# 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. |

## References

- Hoppe, H.; Moldovanu, B.; Sela, A. The theory of assortative matching based on costly signals. Rev. Econ. Stud.
**2009**, 76, 253–281. [Google Scholar] [CrossRef] - Spence, M. Job market signaling. Q. J. Econ.
**1973**, 87, 296–332. [Google Scholar] [CrossRef] - Tullock, G. Efficient rent-seeking. In Toward a Theory of Rent-Seeking Society; Buchanan, J.M., Tollison, R.D., Tullock, G., Eds.; Texas A&M University Press: College Station, TX, USA, 1980. [Google Scholar]
- Clark, D.; Riis, C. A multi-winner nested rent-seeking contest. Public Choice
**1996**, 77, 437–443. [Google Scholar] [CrossRef] - Clark, D.J.; Riis, C. Competition over more than one prize. Am. Econ. Rev.
**1998**, 88, 276–289. [Google Scholar] - Moldovanu, B.; Sela, A. The optimal allocation of prizes in contests. Am. Econ. Rev.
**2001**, 91, 542–558. [Google Scholar] [CrossRef] - Moldovanu, B.; Sela, A. Contest architecture. J. Econ. Theory
**2006**, 126, 70–96. [Google Scholar] [CrossRef] - Akerlof, R.; Holden, R. The nature of tournaments. Econ. Theory
**2012**, 51, 289–313. [Google Scholar] [CrossRef] - Gonzalez-Diaz, J.; Siegel, R. Matching and price competition: Beyond symmetric linear costs. Int. J. Game Theory
**2013**, 42, 835–844. [Google Scholar] [CrossRef] - Siegel, R. All-pay contests. Econometrica
**2009**, 77, 71–92. [Google Scholar] - Xiao, J. Asymmetric all-pay contests with heterogeneous prizes. J. Econ. Theory
**2016**, 163, 178–221. [Google Scholar] [CrossRef] - Xiao, J. Equilibrium analysis of the all-pay contest with two nonidentical prizes: Complete results. J. Math. Ecno.
**2018**, 74, 21–34. [Google Scholar] [CrossRef] - Szidarovszky, F.; Okuguchi, K. On the existence and uniqueness of pure Nash equilibrium in rent-seeking games. Games Econ. Behav.
**1997**, 18, 135–140. [Google Scholar] [CrossRef] - Baye, M.; Hoppe, H. The strategic equivalence of rent-seeking, innovation, and patent-race games. Games Econ. Behav.
**2003**, 44, 217–226. [Google Scholar] - Clark, D.; Konrad, K. Contests with multi-tasking. Scand. J. Econ.
**2007**, 109, 303–319. [Google Scholar] [CrossRef] - Clark, D.; Riis, C. Contest success functions: An extension. Econ. Theory
**1998**, 11, 201–204. [Google Scholar] [CrossRef] - Corchon, L.; Dahm, M. Foundations for contest success functions. Econ. Theory
**2010**, 43, 81–98. [Google Scholar] [CrossRef] - Fu, Q.; Lu, J. The optimal multi-stage contest. Econ. Theory
**2012**, 51, 351–382. [Google Scholar] [CrossRef] - Fu, Q.; Lu, J.; Wang, Z. Reverse nested lottery contests. J. Math. Econ.
**2014**, 50, 128–140. [Google Scholar] [CrossRef] - Einy, E.; Haimenko, O.; Moreno, D.; Sela, A.; Shitovitz, B. Equilibrium existence in Tullock contests with incomplete information. J. Math. Econ.
**2015**, 61, 241–245. [Google Scholar] [CrossRef] - Che, Y.-K.; Gale, I. Caps on political lobbying. Am. Econ. Rev.
**1998**, 88, 643–651. [Google Scholar] - Segev, E.; Sela, A. Multi-stage sequential all-pay auctions. Eur. Econ. Rev.
**2014**, 70, 371–382. [Google Scholar] [CrossRef] - Chen, Z.; Ong, D.; Segev, E. Heterogeneous risk/loss aversion in complete information all-pay auctions. Eur. Econ. Rev.
**2017**, 95, 23–37. [Google Scholar] [CrossRef] - Lazear, E.; Rosen, S. Rank-order tournaments as optimum labor contracts. J. Political Econ.
**1981**, 89, 841–864. [Google Scholar] [CrossRef] - Rosen, S. Prizes and incentives in elimination tournaments. Am. Econ. Rev.
**1996**, 76, 701–715. [Google Scholar] - Chao, H.; Wilson, R. Priority service: Pricing, investments, and market organization. Am. Econ. Rev.
**1987**, 77, 899–916. [Google Scholar] - Wilson, R. Efficient and competitive rationing. Econometrica
**1989**, 57, 1–40. [Google Scholar] [CrossRef] [Green Version] - Fernandez, R.; Gali, J. To each according to…? Markets, tournaments and the matching problem with borrowing constraints. Rev. Econ. Stud.
**1999**, 66, 799–824. [Google Scholar] [CrossRef] - Bhaskar, V.; Hopkins, E. Marriage as a rat race: Noisy pre-marital investments with assortative matching. J. Political Econ.
**2016**, 124, 992–1045. [Google Scholar] [CrossRef] - Hoppe, H.; Moldovanu, B.; Ozdenoren, E. Coarse matching with incomplete information. Econ. Theory
**2011**, 47, 75–104. [Google Scholar] [CrossRef] - Barut, Y.; Kovenock, D. The symmetric multiple prize all-pay auction with complete information. Eur. J. Political Econ.
**1998**, 14, 627–644. [Google Scholar] [CrossRef] - Peters, M. The pre-marital investments game. J. Econ. Theory
**2007**, 137, 186–213. [Google Scholar] [CrossRef] - Dizdar, D.; Moldovanu, B.; Szech, N. The feedback effect in two-sided markets with bilateral investments. J. Econ. Theory
**2019**, 182, 106–142. [Google Scholar] [CrossRef] - Sela, A. All-Pay Matching Contests. International Journal of Game Theory. Forthcoming 2022. Available online: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3696376 (accessed on 1 August 2022).
- Sela, A. Optimal allocations of prizes and punishments in Tullock contests. Int. J. Game Theory
**2020**, 49, 749–771. [Google Scholar] [CrossRef]

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