← Back to explainers
View source on GitHub
Explainer

Risk-Based Pricing Gaps

A model can approve everyone and still treat groups differently.

Learn how to measure a fairness gap in a continuous quoted amount or rate when no one is declined. See German Credit Lending: models quote 7 to 13 percent less to under-30s, with intervals that still include zero at 200 test rows.

A model can approve everyone and still treat groups differently, by deciding how much to offer or how much to charge. Fairness metrics for yes-or-no decisions do not see it.

The One-Sentence Definition

A pricing gap is a difference between groups in the amount, rate or term a model quotes to applicants who were all approved, and it is measured as a difference in a continuous outcome instead of a difference in selection rates.

Why It Matters

Every metric in this repo's benchmark is built on a decision that is yes or no: Demographic Parity, Equalized Odds, Disparate Impact. A lender that approves every applicant passes all of them. It can still offer a smaller line of credit, a higher rate or a shorter term to one group, and that is a decision with the same consequences.

The measurement is simpler than it sounds. The quantity is a number, so the gap is a difference of means, and the uncertainty is a bootstrap interval (Bootstrap Confidence Intervals). The hard parts are choosing the quantity and remembering that the same gap can be fair or unfair depending on whether the difference in price tracks a real difference in risk.

Core Concept: A Regression, Then a Gap in Its Output

1. train a regression for the quoted amount (rate, limit, term) on the non-protected features
2. predict it for the test applicants
3. gap = mean(quoted | disadvantaged) - mean(quoted | advantaged), with a bootstrap interval
4. compare it with the gap in the amounts people actually received, and with the approve or deny gap

If the quoted gap is larger than the real-world gap, the model has widened a difference. If it is zero while the approval gap is not, price is not where the unfairness sits.

Concrete Example: German Credit Lending

German Credit Lending has a continuous credit_amount, which stands in for the amount a lender would offer. Everyone is treated as approved; the model predicts the amount from every other core column, with age left out, using the benchmark's split (200 test applicants, 69 of them under 30).

real amounts: under 30 3,277, 30 and over 3,790 (gap -513)
linear regression  quoted amount, under 30 minus 30 and over: -233 (-7%), 95% interval [-802, +308]
gradient boosting  quoted amount, under 30 minus 30 and over: -477 (-13%), 95% interval [-1,039, +94]

In the data, under-30s received 513 less on average (3,277 against 3,790). Both models quote a smaller amount to the under-30s too: 233 less (7%) from the linear regression and 477 less (13%) from gradient boosting. The intervals say how much to read into that: [-802, +308] and [-1,039, +94] both include zero. With 200 test rows the data cannot separate a real 13% gap from noise, which is the situation Permutation Tests for Fairness Gaps describes.

For comparison, the same audit's approve-or-deny parity gap in results/ (the current results, S2 strategy) is -0.0914 with an interval of -0.218 to +0.041. Both are the same direction and both are inconclusive on this sample, and neither alone would show the other.

Detection/Implementation Code

Run from the repo root (PYTHONPATH=. python3 explainers/code/risk-based-pricing-gaps.py; that file is generated from the block below).

import numpy as np
from sklearn.ensemble import GradientBoostingRegressor
from sklearn.linear_model import LinearRegression
from faircode.benchmark import load_audit        # same loader, encoding and split as the benchmark harness

data = load_audit("German Credit Lending")       # S2 feature set: core features, no protected column
train, test = data.idx_train, data.idx_test
young = data.masks["age"].to_numpy().astype(int)[test]     # 1 = under 30, the disadvantaged group
rng = np.random.default_rng(0)

# Everyone is "approved"; the model decides how much to lend. The quoted amount is predicted from the other
# columns (the one continuous amount a lender sets per applicant in this file).
X = data.X.drop(columns=["credit_amount"])
amount = data.df["credit_amount"].to_numpy(dtype=float)

def bootstrap_gap(values, group, n=2000):
    a, b = values[group == 1], values[group == 0]
    gaps = [rng.choice(a, len(a)).mean() - rng.choice(b, len(b)).mean() for _ in range(n)]
    return np.percentile(gaps, [2.5, 97.5])

print(f"real amounts: under 30 {amount[test][young == 1].mean():,.0f}, 30 and over {amount[test][young == 0].mean():,.0f} "
      f"(gap {amount[test][young == 1].mean() - amount[test][young == 0].mean():+,.0f})")
for name, model in (("linear regression", LinearRegression()),
                    ("gradient boosting", GradientBoostingRegressor(random_state=data.random_state))):
    quoted = model.fit(X.iloc[train], amount[train]).predict(X.iloc[test])
    gap = quoted[young == 1].mean() - quoted[young == 0].mean()
    lo, hi = bootstrap_gap(quoted, young)
    relative = gap / quoted[young == 0].mean()
    print(f"{name:18s} quoted amount, under 30 minus 30 and over: {gap:+,.0f} ({relative:+.0%}), 95% interval [{lo:+,.0f}, {hi:+,.0f}]")

Limitations

  1. A price gap is not automatically unfair. If the quoted amount tracks real differences in risk or need, a gap can be justified. The test is whether the group still gets a worse offer at the same risk, which needs a risk estimate that is itself not biased.
  2. The stand-in outcome is imperfect. credit_amount here is what applicants asked for and received, not an offer. Real pricing data would have the rate and term the lender set.
  3. Small samples hide the answer. Both intervals cross zero. See Disaggregated Evaluation for how fast intervals widen.
  4. One number per group loses the spread. A mean gap can hide a gap that only exists at the top or bottom of the distribution, so quantiles or a full comparison of distributions may be needed.
  5. There is no standard threshold. The four-fifths rule is for selection rates. For prices, the standard is a legal and policy question to answer for your setting, from primary sources.

Further Reading

Part of The Fair Code Project - exposing and fixing algorithmic bias with real data and open code.