Towards Efficient Pareto-optimal Utility-Fairness between Groups in Repeated Rankings

TL;DR

This work tackles balancing user utility and producer fairness in repeated rankings by reframing the exposure and ranking distribution problem on the Expohedron, a geometric polytope of achievable exposures. It identifies the computational bottleneck of traditional Birkhoff–von Neumann decomposition and introduces a geodesic-based Sphere-Expo approximation that maps the Pareto frontier to hypersphere geodesics, enabling scalable profiling via Carathéodory decomposition. The authors prove that Pareto-optimal solutions lie on Expohedron facets and show how to trace the frontier across facets, then demonstrate that Sphere-Expo closely approximates the true Pareto front with controllable accuracy through the number of marked points, while offering practical runtimes on real datasets (TREC Fair Ranking, MRLR). Overall, the method provides a time-efficient, tunable trade-off between utility and fairness for group-based exposure in repeated rankings, with strong empirical support and potential applicability to broader exposure models.

Abstract

In this paper, we tackle the problem of computing a sequence of rankings with the guarantee of the Pareto-optimal balance between (1) maximizing the utility of the consumers and (2) minimizing unfairness between producers of the items. Such a multi-objective optimization problem is typically solved using a combination of a scalarization method and linear programming on bi-stochastic matrices, representing the distribution of possible rankings of items. However, the above-mentioned approach relies on Birkhoff-von Neumann (BvN) decomposition, of which the computational complexity is $\mathcal{O}(n^5)$ with $n$ being the number of items, making it impractical for large-scale systems. To address this drawback, we introduce a novel approach to the above problem by using the Expohedron - a permutahedron whose points represent all achievable exposures of items. On the Expohedron, we profile the Pareto curve which captures the trade-off between group fairness and user utility by identifying a finite number of Pareto optimal solutions. We further propose an efficient method by relaxing our optimization problem on the Expohedron's circumscribed $n$-sphere, which significantly improve the running time. Moreover, the approximate Pareto curve is asymptotically close to the real Pareto optimal curve as the number of substantial solutions increases. Our methods are applicable with different ranking merits that are non-decreasing functions of item relevance. The effectiveness of our methods are validated through experiments on both synthetic and real-world datasets.

Figures (6)

• Figure 1: An example of utility and fairness level set when n=3 and g=2. While the blue line is the utility level set, the target fairness level is the red ones and one set of fairness level are red dashed line. The green line segments form the Pareto-curve in the Expohedron
• Figure 2: An example of utility and fairness level set when n=4. The green line segments form the Pareto-curve in the Expohedron. The red points are “break” points in Pareto set curve
• Figure 3: An example of how a set of connected segment could be approximate with set of geodesics. The dash is the approximate geodesic curve and its projection on Expohedron, while the line is the exact solution.
• Figure 4: Aggregated Pareto fronts in small synthetic dataset $D_s$ showing the drawback of the P-Expo
• Figure 5: Aggregated Pareto fronts 2 real-world datasets. The more marked points, the closer the approximation to the true Pareto front.