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Computational Complexity pp 3234–3257Cite as

Two-Sided Matching Models

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Article Outline

Glossary

Definition of the Subject

Introduction

Discrete Two-Sided Matching Models

Continuous Two-Sided Matching Model with Additively Separable Utility Functions

Hybrid One-to-One Matching Model

Incentives

Future Directions

Bibliography

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Abbreviations

Two-sided matching model:

is a game theoretical model whose elements are (i) twodisjoint and finite sets of agents: F with m elements, and W with n elements, referred to as thesides of the matching model; (ii) the structure of agents' preferences and (iii) the agents'quotas.

The rules of the game determine the feasible outcomes. The main activity of the agents from oneset is to form partnerships with the agents on the other set. Players derive their payoffs fromthe set of partnerships they form. The agents belonging to F and W are called F‑agents andW‑agents, respectively.

Quota of an agent:

in a two-sided matching model is the maximum number of partnerships anagent is allowed to form. When every participant can form one partnership at most the matchingmodel is called one-to-one. If only the players of one of the sides can form more than onepartnership the matching model is said to be many-to-one. Otherwise the matching model ismany-to-many.

Allowable set of partners:

for \( { f \in F } \) with quota r(f) is a family of elements of \( { F\cup W } \) with k distinct W‑agents, \( { 0 \leqslant k \leqslant r(f) } \), and \( { r(f)-k } \) repetitions off.

Discrete two-sided matching model:

In the discrete two-sided matching models agents havepreferences over allowable sets of partners. The allowable sets of partners for f of the type\( { \{ w,f, \ldots ,f \} } \) are identified with the individual agent \( { w \in W } \) and the allowable set ofpartners \( { \{ f \ldots f \} } \) is identified with f. Under this identification, agent w isacceptable to agent f if and only if f likes w as well as him/her/itself. Similar definitions and identifications apply to an agent \( { w \in W } \). These preferences aretransitive and complete, so they can be represented by ordered lists of preferences. The modelcan then be described by \( { (F,W,P,r,s) } \), where P is the profile of preferences and r and sare the arrays of quotas for the F‑agents and W‑agents, respectively.

Continuous two-sided matching model:

In this model the structure of preferences is given by utility functions which are continuous in somemoney variable which varies continuously in the set of real numbers. A particular case is obtained when agents place a monetary value on each possiblepartner or on each possible set of partners.

Hybrid two-sided matching model:

is a unification of the discrete and the continuousmodels. It is obtained by allowing the agents of both markets to trade with each other in thesame market.

Matching μ :

in a two-sided matching model with sides F and W is a function that mapsevery agent into an allowable set of partners for him/her/it, such that f is in \( { \mu(w) } \) ifand only if w is in \( { \mu(f) } \), for every \( { (f,w) \in F \times W } \). If we relax this conditionthe function is called a pre‐matching.

A matching describes the set of partnerships of the type \( { (f,w) } \), \( { (f,f) } \) or \( { (w,w) } \), with \( { f\in F } \) and \( { w \in W } \), formed by the agents. We say that a player that does not enter anypartnership is unmatched. Agents compare two matchings by comparing the two allowable sets ofpartners they obtain.

Feasible assignment:

for a two-sided matching model with sides F and W is an \( m \times n \) matrix \( { x=(x_{fw}) } \) whose entries are zeros or ones such that \( { \sum _{f} x_{fw}\leqslant s(w) } \) for all \( { w \in W } \) and \( { \sum_{w} x_{fw} \leqslant r(f) } \) for all \( f \in F \). We say that \( { x_{fw}=1 } \) if f and w form a partnership and \( { x_{fw}=0 } \)otherwise. A feasible assignment x corresponds to a matching μ which matches f to wif and only if \( { x_{fw}=1 } \). Thus, if \( { \sum _{f} x_{fw} =0 } \) then w is unassigned at xor, equivalently, unmatched at μ, and if \( { \sum _{w} x_{fw}=0 } \), then f is likewiseunassigned at x or, equivalently, unmatched at μ.

Responsive preference:

in a  discrete two-sided matching model with sides F and W. Agent \( { f \in F } \)has a responsive preference relation over allowable sets of partners if whenever (i) A and B are two allowable sets of partners for player f; (ii) j andk are two elements of \( { W \cup \{ f \} }\) and (iii) \( { A = B \cup \{ w \} \setminus\{w^{\prime}\} } \) with \( { w \notin B }\) and \( { w^{\prime} \in B }\), then f prefers A to B if and only if f prefers w to \( { w^{\prime} } \). Similarly we define the responsive preferencefor \( { w \in W } \).

r(f)‐Separable preference:

in a discrete two-sided matching model with sides F and W. Agent \( { f \in F }\) with a quota of r(f) has a r(f)‐separable preference relation over allowable sets of partners if whenever \( { A= B \cup \{ w \} \setminus \{ f \} } \) with \( { w \notin B } \) and \( { f \in B } \), then f prefersA to B if and only if f prefers w to f. Similarly we define s(w)‐separable preference for \( { w \in W }\).

Maximin preferences:

in a discrete two-sided matching model with sides F and W. Agent \( { f \in F } \) with a quota of r(f) has a maximin preference relation over allowable sets of partners if whenever twoallowable sets C and \( { C^{\prime} }\) contained in W, such that f prefers\( { C^{\prime} } \) to C and no w in C is unacceptable to f, then a) all of \( { C^{\prime} }\) are acceptable to f and b) if \( { |C| =r(f) } \), then the least preferred worker in \( { C^{\prime}-C } \) is preferred by f to the least preferred worker in \( { C-C^{\prime} }\). Similarly we define maximin preference for \( { w\in W } \).

Choice set of \( { f \in F } \) from \( {A \subseteq W (Ch_{f}(A)) } \) :

in a discrete two-sidedmatching model with sides F and W. Let \( { B=\{A^{\prime}| A^{\prime} } \) is an allowable set of partnersfor f and \( { A^{\prime} \cap W } \) is contained in \( { A\} } \). Then, \( { A^{\prime} \in Ch_{f}(A) } \) if and only if\( { A^{\prime} \in B } \) and f likes \( { A^{\prime} } \) at least as well as \( { A^{\prime\prime} } \), for all \( { A^{\prime\prime} \in B } \). Similarly we define \( { Ch_{w}(A) } \) for \( { w \in W } \) and \( { A \subseteq F } \).

Substitutable preferences:

in a discrete two-sided matching model with sides F andW. Agent \( { f \in F } \) has a substitutable preference relation over allowable sets of partners ifwhenever \( { A \subseteq W } \) and \( { B \subseteq W } \) are such that \( { A \cap B= \phi } \) then (i) for all \( { S^{\prime}\in Ch_{f}(A \cup B) } \) there is some \( { S \in Ch_{f}(A) } \) such that \( { S^{\prime}\cap A \subseteq S } \) and (ii)for all \( { S \in Ch_{f}(A) } \) there is some \( { S^{\prime} \in Ch_{f}(A \cup B) } \) such that \( S' \cap A \subseteq S \). If an agent's preference is responsive then it is substitutable.

When preferences are strict, conditions (i) and (ii) are equivalent to requiring that if\( { Ch_{f}(A \cup B)=S^{\prime} } \) then \( { S^{\prime} \cap A \subseteq Ch_{f}(A) } \). This concept is similarly definedfor \( { w \in W } \).

Strongly substitutable preferences:

in a discrete two‐sided matching model with sides Fand W. Agent \( { f \in F } \) has a strongly substitutable preference relation over allowable sets ofpartners if for every pair of allowable sets of partners A and B such that \( { A >_{f} B } \), if\(w \in Ch_{f} (W \cap A \cup \{w\})\), then \(w \in Ch_{f}(W \cap B \cup \{ w \})\). This isa stronger condition than substitutability and responsiveness.

Additively separable preferences:

in a discrete two-sided matching model with sides Fand W. Agent \( { f \in F } \) has additively separable preferences if he/she/it assigns a nonnegativenumber a fw to each \( { w \in W } \) and assigns the value \( \smash{ v(A)= \sum_{{w \in A} a_{fw}} } \) toeach allowable set A of partners for f. Agent f compares two allowable sets by comparingthe values of these sets. This concept is similarly defined for \( { w \in W } \). If the agents haveadditively separable preferences we can think that if a partnership \( { (f,w) \in F \times W } \) isformed then the partners participate in some joint activity that generates a payoff a fw forplayer f and b fw for player w. These numbers are fixed, i. e., they are notnegotiable. If the preferences of the agents are additively separable then they areresponsive. The converse is not true (see Kraft, Pratt and Seidenberg [47]).

T‑map:

in a discrete two-sided matching model with sides F and W is defined asfollows. For every pre‐matching μ, let \(T(\mu(f))= Ch_{f} (U(f, \mu))\) for all \(f \in F\), where \(U(f,\mu) = \{ w \in W | f \in Ch_{w} (\mu (w) \cup \{ f \} ) \}\). Similarly, \(T(\mu(w))=Ch_{w} (U(w,\mu)) \) for all \( { f \in F } \), where \(U(w,\mu) = \{f \in F | w \in Ch_{f}(\mu(f) \cup \{w\})\}\).

Lattice property:

A  set L endowed with a partial order relation \( { \geqslant } \) has thelattice property if \( { \sup\{x, y\} \equiv x \vee y } \) and \( { \inf \{x, y\} \equiv x \wedge y } \) are inL, for all \( { x, x^{\prime} \in L } \). The lattice is complete if all its subsets have a supremum and aninfimum. (See Birkhoff [11]).

Pareto‐optimal matching:

A  feasible matching μ is Pareto‐optimal if there is nofeasible matching which is weakly preferred to μ by all players and it is strictlypreferred by at least one of them.

Outcome:

For the discrete two-sided matching models the outcome is a matching or at leastcorresponds to a matching; for the continuous two-sided matching models the outcome specifiesa payoff for each agent and a matching.

Stable outcome:

It is the natural solution concept for a two-sided matching model. It isalso referred as setwise‐stable outcome. See the definition below.

F‑optimal stable matching (respectively, payoff):

for a discrete (respectively,continuous) two-sided matching model is the stable matching (respectively, payoff) which isweakly preferred by every agent in F. Similarly we define the W‑optimal stable matching(respectively, payoff).

Achievable mate:

for agent y in a discrete two-sided matchingmodel is any y's of partner under some stable matching.

Matching mechanism:

For the discrete two-sided matching models, a matching mechanism isa function h whose range is the set of all possible inputs \( { X=(F,W,P,r,s) } \), and whose output \( { h(X } \)) is a matching for X.

Stable matching mechanism:

It is a matching mechanism h such that h(X) is alwaysstable for the market X. If h(X) always produces the F‑optimal stable matching for Xthen it is called an F‑optimal stable matching mechanism, and so on.

Revelation mechanism:

Given the discrete two-sided matching model \( { (F,W, P,r,s) } \),a revelation mechanism is the restriction of a matching mechanism h to the set of discretetwo-sided matching markets \( { (F,W, Q,r,s) } \) where the sets of agents and quotas are fixed.

Revelation game:

It is the strategic game induced by a revelation mechanism for thediscrete two-sided matching \( { (F,W,P,r,s) } \): the set of players is given by the union of F andW; a strategy of player j is any possible list of preferences Q(j) that player j canstate; the outcome function is given by the mechanism h and the preferences of the playersover the set of outcomes are determined by P.

Sincere strategy:

for a player j in a revelation game is the true list of preferencesP(j).

Manipulable mechanism:

A  mechanism h is manipulable or it is not strategy‐proof if in some revelation game induced by h, stating the true preferences isnot a dominant strategy for at least one player. A mechanism h is collectivelymanipulable if in some revelation game induced by h, there is a coalition whose members can be better off bymisrepresenting its preferences.

Rematching proof equilibrium:

is a Nash equilibrium profile from which no pair ofplayers \( { (f,w) \in F \times W } \) can profitably deviate given that the other players do not changetheir strategies.

Truncation:

Let P(a) be the a's preference list over individuals for a discrete two-sidedmatching model. For \( { a \in F \cup W } \), the list of preferences Q(a) over individuals isa truncation of P(a) at some agent b if Q(a) keeps the same ordering as P(a) but ranks asunacceptable all agents which are ranked below b.

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Author information

Authors and Affiliations

  1. Department of Economics, University of São Paulo/SP, São Paulo, Brazil

    Marilda Sotomayor

  2. Department of Economics, Brown University, Providence, USA

    Marilda Sotomayor & Ömer Özak

Authors
  1. Marilda Sotomayor
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  2. Ömer Özak
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Sotomayor, M., Özak, Ö. (2012). Two-Sided Matching Models. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_200

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