📈 economics

Desirable Rankings

This paper proposes a new method for aggregating collective preferences into a single ranking by prioritizing alternatives that agents rank above their assigned matches, demonstrating that this "desirable ranking" approach converges to true quality rankings and outperforms traditional methods like Borda counts and revealed preference rankings.

Original authors: Gaurab Aryal, Thayer Morrill, Peter Troyan

Published 2026-02-11

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CC BY 4.0

Original authors: Gaurab Aryal, Thayer Morrill, Peter Troyan

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 trying to figure out which restaurant in your city is truly the best.

You could look at Star Ratings (like Yelp), but those are easy to fake—restaurants might ask friends to leave five-star reviews just to boost their score. You could look at Popularity (how many people go there), but that’s also tricky—a massive fast-food chain might be "popular" simply because it’s cheap and everywhere, not because the food is actually high quality.

This academic paper proposes a much smarter way to rank things (like colleges, medical programs, or journals) by looking at "Desire" rather than just "Popularity" or "Metrics."

The Core Idea: The "Upgrade" Principle

The researchers argue that the best way to find quality is to look at what people wish they had.

Think of it like this: Imagine a group of people are all eating at different restaurants. Some are at fancy steakhouses, and some are at taco stands.

If a person sitting at a taco stand looks at the menu of the steakhouse and says, "Man, I wish I was eating there instead," that is a signal of desire.
If that happens a lot, it’s a very strong sign that the steakhouse is higher quality.

The researchers call this the Axiom of Desire. They aren't just looking at where people are; they are looking at where people want to be relative to their current situation.

The Problem: The "Noise" of Personal Taste

The tricky part is that people have "noise" in their preferences. You might be at a mediocre pizza place, but you really want to be at a sushi place—not because sushi is "better" for everyone, but just because you personally love fish. If we only looked at your wish, we might wrongly rank sushi as the best food in the world.

To fix this, the authors use a mathematical "filter" called a Shadow Matching.

The Metaphor: The Great Trade-Off
Imagine everyone in the city is allowed to trade meals once. If you have a taco and I have a steak, and we both think the other's meal is better, we swap. We keep swapping until no one wants to trade anymore. This "perfected" state is the Shadow Matching.

Once everyone has reached this "ideal" state where no more trades can be made, the researchers look at the remaining wishes. If people still wish they could move from their "ideal" meal to a different one, that wish isn't just a random personal quirk—it’s a signal of true, universal quality.

The Algorithm: The "Bottom-Up" Cleanup

To turn this into a ranking, they use an algorithm called IRUS (Iterated Removal of Underdemanded Schools).

Think of it like a Tournament of Desirability:

First, they look for the "least wanted" places—the ones that nobody, even after the "Great Trade-Off," wishes they could move to. These are the bottom tier.
They "remove" those places from the map and look at the remaining ones.
Now, they look for the next least wanted group.
They keep climbing up the ladder until they reach the very top.

Why does this matter? (The Results)

The researchers tested this on medical schools in Chile. They found that their method was much better than the two standard ways:

Better than "Revealed Preference": (Looking only at where people actually ended up). This method is often "jittery" and changes wildly every year.
Better than "Borda Counts": (Giving points based on how high a school is on a list). This method is biased toward big, famous schools that everyone puts on their list as a "safety," even if they aren't actually the best.

The Bottom Line:
The "Desire" method is like a truth-detector. It filters out the "noise" of personal quirks and the "cheating" of fake metrics to find the real signal: What do people actually value when they are given the chance to choose?

Technical Summary: Desirable Rankings

Authors: Gaurab Aryal, Thayer Morrill, and Peter Troyan
Date: February 6, 2026
Field: Economics (Social Choice Theory / Market Design)

1. The Problem: Aggregating Preferences in Matched Markets

The paper addresses the challenge of constructing a collective ranking of alternatives (e.g., colleges, medical residency programs, or academic journals) when agents are matched to those alternatives.

Traditional ranking methods often fall into two categories, both of which have significant flaws in matched settings:

Index-based measures: Rely on indicators like acceptance rates or citation counts. These are susceptible to Goodhart’s Law (when a measure becomes a target, it ceases to be a good measure), leading institutions to manipulate data to improve their standing.
Preference-based aggregation: Methods like Borda counts or Revealed Preference rankings. Borda counts are biased toward large programs (safety options), while Revealed Preference methods are data-intensive, require complex utility specifications, and can be highly volatile.

The core difficulty is distinguishing between common components of preference (intrinsic quality/prestige) and idiosyncratic components (personal fit, location, or individual taste).

2. Methodology: The Desirable Ranking Framework

The authors propose a new paradigm based on the concept of "Desirability." The fundamental intuition is that if an agent prefers an alternative $c$ to their current match $\mu(i)$ , then $c$ should be ranked higher in the social ranking.

A. Shadow Matchings and Pareto Improvement
To filter out idiosyncratic noise, the authors utilize shadow matchings ( $\mu^*$ ). A shadow matching is a Pareto-efficient improvement of the observed matching $\mu$ . By assuming that Pareto inefficiencies in the initial matching are driven by idiosyncratic preferences, the authors use the shadow matching as a "cleaner" baseline for preference.

B. Axiomatic Foundation
A ranking is defined as "Desirable" if it satisfies two axioms:

Axiom of Desire (AoD): Any college $c$ that a student desires (i.e., $c$ is strictly preferred to their shadow matching $\mu^*(i)$ ) must be ranked higher than $\mu^*(i)$ .
Justification: A high-ranked college is only justified if it is desired by a student currently matched to a lower-ranked tier. This prevents "unpopular" colleges from occupying high tiers simply because no one explicitly ranked them lower.

C. The IRUS Algorithm
The authors introduce the Iterated Removal of Underdemanded Schools (IRUS) algorithm. It works bottom-up:

Identify "underdemanded" schools (those that no student prefers over their shadow match).
Rank these schools in the lowest tier and remove them (along with their matched students) from the market.
Repeat the process on the remaining sub-market until all schools are ranked.

3. Key Contributions and Theoretical Results

Existence and Uniqueness: The authors prove that the IRUS algorithm always produces a desirable ranking. If the initial matching is already Pareto efficient, the IRUS ranking is the unique desirable ranking.
Convergence (The "Correctness" Result): The paper provides a major theoretical result: as the market size ( $n$ ) grows, desirable rankings converge to the true underlying quality ranking of the alternatives.
Tier Distinction: They prove that while the ranking allows for ties (tiers), the size of these tiers (as a percentage of the market) shrinks to zero as the market grows, meaning the ranking remains highly granular and informative.

4. Results and Empirical Validation

A. Simulations
Simulations confirm that the algorithm is robust to the choice of shadow matching and converges more quickly when students place higher weight on quality ( $\alpha$ ) versus idiosyncratic taste. The Mean Squared Error (MSE) decreases significantly as market size increases.

B. Empirical Application: Chilean Medical Programs
The authors applied IRUS to Chilean medical school admission data (2006–2009).

Data Parsimony: The method worked effectively even with sparse data (students ranked only ~3.5 programs out of 22).
Stability and Validity: The top-ranked programs (Catholic University and University of Chile) remained stable across years and aligned closely with international benchmarks (QS Rankings).
Benchmarking: The IRUS ranking outperformed both Revealed Preference (which was highly volatile) and Borda Counts (which were biased toward larger programs).

5. Significance

This paper introduces a mathematically rigorous yet practically simple method for institutional ranking. Its significance lies in:

Robustness to Manipulation: Unlike index-based rankings, a college cannot "game" a desirable ranking by manipulating admission rates; it can only improve its rank by increasing its actual desirability (quality).
Minimal Data Requirements: It requires only ordinal Rank-Order Lists (ROLs), making it applicable in many real-world markets where detailed utility data is unavailable.
Filtering Noise: It provides a formal mechanism to separate "prestige" from "personal fit," offering a more accurate reflection of institutional quality in large-scale markets.