Fairness and Efficiency in Two-Sided Matching Markets

TL;DR

This work studies MEFE-Matching, a two-sided matching framework that enforces fairness on the TA side via merit-based envy-freeness and ensures course satisfaction via average utility thresholds. It shows that the problem is NP-hard even in highly restricted scenarios, yet identifies a broad landscape of polynomial-time solvable cases and fixed-parameter/approximation algorithms under various structural constraints. The authors connect MEFE-Matching to stable/HR frameworks to derive existence results and provide algorithmic strategies across hardness and tractability regimes. The findings offer both theoretical insights and practical guidance for allocating TAs to courses in a way that respects merit, course needs, and cross-side fairness, with implications for related resource-allocation settings.

Abstract

We propose a new fairness notion, motivated by the practical challenge of allocating teaching assistants (TAs) to courses in a department. Each course requires a certain number of TAs and each TA has preferences over the courses they want to assist. Similarly, each course instructor has preferences over the TAs who applied for their course. We demand fairness and efficiency for both sides separately, giving rise to the following criteria: (i) every course gets the required number of TAs and the average utility of the assigned TAs meets a threshold; (ii) the allocation of courses to TAs is envy-free, where a TA envies another TA if the former prefers the latter's course and has a higher or equal grade in that course. Note that the definition of envy-freeness here differs from the one in the literature, and we call it merit-based envy-freeness. We show that the problem of finding a merit-based envy-free and efficient matching is NP-hard even for very restricted settings, such as two courses and uniform valuations; constant degree, constant capacity of TAs for every course, valuations in the range {0,1,2,3}, identical valuations from TAs, and even more. To find tractable results, we consider some restricted instances, such as, strict valuation of TAs for courses, the difference between the number of positively valued TAs for a course and the capacity, the number of positively valued TAs/courses, types of valuation functions, and obtained some polynomial-time solvable cases, showing the contrast with intractable results. We further studied the problem in the paradigm of parameterized algorithms and designed some exact and approximation algorithms.

Figures (3)

• Figure 1: An instance to show unfairness on the TA's side if we consider grades as utilities from the course side and our satisfaction criteria of courses alone for a fair matching. Here the tuple $(p,q,r)$ for a course $c_i$ (or TA $t_i$), denotes the valuation for TAs $t_1,t_2,t_3$ (courses $c_1,c_2,c_3$), respectively. Edges denote a matching for $k=7$. TA $t_1$ has a higher grade in $c_1$ than TA $t_2$ and also values $c_1$ more than $c_2$. Thus, this matching is unfair for $t_1$.
• Figure 2: An instance of MEFE-Matching is presented in \ref{['fig:mefe1']} corresponding to an instance of (3-3)-com smti presented in \ref{['fig:tcm1']}
• Figure 3: MEFE instance of $3$-dimensional Perfect Matching where $(p_1,q_1,r_1)$ is an edge. Edge for $r_n$ are not drawn

Theorems & Definitions (91)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• proof
• Corollary 1
• Theorem 5
• Theorem 6
• Corollary 2
• proof