# An Accounting Identity for Algorithmic Fairness

arXiv: https://arxiv.org/abs/2601.20217 · HTML: https://mlsanity.com/paper/2601.20217/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 20 references that are in the map.
- First posted on arXiv 28 Jan 2026.

## Builds on (most cited first, 13 of 13)

- [Equality of Opportunity in Supervised Learning](https://mlsanity.com/paper/1610.02413/) (2016)
- [A Survey on Bias and Fairness in Machine Learning](https://mlsanity.com/paper/1908.09635/) (2019)
- [Fair prediction with disparate impact: A study of bias in recidivism prediction instruments](https://mlsanity.com/paper/1610.07524/) (2016)
- [A Reductions Approach to Fair Classification](https://mlsanity.com/paper/1803.02453/) (2018)
- [Inherent Trade-Offs in the Fair Determination of Risk Scores](https://mlsanity.com/paper/1609.05807/) (2016)
- [On Fairness and Calibration](https://mlsanity.com/paper/1709.02012/) (2017)
- [AI Fairness 360: An Extensible Toolkit for Detecting, Understanding, and Mitigating Unwanted Algorithmic Bias](https://mlsanity.com/paper/1810.01943/) (2018)
- [A Convex Framework for Fair Regression](https://mlsanity.com/paper/1706.02409/) (2017)
- [Algorithmic Fairness](https://mlsanity.com/paper/2001.09784/) (2020)
- [The Measure and Mismeasure of Fairness](https://mlsanity.com/paper/1808.00023/) (2018)
- [Fairlearn: Assessing and Improving Fairness of AI Systems](https://mlsanity.com/paper/2303.16626/) (2023)
- [The Possibility of Fairness: Revisiting the Impossibility Theorem in Practice](https://mlsanity.com/paper/2302.06347/) (2023)
- [A Possibility in Algorithmic Fairness: Can Calibration and Equal Error Rates Be Reconciled?](https://mlsanity.com/paper/2002.07676/) (2020)

## Abstract (arXiv)

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.