# 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.**

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Chen, P.; Egesdal, M.; Pycia, M.; Yenmez, M.B.
Quantile Stable Mechanisms. *Games* **2021**, *12*, 43.
https://doi.org/10.3390/g12020043

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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.
https://doi.org/10.3390/g12020043