📈 economics

Extreme Points and Large Contests

This paper characterizes the extreme points of multidimensional monotone functions and applies this result to simplify the analysis of optimal allocation rules and equilibria in large contests, particularly under complete information.

Original authors: Giovanni Valvassori Bolgè

Published 2026-03-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Giovanni Valvassori Bolgè

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are the head of a massive university admissions office, or perhaps the director of a giant grant-giving foundation. You have a huge pool of applicants (let's say millions of them), and you have a limited budget to give out a fixed number of scholarships or grants (say, 10% of the total applicants).

Your goal is to design a rule for who gets the money. But here's the catch: you care about different things. Sometimes you want to reward the absolute best performers. Other times, you want to help those who are struggling to ensure fairness. Sometimes you want to encourage everyone to try their hardest, even if they don't win.

This paper asks a simple but profound question: What is the best possible rule to use?

The author, Giovanni Valvassori Bolgè, discovers that no matter how complex your goals are, the "perfect" rule is surprisingly simple. It's not a complicated formula; it's basically a two-step ladder.

Here is the breakdown of the paper using everyday analogies:

1. The "Two-Step Ladder" Rule

The paper proves that the optimal way to distribute prizes is almost always a step function. Imagine a ladder with only two or three rungs.

The Top Rung: If you are above a certain performance line, you get the prize (or a high chance of it).
The Bottom Rung: If you are below that line, you get nothing.
The Middle Rung (The "Maybe"): If you are exactly on the line, you might get a random chance (a lottery) to win.

The paper shows that you never need a complex system where "if you score 85 you get 10% of the prize, but if you score 86 you get 12%." The math says the best you can do is a sharp cut-off: You either win big, or you don't.

2. Why Two Steps? (The "Extreme Points" Secret)

To find this rule, the author uses some heavy math involving "extreme points." Think of this like a chef trying to find the perfect recipe.

Imagine all possible rules you could write down are ingredients in a giant kitchen. Some rules are "extreme" (like pure salt or pure sugar), and most rules are just mixtures of these extremes. The author proves that the best recipe (the optimal rule) is always made by mixing at most two of these extreme ingredients.

In the real world, these "extreme ingredients" are simple "Yes/No" decisions based on a threshold. So, the best rule is a mix of two simple "Yes/No" thresholds.

3. What Does the Designer Want? (The Three Scenarios)

The paper looks at three different "personalities" of the person designing the contest (the Principal):

The Elitist (Wants the Best): If the designer only cares about the top performers (like a scholarship for the valedictorian), the rule is simple: Set a high bar. Anyone above the bar wins; anyone below loses.
- The Result: This creates a "winner-take-all" race. Everyone tries to be the absolute best, or they give up entirely.
The Egalitarian (Wants Fairness): If the designer is worried that people are working too hard and wasting energy, or wants to give everyone a fair shot, the rule changes. The best approach is to ignore the scores entirely and just pick winners randomly (a lottery).
- The Result: No one tries to "game" the system by working extra hard, because effort doesn't guarantee a win.
The "Goldilocks" (Wants a Balance): Sometimes the designer wants to encourage some effort, but not too much. Maybe they want to reward the "good" students but not the "obsessive" ones.
- The Result: This is where the Two-Step Ladder shines. You set a lower threshold (to get a small prize) and a higher threshold (to get a big prize). This encourages people to aim for the middle ground, creating a stable equilibrium where people don't feel the need to push themselves to the breaking point.

4. Real-World Examples

The paper explains why we see these rules in real life:

Swiss Science Grants: They have a "funding line." If your proposal is above the line, you get money. If you are right on the line, they might flip a coin (lottery) to decide who gets the last few spots. This is exactly the "Two-Step" rule.
Medical School Admissions: In countries like Germany or the Netherlands, after the top students are admitted, the remaining spots are often filled by a lottery. This prevents students from studying 24/7 just to get a few extra points.

5. The "Complete vs. Incomplete" Information Twist

Complete Information: Everyone knows exactly how hard everyone else is working. The paper shows that if the designer wants to reward the best, the only stable outcome is that a small group works at maximum capacity, and everyone else quits. It's a "do or die" situation.
Incomplete Information: In the real world, effort is noisy. You might study hard but still get a bad grade due to a bad day. The paper shows that even with this noise, the Two-Step Ladder remains the best rule. It simplifies the chaos of the real world into a clear, predictable structure.

The Big Takeaway

If you are designing a competition, a hiring process, or a grant system for a large group of people, stop trying to be too clever with complex formulas.

The math says the best system is usually a simple threshold:

Set a line in the sand.
If you cross it, you win (or get a lottery ticket).
If you don't, you lose.

This simple structure is robust, easy to understand, and mathematically proven to be the most efficient way to achieve almost any goal you might have, whether that goal is finding the absolute best, ensuring fairness, or finding a happy medium.

1. Problem Statement

The paper addresses the problem of optimal contest design in environments with a continuum of agents (large contests). The central challenge is to characterize the optimal allocation rule (the mapping from agent performance/effort to the probability of winning a prize) for a contest designer who holds specific distributional preferences.

Key constraints and objectives include:

Budget Constraint: The designer can award prizes to a fixed mass $k \in (0, 1)$ of agents, regardless of the equilibrium effort distribution.
Distributional Preferences: The designer may care about:
- Rewarding high performers (increasing preference function).
- Promoting equality or minimizing wasteful effort (decreasing preference function).
- Complex, non-monotonic goals (e.g., targeting intermediate effort levels).
Information Structures: The analysis covers both complete information (perfectly observable effort) and incomplete information (noisy performance signals).

The core mathematical difficulty lies in optimizing over a set of multidimensional monotone functions (allocation rules) that depend on the aggregate distribution of efforts, a space that is infinite-dimensional and complex.

2. Methodology

The author employs a geometric approach rooted in functional analysis and convexity theory to solve the optimization problem.

A. Characterization of Extreme Points

The paper first establishes a novel mathematical result regarding the set of admissible allocation rules, denoted as $\mathcal{F}^*$ .

Set Definition: $\mathcal{F}^*$ consists of monotone functions $f: X \times \Delta(X) \to [0, 1]$ satisfying a budget constraint $\int f(x, \mu) d\mu(x) \leq k$ .
Main Theorem (Theorem 1): The set of extreme points of $\mathcal{F}^*$ $F^{*}$ , denoted $\text{ext}(\mathcal{F}^*)$ $ext (F^{*})$ , consists of functions that are step functions with at most two jumps.
- Formally, any extreme point $f^*$ takes the form:
  $f^*(x, \mu) = \lambda \mathbb{I}_{A_1}(x) + (1-\lambda) \mathbb{I}_{A_2}(x)$
  where $A_1 \subseteq A_2$ are "up-sets" (sets closed under increasing effort), and $\lambda \in [0, 1]$ .
- This result relies on the work of [Win88] (convex combinations of extreme points) and [YY25] (characterization of monotone functions via nested up-sets).

B. Optimization via Bauer's Maximum Principle

To find the optimal allocation rule, the author maximizes a functional $\Pi(f)$ representing the designer's payoff.

Convexity: The objective functional is shown to be convex (or linear) in the allocation rule $f$ .
Application of Bauer's Principle: Since the feasible set $\mathcal{F}^*$ is compact and convex, and the objective is convex, the maximum must occur at one of the extreme points of the set.
Majorization: The paper utilizes the concept of majorization ( $\succ$ ) and Theorem 2 (from [FL54]) to determine which specific extreme point maximizes the objective based on the monotonicity of the designer's preference function $\pi(x)$ .

3. Key Results

A. Structure of Optimal Allocation Rules (Theorem 3)

The optimal allocation rule is always a two-threshold scheme (a step function with at most two steps), determined by the monotonicity of the designer's preference function $\pi(x)$ :

Increasing $\pi(x)$ (Rewarding Top Performers):
- The optimal rule is a single-threshold (deterministic) rule.
- Agents exerting effort above a cutoff $\tilde{x}$ win with probability 1; those below win with probability 0.
- Interpretation: This aligns with standard meritocratic selection (e.g., funding lines).
Decreasing $\pi(x)$ (Promoting Equality/Reducing Waste):
- The optimal rule is a uniform lottery.
- Every agent wins with probability $k$ , regardless of effort.
- Interpretation: This eliminates incentives for wasteful effort, as effort does not affect winning probability.
Non-Monotonic $\pi(x)$ (Targeting Specific Effort Levels):
- The optimal rule is a two-threshold scheme with a "flat" middle section.
- Structure:
  - $q=1$ for $x > x''$
  - $q=\alpha$ for $x' \leq x \leq x''$
  - $q=0$ for $x < x'$
- Interpretation: This allows the designer to induce agents to exert intermediate effort levels by creating a "plateau" in the winning probability, preventing agents from jumping to the highest effort level.

B. Equilibrium Analysis in Complete Information

The paper demonstrates that the large-contest assumption significantly simplifies equilibrium characterization:

Case: Increasing $\pi(x)$ : The unique equilibrium involves a mass $k$ $k$ of agents exerting maximal effort ( $x_{max}$ $x_{ma x}$ ), while the remaining mass $1-k$ $1 - k$ exerts zero effort (stays out).
- Mechanism: The discontinuity at the threshold creates an incentive for agents to slightly outbid the cutoff. This "bidding war" continues until the marginal cost of effort equals the prize value, driving the payoff of the marginal winner to zero.
Case: Non-monotonic $\pi(x)$ : The designer can use the two-threshold rule to sustain an equilibrium where agents exert intermediate effort, avoiding the "all-or-nothing" outcome.

C. Robustness to Incomplete Information

Corollary 1 extends these results to settings with noisy performance signals.

Even when effort is not perfectly observable, the optimal allocation rule remains a step function (specifically, a Random Performance Function as defined by [AC24]).
The logic holds because the induced Contest Success Function (CSF) preserves the majorization ordering of the underlying allocation rules.

D. Extensions

Aggregate Effort Maximization: The results hold even if the designer's objective is to maximize aggregate effort (non-monotonic in effort).
Multiple Prizes: The paper proves that focusing on a single prize level is without loss of generality due to the concavity of the payoff function with respect to the prize mass.
Diversity Constraints: The model can be extended to handle quotas (e.g., reserving a share of prizes for a specific demographic), effectively splitting the problem into independent sub-problems for each group.

4. Significance and Contributions

Mathematical Contribution: The paper provides a rigorous characterization of the extreme points of multidimensional monotone functions under an integral constraint. This is a non-trivial extension of existing literature (e.g., [KMS21], [YY25]) to settings where the function depends on the distribution of types/efforts, not just the types themselves.
Economic Foundation for Real-World Mechanisms: The results provide a theoretical justification for observed practices in large contests:
- Two-tier selection: Explains why institutions (like the Swiss National Science Foundation or medical school admissions) often use a "cutoff + lottery" mechanism.
- Lotteries: Validates the use of lotteries not just as a fairness tool, but as an optimal mechanism when the designer wishes to minimize effort costs or promote equality.
Simplification of Equilibrium Analysis: By leveraging the "large contest" (continuum) assumption, the paper bypasses the complex equilibrium analysis typical of finite-agent contests. It shows that optimal mechanisms in large contests are structurally simple (step functions), leading to tractable equilibrium predictions (e.g., the "bang-bang" solution where only the top $k$ exert maximal effort).
Unification of Literature: The paper connects the literature on Contest Success Functions (CSFs) with Mechanism Design and Extreme Point Theory, showing that Random Performance Functions (RPFs) are not just axiomatic constructs but are actually optimal under specific distributional goals.

In summary, the paper demonstrates that for a broad class of preferences, the optimal way to allocate prizes in a large contest is a simple, threshold-based rule (potentially with a lottery component), and this simplicity drastically reduces the complexity of analyzing strategic behavior in such environments.