"It's not fair!" "Yes it is!" "Ain't so!" "'Tis too!" "No!" "Yes!" ...The above exchange can be overheard just about any place you happen to be. It is a shouting match between two people, neither of whom likely knows what he or she is talking about. Because I'll bet dollars to donuts neither of them has bothered to consider what they can agree on as a definition for fairness.
Let's take a very familiar example to demonstrate what is fair under certain circumstances. Consider the usual process that will determine which team will kick off a game of football. The outcome is based on a coin toss, and it is fair because, presumably, a fair coin will be tossed by the referee. What makes a coin fair? Clearly it is a coin that is not biased in favor of either of the two possible outcomes, heads or tails. From this we can conclude that an important, arguably the most important, attribute of fairness is a sense of equal treatment of participants whenever multiple participants are involved in a process that lends itself to a fairness assessment.
Let's take another example, much more complicated than the above, but also near and dear to every American with an income, namely, "What makes for a fair income tax structure?" Here we have a much more complex situation than the above, for several reasons. First, we want to define equal treatment for all participants, but with the added complication that there are many significant differences (read inequalities) among the huge population of income earning participants. Another set of complications derive from the complexity of the need for an income tax. The Government requires the income tax to generate funds that in turn enable the Government to function. How to proceed?
Frequently, it is easier to work backwards, which in mathematics is sometimes referred to as the adjoint approach. To begin with, the Government must raise some minimum amount of tax revenue in order to perform its functions for the citizenry. Given the required amount that must be raised, how to apportion the burden on individual taxpayers that would be fair in some sense? First and foremost, it is obvious that every increment of income should be taxed at a specific (though possibly different from other ordinal increments') rate. This insures that every taxpayer's increment of income is taxed fairly, because every ordinal increment of income for every individual taxpayer is taxed identically. Now all that would be required is a formula for specifying the tax rates of ordinal income increments that would guarantee the tax revenue so generated will be sufficient for the Government's needs.
The formula will doubtless have to be specified such that the tax rate will increase with increasing ordinal income increment because that will satisfy the sense of fairness based on the recognition that the ability to pay taxes without exceeding an individual's ability to survive the burden so imposed increases with increasing ordinal income increment. But we must also bear in mind an individual's primary motivation for earning additional income: There aren't many people willing to earn income unless some of it can be kept. Hence, the maximum tax rate must remain below 100%.
At this point we have almost reached the end of our definition of a fair income tax structure. There remains the need to specify an algorithm for calculating an individual's income tax. Ideally, an algorithm would correspond to a graph of Tax Rate[%] as a function of Income Increment[$] that begins at a point in the neighborhood of ($0,0%), and monotonically increases toward ($Gates-Highest-Increment,Max%), such that, for each individual taxpayer, his own last increment of income corresponds to his own highest percent tax. The overall constraint would be that the total of all taxpayers' payments met the Government's minimum revenue requirement.
Each taxpayer's payment would correspond to the area under the tax graph of Tax Rate[%] versus Income Increment[$], between the income limits of zero and final dollar earned (by the individual). This computed area is expressed in units of "%$", which is equivalent to "$" because "%" is a dimensionless (i.e., scalar) quantity. Computation of such an area is equivalent to mathematical integration. It is commonly approximated numerically by summing a histogram of contiguous tax brackets.
I believe that the crucial consideration of fairness, which is at the heart of most controversy, concerns the specification of bracket boundaries and their corresponding tax rates. These are the free parameters that are most likely to be based on political and pragmatic considerations. In order to introduce at least a perceived spirit of fairness in choosing these parameters, a requirement could be imposed that each bracket contains an equal portion of the total area under the tax curve. In so doing, each bracket of income would, in principal, contribute an equal share of the tax burden for each individual taxpayer. This might satisfy the demands of those who require that everyone pays his "fair share" of tax revenue.
The final bow to fairness is necessarily arbitrary to the extent that the area calculation is only approximate, though the approximation can be made as precise as desired by using as many tax brackets as required. In a very real sense, there is a trade off between the degree of "fairness" and the degree of the individual tax preparer's burden. Considering that tax preparation is already so harrowing that it prompted Albert Einstein to quip 'The hardest thing in the world to understand is the income tax', my guess is that about a half dozen tax brackets is all anyone can handle before they would be persuaded to blow their own brains out.