µ 2 ; but m 2 prefers µ 2 to µ 1
The Characterization of the Core in Marriage Problems • Sasaki and Toda (1992, JET) • Model with Strict Preferences, No agent stays single, there are equal number of men and women . • Axioms: • Axioms: • The core is the unique correspondence which satiesfies Pareto Optimaliy, Anonymity, Consistency and Converse Consistency.
Consistency • 100 Euro among three agents: (A,50), (B,30), (C,20) • Suppose A leaves the society having 50 Euro. A consistent rule allocates 50 Euro among agents B and C as (B,30) and (C,20). • DEFINITION: Consider some problem p and some solution µ. Let µ be a matching recommended by the rule Ω at p. The consistency axiom requires that the restriction of µ to each subgroup of matched pairs is among the recommendations made by the solution Ω for the reduced problem of p with respect to this subgroup.
The men optimal solution does not satisfies consistency P m1 P m2 P m3 P w1 P w2 P w3 w 1 w 2 w 1 m 2 m 1 m 1 w 2 w 1 w 2 m 3 m 2 m 2 w 3 w 3 w 3 w 3 w 3 w 3 m 1 m 1 m 3 m 3 m 3 m 3 m 1 m 2 m 3 w 1 w 2 w 3 • Next, let the matched pair (m 3 , w 3 ) leave the society. For this • The men optimal solution subproblem, the men optimal assigns w 2 to m 1 , w 1 to m 2 , and solution assigns w 1 to m 1 , and w 3 to m 3 . w 2 to m 2 .
Converse Consistency • 80 Euro among agents A and B; (A,50), (B,30) 70 Euro among agents A and C; (A,50), (C,20) 50 Euro among agents B and C; (B,30), (C,20) A converse consistent rule allocates 100 Euro among three agents as (A,50), (B,30) A converse consistent rule allocates 100 Euro among three agents as (A,50), (B,30) and (C,20). DEFINITION: Let p a problem. Take a matching µ. Consider any subproblem containing exactly two matched pairs who are matched at µ. The converse consistency axiom requires that µ must be a solution to the original problem p, if the solution Ω provides for any such subproblems the same matching pairs as the ones at µ.
The men optimal solution does not satisfies converse consistency P m1 P m2 P m3 P w1 P w2 P w3 w 1 w 2 w 3 m 2 m 3 m 1 w 2 w 3 w 1 m 3 m 1 m 2 w 3 w 3 w 1 w 1 w 2 w 2 m 1 m 1 m 2 m 2 m 3 m 3 m 1 m 2 m 3 w 1 w 2 w 3 Considering all the subproblems: Consider the matching µ x which assigns A 1 =(m 1 ,m 2 ,w 2 ,w 3 ), A 2 =(m 1 ,m 3 ,w 1 ,w 3 ), w 2 to m 1 , w 3 to m 2 , and w 1 to m 3 . A 3 =(m 2 ,m 3 , w 1 , w 3 ). The men optimal Clearly, it is not men optimal. optimal solution of each subproblem gives exactly the same matching pairs as under µ x.
Other Core Characterization Results in Marriage Problems • Nizamoğulları and Özkal-Sanver (2014) (strict preferences, but agents are allowed to stay single) The core solution is the unique correspondence which satisfies individual rationality, Pareto optimality, genderfairness, consistency and converse consistency. • Nizamoğulları and Özkal-Sanver (2014) (weak preferences) The core solution is the unique correspondence which satisfies individual rationality, weak Pareto optimality, monotonicity, genderfairness, consistency and converse consistency.
Some More Results in Roommate Problems • Özkal-Sanver (2010) There exists no solution which satisfies Pareto optimality, anonymity sand converse consistency. • Özkal-Sanver (2010) No core extension satisfies consistency. • Özkal-Sanver (2010), Klaus (2017) On the domain of solvable markets, • Özkal-Sanver (2010), Klaus (2017) On the domain of solvable markets, the core satisfies consistency, but not converse consistency. • Klaus (2017) On the domain of solvable markets without 3-rings, the core solution is the unique correspondence which satisfies individual rationality, Pareto optimality, anonymity, consistency and converse consistency.
Many-to-one matching model • in which firms may employ many workers, or colleges admit students A matching µ is a function from the set CUS into the set of unordered families of elements of CUS such that i. Iµ(s)I =1 for every student s and µ(s)=s if µ(s) is not in C i. Iµ(s)I =1 for every student s and µ(s)=s if µ(s) is not in C ii. Iµ(C)I =q c for every college C, and if the number of students in µ(C), say r, is less than the q c , then µ(C) contains q c -r copies of C, iii. µ(s)=C if and only if s is in µ(C). • Under responsive preferences; stable and optimal matchings exist
Other models: • Married couples • College admission problems with or without money and complex preferences (Boston college admission problems thanks to Abdulkadirogulları, Roth, Sonmez, …) • Many-to-many problems with or without budget constraints • Assignment models (e.g. House allocation models, top trading cycle) Assignment models (e.g. House allocation models, top trading cycle) • Multiobject Auction Mechanism • A recent and important field of matching theory and practice: Kidney Exchange (Roth, Sonmez and Unver) and more to be studied …..
References and Recommended Reading List: • Gale, D., Shapley, L. (1962) College admissions and the stability of marriage, American Mathematical Monthly , 69, 9-15. • Roth, A.E. , Sotomayor, M. (1990) Two-Sided Matching: A Study in • Roth, A.E. , Sotomayor, M. (1990) Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis , Econometric Society Monograph Series, Cambridge University. • Sasaki, H., Toda, M. (1992) Consistency and characterization of the core of two-sided matching problems, Journal of Economic Theory , 56, 218-237.
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