An Accounting Identity for Algorithmic Fairness

TL;DR

An accounting identity ties calibration-based fairness measures to a total unfairness budget under global calibration, showing that for binary Y the left-hand side $\delta_B + \delta_C$ equals $\text{MSE}(Z)[P(G=1|Y=1)-P(G=1|Y=0)]$, thereby making accuracy and fairness complements in binary prediction. The general non-binary extension replaces the budget with $\text{Cov}(\mathbb{E}[G|Y], Y-\mathbb{E}[Z|Y])$ and provides a bound $|\delta_B+\delta_C| \le \sqrt{\mathrm{Var}(\mathbb{E}[G|Y])}\sqrt{\text{MSE}(Z)}$, clarifying when binary impossibility results extend to regression. Empirically, the identity is validated on COMPAS, Adult, German Credit, and Bank Marketing, showing that fairness interventions tend to reallocate unfairness across dimensions and may raise the total budget if accuracy declines. The framework offers a principled lens on accuracy-fairness tradeoffs and indicates how additional outcome information in non-binary tasks can relax incompatibilities under certain conditions.

Abstract

We derive an accounting identity for predictive models that links accuracy with common fairness criteria. The identity shows that for globally calibrated models, the weighted sums of miscalibration within groups and error imbalance across groups is equal to a "total unfairness budget." For binary outcomes, this budget is the model's mean-squared error times the difference in group prevalence across outcome classes. The identity nests standard impossibility results as special cases, while also describing inherent tradeoffs when one or more fairness measures are not perfectly satisfied. The results suggest that accuracy and fairness are best viewed as complements in binary prediction tasks: increasing accuracy necessarily shrinks the total unfairness budget and vice-versa. Experiments on benchmark data confirm the theory and show that many fairness interventions largely substitute between fairness violations, and when they reduce accuracy they tend to expand the total unfairness budget. The results extend naturally to prediction tasks with non-binary outcomes, illustrating how additional outcome information can relax fairness incompatibilities and identifying conditions under which the binary-style impossibility does and does not extend to regression tasks.

Figures (5)

• Figure 1: COMPAS experiments. (a) Identity validation: $\delta_B + \delta_C$ vs MSE for four classifiers with number of varying features included. (b) Decomposition of unfairness into constituent parts ($\delta_C$, $\delta_B(0)$ weighted by $\omega(0)$ and $\delta_C(0)$ scaled by $\omega(1)$ as estimated under each model) by feature count for logistic regression. (c) Decomposition of total unfairness among Fairness methods from AIF360/FairLearn. (d) Decomposition of unfairness from penalized regression varying $\lambda$ on $\delta_B$.
• Figure 2: Adult experiments. (a) Identity validation: $\delta_B + \delta_C$ vs MSE for four classifiers with varying features. (b) Decomposition by feature count for logistic regression. (c) Fairness methods from AIF360/FairLearn. (d) Penalized regression varying $\lambda$ on $\delta_B(1)$.
• Figure 3: Bank experiments. (a) Identity validation: $\delta_B + \delta_C$ vs MSE for four classifiers with varying features. (b) Decomposition by feature count for logistic regression. (c) Fairness methods from AIF360/FairLearn. (d) Penalized regression varying $\lambda$ on $\delta_B(1)$.
• Figure 4: German experiments. (a) Identity validation: $\delta_B + \delta_C$ vs MSE for four classifiers with varying features. (b) Decomposition by feature count for logistic regression. (c) Fairness methods from AIF360/FairLearn. (d) Penalized regression varying $\lambda$ on $\delta_B(1)$.
• Figure 5: Unfairness decomposition for penalized regression approach, where the penalty is on $\delta_B(1)$ only. The height of each bar is the unfairness, split into $\delta_C$, $\delta_B(0)\omega(0)$, and $\delta_B(1)\omega(1)$, and the location of each bar corresponds to a different choice of lambda. The cross marks are the total unfairness predicted by equation \ref{['eq:accounting_bin']}.

Theorems & Definitions (18)

• Lemma 1
• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4: Sufficient conditions for pointwise fairness
• Proposition 5
• Proposition 6
• proof
• proof : Proof of Proposition \ref{['prop:accounting_general']}
• Lemma 2