Abstract
For a many-to-many matching model with contracts in which the preferences of all hospitals satisfy substitutability and the preferences of all doctors satisfy substitutability, the law of aggregate demand and q-congruence, we show the existence of a convergent blocking path. In other words, we start from an arbitrary allocation and build a finite sequence of allocations leading to a stable outcome, with the special feature that each allocation can be obtained from the previous one by satisfying a unilateral or a bilateral blocking contract. As a consequence, we prove that the process of allowing randomly selected blocks to be satisfied eventually leads to a stable outcome. This explains the fact that some markets with contracts reach stable assignments by means of decentralized decisions.
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Notes
Gale and Shapley (1962) defined the pioneering Deferred Acceptance Algorithm.
In terms of contracts, the preferences of an agent \(f\in F\) satisfy \(q_{f}\)-responsiveness, where \(q_{f}\) is a nonnegative integer number, if:
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(i)
For every pair of contracts \(x,y\in {\mathbf {X}}_{f},\) and every \( S\subseteq {\mathbf {X}}_{f}{\setminus } \{x,y\}\) with \(\mid S\mid <q_{f}\), we have
$$\begin{aligned} S\cup \{x\}\succeq _{f}S\cup \{y\}\text { if and only if }\{x\}\succeq _{f}\{y\} \end{aligned}$$whenever \(S\cup \{x\}\) and \(S\cup \{y\}\) are allocations.
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(ii)
For every contract \(x\in {\mathbf {X}}_{f},\) and every \(S\subseteq \mathbf { X}_{f}{\setminus } \{x\}\) with \(\mid S\mid <q_{f}\), we have,
$$\begin{aligned} S\cup \{x\}\succeq _{f}S\text { if and only if }\{x\}\succeq _{f}\varnothing ; \end{aligned}$$whenever \(S\cup \{x\}\) is an allocation.
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(iii)
For all \(S\subseteq {\mathbf {X}}_{f}\) such that \(\mid S\mid >q_{f}\), we have \(\varnothing \succeq _{f}S\).
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(i)
Note that the allocation \(Y\subseteq {\mathbf {X}},\) is individually rational if \(C_{f}\left( Y\right) =Y_{f}\) for every \(f\in F\).
\(Y\in A({\mathbf {X}})\) is a pairwise-stable allocation if
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(i)
Y is individually rational;
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(ii)
There does not exist \(x\in {\mathbf {X}}\backslash Y\) such that \(x\in C_{x_{D}}\left( Y\cup \left\{ x\right\} \right) \cap C_{x_{H}}\left( Y\cup \left\{ x\right\} \right) \).
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(i)
Symmetrically, or each \(h\in H,\) \(J\subseteq F\) and \(Y\in A\left( {\mathbf {X}} \right) \) let \(I_{J}(h,Y)\) denote the set of all contracts connecting h with a doctor contained in J, wanted by the corresponding doctor if the contracts in Y are also available. This is
$$\begin{aligned} I_{J}(h,Y)=\left\{ x\in {\mathbf {X}}_{h}:x_{D}\in J \ and \ x\in C_{x_{D}}(Y\cup \left\{ x\right\} )\right\} \ . \end{aligned}$$Because of substitutability, if Z is a blocking set of the allocation Y, then \(\left\{ z\right\} \) is a blocking set of the allocation Y for all \( z\in Z\backslash Y\) .
Gale and Shapley (1962)
Doctor quasi-stability is called Inclusion Property for Doctors in Pepa Risma (2015).
By definition of DOA, the output the last iteration is equal to the output of the last-but-one iteration.
Observe that \(\left\{ z_{i}\right\} \) is a blocking set for \(X^{i-1}\). Indeed, from the construction of \(\left\{ X^{i}\right\} _{i=0}^{n},\) it follows that \(z_{i}\notin X^{i-1}\); and \(z_{i}\in C_{D}(X\cup Z)\cap C_{H}\left( X\cup Z\right) \) implies \(z_{i}\in C_{D}(X^{i-1}\cup \left\{ z_{i}\right\} )\cap C_{H}\left( X^{i-1}\cup \left\{ z_{i}\right\} \right) \) because all agents have substitutable preferences.
If \(y\in Y^{1}{\setminus } Y\) and \(f\in H,\) it follows that \(y\in S\subseteq I_{ {\overline{J}}}(f,Y)\). If \(y\in Y^{1}{\setminus } Y\) and \(f\in D,\) we have \(y\in C_{d}\) \(\left( Y\cup Y\right) ,\) and hence \(y\in C_{d}\) \(\left( Y\cup \left\{ y\right\} \right) \) due to substitutability.
References
Blum Y, Roth A, Rothblum U (1997) Vacancy chains and equilibration in senior-level. Labor markets. J Econ Theory 76:362–411
Cantala D (2003) Restabilizing matching markets at senior level. J Econ Theory 103:429–443
Gale D, Shapely LS (1962) College admissions and the stability of marriage. Am Math Mon 69:9–15
Hatfield J, Milgrom P (2005) Matching with contracts. Am Econ Rev 95(4):913–935
Hatfield J, Kominers S (2016) Contract design and stability in many-to-many matching. Games Econ Behav. https://doi.org/10.1016/j.geb.2016.01.002
Klaus B, Walzl M (2009) Stable many-to-many matching with contracts. J Math Econ 45(7–8):422–434
Knuth D (1976) Marriages stables. Les Presses de l’Universite de Montréal, Montréal
Kojima F, Ünver U (2008) Random paths to pairwise stability in many to many matching problems: a study on market equilibration. Int J Game Theory 36:473–488
Ma J (1996) On randomized matching mechanisms. Econ Theory 8:377–381
Pepa Risma E (2015) A deferred-acceptance algorithm with contracts. J Dyn Games 2(2). https://doi.org/10.3934/jdg.2015005
Roth A, Vande Vate J (1990) Random paths to stability in two sided matching. Econometrica 58(6):1475–1480
Roth A (1986) On the allocation of residents to rural hospitals: a general property of two sided matching markets. Econometrica 24:425–427
Sotomayor M (1996) A non-constructive elementary proof of the existence of stable marriages. Games Econ Behav 3:5–137
Acknowledgements
We are grateful to the Advisory Editor and the Reviewers for their useful recommendations in order to improve the exposition of this paper; to the members of the Game Theory Group of IMASL for their queries and comments; and to the members of GAECI for their assistant with the English writing.
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This work is partially supported by the Universidad Nacional de San Luis, through grant 31012, and by the Consejo Nacional de Investigaciones Cient íficas y Técnicas (CONICET), through Grant PIP 112-200801-00464.
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Millán, B., Pepa Risma, E. Random path to stability in a decentralized market with contracts. Soc Choice Welf 51, 79–103 (2018). https://doi.org/10.1007/s00355-018-1108-6
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DOI: https://doi.org/10.1007/s00355-018-1108-6