Individual and Counterfactual Fairness

A lender can equalise approval rates across two groups by approving the strongest applicants in one group and a random draw from the other. Demographic parity holds exactly. Two applicants with identical files, one from each group, still face different odds, and a rejected applicant in the second group was never assessed at all. Dwork and colleagues used examples like this to argue that group criteria, the subject of group fairness criteria and why they conflict, can be satisfied while individuals are treated arbitrarily (Dwork et al., 2012, Fairness Through Awareness, ITCS, arXiv:1104.3913).

Similar individuals, similar outcomes

Their definition works on a randomised classifier \(M: V \to \Delta(A)\) that maps each individual \(x\) in a population \(V\) to a probability distribution over outcomes \(A\). Given a task-specific metric \(d(x, y)\) on individuals and a distance \(D\) between distributions, \(M\) is individually fair if it is Lipschitz:

With \(D\) the total variation distance and two outcomes, this says the approval probabilities of \(x\) and \(y\) may differ by at most \(d(x, y)\). If two applicants sit at distance 0.05, their approval probabilities cannot differ by more than five points. Finding the most accurate fair classifier is a linear program: minimise expected loss subject to one constraint per pair. On 10,000 individuals that is about \(5 \times 10^7\) pairwise constraints, which is why practical methods relax or sample them.

The paper's structural result connects the individual view back to groups. For groups with distributions \(S\) and \(T\), the largest statistical-parity gap any Lipschitz classifier can produce is at most the earthmover distance \(d_{\mathrm{EM}}(S, T)\) between the groups under \(d\), with equality when \(d \le 1\). So individual fairness implies approximate demographic parity exactly when the metric says the two groups are close. If the metric says they differ, individual fairness permits the disparity.

The metric is the whole problem

Every value judgement has moved into \(d\). A metric that measures similarity on credit history treats applicants whose histories were shaped by past exclusion as genuinely different; a metric that discounts those features does not. Dwork and colleagues left the metric as a hypothetical input supplied by society or a regulator, and no general procedure for building one has emerged since.

Two lines of work try. Ilvento learns an approximate metric from a human arbiter's answers to a limited number of similarity queries, assuming the arbiter is free of explicit bias and has enough domain knowledge (Ilvento, 2020, Metric Learning for Individual Fairness, FORC, arXiv:1906.00250). Yurochkin, Bower and Sun instead learn a sensitive subspace of directions, such as those correlated with gender, along which the model's performance should be invariant, and enforce it with distributionally robust training (Yurochkin et al., 2020, ICLR, arXiv:1907.00020). Both make the problem tractable by assuming someone can say which differences are irrelevant, which is the question that was hard in the first place.

Counterfactual fairness

Kusner, Loftus, Russell and Silva take a causal route (Kusner et al., 2017, Counterfactual Fairness, NeurIPS, arXiv:1703.06856). Given a structural causal model with protected attribute \(A\), observed features \(X\) and background variables \(U\), a predictor \(\hat{Y}\) is counterfactually fair if

for every \(y\) and every alternative value \(a'\): holding the individual's background fixed, intervening on their protected attribute leaves the prediction's distribution unchanged. Their Lemma 1 gives the simplest route: a predictor that uses only non-descendants of \(A\) in the causal graph is counterfactually fair. Because GPA and test scores are typically descendants of race and sex, the practical method infers latent variables, such as a student's underlying knowledge, and predicts from those.

On a Law School Admission Council survey of 21,790 students at 163 law schools, predicting first-year grades with logistic regression, the full model using race and sex reached RMSE 0.873, the model that merely dropped them 0.894, and the two counterfactually fair models 0.929 and 0.918. The unaware model is not fair under the paper's causal assumptions: dropping the attribute leaves its effect flowing through GPA and LSAT. The accuracy cost is real but modest, and it buys a property that fairness through unawareness only imitates.

When it breaks

Counterfactuals on social categories may be incoherent. Kasirzadeh and Smart argue that race and gender are not variables that can be toggled with everything upstream held fixed, since they are constituted partly by the very social position the model conditions on (Kasirzadeh & Smart, 2021, The Use and Misuse of Counterfactuals in Ethical Machine Learning, FAccT, arXiv:2102.05085). On that view "same person, different race" has no well-defined answer. The modeller's reply is that the causal model is a device for tracing which pathways from the attribute are acceptable, not a metaphysical claim about persons; which reading is right remains contested.

The causal graph is an assumption with no empirical check. Which features descend from \(A\) decides what the predictor may use, and the data cannot confirm the graph. Two analysts with different graphs produce different "fair" models from identical data.

Similar treatment is not sufficient. Fleisher notes that a Lipschitz classifier that treats everyone similarly badly is individually fair, and that a metric learned from human judgement can encode the judges' implicit bias (Fleisher, 2021, What's Fair about Individual Fairness?, AIES).

Neither criterion is auditable from outputs alone. Checking a Lipschitz condition needs the metric; checking counterfactual fairness needs the causal model. An external auditor who has neither can test group disparities but not these properties.