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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

Applications: Marriage problems (one to one)

• One-sided matching problems

Applications: Roommate problems Marriage problems (one to one) Matching of workers to firms (many to one) Matching of doctors to hospitals (many to many) Kidney exchange

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 Pi

• P= (Pi)A is a preference profile.
• A matching problem is a pair p=(A,P).

Illustration of a Matching

.m1 .m2 .w1 .w

2

.m3 .m4 .w2 .w3

Pm1 Pm2 Pm3 w2 w3 w3 w3 w1 w2 w w w Pw1 Pw2 Pw3 m2 m2 m1 m3 w2 m2 m m m w1 w2 w1 m1 m2 m3 m1 m1 m3 w1 m3 w3

A mild axiom: STABILITY

• Consider the matching µ1 where
• µ1(m1) = w2
• µ1(m2) = w3

µ (m ) = w

• Consider the matching µ2 where
• µ2(m1) = w3
• µ2(m2) = w2

µ (m ) = w

• µ1(m3) = w1

NOT INDIVIDUALLY RATIONAL

• µ2(m3) = w1
• BLOCKED BY THE PAIR (m2,w1)

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 Pi 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

Pm1 Pm2 Pm3 w2, w3 w2 w3 w1 w1 w1 Pw1 Pw2 Pw3 m1 m1 m1 m2 m2 m3 w1 w1 w1 Consider µ1 where µ1(m1) = w2 µ1(m2) = w1 and µ1(m3) = w3 .

• m3 prefers µ1 to µ2 ;

m2 m2 m3 m3 Consider µ2 where µ2(m1) = w3 µ2(m2) = w2 and µ2(m3) = w1 . but m2 prefers µ2 to µ1

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

Pm1 Pm2 Pm3 w1 w2 w1 w2 w1 w2 w3 w3 w3 Pw1 Pw2 Pw3 m2 m1 m1 m3 m2 m2 m1 m3 m3 w3 w3 w3 m1 m2 m3

• The men optimal solution

assigns w2 to m1, w1 to m2, and w3 to m3. m1 m3 m3 w1 w2 w3

• Next, let the matched pair (m3,

w3) leave the society. For this subproblem, the men optimal solution assigns w1 to m1, and w2 to m2.

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

• nes at µ.

The men optimal solution does not satisfies converse consistency

Pm1 Pm2 Pm3 w1 w2 w3 w2 w3 w1 w3 w1 w2 Pw1 Pw2 Pw3 m2 m3 m1 m3 m1 m2 m1 m2 m3 w3 w1 w2 m1 m2 m3

Consider the matching µx which assigns w2 to m1, w3 to m2, and w1 to m3. Clearly, it is not men optimal.

m1 m2 m3 w1 w2 w3

Considering all the subproblems: A1=(m1,m2,w2,w3), A2=(m1,m3,w1,w3), A3=(m2,m3, w1, w3). The men optimal

• ptimal solution of each subproblem

gives exactly the same matching pairs as under µx.

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

• ptimality, 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
• ptimality, 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 =qc for every college C, and if the number of students in µ(C),

say r, is less than the qc, then µ(C) contains qc-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.

• Abdulkadiroğlu, A., Sönmez, T. (1999) House allocation with existing tenants, Journal of Economic

Theory, 88, 233-260.

• Abdulkadiroğlu, A., Sönmez, T. (2003) School choice: a mechanism design approach, American

Economic Review, 93 (3), 729-747.

• Chung, K.S. (2000) On the existence of stable roommate matchings, Games and Economic

Behavior, 33, 206-230.

• Crawford, V.P., Knoer, E.M. (1981) Job matching with heterogeneous firms and workers,

Econometrica, 49, 437-450.

• Gusfield, D., Irving, R. W. (1989) The Stable Marriage Problem: Structure and Algorithms, MIT

Press, Boston, Ma. Press, Boston, Ma.

• Kelso, A.S., Crawford, V.P. (1982) Job matching, coalition formation, and gross substitutes,

Econometrica, 50 (6), 1483-1504.

• Klaus, B. (2011) Competition and resource sensitivity in marriage and roommates markets,

Games and Economic Behavior, 72, 172-186.

• Knuth, D.E. (1976) Marriages Stables. Montreal: Les Presses de l’Université de Montreal.
• Nizamogulları, D., Özkal-Sanver, İ. (2014) Characterization of the Core in Full Domain Marriage

Problems, Mathematical Social Sciences, 69, 34-42.

• Nizamogulları, D., Özkal-Sanver, İ. (2015) Consistent enlargements of the core in roommate

problems, Theory and Decision, 79 (2).

• Özkal-Sanver, İ. (2010) Impossibilities for roommate problems, Mathematical Social Sciences, 59,

360-363.

• Pathak, P., Sönmez T. (2013) School admissions reform in Chicago and England: comparing

mechanisms by their vulnerability to manipulation, American Economic Review, 103 (1), 80-106.

• Roth, A.E. (1984) Stability and polarization of interests in job matching, Econometrica, 52, 47-57.
• Roth, A., Sönmez, T., Ünver, U. (2004) Kidney exchange, Quarterly Journal of Economics, 119 (2),

457-488.

• Roth, A., Sönmez, T., Ünver, U. (2005) Pairwise kidney exchange, Journal of Economic Theory, 125,

151-188.

• Shapley, L., Shubik, M. (1972) The assignment game I: the core, International Journal of Game

Theory, 1, 111-130. Theory, 1, 111-130.

• Sönmez, T., Ünver, U. (2005) House allocation with existing tenants: an equivalence, Games and

Economic Behavior, 52: 153-185.

• Tan, J.J.M. (1990) A maximum stable matching for the roommate problems, BIT 29, 631-640.
• Tan, J. J. M. (1991) A necessary and sufficient condition for the existence of a complete stable

matching, Journal of Algorithms, 12, 154–178.

• Thomson, W. (1990) The Consistency Principle. In: Ichiishi, T., Neyman A., Tauman, Y., (Eds.), Game

Theory and Applications. Proceedings of the 1987 International Conference held at Ohio State University, Columbus, Ohio. Academic Press, New York, 187-215.

• Thomson, W. (2005), Consistent allocation rules, unpublished monograph.