Fairness Metrics & Their Conflicts

Quantitative Definitions of Algorithmic Fairness

How do you mathematically define whether a classification model (for example a credit scoring model) is Fair across protected demographic groups $A$ and $B$?

There is no single definition of fairness. Computer science literature defines Three Core Fairness Metrics:

┌──────────────────────────┬──────────────────────────┬──────────────────────────┐
│ 1. DEMOGRAPHIC PARITY    │ 2. EQUAL OPPORTUNITY     │ 3. PREDICTIVE PARITY     │
├──────────────────────────┼──────────────────────────┼──────────────────────────┤
│ Equal Selection Rates:   │ Equal True Positive Rates│ Equal Precision /        │
│ P(Y_hat=1|A) =           │ (Recall):                │ Calibration:             │
│ P(Y_hat=1|B)             │ P(Y_hat=1|Y=1,A) =       │ P(Y=1|Y_hat=1,A) =       │
│                          │ P(Y_hat=1|Y=1,B)         │ P(Y=1|Y_hat=1,B)         │
└──────────────────────────┴──────────────────────────┴──────────────────────────┘

1. Demographic Parity (Independence)

Requires the overall positive prediction rate to be identical across groups, regardless of true ground truth label distributions:

$$P(\hat{Y} = 1 \mid A) = P(\hat{Y} = 1 \mid B)$$

• Use Case: University admissions or job interview invitations where structural equality of outcome is mandated.
• Trade-off: Ignores true ground truth base rate differences between groups.

2. Equal Opportunity (Separation)

Requires the True Positive Rate (Recall) to be identical across groups:

$$P(\hat{Y} = 1 \mid Y = 1, A) = P(\hat{Y} = 1 \mid Y = 1, B)$$

• Use Case: Credit loan approvals, ensuring qualified applicants ($Y = 1$) have equal probability of loan approval regardless of demographic group.

3. Predictive Parity (Sufficiency / Calibration)

Requires predicted probabilities to have equal Precision / Calibration across groups:

$$P(Y = 1 \mid \hat{Y} = 1, A) = P(Y = 1 \mid \hat{Y} = 1, B)$$

• Use Case: Risk assessment tools where a score of $80%$ must mean an $80%$ real world probability for all groups.

The Impossibility Theorem of Fairness

Can we design a model that satisfies Demographic Parity, Equal Opportunity, AND Predictive Calibration simultaneously?

Kleinberg et al. (2016) proved mathematically that this is IMPOSSIBLE!

$$\text{If base rates } P(Y=1 \mid A) \neq P(Y=1 \mid B), \text{ you CANNOT satisfy all 3 metrics simultaneously!}$$

You must choose which fairness metric aligns best with your product goals and regulatory requirements.

Say this out loud

Algorithmic fairness definitions include Demographic Parity, Equal Opportunity, and Predictive Calibration. Demographic Parity requires equal selection rates. Equal Opportunity requires equal True Positive Rates. The Impossibility Theorem of Fairness proves that when base rates differ across groups, a classifier cannot satisfy all fairness metrics simultaneously, requiring explicit tradeoffs.

Followups to expect

1. What is the 80% Rule (Four-Fifths Rule) in legal employment compliance? A legal guideline stating that selection rates for any protected group must be at least 80% of the selection rate for the highest group to avoid disparate impact findings.
2. What is Individual Fairness? The principle that similar individuals should receive similar predictions ($d(\hat{y}_i, \hat{y}_j) \le d(x_i, x_j)$), complementing group fairness metrics.