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Contest Design with Threshold Objectives

Edith Elkind, Abheek Ghosh, Paul W. Goldberg

TL;DR

This paper introduces threshold-based objective functions for contest design under incomplete information, extending the classic total-output objective. It analyzes two modeling paradigms—rank-order prize allocation and general all-pay contests—under unit-sum and unit-range constraints, and proves that optimal threshold-contests retain simple structures: a single simple contest is optimal for binary thresholds in rank-order settings, while a convex combination of at most three simple contests is optimal for linear thresholds; in general all-pay contests, optimal mechanisms exhibit reserve/saturation features with efficient computation under regular distributions and ironing for irregulars. The results clarify why multiple prizes are common in practice and provide computational methods to implement optimal thresholds. Collectively, the work generalizes total-output contest design to threshold-focused objectives, offering practical guidelines for designing incentives in crowdsourcing, education, and competitive environments. The framework unifies and extends prior results on Bayes–Nash equilibria, Myersonian auction design, and single-crossing properties, with clear implications for mechanism design under bounded prizes and strategic effort.

Abstract

We study contests where the designer's objective is an extension of the widely studied objective of maximizing the total output: The designer gets zero marginal utility from a player's output if the output of the player is very low or very high. We consider two variants of this setting, which correspond to two objective functions: binary threshold, where the designer's utility is a non-decreasing function of the number of players with output above a certain threshold; and linear threshold, where a player's contribution to the designer's utility is linear in her output if the output is between a lower and an upper threshold, and becomes constant below the lower and above the upper threshold. For both of these objectives, we study rank-order allocation contests and general contests. We characterize the contests that maximize the designer's objective and indicate techniques to efficiently compute them.

Contest Design with Threshold Objectives

TL;DR

This paper introduces threshold-based objective functions for contest design under incomplete information, extending the classic total-output objective. It analyzes two modeling paradigms—rank-order prize allocation and general all-pay contests—under unit-sum and unit-range constraints, and proves that optimal threshold-contests retain simple structures: a single simple contest is optimal for binary thresholds in rank-order settings, while a convex combination of at most three simple contests is optimal for linear thresholds; in general all-pay contests, optimal mechanisms exhibit reserve/saturation features with efficient computation under regular distributions and ironing for irregulars. The results clarify why multiple prizes are common in practice and provide computational methods to implement optimal thresholds. Collectively, the work generalizes total-output contest design to threshold-focused objectives, offering practical guidelines for designing incentives in crowdsourcing, education, and competitive environments. The framework unifies and extends prior results on Bayes–Nash equilibria, Myersonian auction design, and single-crossing properties, with clear implications for mechanism design under bounded prizes and strategic effort.

Abstract

We study contests where the designer's objective is an extension of the widely studied objective of maximizing the total output: The designer gets zero marginal utility from a player's output if the output of the player is very low or very high. We consider two variants of this setting, which correspond to two objective functions: binary threshold, where the designer's utility is a non-decreasing function of the number of players with output above a certain threshold; and linear threshold, where a player's contribution to the designer's utility is linear in her output if the output is between a lower and an upper threshold, and becomes constant below the lower and above the upper threshold. For both of these objectives, we study rank-order allocation contests and general contests. We characterize the contests that maximize the designer's objective and indicate techniques to efficiently compute them.

Paper Structure

This paper contains 47 sections, 24 theorems, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Designer's Objective
  3. Other Modeling Choices
  4. Rank-Order Allocation and General All-Pay Allocation
  5. Unit-Sum and Unit-Range
  6. Our Results
  7. Contests with Rank-Order Allocation
  8. General All-Pay Contests
  9. Related Work
  10. Notation and Preliminaries
  11. Contests with a Rank-Order Allocation of Prizes
  12. General All-Pay Contests
  13. Objective Functions
  14. Rank-Order Allocation of Prizes
  15. Unit-Sum
  16. ...and 32 more sections

Key Result

Theorem 2.1

moldovanu2001optimalchawla2013auctions Consider the game that models the rank-order allocation contest with the values of placement prizes $w_1 \ge w_2 \ge \ldots \ge w_n \ge 0$. The unique Bayes--Nash equilibrium is given by where $\beta(v)$ is the output generated by a player with ability $v$.

Figures (2)

  • Figure 1: Designer's objectives (left to right): total output, binary threshold, and linear threshold.
  • Figure 2: Plot of $\beta_i, \beta_k, \beta$ where $i < k$.

Theorems & Definitions (40)

  • Theorem 2.1
  • Definition 2.1: Simple Contest
  • Definition 2.2: Single-Crossing
  • Theorem 2.2
  • Definition 2.3: Regular Distributions
  • Definition 2.4: Binary Threshold Objective
  • Lemma 2.3
  • Definition 2.5: Linear Threshold Objective
  • Theorem 3.1
  • Theorem 3.2
  • ...and 30 more