The Price of Opportunity Fairness in Matroid Allocation Problems

TL;DR

This work studies the price of opportunity fairness for matroid allocation problems with groups defined by sensitive attributes. By leveraging a polymatroid representation, it derives a concise PoF characterization and tight bounds across adversarial and random settings, including a semi-random coloring regime and random-graph models. The key results show PoF is at most $C-1$ in the worst case, but often collapses to 1 when no group dominates or in large markets under random coloring or Erdős–Rényi graph models, illustrating that opportunity fairness can achieve near-optimal social welfare in many practical scenarios. The findings offer structural insights into when fairness constraints minimally impact welfare and highlight directions for extending to weighted allocations and other fairness notions.

Abstract

We consider matroid allocation problems under opportunity fairness constraints: resources need to be allocated to a set of agents under matroid constraints (which include classical problems such as bipartite matching). Agents are divided into $C$ groups according to a sensitive attribute, and an allocation is opportunity-fair if each group receives the same share proportional to the maximum feasible allocation it could achieve in isolation. We study the Price of Fairness (PoF), i.e., the ratio between maximum size allocations and maximum size opportunity-fair allocations. We first provide a characterization of the PoF leveraging the underlying polymatroid structure of the allocation problem. Based on this characterization, we prove bounds on the PoF in various settings from fully adversarial (worst-case) to fully random. Notably, one of our main results considers an arbitrary matroid structure with agents randomly divided into groups. In this setting, we prove a PoF bound as a function of the (relative) size of the largest group. Our result implies that, as long as there is no dominant group (i.e., the largest group is not too large), opportunity fairness constraints do not induce any loss of social welfare (defined as the allocation size). Overall, our results give insights into which aspects of the problem's structure affect the trade-off between opportunity fairness and social welfare.

Figures (22)

• Figure 1: Examples of the set of fractional feasible allocations $M$ (dark blue solid region) and the Pareto frontier $P$ (light blue region) for $C=2$ and $C=3$.
• Figure 2: PoF upper bounds stated in \ref{['prop:rho_bound']} for $5$ groups with equal rank, with variable value of the independence index $\rho$.
• Figure 3: Shape of $M$ for extreme values of independence index. Left: $\rho = 1/C$, right: $\rho = 1.$
• Figure 4: \ref{['thm:semi_random']} for $C=5$ groups: worst-case PoF, $\psi_{\lambda}$ for $\lambda \in \{3,4\}$, and the relaxed bound $C - 1/\pi$.
• Figure 5: Graphic matroid example showing that integrality can lead to null opportunity fair allocations. (Left) Graph defining a colored graphic matroid with two groups. Group $1$ is denoted by the green edges while Group $2$ by the red edges. It follows that $r(1)=5$, $r(2)=3$, and $r(\{1,2\})=7$. (Right) Set of integer feasible allocations (blue dots) and set of opportunity fair allocations (orange line), whose only intersection is at the origin.

Theorems & Definitions (57)

• Definition 2.1
• Definition 2.2
• Example
• Definition 2.3
• Proposition 2.4
• proof : Proof Sketch
• Proposition 2.5
• Definition 3.1
• Proposition 3.2
• Corollary 3.3