Why do these two measures of monetary loss differ, numerically and logically? [closed]

When calculating damages in an employment discrimination case, two approaches are used: lost opportunity and shortfall in hires (or promotions). These approaches are used to calculate lost wages for the protected group, say women.

For the sake of describing the two approaches, let's assume N employees competed for promotion and a court decided the promotion process used by the employer was illegal because it was not fair. Let's assume that the lost wages (L) due to non-promotion are known. Let's use w for the number of women promoted, W for the total number of women, and p for the overall promotion rate (for both men and women).

The opportunity lost approach would calculate lost salary (S) for one not-promoted employee as follows:
S = pL

The logic is that any one employee was not certain of being promoted, so the loss of L was not a certain loss, only a potential loss. This approach holds that the loss was only based on the opportunity to be promoted (p).

The opportunity loss approach would calculate the total lost salary (T1) for all women as:
T1=(W-w)pL
This is the number of not promoted women times the prob of promotion times the loss.

The shortfall approach would calculate lost wages based on the difference between the number of women promoted (w) and the number that would be expected to be promoted based on chance. The expected number of promotions for women (E) would be:
E=pW

The shortfall (F) would be the difference between pW and the actual number of women promoted (w).
F = E-w.

Using the shortfall approach, the lost income (T2) would be FL.
T2=FL

In summary, these approaches give different dollar numbers because the total loss (T) formulas are different.

We have:

T1 = L (W-w)p

T2 = L (pW - w )

To compare T1 and T2 we can cancel L and expand T1 to get:

T1 = Wp - wp

T2 = pW -w

Since p < 1, T1 will always be bigger than T2.

The question is, is this algebraic analysis correct and which approach is more sound or logical, from a mathematical perspective?

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• This question may be better suited for math.stackexchange.com. Have you used W consistently? you originally defined it as the number of women promoted. but it sounds like it comes to represent the number of women in total – Ryan Jul 3 at 15:37
• @Ryan TY. I tried to clarify the notation, which led to changes throughout. I am unfamiliar with math.stackexchange.com. TY for the link. – Joel W. Jul 3 at 15:56

Let's start by defining some terms.

let $$N$$ denote the number of people applying for a promotions. let $$W$$ denote the number of women applying for a promotion. let $$n$$ denote the number of people selected for promotion under both recruitment processes. let $$w$$ denote the expected number of women selected for promotion under the unfair recruitment process. let $$L$$ denote the value of a promotion (assumed to be equal for all candidates)

Next lets define the expected value each person would gain under both a fair and unfair recruitment processes given no information about the candidate other than their gender, and assuming gender has no fair impact on probability of recieving a promotion.

Fair recruitment

$$\mathtt{ExpectedValueOfPromotion} = \mathtt{ValueOfaPromotion} * \mathtt{ProbabilityOfReceivingAPromotion} = L \frac{n}{N}$$

unfair recruitment

$$\mathtt{ExpectedValueOfPromotion(Women)} = \mathtt{Value Of A Promotion} * \mathtt{Probability Of Receiving A Promotion} = L \frac{w}{W}$$

$$\mathtt{Expected Value Of Promotion (Not Women)} = \mathtt{Value Of A Promotion} * \mathtt{probability Of Receiving a promotion} = L \frac{(n-w)}{(N-W)}$$

expected difference between fair and unfair (fair - unfair )

$$\mathtt{difference (women)} = L \frac{n}{N} - L \frac{w}{W} = L (\frac{n}{N} - \frac{w}{W})$$

$$\mathtt{difference (not women)} = L \frac{n}{N} - L \frac{(n-w)}{(N-W)} = L ( \frac{n}{N} - \frac{(n-w)}{(N-W)} )$$

Finally lets see how these compare to your two approaches.

Approach 1 - opportunity loss

The opportunity lost approach would calculate lost salary (S) for one not-promoted employee as follows: S = pL.

Here $$p$$ represents the overall promotion rate (i.e. $$\frac{n}{N}$$). Next we notice that $$S = L (\frac{n}{N} - \frac{w}{W})$$ when $$w = 0$$. in otherwords S = the expected difference in value for a woman between fair and unfair recruitment if no women are recruited via the unfair recruitment practice.

The opportunity loss approach would calculate the total lost salary (T1) for all women as: T1=(W-w)pL This is the number of not promoted women times the prob of promotion times the loss

This is not quite true as using p for the probability of promotion is not really valid when you have systematically excluded women who were able to achieve a promotion despite the unfair recruitment processes.
You would be better to interpret this as $$T1 = (W-w)S$$ which is the number of not promoted women * the expected lost earnings (assuming no women are promoted).

Approach 2 - The shortfall approach

The shortfall approach would calculate lost wages based on the difference between the number of women promoted (w) and the number that would be expected to be promoted based on chance. The expected number of promotions for women (E) would be: E=pW

The shortfall (F) would be the difference between pW and the actual number of women promoted (w). F = E-w.

Using the shortfall approach, the lost income (T2) would be FL. T2=FL

notice that $$T2 = FL = (pW - w)L$$ and that this represents the Total loss for all women. dividing through by the number of women W gives us the expected loss for each women

$$\mathtt{expected loss} = (p - \frac{w}{W})*L = (\frac{n}{N} - \frac{w}{W}) * L.$$ This is consistent with our original definition.

in summary, as you have described them, opportunity loss calculates expected loss as if no women were promoted. The shortfall approach calculates expected loss as if the observed number of women were promoted.

• TY Ryan. Note that the less formatted version of your answer copied easily into a word processor. The recently formatted version does not copy correctly. Your last equation pastes into my word processor as: ????????????=(?−??)∗?=(??−??)∗?. – Joel W. Jul 3 at 19:26
• How do the two approaches compare when w is not zero, say when 10% of the women were promoted and 20% of the total group were promoted? If the number of men is important, assume that the men outnumber the women by 3 to 1. – Joel W. Jul 3 at 23:27