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Extreme Points and Large Contests

Giovanni Valvassori Bolgè

TL;DR

The paper addresses optimal contest design for a continuum of agents with a budget to award prizes to a mass $k\in(0,1)$. It develops an extreme-points analysis for a class of multidimensional monotone allocation rules, showing extreme points are convex combinations of at most two indicator up-sets and that optimal rules are two-threshold step functions with at most two jumps. The approach yields simple equilibrium characterizations in large-contest settings and connects to the Random Performance Function (RPF) representation under increasing distributional concerns. The results justify two-tier prize schemes in admissions, funding allocations, and lottery-based selections, providing a tractable foundation for practical contest design.

Abstract

In this paper, we characterize the extreme points of a class of multidimensional monotone functions. This result is then applied to large contests, where it provides a useful representation of optimal allocation rules under a broad class of distributional preferences of the contest designer. In contests with complete information, the representation significantly simplifies the characterization of the equilibria.

Extreme Points and Large Contests

TL;DR

The paper addresses optimal contest design for a continuum of agents with a budget to award prizes to a mass . It develops an extreme-points analysis for a class of multidimensional monotone allocation rules, showing extreme points are convex combinations of at most two indicator up-sets and that optimal rules are two-threshold step functions with at most two jumps. The approach yields simple equilibrium characterizations in large-contest settings and connects to the Random Performance Function (RPF) representation under increasing distributional concerns. The results justify two-tier prize schemes in admissions, funding allocations, and lottery-based selections, providing a tractable foundation for practical contest design.

Abstract

In this paper, we characterize the extreme points of a class of multidimensional monotone functions. This result is then applied to large contests, where it provides a useful representation of optimal allocation rules under a broad class of distributional preferences of the contest designer. In contests with complete information, the representation significantly simplifies the characterization of the equilibria.
Paper Structure (10 sections, 13 theorems, 32 equations, 1 figure)

This paper contains 10 sections, 13 theorems, 32 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Extreme Points
  3. Large Contests
  4. Complete Information
  5. Incomplete Information
  6. More General Objective Function
  7. Conclusion
  8. Proofs
  9. Proof of Theorem \ref{['thrm']}
  10. Alternative Proof of Proposition \ref{['optimalq']}.

Key Result

Theorem 1

The set of extreme points of $\mathcal{F}^*$ is given by for each $\mu \in \Delta(\mathcal{X})$ and some $A_1 \subseteq A_2$.

Figures (1)

  • Figure 1: An optimal allocation rule with two steps.

Theorems & Definitions (27)

  • Theorem 1
  • Definition 1
  • Theorem 2: FanLorentz
  • Definition 2
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 17 more