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Maximin Relative Improvement: Fair Learning as a Bargaining Problem

Jiwoo Han, Moulinath Banerjee, Yuekai Sun

TL;DR

This work reframes fairness across multiple subpopulations as a cooperative bargaining problem, introducing maximin relative improvement where each group gains a proportional fraction $\rho_g(f)$ of its potential risk reduction from baseline $f_0$ to its group optimum $f_g^*$. The framework connects to the Kalai–Smorodinsky solution in the two-group case and extends to a leximin refinement for $m>2$, yielding a unique, scale-invariant, no-harm fairness criterion under mild regularity assumptions. It also shows how existing robustness-based fairness criteria (e.g., GDRO, MMV, MMR) map to different bargaining solutions, and provides an empirical estimator with finite-sample convergence guarantees. The approach is particularly suitable for heterogeneous group predictability and offers a principled, axiomatic basis for fair learning, with broad implications for multi-objective optimization beyond fairness alone.

Abstract

When deploying a single predictor across multiple subpopulations, we propose a fundamentally different approach: interpreting group fairness as a bargaining problem among subpopulations. This game-theoretic perspective reveals that existing robust optimization methods such as minimizing worst-group loss or regret correspond to classical bargaining solutions and embody different fairness principles. We propose relative improvement, the ratio of actual risk reduction to potential reduction from a baseline predictor, which recovers the Kalai-Smorodinsky solution. Unlike absolute-scale methods that may not be comparable when groups have different potential predictability, relative improvement provides axiomatic justification including scale invariance and individual monotonicity. We establish finite-sample convergence guarantees under mild conditions.

Maximin Relative Improvement: Fair Learning as a Bargaining Problem

TL;DR

This work reframes fairness across multiple subpopulations as a cooperative bargaining problem, introducing maximin relative improvement where each group gains a proportional fraction of its potential risk reduction from baseline to its group optimum . The framework connects to the Kalai–Smorodinsky solution in the two-group case and extends to a leximin refinement for , yielding a unique, scale-invariant, no-harm fairness criterion under mild regularity assumptions. It also shows how existing robustness-based fairness criteria (e.g., GDRO, MMV, MMR) map to different bargaining solutions, and provides an empirical estimator with finite-sample convergence guarantees. The approach is particularly suitable for heterogeneous group predictability and offers a principled, axiomatic basis for fair learning, with broad implications for multi-objective optimization beyond fairness alone.

Abstract

When deploying a single predictor across multiple subpopulations, we propose a fundamentally different approach: interpreting group fairness as a bargaining problem among subpopulations. This game-theoretic perspective reveals that existing robust optimization methods such as minimizing worst-group loss or regret correspond to classical bargaining solutions and embody different fairness principles. We propose relative improvement, the ratio of actual risk reduction to potential reduction from a baseline predictor, which recovers the Kalai-Smorodinsky solution. Unlike absolute-scale methods that may not be comparable when groups have different potential predictability, relative improvement provides axiomatic justification including scale invariance and individual monotonicity. We establish finite-sample convergence guarantees under mild conditions.
Paper Structure (30 sections, 11 theorems, 85 equations, 6 figures, 1 table)

This paper contains 30 sections, 11 theorems, 85 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Related Work
  4. Bargaining Problem Perspective
  5. Fairness as Bargaining: A General Framework
  6. The Kalai-Smorodinsky Solution and Relative Improvement
  7. Comparison with Alternative Fairness Approaches
  8. Risk Set Properties and Well-Posedness
  9. Fairness Guarantees of Relative Improvement
  10. No-Harm Guarantee
  11. Axiomatic Characterization: $m=2$ case
  12. Axiomatic Characterization: $m>2$ case
  13. Comparison with Alternative Bargaining Solutions
  14. Empirical Estimator
  15. Discussion
  16. ...and 15 more sections

Key Result

Theorem 3.4

Under either Assumption assumps_parametric or Assumption assumps_nonparam, the risk set $\mathcal{R}(\mathcal{F})$ satisfies: When the loss function is additionally strictly convex (in $\theta$ or $t$, respectively), the following properties also hold:

Figures (6)

  • Figure 1: Minimax regret versus maximin relative improvement in a two-group simple linear regression example with heterogeneous predictability ($\beta_1 = 2, \sigma_1^2 = 1$ versus $\beta_2 = 7, \sigma_2^2 = 9$), yielding achievable risk reductions of 4 and 49 units, respectively. Left: risk landscape. Right: relative improvement space, where minimax regret yields $(-56\%, 87\%)$ while maximin relative improvement achieves $(69\%, 69\%)$.
  • Figure 2: Transformation from risk space to utility space. Left: risk space $\mathcal{R}(\mathcal{F})$ where groups aim to minimize risks. Right: utility space $\mathcal{U} = -\mathcal{R}(\mathcal{F})$ where players aim to maximize utilities.
  • Figure 3: Schematic diagram comparing group fairness methods in risk space, after assuming the Pareto Frontier. Each method selects a different solution on the Pareto frontier. Detailed characterization are provided in Appendix \ref{['app:fairness_methods']}.
  • Figure 4: Risk set $\mathcal{R}(\mathcal{F})$, its convex hull (left), and comprehensive closure (right) for the linear regression setting (in Section \ref{['sec:formulation']}) with $\Theta = \{\theta:\|\theta\| \leq 1\}$, $\beta_1 = (0.4, 0)$, $\beta_2 = (0.4, 0.6)$, $\Sigma_1 = \left(10.50.51\right)$, $\Sigma_2 = \left(1001\right)$, and $\sigma_g = 1$. Note that $\mathrm{conv}(\mathcal{R}(\mathcal{F})) \setminus \mathcal{R}(\mathcal{F})$ and $\mathrm{comp}(\mathcal{R}(\mathcal{F})) \setminus \mathcal{R}(\mathcal{F})$ contain no Pareto optimal points.
  • Figure 5: Transformation between risk space and relative improvement space. Left: Risk space with group-optimal risks $r_1^* = 0.1$, $r_2^* = 0.4$ (utopia point) and baseline disagreement point $(1, 1)$. Right: Relative improvement space where the disagreement point maps to $(0,0)$ and group optimal points to $(1, \rho_2^{(1)})$ and $(\rho_1^{(2)}, 1)$. The fairness diagonal $\rho_1 = \rho_2$ intersects the Pareto frontier at the KS solution (orange star).
  • ...and 1 more figures

Theorems & Definitions (46)

  • Remark 2.1: Beyond Fairness
  • Remark 3.1: Uniqueness of risk vectors vs. predictors
  • Theorem 3.4: Properties of the risk set
  • Theorem 4.1: Bounded Loss Relative to Baseline
  • Proposition 4.2: Relative improvement as the KS solution ($m=2$), adapted from kalai_1975_other
  • Theorem 4.3: Equivalence under comprehensive closure
  • Proposition 4.4: Relative improvement as the KS solution ($m>2$), adapted from imai1983individual
  • Lemma 5.2: Sufficient conditions for Assumption \ref{['asmp:cond1']}
  • Theorem 5.4: Convergence Rate of Relative Improvement
  • Definition 1.1: Risk vector
  • ...and 36 more