# Quantile Stable Mechanisms

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## Abstract

**:**

## 1. Introduction

## 2. Model

**feasible**if for every firm-worker pair $(f,w)\in F\times W$, $|{X}_{f}^{\prime}\cap {X}_{w}^{\prime}|\le 1$, i.e., each firm-worker pair can sign at most one joint contract. A

**matching**is a feasible set of contracts.

**acceptable**to agent a if there exists a set of contracts ${X}_{a}\ni x$ such that ${X}_{a}{\succ}_{a}\varnothing $; otherwise contract x is unacceptable to a.

**Definition**

**1.**

**stable**if

- 1.
- for all a, ${C}_{a}\left(Y\right)={Y}_{a}$
**(individual rationality)**and - 2.
- there does not exist a nonempty set of contracts $Z\overline{)\subseteq}Y$ such that for all a, ${Z}_{a}\subseteq {C}_{a}(Y\cup Z)$
**(no blocking)**.

**Definition**

**2.**

**substitutes**in preferences of agent a if for any contract $x\in X$ and sets of contracts $Y,{Y}^{\prime}\subseteq X$ such that $Y\subseteq {Y}^{\prime}$,

**Definition**

**3.**

**strong substitutes**in preferences of agent a if for any sets of contracts $Y,{Y}^{\prime}\subseteq X$ such that ${C}_{a}\left({Y}^{\prime}\right){\u2ab0}_{a}{C}_{a}\left(Y\right)$,

**Definition**

**4.**

**law of aggregate demand**in preferences of agent a if for all $Y,{Y}^{\prime}\subseteq X$ such that $Y\subseteq {Y}^{\prime}$

**Example**

**1.**

## 3. Quantile Stable Matchings

**worker-optimal stable matching**and ${X}_{W}^{k}$ is the

**worker-pessimal stable matching**. In the analogous case, when contracts are strong substitutes for firms, we define the quantile, firm-optimal, and firm-pessimal stable matchings for firms. Furthermore, the quantile stable matchings for firms and workers are exactly the same with the polarization of interests property, ${X}_{W}^{i}={X}_{F}^{k+1-i}$, when contracts are strong substitutes for both workers and firms. Thus, the worker-optimal stable matching is the firm-pessimal stable matching, the $\left(2\right)$-nd quantile stable matching for workers is the $(k-1)$-th quantile stable matching for firms, etc. In particular, when k is odd, there exists a stable matching that assigns all agents their median stable matching outcomes since ${X}_{F}^{(k+1)/2}={X}_{W}^{(k+1)/2}$.

**Definition**

**5.**

**responsive**preferences over contracts if there exist a quota ${q}_{a}$ and a strict ordering ${\u22d7}_{a}$ on contracts ${X}_{a}$ and the empty contract ∅ such that (1) agent a prefers the empty set of contracts to any set of contracts $\left|Y\right|>{q}_{a}$, and (2) agent a prefers a set of contracts $Y\equiv \{{y}_{1},\dots ,{y}_{\left|Y\right|}\}\subseteq {X}_{a}$ (indexed so that ${y}_{1}{\u22d7}_{a}\dots ,{\u22d7}_{a}{y}_{\left|Y\right|}$) to another set of contracts $Z\equiv \{{z}_{1},\dots ,{z}_{\left|Z\right|}\}\subseteq {X}_{a}$ (indexed so that ${z}_{1}{\u22d7}_{a}\dots ,{\u22d7}_{a}{z}_{\left|Y\right|}$) such that $Y\ne Z$ whenever $\left|Y\right|\le {q}_{a}$ and either

- $\left|Y\right|\ge \left|Z\right|$, ${y}_{i}{\u22d7}_{a}{z}_{i}$ or ${y}_{i}={z}_{i}$ for every $1\le i\le \left|Z\right|$, and ${y}_{i}{\u22d7}_{a}\varnothing $ for every $\left|Z\right|+1\le i\le \left|Y\right|$, or
- $\left|Y\right|\le \left|Z\right|$ and ${y}_{i}{\u22d7}_{a}{z}_{i}$ or ${y}_{i}={z}_{i}$ for every $1\le i\le \left|Y\right|$ and $\varnothing {\u22d7}_{a}{z}_{i}$ for every $\left|Y\right|+1\le i\le \left|Z\right|$.

**Proposition**

**1.**

## 4. Quantile Stable Mechanisms

**quantile stable mechanism**${\phi}^{q}$ is the mapping from agents’ preference profiles to matchings such that for every preference profile ≻, the mechanism ${\phi}^{q}(\succ )$ selects the $\lceil kq\rceil $-th quantile stable matching for firms where k is the number of stable matchings under ≻. Here, $\lceil x\rceil $ denotes the lowest integer equal to or larger than x (all our results remain valid for mechanisms that always select the $\lfloor kq\rfloor $-th quantile stable matching, where $\lfloor x\rfloor $ is the highest integer smaller than or equal to x).

**Proposition**

**2.**

**as manipulable as**mechanism $\varphi $ for agent a if

**more manipulable than**mechanism $\varphi $ for agent a if $\psi $ is as manipulable as $\varphi $ for agent a and there exists a preference profile $\succ \in P$ such that

**Proposition**

**3.**

- ${\phi}^{q}={\phi}^{{q}^{\prime}}$, or
- ${\phi}^{q}$ is more manipulable than ${\phi}^{{q}^{\prime}}$ for all firms and ${\phi}^{{q}^{\prime}}$ is more manipulable than ${\phi}^{q}$ for all workers.

**closed**if for all $\succ \in P$ and for all matchings Y that are stable with respect to ≻, if the preference relation ${\succ}_{a}^{\prime}$ ranks sets of contracts in the same way as ${\succ}_{a}$ except that only contracts in ${Y}_{a}$ are acceptable to agent a, then $({\succ}_{a}^{\prime},{\succ}_{-a})\in P$.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Necessity Examples

**Example**

**A1.**

**Example**

**A2.**

## References

- Roth, A.E. Stability and Polarization of Interests in Job Matching. Econometrica
**1984**, 52, 47–57. [Google Scholar] [CrossRef] - Hatfield, J.; Milgrom, P. Matching with Contracts. Am. Econ. Rev.
**2005**, 95, 913–935. [Google Scholar] [CrossRef][Green Version] - Fleiner, T. A Fixed-Point Approach to Stable Matchings and Some Applications. Math. Oper. Res.
**2003**, 28, 103–126. [Google Scholar] [CrossRef][Green Version] - Klaus, B.; Walzl, M. Stable many-to-many matchings with contracts. J. Math. Econ.
**2009**, 45, 422–434. [Google Scholar] [CrossRef][Green Version] - Hatfield, J.W.; Kominers, S.D. Contract Design and Stability in Many-to-Many Matching. Games Econ. Behav.
**2017**, 101, 78–97. [Google Scholar] [CrossRef][Green Version] - Roth, A.E. The Evolution of the Labor Market for Medical Interns and Residents: A Case Study in Game Theory. J. Political Econ.
**1984**, 92, 991–1016. [Google Scholar] [CrossRef] - Abdulkadiroğlu, A.; Sönmez, T. School Choice: A Mechanism Design Approach. Am. Econ. Rev.
**2003**, 93, 729–747. [Google Scholar] [CrossRef][Green Version] - Echenique, F.; Yariv, L. An Experimental Study of Decentralized Matching; Working Paper; Caltech: Pasadena, CA, USA, 2013. [Google Scholar]
- Anbarci, N.; Feltovich, N. How Fully do People Exploit Their Bargaining Position? The Effects of Bargaining Institution and the 50–50 Norm. J. Econ. Behav. Organ.
**2018**, 145, 320–334. [Google Scholar] [CrossRef][Green Version] - Yan, H.; Friedman, D.; Munro, D. An Experiment on a Core Controversy. Games Econ. Behav.
**2016**, 96, 132–144. [Google Scholar] [CrossRef][Green Version] - Teo, C.P.; Sethuraman, J. The geometry of fractional stable matchings and its applications. Math. Oper. Res.
**1998**, 23, 874–891. [Google Scholar] [CrossRef] - Fleiner, T. Some Results on Stable Matchings and Fixed Points; Working Paper; Egrervary Research Group: Budapest, Hungary, 2002. [Google Scholar]
- Schwarz, M.; Yenmez, M.B. Median Stable Matching for Markets with Wages. J. Econ. Theory
**2011**, 146, 619–637. [Google Scholar] [CrossRef] - Klaus, B.; Klijn, F. Median stable matching for college admissions. Int. J. Game Theory
**2006**, 34, 1. [Google Scholar] [CrossRef][Green Version] - Sethuraman, J.; Teo, C.P.; Qian, L. Many-to-One Stable Matching: Geometry and Fairness. Math. Oper. Res.
**2006**, 31, 581–596. [Google Scholar] [CrossRef][Green Version] - Klaus, B.; Klijn, F. Smith and Rawls share a room: Stability and medians. Soc. Choice Welf.
**2010**, 35, 647–667. [Google Scholar] [CrossRef][Green Version] - Chen, P.; Egesdal, M.; Pycia, M.; Yenmez, M.B. Median Stable Matchings in Two-Sided Markets. Games Econ. Behav.
**2016**, 97, 64–69. [Google Scholar] [CrossRef][Green Version] - Chen, P.; Egesdal, M.; Pycia, M.; Yenmez, M.B. Quantile Stable Mechanisms; SSRN Working Paper No. 2526505; SSRN: Rochester, NY, USA, 2015. [Google Scholar]
- Chen, P.; Egesdal, M.; Pycia, M.; Yenmez, M.B. Manipulability of Stable Mechanisms. Am. Econ. J. Microecon.
**2016**, 8, 202–214. [Google Scholar] [CrossRef][Green Version] - Chen, P.; Egesdal, M.; Pycia, M.; Yenmez, M.B. Ranking by Manipulability and Quantile Stable Mechanisms; Working Paper; UCLA: Los Angeles, CA, USA, 2012. [Google Scholar]
- Echenique, F.; Oviedo, J. A Theory of Stability in Many-to-Many Matching Markets. Theor. Econ.
**2006**, 1, 233–273. [Google Scholar] [CrossRef][Green Version] - Eeckhout, J.; Kircher, P. Assortative Matching with Large Firms: Span of Control over More Versus Better Workers. Econometrica
**2018**, 86, 85–132. [Google Scholar] [CrossRef][Green Version] - Day, R.; Milgrom, P. Core-selecting Package Auctions. Int. J. Game Theory
**2008**, 36, 393–407. [Google Scholar] [CrossRef] - Pathak, P.; Sönmez, T. School Admissions Reform in Chicago and England: Comparing Mechanisms by Their Vulnerability to Manipulation. Am. Econ. Rev.
**2013**, 1031, 80–106. [Google Scholar] [CrossRef][Green Version] - Kelso, A.S.; Crawford, V.P. Job Matching, Coalition Formation, and Gross Substitutes. Econometrica
**1982**, 50, 1483–1504. [Google Scholar] [CrossRef] - Aygün, O.; Sönmez, T. Matching with Contracts: Comment. Am. Econ. Rev.
**2013**, 103, 2050–2051. [Google Scholar] [CrossRef][Green Version] - Alkan, A. A class of multipartner matching models with a strong lattice structure. Econ. Theory
**2002**, 19, 737–746. [Google Scholar] [CrossRef][Green Version] - Alkan, A.; Gale, D. Stable schedule matching under revealed preference. J. Econ. Theory
**2003**, 112, 289–306. [Google Scholar] [CrossRef][Green Version] - Kojima, F. The Law of Aggregate Demand and Welfare in Two-sided Matching. Econ. Lett.
**2007**, 99, 581–584. [Google Scholar] [CrossRef] - Delacretaz, D.; Loertscher, S.; Marx, L.M.; Wilkening, T. Two-sided allocation problems, decomposability, and the impossibility of efficient trade. J. Econ. Theory
**2019**, 179, 416–454. [Google Scholar] [CrossRef] - Roth, A.E.; Sotomayor, M.A.O. Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis; Econometric Society Monographs: Cambridge, UK, 1990. [Google Scholar]
- Fernandez, M.A. Deferred Acceptance and Regret-Free Truth-Telling; Working Paper; John Hopkins University: Baltimore, MD, USA, 2020. [Google Scholar]

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Chen, P.; Egesdal, M.; Pycia, M.; Yenmez, M.B. Quantile Stable Mechanisms. *Games* **2021**, *12*, 43.
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Chen P, Egesdal M, Pycia M, Yenmez MB. Quantile Stable Mechanisms. *Games*. 2021; 12(2):43.
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Chen, Peter, Michael Egesdal, Marek Pycia, and M. Bumin Yenmez. 2021. "Quantile Stable Mechanisms" *Games* 12, no. 2: 43.
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