# Rent Dissipation in Simple Tullock Contests

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

## 3. Rent Dissipation in Share Contests

#### 3.1. Constant Marginal Costs

**Proposition 1.**

**Proposition 2.**

**Proof.**

**Example 1.**

#### 3.2. More General Costs

**Proposition 3.**

**Proposition 4.**

- 1.
- ${D}^{*}\to 0$ as $Z\to \infty $, implying there is a $\overline{Z}$ such that in a contest with $Z>\overline{Z}$ under-dissipation occurs; and
- 2.
- ${D}^{*}\to \infty $ as $Z\to 0$, implying there is a $\underline{Z}$ such that in a contest with $Z<\underline{Z}$ over-dissipation occurs.

**Proof.**

## 4. Rent Dissipation in Winner-Take-All Contests

#### 4.1. Constant Marginal Costs

**Proposition 5.**

**Proof.**

**Example 2.**

#### 4.2. More General Costs

**Proposition 6.**

- 1.
- ${D}^{\u2020}\to 0$ as $Z\to \infty $, implying there is a $\hat{\overline{Z}}$ such that in a contest with $Z>\hat{\overline{Z}}$ under-dissipation occurs; and
- 2.
- ${D}^{\u2020}\to \infty $ as $Z\to 0$, implying there is a $\hat{\underline{Z}}$ such that in a contest with $Z<\hat{\underline{Z}}$ over-dissipation occurs.

**Proof.**

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Notes

1 | Konrad and Schlesinger [9] establish that risk-averse players may nevertheless expend more effort than risk-neutral players in equilibrium, a finding that is of course totally compatible with the absence of overdissipation. |

2 | See [17] for a recent survey. |

3 | Indeed, even in the early literature, many of the rent-seeking applications can be interpreted as share contests. For example [2] provides examples of rent seeking of dairy farmers to obtain a share of government funds as well as lobbying for Korean steel import bans, which can be viewed as rent-seeking for an increased market share by domestic producers. |

4 | |

5 | An exception is Dickson et al. [27] that considers a general formulation of utility in a share contest. |

6 | If $\eta \left(x\right)$ is monotonically increasing in x, then the fact that with linear evaluation of the rent ${x}^{*}$ is monotonically increasing in Z would allow us to identify that there is a threshold rent $\stackrel{\u02c7}{Z}$, say, such that the associated equilibrium effort $\stackrel{\u02c7}{x}$ satisfies $\frac{n-1}{n}\frac{\beta}{\eta \left(\stackrel{\u02c7}{x}\right)}=1$, that allows us to understand the pattern of dissipation depending on Z. In particular, we can conclude that ${D}^{*}<(>)1$ if $Z>(<)\stackrel{\u02c7}{Z}$, consistent with the pattern of dissipation found elsewhere of over-dissipation of small rents and under-dissipation of large rents. |

7 | We could capture risk preferences in a winner-take-all contest by introducing an increasing function $f(\xb7)$ and specifying the payoff to be $\varphi ({x}_{i},{\mathbf{x}}_{-i})f(v\left(Z\right)-c\left({x}_{i}\right))+[1-\varphi ({x}_{i},{\mathbf{x}}_{-i})]f(-c\left({x}_{i}\right))$. The same function could be applied, for consistency, to the share contest to give a payoff $f(v\left({z}_{i}\right)-c\left({x}_{i}\right))$ but since this is just a monotonic transformation of the original payoff nothing in the analysis of share contests would change. |

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**Figure 1.**This illustrates the critical threshold of rent, $\tilde{Z}$, under which over-dissipation of the rent will occur, plotted as a function of $\alpha $ which controls the degree of concavity of preferences.

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## Share and Cite

**MDPI and ACS Style**

Dickson, A.; MacKenzie, I.A.; Sekeris, P.G.
Rent Dissipation in Simple Tullock Contests. *Games* **2022**, *13*, 83.
https://doi.org/10.3390/g13060083

**AMA Style**

Dickson A, MacKenzie IA, Sekeris PG.
Rent Dissipation in Simple Tullock Contests. *Games*. 2022; 13(6):83.
https://doi.org/10.3390/g13060083

**Chicago/Turabian Style**

Dickson, Alex, Ian A. MacKenzie, and Petros G. Sekeris.
2022. "Rent Dissipation in Simple Tullock Contests" *Games* 13, no. 6: 83.
https://doi.org/10.3390/g13060083