Wisdom and Foolishness of Noisy Matching Markets

TL;DR

This paper analyzes a two-sided stable matching market with a continuum of students and a finite set of colleges, where colleges rank students by independent noisy estimates of true value $v$ drawn from distribution $\mathcal{D}$. It reveals two stark large-market regimes: if $\mathcal{D}$ is light-tailed (max-concentrating), noise is attenuated and matches align with a sharp cutoff at $v_S$; if $\mathcal{D}$ is long-tailed, noise is amplified and the matching probability becomes essentially independent of $v$, yielding near-random allocations proportional to college capacity. The authors develop a cutoff-based analysis that avoids explicit computation of market-clearing cutoffs and extend the framework to coalitions of colleges sharing true preferences, showing analogous attenuation/amplification effects within coalitions. They also propose a general methodological framework for studying imperfect preference formation in markets, including computational experiments illustrating the extreme regimes. Overall, the work highlights how local noise in evaluation can produce either “wisdom of crowds” or “foolishness of crowds” effects in large-scale matching markets and provides a tractable approach to quantify these effects.

Abstract

We consider a many-to-one matching market where colleges share true preferences over students but make decisions using only independent noisy rankings. Each student has a true value $v$, but each college $c$ ranks the student according to an independently drawn estimated value $v + X_c$ for $X_c\sim \mathcal{D}.$ We ask a basic question about the resulting stable matching: How noisy is the set of matched students? Two striking effects can occur in large markets (i.e., with a continuum of students and a large number of colleges). When $\mathcal{D}$ is light-tailed, noise is fully attenuated: only the highest-value students are matched. When $\mathcal{D}$ is long-tailed, noise is fully amplified: students are matched uniformly at random. These results hold for any distribution of student preferences over colleges, and extend to when only subsets of colleges agree on true student valuations instead of the entire market. More broadly, our framework provides a tractable approach to analyze implications of imperfect preference formation in large markets.

Figures (5)

• Figure 1: We compare $p_\mu(v)$, the probability that a student with true value $v$ matches, across three economies that differ only in their noise distribution $\mathcal{D}$. For uniform noise, which is max-concentrating, $p_\mu(v)$ approaches a step function as $C$ grows large (\ref{['thm:attenuating']}). For Pareto noise, which is long-tailed, $p_\mu(v)$ approaches a constant as $C$ grows large (\ref{['thm:amplifying']}). For exponential noise, which lies between our two regimes of focus, $p_\mu(v)$ does not change significantly as $C$ grows large. The economy here has $2000$ students and the total capacity of colleges is $1000$. Students have uniformly random preferences over colleges. Plots display averages over 100 simulations.
• Figure 2: $P^*$ is the smallest real number such that $[P^*, P^*+\delta]$ contains at least $M$ cutoffs.
• Figure 3: An economy with 2000 students and two coalitions of colleges. Coalition $1$ (red) has total capacity $500$ and coalition $2$ (blue) has total capacity $1000$. All students strictly prefer colleges in coalition $1$ to those in coalition $2$. Thus, in the noiseless case (left), the students with the highest true values at coalition 1 ($v_1>\frac{3}{4}$) match to a college in coalition 1, those out of the remaining students with the highest true values at coalition 2 ($v_2 > \frac{1}{3}$) match to a college in coalition 2, and the remaining students are unmatched. The top row on the right illustrates \ref{['thm:attenuating-extended']}: as coalition size grows large with a max-concentrating distribution, the market approaches the noiseless setting. The bottom row illustrates \ref{['thm:amplifying-extended']}: with a long-tailed distribution, noise is amplified as coalition size increases. In particular, the bottom right matching is noisier than the bottom left matching, and is approaching uniform at random matching.
• Figure 4: A framework for studying the effect of imperfect preference formation.
• Figure 5: $P^*$ is the smallest real number such that $[P^*, P^*+\delta]$ contains at least $C^{\phi_2}$ cutoffs.

Theorems & Definitions (33)

• Theorem 1: Attenuation
• Theorem 2: Amplification
• Theorem 3: Attenuation in Coalitions
• Theorem 4: Amplification in Coalitions
• Proposition 1
• Proposition 2
• proof
• Proposition 3
• proof : Proof of (i)
• proof : Proof of (ii).