[Paper Review] Undergraduate Course Allocation through Competitive Markets

Daniel Kornbluth, Alexey Kushnir|arXiv (Cornell University)|Dec 7, 2024

Innovations in Educational Methods4 citations

TL;DR

This paper proposes the Pseudo-Market with Priorities (PMP) mechanism, a competitive equilibrium-based course allocation system that uses fake money and priority-specific prices to assign seats while respecting student priorities. It achieves approximate stability, efficiency, envy-freeness, and strategy-proofness in large markets, reducing student envy by 8% (≈500 students) compared to traditional reserved-seat systems.

ABSTRACT

Prevailing methods of course allocation at undergraduate institutions involve reserving seats to give priority to designated groups of students. We introduce a competitive equilibrium-based mechanism that assigns course seats using student preferences and course priorities. This mechanism satisfies approximate notions of stability, efficiency, envy-freeness, and strategy-proofness. We evaluate its performance relative to a mechanism widely used in practice using preferences estimated from university data. Our empirical findings demonstrate an improvement in student satisfaction and allocation fairness. The number of students who envy another student of weakly lower priority declines by 8 percent, or roughly 500 students.

Motivation & Objective

• To address inefficiencies and fairness issues in traditional undergraduate course allocation systems that rely on reserved seats and rigid priority hierarchies.
• To design a deterministic, scalable mechanism that respects student priorities while enabling efficient, stable, and approximately strategy-proof allocations.
• To evaluate the performance of the proposed mechanism against widely used real-world course allocation practices using empirical data from universities.
• To demonstrate that competitive equilibrium mechanisms can be adapted to two-sided, many-to-many matching problems with heterogeneous priorities.
• To show that approximate competitive equilibrium with equal incomes can be extended to settings with course-specific priorities without sacrificing key fairness and efficiency properties.

Proposed method

• The PMP mechanism uses student-reported preferences over course schedules and course-specific priority cutoffs to set prices that reflect institutional priorities.
• Each student is assigned a random budget drawn from a uniform distribution, ensuring approximate budget equality while preserving priority-based access.
• An approximate competitive equilibrium is computed where each student selects her most-preferred affordable schedule given her budget and course prices.
• Course prices are set such that students at or above a cutoff priority level can afford a course, while those below cannot, embedding priority constraints into the market-clearing process.
• The mechanism is semi-anonymous: students within the same priority group are treated identically in terms of prices and budget lotteries, but can have different preferences.
• Theoretical analysis establishes that the mechanism is approximately strategy-proof in large markets and envy-free except for random tie-breaking.

Experimental results

Research questions

• RQ1 Can a competitive equilibrium mechanism be adapted to undergraduate course allocation with heterogeneous student priorities?
• RQ2 How does the PMP mechanism compare to traditional reserved-seat mechanisms in terms of student satisfaction and fairness?
• RQ3 To what extent does the PMP mechanism achieve approximate strategy-proofness in large markets?
• RQ4 What is the impact of introducing priority-specific prices on allocation efficiency and envy?
• RQ5 Can approximate competitive equilibrium be used to design a fair, stable, and scalable course allocation mechanism in a many-to-many matching setting?

Key findings

• The PMP mechanism reduces the number of students who envy another student of lower priority by 8%, equivalent to approximately 500 students, compared to the reserved-seat mechanism.
• The mechanism achieves approximate stability, efficiency, and envy-freeness, with theoretical guarantees under large-market conditions.
• Student satisfaction improves under the PMP mechanism due to better alignment of course assignments with individual preferences.
• The mechanism is approximately strategy-proof in large markets, meaning truthful preference reporting is nearly optimal for individuals.
• The PMP mechanism generalizes the approximate competitive equilibrium from equal incomes (ACEI) mechanism to settings with course-specific priorities.
• Empirical evaluation using real university data confirms that the PMP mechanism outperforms traditional reserved-seat systems in fairness and efficiency metrics.