# Assortative Matching by Lottery Contests

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### Related Literature

## 2. The Assortative Matching Contest

## 3. The n × n Assortative Matching Contests

**Proposition**

**1.**

**Proposition**

**2.**

**Proof.**

#### The 2 × 2 Assortative Matching Contests

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

## 4. The m × n Assortative Matching Contests

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

**Proposition**

**7.**

**Proof.**

**Proposition**

**8.**

**Proof.**

#### The 3 × 2 Assortative Matching Contest

**Proposition**

**9.**

**Proof.**

**Example**

**1.**

**Example**

**2.**

## 5. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Proposition 2

#### Appendix A.2. Proof of Proposition 3

#### Appendix A.3. Proof of Proposition 5

#### Appendix A.4. Proof of Proposition 6

#### Appendix A.5. Proof of Proposition 7

#### Appendix A.6. Proof of Proposition 8

#### Appendix A.7. Proof of Proposition 9

## Notes

1 | In $2x2$ matching contests there is no partial interior equilibrium in which only some of the agents exert positive efforts. |

2 | The agents may be different by their marginal costs. |

3 | If ${x}_{i}=0$ for all $1\le i\le m,$ each firm’s probability of winning is assumed to be $1/m$. Similarly, if ${y}_{j}=0$ for all $1\le j\le n$, each worker’s probability of winning is assumed to be $1/n$. |

4 | Note that our results in this section can be immediately extended to match-value functions of the form f (m _{i},w_{j}) = δ(m_{i})ρ(w_{j}), where δ and ρ are strictly increasing and differentiable. |

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Size | Multiplicative Form | Additive Form |
---|---|---|

2 × 2 | Agents exert efforts | Agents do not exert efforts |

3 × 2 | At least two firms exert efforts | All the three firms exert efforts |

$m\phantom{\rule{0.166667em}{0ex}}\times $ 2 | Both types of worker may exert efforts | Both types of worker exert the same effort |

$m\times n$ | At least n firms exert efforts | At least $n+1$ firms exert efforts |

$n\times n$ | All agents might not exert efforts | All agents might not exert efforts |

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**MDPI and ACS Style**

Cohen, C.; Rabi, I.; Sela, A.
Assortative Matching by Lottery Contests. *Games* **2022**, *13*, 64.
https://doi.org/10.3390/g13050064

**AMA Style**

Cohen C, Rabi I, Sela A.
Assortative Matching by Lottery Contests. *Games*. 2022; 13(5):64.
https://doi.org/10.3390/g13050064

**Chicago/Turabian Style**

Cohen, Chen, Ishay Rabi, and Aner Sela.
2022. "Assortative Matching by Lottery Contests" *Games* 13, no. 5: 64.
https://doi.org/10.3390/g13050064