# Level-k Models and Overspending in Contests

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model

## 3. Benchmark: Nash Equilibrium and Two-Player Symmetric Contests with Level-k Reasoning

**Nash equilibrium under full rationality.**With two possibly heterogeneous players, the well-known unique Nash equilibrium of the full rationality model for the two players is

**Two-player symmetric contests with level-k reasoning.**We now consider the symmetric two-player contest with level-k reasoning and prove the impossibility of overspending result. This benchmark thus generalizes the impossibility found in [26] to any $(c,\overline{e})$, where c is the marginal cost of effort for a player, as he studied the special case of $c=\overline{e}=1$.

**Proposition**

**1.**

## 4. Overspending in Two-Players Asymmetric Contest

**Proposition**

**2.**

## 5. Overspending in Three-Players Symmetric Contest

**Proposition**

**3.**

## 6. Discussion and L2

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Lemma**

**A1.**

**Proof**

**of**

**Lemma A1.**

**Proof**

**of**

**Lemma A2.**

**Proof**

**of**

**Lemma A3.**

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**Figure 1.**Region in the $({c}_{1}\overline{e},a)-$ space where in a two-player contest ${e}^{L1}>{e}^{NE}$—that is, the L1-level of effort is greater than the Nash equilibrium level.

**Figure 2.**Region in the $(c,\overline{e})-$ space where in a three-player contest ${e}^{L1}>{e}^{NE}$—that is, the L1 level of effort is greater than the Nash equilibrium level. The dashed line is condition $\overline{e}c=\frac{1}{4}$.

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Arve, M.; Serena, M. Level-*k* Models and Overspending in Contests. *Games* **2022**, *13*, 45.
https://doi.org/10.3390/g13030045

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Arve M, Serena M. Level-*k* Models and Overspending in Contests. *Games*. 2022; 13(3):45.
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**Chicago/Turabian Style**

Arve, Malin, and Marco Serena. 2022. "Level-*k* Models and Overspending in Contests" *Games* 13, no. 3: 45.
https://doi.org/10.3390/g13030045