Fairness parity

Fairness parity is an umbrella term: it means group-level outcomes or error rates are approximately equal under some stated criterion, within a documented tolerance. The criterion has to be named, because there are several and they measure different harms.

• Demographic parity — each group is selected at the same rate. Ignores the outcome entirely.
• Equal opportunity — among those who should be selected, each group is selected at the same rate. Equal true positive rates.
• Equalized odds — equal true positive rates and equal false positive rates.
• Predictive parity — among those selected, the same proportion in each group are actually positive. Equal precision.
• Calibration within groups — a score of 0.7 means a 70% chance in every group. Related to predictive parity but not the same condition, as the article on calibration sets out.

Some combinations conflict under specified conditions. The conditions include which criteria are exact, the operating point and whether predictive values lie strictly between zero and one. Unequal base rates alone neither determine the size of every gap nor make every pair of fairness criteria incompatible.

Read the denominators before comparing rates

Consider an illustrative screening system with 1,000 people in each of two groups. A positive label means a person has the condition being screened for; a positive decision means the system flags that person for follow-up. In group A, 200 people have the condition. In group B, 500 do. These base rates are 20% and 50%. This constructed example isolates the arithmetic; it is not evidence about any real population.

Suppose both groups have an 80% true positive rate and a 10% false positive rate. In A, the system flags 160 of the 200 people with the condition and 80 of the 800 people without it. It misses 40 positives. In B, it flags 400 of the 500 people with the condition and 50 of the 500 people without it, missing 100 positives. Equalized odds holds because the error rates conditional on the actual label agree across groups.

Now change the question to the people who were flagged. In A, 160 out of 240 flags are true positives, giving precision of 66.7%. In B, 400 out of 450 are true positives, giving precision of 88.9%. The precision gap appears because the flagged populations combine different numbers of actual positives and negatives. Equal error rates do not make those mixtures identical.

Selection rates differ too: 240/1,000 = 24% in A and 450/1,000 = 45% in B. Demographic parity asks about this fraction of the whole group. Equal opportunity asks about the fraction selected among actual positives. Predictive parity asks about the fraction actually positive among those selected. All three use counts from the same decisions, but answer different questions because their denominators differ.

Those differences change which harm a metric reveals. A false negative in screening can delay needed care; a false positive can impose unnecessary follow-up. In a beneficial allocation, failing to select an eligible person has a different meaning from falsely flagging someone for investigation. Specify what the positive label and positive action mean before choosing which rate to constrain.

Calibration is a property of probability scores: among members of a group assigned probability 0.7, approximately 70% should have the positive outcome. Selecting everyone above 0.7 pools multiple score values. The average score among selected people can differ across groups even when calibration holds, so calibration does not by itself guarantee equal precision after thresholding. Nor does calibration establish that labels or downstream actions are free from bias.

Why it is arithmetic and not a modeling failure

Chouldechova gives the relation directly. Let p be the fraction with the positive label, PPV the precision among selected people, FNR the missed-positive fraction, and FPR the selected fraction among actual negatives. For 0 < p < 1 and defined positive PPV:

\[ FPR = \frac{p}{1-p} \cdot \frac{1-PPV}{PPV} \cdot (1 - FNR) \]

For different prevalences, equal PPV strictly between 0 and 1 and equal positive TPR force different FPRs through this identity. Boundary cases need separate treatment: FPR=0 and TPR=0.5 in both groups give PPV=1 and equalized odds even when prevalence differs. PPV also requires a nonempty selected set. In the constructed example, equal TPR and FPR instead yield different PPVs. Both readings follow from the same relation.

Kleinberg, Mullainathan and Raghavan study a related but distinct set of score conditions: calibration within groups and balance of mean scores within each outcome class. Their exceptions include equal base rates or perfect prediction. Those score conditions should not be substituted for every combination of binary-decision criteria or approximate guardrails.

In the recidivism-risk setting discussed by Chouldechova, a predictive-parity claim and an error-rate critique concern different properties of the same instrument. Establishing one property does not refute a complaint about another. The identities explain a constraint on metric combinations; they do not settle which decisions are justified or whether the recorded outcomes are suitable labels.

What a parity claim has to say

A claim of fairness parity needs its terms attached. Specify the following so the requirement and its feasibility can be assessed:

• Which criterion, and which harm it is meant to bound — a false positive and a false negative are different injuries and the choice is an argument about which one to prioritize.
• Which groups, and against which reference — the highest-rate group, a designated majority, or the worst pairwise gap.
• The tolerance, stated in advance. “Within 3 percentage points” is an operational threshold, not evidence that three points are immaterial to the people in them.
• Sample sizes and uncertainty. Report the underlying counts and uncertainty. A small denominator can make an apparently large gap unstable; a wide interval is not evidence of equality.
• What was given up. Measure whether satisfying one criterion changes the others at the chosen operating points. Trade-offs are possible, but not universal across criteria and boundary cases.
• Label quality. Among the criteria above, all except demographic parity rely on the recorded outcome Y. If Y reflects unequal measurement or treatment, parity relative to Y can conceal harm. Inspect how labels are collected, including whether outcomes are observed only for people previously selected.

The four-fifths rule, precisely

The US Uniform Guidelines on Employee Selection Procedures use the four-fifths rule as a rule of thumb for identifying adverse impact: compare a group’s selection rate with that of the highest-rate group. A rate below 80% of the highest rate generally indicates adverse impact under this rule. For example, rates of 30% and 50% give a ratio of 0.60, not a 20% ratio. The 20-percentage-point difference is a separate measure.

The EEOC’s explanatory guidance also discusses cases in which smaller differences matter and cases in which small samples make a large ratio difference inconclusive. The ratio is one part of an adverse-impact analysis, not a general fairness certificate or an automatic legal finding. Its employment context should not be converted into a universal threshold for every model or application.

Working with the trade-off

Choose the criterion from the decision’s purpose and the harms at stake, then examine the other rates to see what that choice leaves unresolved. Report selection rate, TPR, FPR and precision by group at the operating point actually used, with their denominators. Compare feasible policies, including changes to data collection or the decision process, rather than only optimizing a single parity number. Agreement and conflict both matter, and group averages can hide disparities within intersecting subgroups.

Where the base rates differ because of measurement rather than reality — a group under-sampled, an outcome recorded less reliably — the gap may be an artifact worth fixing at the source, and that is a better use of effort than choosing between criteria. Where rates differ for reasons outside the model, assess the actual feasible policies and their consequences rather than assuming that every threshold rule must fail.

References

• Chouldechova, Fair prediction with disparate impact: A study of bias in recidivism prediction instruments, 2017
• Kleinberg, Mullainathan & Raghavan, Inherent Trade-Offs in the Fair Determination of Risk Scores, ITCS 2017
• EEOC, Questions and Answers on the Uniform Guidelines — adverse impact and the four-fifths rule