Revisiting Fairness Impossibility with Endogenous Behavior

Fairness in algorithmic decision-making is often defined in the predictive space, where predictive performance - used as a proxy for decision-maker (DM) utility - is traded off against prediction-based fairness notions, such as demographic parity or equality of opportunity. This perspective, however, ignores how predictions translate into decisions and ultimately into utilities and welfare for both DM and decision subjects (DS), as well as their allocation across social-salient groups. In this paper, we propose a multi-stakeholder framework for fair algorithmic decision-making grounded in welfare economics and distributive justice, explicitly modeling the utilities of both the DM and DS, and defining fairness via a social planner's utility that captures inequalities in DS utilities across groups under different justice-based fairness notions (e.g., Egalitarian, Rawlsian). We formulate fair decision-making as a post-hoc multi-objective optimization problem, characterizing the achievable performance-fairness trade-offs in the two-dimensional utility space of DM utility and the social planner's utility, under different decision policy classes (deterministic vs. stochastic, shared vs. group-specific). Using the proposed framework, we then identify conditions (in terms of the stakeholders' utilities) under which stochastic policies are more optimal than deterministic ones, and empirically demonstrate that simple stochastic policies can yield superior performance-fairness trade-offs by leveraging outcome uncertainty. Overall, we advocate a shift from prediction-centric fairness to a transparent, justice-based, multi-stakeholder approach that supports the collaborative design of decision-making policies.

As machine learning (ML) systems increasingly shape access to credit, jobs, and other opportunities, the fairness of algorithmic decisions has become a central concern. Yet it remains unclear when enforcing fairness constraints in these systems genuinely improves outcomes for affected groups or instead leads to "leveling down," making one or both groups worse off. We address this question in a unified, population-level (Bayes) framework for binary classification under prevalent group fairness notions. Our Bayes approach is distribution-free and algorithm-agnostic, isolating the intrinsic effect of fairness requirements from finite-sample noise and from training and intervention specifics. We analyze two deployment regimes for ML classifiers under common legal and governance constraints: attribute-aware decision-making (sensitive attributes available at decision time) and attribute-blind decision-making (sensitive attributes excluded from prediction). We show that, in the attribute-aware regime, fair ML necessarily (weakly) improves outcomes for the disadvantaged group and (weakly) worsens outcomes for the advantaged group. In contrast, in the attribute-blind regime, the impact of fairness is distribution-dependent: fairness can benefit or harm either group and may shift both groups' outcomes in the same direction, leading to either leveling up or leveling down. We characterize the conditions under which these patterns arise and highlight the role of "masked" candidates in driving them. Overall, our results provide structural guidance on when pursuing algorithmic fairness is likely to improve group outcomes and when it risks systemic leveling down, informing fair ML design and deployment choices.

Designing fair algorithmic decision systems requires balancing model performance with fairness toward affected individuals: More fairness might require sacrificing some performance and vice versa, yet the space of possible trade-offs is still poorly understood. We investigate fairness in binary prediction-based decision problems by conceptualizing decision making as a multi-objective optimization problem that simultaneously considers decision-maker utility and group fairness. We investigate the set of Pareto-optimal decision rules for arbitrary utility functions for decision maker, arbitrary population distributions, and a wide range of group fairness metrics. We find that the Pareto frontier consists of deterministic, group-specific threshold rules applied to individuals' success probability. This complements existing optimality theorems from literature which, for specific fairness constraints, posit lower-bound threshold rules only. However we also show that, depending on the used fairness metric, the Pareto frontier may include upper-bound threshold rules, thus preferring individuals with lower success probabilities. We show that the location of the Pareto frontier depends only on population characteristics, utility functions and fairness score, but not on the technical design of the algorithm - our findings hold for pre-, in-, and post-processing approaches alike. Our results generalize existing optimality theorems for fairness-constrained classification and extend them to generalized fairness metrics and fairness principles, and to partial fairness regimes. This paper connects formal fairness research with legal and ethical requirements to search for less discriminatory alternatives, offering a principled foundation for evaluating and comparing algorithmic decision systems.