# Optimal Accuracy of Unbiased Tullock Contests with Two Heterogeneous Players

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## Abstract

**:**

## 1. Introduction

## 2. Set-Up and Notation

- there is a unique Nash equilibrium, which is in pure strategies, if $0\le r\le \overline{r}$,
- there is a unique Nash equilibrium, which is in semi-mixed strategies, if $\overline{r}<r\le 2$,
- any Nash equilibrium is an all-pay auction equilibrium (i.e., it yields the same expected efforts, winning probabilities and expected payoffs as well as the same expected revenue $\mathcal{R}$ for the contest designer as the unique equilibrium of the corresponding all-pay auction) in mixed strategies if $2<r$,

## 3. Maximization of Selection Quality

**Proposition**

**1.**

## 4. Effort Maximization

#### 4.1. Maximization of Aggregate Effort

**Proposition**

**2.**

**Corollary**

**1.**

- (a)
- $r={r}_{A}$ if $0<\omega <{\overline{\omega}}_{A}$,
- (b)
- $r=\overline{r}$ if ${\overline{\omega}}_{A}\le \omega <1$.

#### 4.2. Maximization of the Winner’s Expected Effort

**Proposition**

**3.**

**Corollary**

**2.**

- (a)
- $r={r}_{B}$ if $0<\omega <{\overline{\omega}}_{B}$,
- (b)
- $r=\overline{r}$ if ${\overline{\omega}}_{B}\le \omega <1$.

## 5. Conflicting Objectives

#### 5.1. Tradeoff between Selection Quality and Minimum Effort

#### 5.2. Tradeoff between Selection Quality and Maximum Effort

## 6. Conclusions

## Funding

## Conflicts of Interest

## Notes

1 | In many countries (like France or Germany), the size of the jury depends on the importance of the case (amount in dispute, public interest, severeness) and increases from level to level of jurisdiction. Moreover, the legislator can adjust it according to the contemporary priorities: fore example, in 2012, France reduced the number of jurors from 9 to 6 for first instance proceedings and from 12 to 9 for appeal proceedings [1]. |

2 | Mastromarco and Runkel [2] provide an overview and suggest an alternative argument for the numerous rule changes in Formula One motor racing. |

3 | In October 2000, the International Table Tennis Federation replaced the older 38 mm (1.50 in) balls by 40 mm (1.57 in) balls to reduce the speed (and thus inherent noise) of the game [3]. |

4 | For example, the golden goal (sudden death)—a tie breaking rule by which the first team to score during extra-time was declared to be the winner—was introduced experimentally in 1993, used at the 1998 and 2002 FIFA World Cup tournaments, and abolished again in 2004 [4]. |

5 | Skaperdas [11] provides an axiomatic foundation for this type of contest success function. |

6 | |

7 | I used the software Mathematica to verify that $dg/d{r}_{B}>0$ for all ${r}_{B}\in (1,2)$. |

## References

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**MDPI and ACS Style**

Sahm, M. Optimal Accuracy of Unbiased Tullock Contests with Two Heterogeneous Players. *Games* **2022**, *13*, 24.
https://doi.org/10.3390/g13020024

**AMA Style**

Sahm M. Optimal Accuracy of Unbiased Tullock Contests with Two Heterogeneous Players. *Games*. 2022; 13(2):24.
https://doi.org/10.3390/g13020024

**Chicago/Turabian Style**

Sahm, Marco. 2022. "Optimal Accuracy of Unbiased Tullock Contests with Two Heterogeneous Players" *Games* 13, no. 2: 24.
https://doi.org/10.3390/g13020024