A Family of Convex Models to Achieve Fairness through Dispersion Control

TL;DR

This work addresses enforcing fairness in centralized optimization by controlling dispersion of decision variables through a unifying convex family parameterized by $\varepsilon \in [0,1]$. Building on finite-dimensional norm equivalence, it defines $(\varepsilon,p)$-fairness and a normalized set $\mathcal{Y}(\varepsilon,p)$ that tightens dispersion as $\varepsilon$ increases, with a provable bound on the coefficient of variation: $(\mathrm{CV}(\bm x))^2 \le B_p(\varepsilon) = \frac{(D_p+1)^2}{(1+\varepsilon D_p)^2}-1$. The paper establishes that, for $\varepsilon$ at the extremes, feasible sets are invariant across $p$, while for $\varepsilon\in(0,1)$ the sets strictly shrink as $p$ grows, via a strict inclusion result. Empirical results on a fairness-constrained multi-agent assignment problem show monotone dispersion reduction with $\varepsilon$, a clear price of fairness trade-off, and computational advantages for the linear $p=\infty$ variant, highlighting practical applicability across domains with minimal integration effort.

Abstract

Controlling the dispersion of a subset of decision variables in an optimization problem is crucial for enforcing fairness or load-balancing across a wide range of applications. Building on the well-known equivalence of finite-dimensional norms, the note develops a family of parameterized convex models that regulate the dispersion of a vector of decision-variable values through its coefficient of variation. Each model contains a single parameter that takes a value in the interval $[0,1]$. When the parameter is set to zero, the model imposes only a trivial constraint on the optimization problem; when set to one, it enforces equality of all the decision variables. As the parameter varies, the coefficient of variation is provably bounded above by a monotonic function of that parameter. The note also presents theoretical results that relate the space of feasible solutions to all the models. Finally it compares the models' solution quality on a variant of the assignment problem that regulates the dispersion in the assignment costs.

Figures (4)

• Figure 1: Graphical illustration for $\varepsilon_{p_1}(\bm x) > \varepsilon_{p_2}(\bm x)$.
• Figure 2: Coefficient of variation $\mathrm{CV}(\bm t^*(\varepsilon, p))$ versus $\varepsilon$ for different $p$ for one instance.
• Figure 3: $\mathrm{PoF}(\varepsilon,p)$ versus $\varepsilon$ for various $p$.
• Figure 4: Box-plot of computation times in seconds for all the 50 instances for $\varepsilon \in \{0.5, 0.7, 0.9, 0.99\}$.

Theorems & Definitions (19)

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