You Cannot Have All Three: COMPAS, Calibration and the Impossibility Theorems of Fair Classification

Brisha Borden, 18, briefly took an unlocked child's bicycle in Florida. Vernon Prater, 41, a former armed robber, shoplifted tools. A commercial risk tool called COMPAS scored Borden 8 and Prater 3. Two years later Borden had no new charges and Prater was serving eight years for a warehouse burglary. That pairing opened ProPublica's "Machine Bias" on 23 May 2016, which reported that among defendants who did not reoffend, 44.9% of Black defendants had been labelled higher risk against 23.5% of white defendants (Angwin et al., 2016; Larson et al., 2016).

Northpointe, the vendor, replied with a 37-page rebuttal: COMPAS was fair by the standard its field had always used, because a given score carried the same probability of reoffending whatever the defendant's race (Dieterich, Mendoza & Brennan, 2016). Within months, researchers proved that neither side had made an arithmetic mistake. When two groups reoffend at different rates, a score that means the same thing for both must make different kinds of errors for each. The argument was never about the numbers. It was about which number should count.

Why this matters: Every scoring model deployed across groups with different base rates sits inside the same theorem. Choosing a fairness metric is choosing which error to distribute unequally, and the choice is usually made by default. Knowing the identity turns that default into a decision someone can defend.

TL;DR

• In ProPublica's Broward tables the COMPAS false positive rate was 44.85% for Black defendants and 23.45% for white defendants, while PPV was 0.63 and 0.59. Critique and defence read the same confusion matrices.
• Chouldechova's identity, \(\mathrm{FPR} = \frac{p}{1-p}\cdot\frac{1-\mathrm{PPV}}{\mathrm{PPV}}\cdot(1-\mathrm{FNR})\), is pure algebra. With PPV and FNR held equal, the FPR ratio equals the prevalence odds ratio; for COMPAS (51% vs 39%) that alone forces a 1.63x gap.
• Kleinberg, Mullainathan and Raghavan proved calibration plus balance for both classes requires equal base rates or perfect prediction, and approximate satisfaction requires approximately degenerate data.
• Calibration does not guarantee equal precision. In the worked example a perfectly calibrated score gives PPV 0.77 in one group and 0.67 in the other.
• Imposing statistical parity on Broward detention was estimated to detain 17% low-risk people and raise violent recidivism among the released by about 9% (Corbett-Davies et al., 2017).
• COMPAS scored 65.2% accuracy against 62.8% for untrained crowd workers on the same cases, and a two-feature regression matched it (Dressel & Farid, 2018).

At a Glance

flowchart LR
    BR["Base rates differ: 51% vs 39%"] --> S["Risk score s(x)"]
    S --> T["Threshold: high risk if score above t"]
    T --> CM["Confusion matrix per group"]
    CM --> SUF["Sufficiency: equal PPV, calibration"]
    CM --> SEP["Separation: equal FPR and FNR"]
    CM --> IND["Independence: equal selection rate"]
    SUF -.->|"identity forbids both"| SEP
    SEP -.->|"forbids unless p equal"| IND
    SUF -.->|"forbids unless p equal"| IND
    classDef blue fill:#1e40af,stroke:#3b82f6,stroke-width:1px,color:#fff
    classDef purple fill:#6d28d9,stroke:#a78bfa,stroke-width:1px,color:#fff
    classDef teal fill:#0e7490,stroke:#22d3ee,stroke-width:1px,color:#fff
    classDef amber fill:#b45309,stroke:#fbbf24,stroke-width:1px,color:#fff
    class BR amber
    class S,T purple
    class CM blue
    class SUF,SEP,IND teal

One score, one threshold, one confusion matrix per group, and three families of criteria reading different cells. Once base rates differ, any two families pull against each other, and an imperfect model cannot satisfy them together.

Before the Theorems: A Dispute About One Spreadsheet

Actuarial risk assessment long predates machine learning, and its fairness standard came from psychological testing, where an instrument whose scores predict the same outcome regardless of group is "free from predictive bias" (Chouldechova, 2017, Big Data). US employment law had settled on a different test, the 1978 Uniform Guidelines' "four-fifths rule", which looks only at selection rates (29 CFR 1607.4(D)).

timeline
    title From one investigation to a field of theorems
    1978 : Uniform Guidelines adopt the four-fifths rule
    May 2016 : ProPublica publishes Machine Bias
             : FPR 44.9% vs 23.5% on 7,214 Broward defendants
    Jul 2016 : Northpointe rebuttal argues predictive parity
             : Wisconsin Supreme Court decides State v. Loomis
    Sep 2016 : Kleinberg et al. prove the calibration trade-off
    Oct 2016 : Chouldechova circulates the PPV-FPR-FNR identity
             : Hardt et al. define equalized odds
    2017 : Corbett-Davies et al. price fairness constraints
         : Pleiss et al. show calibration tolerates one error constraint
    2018 : Dressel and Farid match COMPAS with crowd workers
    2023 : Measure and Mismeasure of Fairness appears in JMLR
         : Barocas, Hardt and Narayanan textbook published

ProPublica obtained scores for all 18,610 people assessed in Broward County in 2013 and 2014, kept pretrial assessments, and checked public records for new charges within two years. Following Northpointe's practitioner guide, they collapsed "medium" and "high" into "higher risk" (Larson et al., 2016). Their tables for the 7,214 defendants with full follow-up are reproduced below.

Northpointe claimed accuracy equity (near-identical ROC curves) and predictive parity (similar reoffence rates among those labelled high risk), and charged that ProPublica had "neglected to consider the base rate" (Larson & Angwin, 2016). That was the correct diagnosis pointed the wrong way: base rates were precisely why both sets of numbers could be true.

The theory followed within weeks: Kleinberg et al. on 19 September 2016, Hardt et al. on 7 October, and Chouldechova's workshop version on 24 October (arXiv:1610.07524). By 30 December ProPublica was running the headline "Bias in Criminal Risk Scores Is Mathematically Inevitable, Researchers Say" (ProPublica, 2016).

[IMAGE: Two mirrored cards, left headed "ProPublica, May 2016" showing FPR 44.9% vs 23.5% as bars, right headed "Northpointe, July 2016" showing PPV 0.63 vs 0.59 as near-equal bars, both drawn from one shared confusion-matrix icon in the middle. Caption: "The same four cells per group, read row-wise by one side and column-wise by the other."]

How the Impossibility Actually Works

Three families, three conditional independences

Write \(A\) for group, \(Y \in \{0,1\}\) for the outcome, \(R\) for a score and \(\hat{Y} = \mathbb{1}[R > t]\) for the decision. Barocas, Hardt and Narayanan sort nearly every group criterion into three conditional independence statements (Barocas, Hardt & Narayanan, 2023):

Independence equalises selection rates; the four-fifths rule is a tolerance on it. Separation says that among people with the same true outcome, scores are distributed alike across groups; for a binary decision that is equal FPR and FNR, Hardt et al.'s equalized odds, with equal opportunity equalising only the true positive rate. Sufficiency says that once you know the score, group adds nothing; for a continuous score that is calibration within groups, for a binary decision equal PPV and NPV.

Separation conditions on the truth; sufficiency conditions on the prediction. Bayes' rule links the two through the base rate \(p_a = P(Y=1 \mid A=a)\). When it differs by group, both cannot be group-invariant.

Chouldechova's identity, derived

Take one group with \(N\) people, prevalence \(p\), and cells \(TP\), \(FP\), \(FN\). By definition \(\mathrm{FPR} = FP/((1-p)N)\), \(\mathrm{FNR} = FN/(pN)\) and \(\mathrm{PPV} = TP/(TP+FP)\). Two rearrangements do the work. PPV gives false positives per true positive,

and FNR gives the true positives as the detected share of positives, \(TP = (1-\mathrm{FNR})\,pN\). Substituting both into the FPR definition:

This is equation (2.6) in Chouldechova's paper, from which she concludes that if PPV is equal across groups but prevalence differs, the instrument "cannot achieve equal false positive and false negative rates across those groups" (Chouldechova, 2017). The derivation mentions no model or feature. It holds for any confusion matrix, including a judge's.

It is most useful as a ratio between two groups:

Insist on predictive parity and equal FNR (both terms 1), and the FPR ratio is pinned to the base-rate odds ratio. No training trick moves it; only equal prevalence or a perfect classifier escapes.

Kleinberg's version: two lines that meet only at perfection

Kleinberg, Mullainathan and Raghavan work with continuous scores and three conditions: (A) calibration within groups; (B) balance for the negative class, meaning true negatives receive the same average score in both groups; (C) balance for the positive class, the same for true positives. Their Theorem 1.1: a score satisfying all three implies either perfect prediction or equal base rates (Kleinberg et al., 2016).

Calibration makes each bin's scores sum to its expected positives, so a group's total score equals its number of positives, \(\mu_t = p_t N_t\). Let \(x\) and \(y\) be the average scores of negatives and positives, shared across groups by balance. Then for each group

Each group is a line in the \((x, y)\) plane. Equal base rates make the lines coincide. Unequal base rates give different slopes, and the lines meet only at \((0,1)\): negatives scored 0, positives scored 1, perfect prediction. The authors also prove that satisfying all three conditions approximately forces the data to be approximately degenerate, so "nearly fair on all three" is no loophole.

[IMAGE: Plot with average true-negative score on the x-axis and average true-positive score on the y-axis. Two lines through (0, 1) with different slopes, for base rates 0.5 and 0.3; dots mark the worked example's scores at (0.244, 0.756) and (0.174, 0.593). Caption: "Calibration pins each group to its own line. Balance needs one shared point; the only one is perfect prediction."]

The pairwise version, and why optimal scores violate separation

The textbook generalises this pairwise. If \(A\) and \(Y\) are dependent: sufficiency and independence cannot both hold; for binary \(Y\) and an informative score, independence and separation cannot both hold; and if every cell of the joint distribution of \((A, R, Y)\) has positive probability, separation and sufficiency cannot both hold, because \(A \perp R \mid Y\) and \(A \perp Y \mid R\) together imply \(A \perp Y\) (Barocas, Hardt & Narayanan, 2023, ch. 3).

Then the consequence practitioners miss. Sufficiency holds for the Bayes-optimal score \(P(Y=1 \mid X, A)\) by construction, so the best possible predictor generally violates separation. Unequal error rates are not evidence of bad training. They are what good training produces when base rates differ.

Calibration is not predictive parity

A well-calibrated score can still fail predictive parity, because PPV above the cut depends on how each group's scores are distributed above it (Chouldechova, 2017). The worked example shows a ten-point PPV gap from a perfectly calibrated score.

Repairing error rates, and the price

Hardt, Price and Srebro gave the constructive answer. Plot each group's ROC curve. Group-specific thresholds, randomised between two thresholds where needed, reach any point in the convex hull under that curve. Equalized odds needs one (FPR, TPR) point inside every group's hull, and the loss-minimising choice is a linear program in four variables (Hardt et al., 2016).

Three costs follow. The better-separated group is pulled down toward the worse group's curve. Randomisation means identical people can receive different decisions by coin flip. And group-specific thresholds use the protected attribute at decision time, which some legal regimes forbid. Pleiss et al. added that keeping calibration permits only a single error constraint, such as equal FNR, and any algorithm achieving it is no better than randomly withholding predictive information from part of one group (Pleiss et al., 2017).

One threshold, and inframarginal statistics

Corbett-Davies et al. recast fairness as constrained optimisation. Maximising public safety at a fixed detention rate needs one threshold on calibrated risk, holding everyone to the same standard; each popular fairness constraint instead produces race-specific thresholds (Corbett-Davies et al., 2017). The 2023 follow-up names the root cause inframarginality: error rates and PPV average over everyone above or below the cut, not the person at the margin, so when risk distributions differ they are "often poor proxies for individual equity or social well-being" (Corbett-Davies et al., 2023, JMLR). It argues further that both families of formal definitions typically yield strongly Pareto-dominated policies.

This is the field's live disagreement. Hardt and colleagues treat separation as a legitimate target that shifts the cost of poor classification onto the decision maker; Corbett-Davies and colleagues treat parity constraints as distortions to be judged by consequences. Both accept the theorems.

Seeing It in Motion

The same data, read in two directions:

sequenceDiagram
    participant PP as ProPublica
    participant NP as Northpointe
    participant TH as Researchers
    participant CT as Courts
    PP->>PP: Match 7,214 scores to 2-year rearrests
    PP->>CT: FPR 44.9% vs 23.5%, error rates unequal
    NP->>PP: PPV 0.63 vs 0.59, predictive parity holds
    NP->>PP: Base rates ignored, cut point wrong
    PP->>NP: Disparity persists at other cut points
    CT->>CT: Loomis requires written cautions
    TH->>TH: Prove calibration and balance incompatible
    TH->>CT: Unequal base rates force a trade-off
    Note over PP,CT: Both readings true, the choice is normative

The theorem hands back a routing decision, not an optimisation: which error harms whom?

flowchart TB
    Q1{"Do the labels encode bias?"}
    Q1 -->|"yes, arrest not crime"| IND["Independence, and fix the label"]
    Q1 -->|"no, outcome trusted"| Q2{"Who consumes the output?"}
    Q2 -->|"human reading a probability"| SUF["Sufficiency: calibrate by group"]
    Q2 -->|"automatic cut-off"| Q3{"Which error is the harm?"}
    Q3 -->|"false positive burdens"| FPR["Equalise FPR, accept PPV gap"]
    Q3 -->|"false negative denies benefit"| TPR["Equal opportunity"]
    Q3 -->|"both, attribute usable"| EO["Equalized odds post-processing"]
    SUF --> AUD["Report every group metric anyway"]
    FPR --> AUD
    TPR --> AUD
    EO --> AUD
    IND --> AUD
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    classDef amber fill:#b45309,stroke:#fbbf24,stroke-width:1px,color:#fff
    classDef slate fill:#334155,stroke:#64748b,stroke-width:1px,color:#e2e8f0
    class Q1,Q2,Q3 amber
    class IND,SUF,FPR,TPR,EO teal
    class AUD slate

The last box matters: the unchosen metrics will move, and the people affected deserve to see where.

[IMAGE: Small multiples of FPR, FNR and PPV by group plotted against COMPAS decile cut points 1 to 10. FPR and FNR curves stay apart at every cut; PPV curves nearly overlap from cut point 4 up. Caption: "Moving the threshold moves every metric, but no cut point closes the error-rate gap while keeping PPV equal."]

By the Numbers

The COMPAS figures are ProPublica's published contingency cells, recomputed. The identity row evaluates Chouldechova's formula on prevalence, PPV and FNR, and reproduces the observed FPR to four decimals, as algebra must.

Sources: ProPublica (Larson et al., 2016); Chouldechova, 2017 (reports 51% vs 39%); Dressel & Farid, 2018; Corbett-Davies et al., 2017; Hardt et al., 2016. Prevalence, selection, accuracy and identity rows are computed here. Corbett-Davies figures are model-based estimates. The four-fifths rule governs US employment, not detention.

Decomposing the observed ratio with the identity is revealing. The Black-to-white FPR ratio is 1.91. Base rates contribute a factor of 1.63. The precision term contributes 0.85: COMPAS's slightly higher PPV for Black defendants narrows the gap. The recall term contributes 1.38: the cut captures 72% of Black recidivists but only 52% of white ones, and that uneven reach pushes the ratio beyond what base rates dictate.

A Concrete Example

Two groups of 1,000. Group A reoffends at 50%, group B at 30%. The score has three bins, 0.1, 0.5 and 0.9, and is perfectly calibrated in both groups. Anyone at 0.5 or above is flagged.

Step 1. The bins.

Calibration checks by hand: \(0.1 \times 400 = 40\), \(0.5 \times 200 = 100\), \(0.9 \times 400 = 360\), and likewise for B. Group B simply has more people in the bottom bin.

Step 2. Confusion matrices at the threshold. A flags 600: \(TP = 460\), \(FP = 140\), \(FN = 40\), \(TN = 360\). B flags 350: \(TP = 235\), \(FP = 115\), \(FN = 65\), \(TN = 585\).

Step 3. The metrics.

Accuracy, NPV and calibration match, yet non-reoffenders in A are flagged 1.7 times as often, and PPV differs by ten points because B's flagged set leans on the 0.5 bin.

Step 4. Check the identity. For A: \(1 \times \frac{140}{460} \times 0.92 = 0.3043 \times 0.92 = 0.280\). For B: \(\frac{0.3}{0.7}\cdot\frac{115}{235}\cdot\frac{235}{300} = 0.4286 \times 0.4894 \times 0.7833 = 0.1643\). Both match the counts.

Step 5. Check Kleinberg's balance. Average true-negative score in A: \((360 \times 0.1 + 100 \times 0.5 + 40 \times 0.9)/500 = 0.244\). In B: \((585 \times 0.1 + 100 \times 0.5 + 15 \times 0.9)/700 = 0.174\). B's line predicts \(x = \frac{0.3}{0.7}(1 - 0.593) = 0.174\), where \(0.593 = 178/300\) is B's average positive score. Balance fails by 0.07, as the theorem says it must.

Step 6. What would equal PPV cost? Give B the same PPV (0.767) and FNR (0.08) as A. The identity fixes B's FPR at \(0.4286 \times (0.233/0.767) \times 0.92 = 0.120\), under half of A's 0.280. Predictive parity plus equal recall guarantees unequal false positive rates.

Step 7. Force equal FPR. B needs 196 false positives, 81 more. The only unflagged people are in the 0.1 bin, so flag a random \(81/585 = 13.8\%\) of it: 90 people, 9 of them positive. B becomes \(TP = 244\), \(FP = 196\), \(FN = 56\), \(TN = 504\).

FPRs match; three things got worse. A flag now means different things by group. FNR is still unequal, so this is equal FPR, not equalized odds. And 90 people at 10% risk were flagged by lottery, the cost Pleiss et al. identified and Corbett-Davies et al. measured as detained low-risk defendants.

[IMAGE: A three-term waterfall bar for COMPAS: start at 1.00, multiply by 1.63 (base-rate odds), by 0.85 (precision term), by 1.38 (recall term), ending at the observed FPR ratio 1.91. Caption: "Where the 1.91x false positive gap comes from."]

[IMAGE: Stacked horizontal bars per group, one per score bin, coloured positives vs negatives, cut by a vertical line at 0.5; a second panel shows a speckled 13.8% slice of group B's 0.1 bin moved across the line. Caption: "Calibrated bins, one threshold, unequal FPR; equalising FPR means flagging a random slice of the lowest-risk bin."]

Where It Breaks

The label is a proxy, and the proxy differs by group

Every quantity here treats \(Y\) as truth. In COMPAS, \(Y\) is a new charge, not a new crime. If one group is policed more heavily, its measured base rate is inflated, and the "unequal base rates" driving the theorem are partly measurement. Bao et al. document measurement errors and discretionary disparities across pretrial datasets and argue that the ProPublica COMPAS data is a poor benchmark for fairness methods (Bao et al., 2021). Friedler et al. formalise this: the right mechanism depends on how observed data maps to an unobservable "construct space" (Friedler et al., 2016). The theorem is exact about your labels and silent about whether they are the right ones.

Calibration can be gamed

Corbett-Davies et al. construct a malicious rule: split one group into 10%, 20% and 40% risk tiers using prior arrests, but label every member of the other group an "average" 20%. Both groups' 20% scores reoffend 20% of the time, so the score is calibrated, yet under a 30% threshold nobody in the second group is ever detained (Corbett-Davies et al., 2017). Discarding information about one group preserves calibration and discriminates. An audit that checks only calibration misses it.

Threshold choice is not neutral

ProPublica cut at "medium or higher"; critics argued for "high" only. Chouldechova shows error-rate imbalance persists across all cut points, and ProPublica found the FPR disparity "starker" at Northpointe's suggested ones. Every fairness number is conditional on a threshold the reader rarely chooses.

Small cells and shifting populations

An FPR estimated from 60 negatives at a true rate of 30% has a standard error near 6 points, enough to manufacture or hide a four-fifths violation, and intersectional subgroups quickly shrink to that size. Calibration is a property of a score and a population: when policing, lending policy or the applicant pool changes, a score calibrated on 2013 data stops being calibrated, and group-specific thresholds solved on old ROC curves go stale with it.

Unawareness does not escape the theorem

Dropping the attribute leaves base rates unchanged, so the identity applies to the resulting matrices as before. It can also miscalibrate the score for a group whose outcomes genuinely differ: in ProPublica's data, women rated high risk recidivated at 47.5% against 61.2% for men rated high risk (Larson et al., 2016). Excluding a predictive attribute can harm the group it was meant to protect (Corbett-Davies et al., 2023).

Alternative Designs

The rows are not ranked. Each design is optimal for a different specification of harm, and that specification is what to argue about.

How It Is Used in Practice

Courts took the least technical route. In State v. Loomis, decided 13 July 2016, the Wisconsin Supreme Court allowed COMPAS at sentencing but required a written advisement in presentence reports covering the tool's proprietary nature, that scores are based on group data, and that studies have questioned whether it disproportionately classifies minority offenders as higher risk (State v. Loomis, 2016 WI 68). It is a disclosure remedy that leaves the trade-off to each judge.

US employment screening defaults to independence because the law does: a selection rate below four-fifths of the highest group's "will generally be regarded" as evidence of adverse impact (29 CFR 1607.4(D)). A hiring model audited only on that ratio will usually fail separation and sufficiency when applicant base rates differ.

Lending shows the price explicitly. In Hardt et al.'s FICO study on 301,536 TransUnion scores, at the 82% profitability line a race-blind threshold earned 99.3% of maximum profit, equal opportunity 92.8%, equalized odds 80.2% and demographic parity 69.8%, so equal opportunity cost under a quarter of what demographic parity did (Hardt et al., 2016). Lenders barred from using the attribute at decision time cannot implement group thresholds at all.

The operational lesson is reporting discipline: compute all three families per group, at production thresholds, with confidence intervals; record which criterion governs and why; recalibrate when the population shifts. And heed Dressel and Farid: years of dispute concerned a tool no better than a two-feature regression, so whether to automate, and whether the label is right, deserve equal scrutiny.

[IMAGE: One-page fairness report: groups as rows; columns for selection rate, FPR, FNR, PPV, calibration slope and four-fifths ratio, each with a confidence bar; one column highlighted "governing criterion" and a text box "harm rationale". Caption: "Report everything, govern by one criterion, and write down why."]

Insights Worth Remembering

1. The COMPAS dispute was about the direction of conditioning, not the data. ProPublica conditioned on outcomes; Northpointe on predictions. Bayes' rule links them through base rates, which differed.

2. Chouldechova's identity is accounting, not statistics. It holds for any confusion matrix, human or machine, so no better model escapes it.

3. Unequal error rates are what a good calibrated model produces. Sufficiency holds for the Bayes-optimal score, so disparate FPR under unequal base rates is not a training bug.

4. Calibration is necessary for honest scores and far from sufficient for fair ones. It coexists with ten-point PPV gaps and survives deliberate discarding of information about one group.

5. Every separation repair spends something visible. Equal FPR in the worked example cost a 13.8% lottery in the lowest-risk bin, PPV down from 0.67 to 0.56, and seven points of accuracy.

6. Decompose the gap before arguing about it. In COMPAS, base rates explain 1.63x; the threshold's uneven recall (72% vs 52%) lifts it to 1.91x. The two parts call for different remedies.

Open Questions

How much of the base-rate gap is measurement? That arrest labels reflect policing intensity is established; its size in any given jurisdiction is largely unmeasured, and it decides whether the impossibility reflects real differences or the label.

Can causal criteria replace group statistics? Counterfactual definitions avoid some pathologies, but the Mismeasure paper argues they too typically yield Pareto-dominated policies, and they need causal models practitioners rarely have.

What should approximate targets be? Kleinberg et al. show near-satisfaction of all three conditions needs near-degenerate data, yet deployed tolerances ("within 5 points", four-fifths) have no theoretical grounding tied to sample size or harm.

Do fairness constraints survive feedback loops? Detention changes who can be rearrested, and so next year's labels. Evidence across repeated deployment cycles is thin.

Sources and Further Reading

1. Angwin, J., Larson, J., Mattu, S., & Kirchner, L. (2016, May 23). "Machine Bias." ProPublica. propublica.org
2. Larson, J., Mattu, S., Kirchner, L., & Angwin, J. (2016, May 23). "How We Analyzed the COMPAS Recidivism Algorithm." ProPublica. propublica.org
3. Dieterich, W., Mendoza, C., & Brennan, T. (2016). "COMPAS Risk Scales: Demonstrating Accuracy Equity and Predictive Parity." Northpointe Inc. Semantic Scholar
4. Larson, J., & Angwin, J. (2016, July 29). "Technical Response to Northpointe." ProPublica. propublica.org
5. ProPublica (2016, December 30). "Bias in Criminal Risk Scores Is Mathematically Inevitable, Researchers Say." propublica.org
6. Kleinberg, J., Mullainathan, S., & Raghavan, M. (2016). "Inherent Trade-Offs in the Fair Determination of Risk Scores." ITCS 2017. arXiv:1609.05807
7. Chouldechova, A. (2017). "Fair Prediction with Disparate Impact: A Study of Bias in Recidivism Prediction Instruments." Big Data, 5(2), 153-163. arXiv:1703.00056; workshop version arXiv:1610.07524
8. Hardt, M., Price, E., & Srebro, N. (2016). "Equality of Opportunity in Supervised Learning." NeurIPS 2016. arXiv:1610.02413
9. Corbett-Davies, S., Pierson, E., Feller, A., Goel, S., & Huq, A. (2017). "Algorithmic Decision Making and the Cost of Fairness." KDD 2017. arXiv:1701.08230
10. Pleiss, G., Raghavan, M., Wu, F., Kleinberg, J., & Weinberger, K. Q. (2017). "On Fairness and Calibration." NIPS 2017. arXiv:1709.02012
11. Corbett-Davies, S., Gaebler, J. D., Nilforoshan, H., Shroff, R., & Goel, S. (2023). "The Measure and Mismeasure of Fairness." JMLR, 24(312), 1-117. arXiv:1808.00023
12. Barocas, S., Hardt, M., & Narayanan, A. (2023). Fairness and Machine Learning: Limitations and Opportunities. MIT Press. fairmlbook.org
13. Friedler, S. A., Scheidegger, C., & Venkatasubramanian, S. (2016). "On the (im)possibility of fairness." arXiv:1609.07236
14. Dressel, J., & Farid, H. (2018). "The accuracy, fairness, and limits of predicting recidivism." Science Advances, 4(1), eaao5580. doi:10.1126/sciadv.aao5580
15. Bao, M., et al. (2021). "It's COMPASlicated: The Messy Relationship between RAI Datasets and Algorithmic Fairness Benchmarks." NeurIPS 2021 Datasets and Benchmarks. arXiv:2106.05498
16. Uniform Guidelines on Employee Selection Procedures (1978), 29 CFR 1607.4(D). law.cornell.edu
17. State v. Loomis, 2016 WI 68, Wisconsin Supreme Court, July 13, 2016. FindLaw

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