So far, we have been able to decompose utilitarian quadratic inequality metric(QIM) into bias and variances. One might wonder where this fits in the greater scheme of things.
Recall that the analysis is being performed on the benefit of a service, provided by a service provider such as company or government, to a customer at cost to the latter party. Equality, as measured by QIM, is being demanded of the provider between repeated experience by the same or different customers. Therefore, it is obvious that the approach does not itself insist on equality of opportunity. It may, if we insist that population we draw from are inclusive and representative, however that is an addition to be made the model and require additional specification.
The bias term really measures the equality of outcome. When the QIM is high, it can be caused by a difference in the beneficial outcome of the same experience to different customers.
But, what, in traditional social, political and economic sciences do we call the variances part of QIM? I initially considered univariate variances a measure of quality of service to each consumer. This makes sense as long as the services is not pathologically constructed situations: e.g. a random experience generator where the desired outcome is high variance. Later, I feel that this question must be punted to the user of the QIM. Let it be a demand to the user that he only parameterize QIM for utility measurements whose high and consistent values are desired. In so far as I have observed, most metric that we measure are like this: revenue, profit, ROI, unemployment rate, even binary variables such as attendance, satisfies this demand. Finally, sign of inequality metric is actually not important. QIM will be large when the two are unequal.
In all cases of metrics, both bias and univariate variance terms of QIM can only increase this measure of inequality. Under QIM, inequality can only be overcome by the covariance term. The only way for QIM to be decreased from highly different outcome or inconsistent behavior to either, or both, consumer comparands, is if the benefit of service or harm of diservice anti-correlates. If they are anti-correlated, the covariance term improves the QIM.
So, if there are all-male school A, and all-female school B. Send randomly selected students from male and female populations to both schools. If students fail or succeed in both schools simultaneously but never fail in one school and succeeds in the other school, then it removes suspicion that the two school provides unequal educational service. In fact, if each of the students succeeds in exactly one of the two schools, the QIM become minimized–the strongest contribution of inequality due to inconsistent experience.
An alternative sampling model to compare two populations (male and female students) at another school (a modern co-ed University) we can simply sample pairs of one male and one female students according to natural distribution. (consider a simple random sample from all pairs of man-women generated from class rosters, or grade levels, dependent on whether the utility metric is class grade or overall GPA or other metrics. These measurements are then used to estimate the QIM between the sexes at said institution.
In both of these cases, the decomposition of the QIM can be computed.
It is my continuing concern that we have not resolved the X of equality-of-X models. Equality of outcome is certainly not the most popular approach right now, where as equality of opportunity is de facto law of the land in United States, at least governments and public corporations are required to be equal opportunity employers.
But whether and how the law applies to the actual experience of employment, and furthermore the experience of actual service rendered by the governed entity, is unclear to me. How equality is monitored is certainly beyond my education. I hope to later find answers to these thoughts in ancient political science texts, that I already live in an as-equitable-as-can-be society…
And if not, here is one attempt at improving all that.