Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The effect of competition in contests: A unifying approach

Andrzej Baranski, Sumit Goel

TL;DR

This paper analyzes how increasing contest competitiveness, implemented via prize inequality, affects effort when agents have privately known abilities drawn from a finite, ordered type-space. The authors fully characterize the unique symmetric Bayes-Nash equilibrium, showing more efficient types mix over contiguous intervals and that competition reshapes expected effort through changes in equilibrium utilities as well as costs. They prove that under linear and concave costs, transferring value to the top prize unambiguously raises effort, making the winner-takes-all design robustly optimal, while transfers to intermediate prizes yield nuanced effects depending on type-wealth and distribution under incomplete information. A convergence result then links finite-type equilibria to the continuum-type benchmark, reinforcing the unifying nature of the finite-type framework. The findings bridge complete and incomplete information literature and have implications for optimal budget allocation and contest design in settings with private information.

Abstract

We study how increasing competition, by making prizes more unequal, affects effort in contests. In a finite type-space environment, we characterize the equilibrium, analyze the effect of competition under linear costs, and identify conditions under which these effects persist under general costs. Our findings reveal that competition may encourage or deter effort, depending on the relative likelihood of efficient versus inefficient types. We derive implications for the classical budget allocation problem and establish that the most competitive winner-takes-all contest is robustly optimal under linear and concave costs, thereby resolving an open question. Methodologically, our analysis of the finite type-space domain -- which includes complete information as a special case and can approximate any continuum type-space -- provides a unifying approach that sheds light on the contrasting results in these extensively studied environments.

The effect of competition in contests: A unifying approach

TL;DR

This paper analyzes how increasing contest competitiveness, implemented via prize inequality, affects effort when agents have privately known abilities drawn from a finite, ordered type-space. The authors fully characterize the unique symmetric Bayes-Nash equilibrium, showing more efficient types mix over contiguous intervals and that competition reshapes expected effort through changes in equilibrium utilities as well as costs. They prove that under linear and concave costs, transferring value to the top prize unambiguously raises effort, making the winner-takes-all design robustly optimal, while transfers to intermediate prizes yield nuanced effects depending on type-wealth and distribution under incomplete information. A convergence result then links finite-type equilibria to the continuum-type benchmark, reinforcing the unifying nature of the finite-type framework. The findings bridge complete and incomplete information literature and have implications for optimal budget allocation and contest design in settings with private information.

Abstract

We study how increasing competition, by making prizes more unequal, affects effort in contests. In a finite type-space environment, we characterize the equilibrium, analyze the effect of competition under linear costs, and identify conditions under which these effects persist under general costs. Our findings reveal that competition may encourage or deter effort, depending on the relative likelihood of efficient versus inefficient types. We derive implications for the classical budget allocation problem and establish that the most competitive winner-takes-all contest is robustly optimal under linear and concave costs, thereby resolving an open question. Methodologically, our analysis of the finite type-space domain -- which includes complete information as a special case and can approximate any continuum type-space -- provides a unifying approach that sheds light on the contrasting results in these extensively studied environments.
Paper Structure (9 sections, 3 theorems, 69 equations)

This paper contains 9 sections, 3 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Model
  3. Equilibrium
  4. Effect of competition on expected effort
  5. Complete information
  6. Incomplete information: Linear cost
  7. Incomplete information: General cost
  8. Convergence to continuum type-space equilibrium
  9. Conclusion

Key Result

Corollary 1

Consider a contest environment $(N+1, \mathcal{C}, p)$ where $\mathcal{C}=\{c_1\}$ and $c_1\in \mathcal{F}$. Suppose any contest $v\in \mathcal{V}$ such that $\sum_{m=0}^N v_m\leq V$ is feasible.

Theorems & Definitions (15)

  • Definition 1: Ordered type-space
  • Definition 2: Parametric type-space
  • Definition 3
  • proof
  • proof
  • proof
  • proof
  • proof
  • Corollary 1
  • proof
  • ...and 5 more