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Fairness Behind a Veil of Ignorance: A Welfare Analysis for Automated Decision Making
Hoda Heidari, Claudio Ferrari, Krishna P. Gummadi, Andreas Krause
TL;DR
Fairness evaluations for consequential automated decisions often emphasize equality or group comparisons while overlooking risk and welfare. The paper proposes convex welfare-based measures grounded in the veil of ignorance, and finds trade-offs with accuracy and other fairness notions while suggesting that welfare lower bounds often bound outcome inequality.
Problem
Fairness research has largely emphasized equality, discrimination, or individual comparisons, while risk and welfare considerations in consequential automated decisions remain comparatively overlooked.
Method
The paper compares expected utility for a randomly selected risk-averse decision subject using cardinal social welfare measures justified by the veil of ignorance and formulates them as convex constraints.
Results
The measures exhibit trade-offs with prediction accuracy, group discrimination, and Dwork et al.’s individual fairness, while lower welfare bounds often lead to bounded inequality in algorithmic outcomes.
Takeaways & Limitations
Welfare-based fairness constraints provide a computationally feasible way to evaluate risk-sensitive fairness and constrain inequality in algorithmic outcomes.
Takeaways & Limitations
Maintaining convexity restricts the benefit functions to positive functions linear in the predicted label, limiting the benefit landscapes expressible in some regression and multiclass-classification settings.
Abstract
from arXiv · showhide
We draw attention to an important, yet largely overlooked aspect of evaluating fairness for automated decision making systems---namely risk and welfare considerations. Our proposed family of measures corresponds to the long-established formulations of cardinal social welfare in economics, and is justified by the Rawlsian conception of fairness behind a veil of ignorance. The convex formulation of our welfare-based measures of fairness allows us to integrate them as a constraint into any convex loss minimization pipeline. Our empirical analysis reveals interesting trade-offs between our proposal and (a) prediction accuracy, (b) group discrimination, and (c) Dwork et al.'s notion of individual fairness. Furthermore and perhaps most importantly, our work provides both heuristic justification and empirical evidence suggesting that a lower-bound on our measures often leads to bounded inequality in algorithmic outcomes; hence presenting the first computationally feasible mechanism for bounding individual-level inequality.
1 Introduction
The paper argues that fairness in automated decision making should account for risk and welfare, not only equality or group comparisons. It proposes welfare-based measures justified by the veil of ignorance and studies their computational properties and trade-offs with existing fairness criteria.
- Automated decisions in employment, lending, policing, and criminal justice can have lasting effects on individuals and groups, motivating fairness guarantees beyond system-wide performance.
- The paper evaluates fairness through risk and welfare, arguing that equality alone can prefer uniformly lower benefits over an unequal distribution that makes almost everyone substantially better off.In the example, model D is preferred to A because it provides higher welfare and lower risk despite unequal benefits.
- Its proposed measures correspond to cardinal social welfare and are justified by choosing the alternative preferred by a subject who does not know which individual they will be.This veil-of-ignorance perspective detaches evaluation from the needs of any particular person or group.
- The welfare-based measures can be imposed as constraints in convex loss-minimization pipelines, allowing the resulting optimization problem to be solved efficiently and exactly.
- Experiments identify trade-offs with accuracy, group discrimination, and individual fairness: greater risk aversion or welfare lower bounds can reduce accuracy while generally reducing inequality and false-positive/negative-rate differences.The effects on demographic parity and Dwork et al.’s individual fairness depend on whether the task is regression or classification.
- The paper presents evidence that lower-bounding social welfare often bounds inequality in algorithmic outcomes, addressing a gap in computationally efficient fairness mechanisms.
2 Our Proposed Family of Measures
The paper evaluates fairness through risk-sensitive social welfare, using positive linear benefits and convex constraints that can be integrated into loss minimization. Its framework characterizes trade-offs among welfare, accuracy, and other fairness notions while linking welfare bounds to inequality under fixed mean benefits.
- Benefit functions are restricted to be positive and linear in the predicted label so fairness constraints remain convex, although this limits expressible benefit landscapes outside binary classification.For binary classification, arbitrary benefits for the four (y, ŷ) pairs can be represented by a linear benefit function.
- The framework measures fairness through expected utility for a randomly chosen individual, using CRRA utility u(b) = b^α with risk aversion represented by 0 < α < 1.The utility function is applied to benefits produced by predictions, and all individuals are assumed to share the same risk parameter.
- Fairness is enforced by minimizing prediction loss subject to a lower bound on the welfare measure, with a threshold that controls the accuracy–fairness trade-off.Both the objective and constraint are convex in the model parameters, allowing efficient and exact optimization.
- For equal-mean positive benefit vectors and 0 < α < 1, higher Atkinson inequality corresponds exactly to lower proposed welfare, supporting welfare lower bounds as inequality controls.The paper states that lower-bounding the measures often leads in practice to low inequality, while Proposition 2 establishes the fixed-mean ordering equivalence.
- No single predictor simultaneously optimizes social welfare, accuracy, individual fairness, and statistical fairness across the examined realizable and unrealizable settings.In the unrealizable classification case, accuracy follows the Bayes classifier, welfare favors desirable predictions for many people, and low positive residual difference requires negative predictions for all individuals.
3 Experiments
The experiments compare welfare-based fairness with accuracy, group discrimination, and individual fairness across classification and regression settings. Increasing the welfare threshold generally reduces inequality but can reduce accuracy and produce dataset-dependent trade-offs.
- Experimental Setup: The experiments use COMPAS for classification and Crime and Communities for regression, with benefits defined from prediction outcomes.The study evaluates welfare-based measures against accuracy and existing group- and individual-fairness definitions.
- Welfare as a Measure of Fairness: Welfare and Atkinson rankings are identical when models have similar mean benefits, as expected from the reported comparison.Figure 2 compares accuracy, social welfare, and Atkinson index across LogReg, NN, AdaBoost, KNN, and Tree.
- Impact on Model Parameters: More restrictive fairness constraints—larger τ and smaller α—produce higher accuracy loss.On COMPAS, increasing τ can also change the sex coefficient from negative to positive as the model seeks higher benefits.
- Trade-offs with Individual Notions: Higher welfare thresholds consistently reduce inequality, while higher risk aversion accelerates the decline; classification cannot support arbitrarily large thresholds.The effect on Dwork et al.’s constraints differs by task: classification violations eventually reach zero, whereas regression violations can rise again.
- Trade-offs with Statistical Notions: Group-discrimination measures generally decrease with τ, reaching zero for classification but not for regression’s positive residual difference.The statistical-fairness trends resemble those for Dwork et al.’s constraints and mean difference.
4 Summary and Future Directions
The paper connects welfare-based fairness in machine learning to cardinal social welfare and argues that convexity enables efficient integration into convex optimization. It reports evidence that welfare constraints often bound outcome inequality while identifying behavioral fairness theories as future work.
- Summary and Future Directions: Convex welfare measures can be imposed as constraints in convex loss-minimization programs and solved efficiently and exactly.The conclusion frames this as a connection between machine-learning fairness and cardinal social welfare in economics.
- Summary and Future Directions: Constraining the proposed measures often leads to bounded inequality in algorithmic outcomes.The paper presents this as evidence supporting a computationally feasible mechanism for bounding individual-level inequality.
- Summary and Future Directions: The authors plan to extend the normative framework to descriptive behavioral theories such as prospect theory.The stated goal is to compare human perceptions of fairness with normative prescriptions.
- Summary and Future Directions: Figure 4 shows group-discrimination patterns as τ changes across different α values, including declining negative residual differences and asymptotic positive residual differences.Its third panel compares patterns for Dwork et al.’s constraint violations and mean difference.
A Related Work (Continued)
Related work connects the paper’s welfare perspective to decision objectives, risk and inequality aversion, and empirical studies of preferences over risk and inequality.
- Decision Objectives: Immediate utility combines expected true positives with the expected cost of positive labels to represent a decision rule’s societal impact.This related approach maximizes immediate utility while satisfying existing fairness constraints.
- Risk and Inequality Aversion: Economics research links greater relative risk aversion with greater willingness to trade expected benefit for a more equal distribution behind a veil of ignorance.The cited literature studies the relationship between inequality aversion and risk aversion.
- Behavioral Evidence: Human-subject experiments have empirically measured perceptions and preferences concerning risk and inequality.Related work also connects standard social-welfare functions with reduced-form expressions involving inequality and mean income.
B Omitted Technical Material
The technical material supplies optimization conditions, establishes the relationship between generalized entropy and the Atkinson index, and characterizes how high welfare can be achieved through an intercept shift.
- Technical Proofs: For binary labels and predictions, arbitrary benefit values can be represented by a linear benefit function of the form c_yŷ + d_y.The proof derives coefficients by solving stationarity, dual feasibility, complementary slackness, and primal feasibility conditions.
- Generalized Entropy vs. Atkinson Index: For 0 < α < 1, generalized-entropy and Atkinson orderings are equivalent across any two benefit distributions.The relationship follows from the stated transformation between A_1−α(b) and G_α(b).
- The Role of the Intercept: When labels are linear in features, the optimal constrained solution adds an intercept shift τ′ = τ^(1/α) − 1 to the unconstrained model.This formalizes the mechanism by which high social welfare can be obtained through a large intercept.
C Omitted Experimental Details
The experiments use Crime and Communities for regression and COMPAS for classification, with convex optimization formulations and fairness measures spanning group and individual comparisons. Results examine how welfare constraints trade off against accuracy as τ and α vary.
- Data sets: The regression experiments use the Crime and Communities dataset, while classification experiments use the COMPAS recidivism dataset.
- Optimization program for classification: Classification optimization replaces a non-convex sign-function constraint with a convex alternative for efficient solution.
- Optimization program for classification: The benefit formulation constrains model parameters, including unit-length θ and a nonnegative benefit range determined by c = 5.
- Fairness Measures: The evaluation compares welfare-based fairness with demographic parity, false-positive and false-negative rate differences, mean difference, residual differences, and Dwork et al.’s individual fairness measure.
- Trade-offs with Accuracy: Stricter fairness constraints—larger τ and smaller α—produce higher loss of accuracy.