5.2 Ranking Homogeneous Populations
I will now go on to articulate and apply a revised maximizing principle that takes account of our Postulate 3. Once again, this principle is intended solely for application to problems involving a variable population base. To illustrate the principle, I will begin by using it to rank a number of theoretical populations in terms of moral desirability. I will do this initially for the simplest kind of scenario. This scenario is one in which each population to be ranked is homogeneous. In each of these populations, every member of the population has the same utility as all other members of the same population.
Diagram 1 – Calculating utilities of homogeneous populations
Consider the six homogeneous populations p_{1} to p_{6} depicted in Diagram 1 above. The average utility for each population (a_{1}, a_{2}, a_{3} . . . ) is represented in the diagram on the vertical axis, while the number of beings in each population (n_{1}, n_{2}, n_{3} . . . ) is represented on the horizontal axis. Let us say that on an arbitrary scale of happiness, a being with a life that is neither worth living nor not worth living has a utility of zero units, while a being that has a happiness level equivalent to that of a very happy, normal human being has a utility of 10 units.
To calculate the moral desirability, or mixed utility, of each population under consideration, first rank the populations in order of average utility. Rank the population with the highest average utility first and that with the lowest average utility last (from left to right and from top to bottom), as shown in the diagram. If there is more than one population in which the average utility is not exceeded by the average utility of any other population (that is, if there is more than one contender for the highest position), then rank these populations in order of their values of n, with the population with the lowest value of n ranked first. For populations other than these that also have identical average utilities, their ordering with respect to each other is immaterial. By convention, the population with the highest ranking is designated p_{1}.
Next, calculate the moral desirability m_{x} for each homogeneous population p_{x}. The equation to apply for each population under consideration depends on whether the average utility for that population is greater than or equal to 0.7 of the average utility of the first ranked population. Where this condition is satisfied, apply Equation 1; otherwise, apply Equation 2.
Equation 1: where 
ax 
≥ 0.7, 
a1 
mx = ax^{2}.nx( 
ax 
) 
^{3} 
a1 
Equation 2: where 
ax 
< 0.7, 
a1 
mx = 
my 
( 
ay^{2}.ny 
) 
+1 
ax^{2}.nx 




where 
'mx' = 'moral desirability of population' 




'ax' = 'average utility of population px' 




'nx' = 'number of individuals in population px' 




'a1' = 'average utility of population p1' 




'n1' = 'number of individuals in population p1' 




'ay' = 'average utility of population py, where population py
is population with lowest mixed utility in the set of
populations that have average utilities >
' 




'ny' = 'number of individuals in population py' 




'my' = 'mixed utility of population py' 
Considering Equation 1 first, this equation is identical to my original equation, m = t.a = a^{2}.n, except that I have now added an extra weighting factor
The purpose of this weighting is to give a more rapid falloff in mixed utility as the average utility falls to 0·7a1. A more rapid falloff is required in order to make the new equation consonant with the insight that below 0.7a1 no increase in nx or tx (total utility) will compensate for the drop in average utility.
Equation 2 is applicable to populations in which the average utility is less than 0.7a1. This equation is designed to satisfy the requirement that as the utility value ax^{2}.nx of the population under consideration approaches infinity units, its mixed utility approaches the mixed utility of some comparison population in the limit. This comparison population, py, could not remain fixed at p1 for all populations under consideration, for this would have allowed some population pz with a low average utility to possess a mixed utility greater than the mixed utility of a population in which the average utility is greater than az/0.7. To have allowed this would have been in contravention of Postulate 3 and, secondly, would have led to interpopulation comparisons of mixed utility being intransitive. Postulate 3 and the requirement for transitivity demands that the comparison population not be fixed and that it be one of the set of populations in which the average utilities are greater than ax/0.7. Since Postulate 3 stipulates that mx cannot be greater than the mixed utility of any of these populations, the comparison population py must be that population in which the mixed utility of the set is lowest.
Continuing with our scenario, it is now possible to calculate the mixed utilities of the populations depicted above. The average utility a1 of population p1 is 10. The next step is to apply Equation 1 to the populations with average utilities greater than 0.7a1 (or 7); that is to populations p1, p2 and p3. The results are:
m1 = a1^{2}.n1( 
a1 
) 
^{3} 
a1 
m2 = a2^{2}.n2( 
a2 
) 
^{3} 
a1 
m3 = a3^{2}.n3( 
a3 
) 
^{3} 
a1 
= 10^{2}.20( 
10 
) 
^{3} 
10 
= 10^{2}.50( 
10 
) 
^{3} 
10 
For population p4, a4/a1 is less than 0.7. So, to solve for m4, apply Equation 2. First find population py. This is the population with the lowest mixed utility my in the set of populations that have average utilities greater than a4/0.7 (or 8.6). Only populations p1 and p2 have average utilities greater than 8.6, with population p1 having the lowest mixed utility m1 at 2,000 units. Now applying Equation 2, solve for the mixed utility of population p4:
m4 = 
my 
( 
ay^{2}.ny 
) 
+1 
a4^{2}.n4 

To calculate m5 and m6, Equation 2 likewise applies. In the case of mixed utility m5, my is the lowest value in the set {m1, m2, m3}. This value accords again with the mixed utility of population p1 at 2,000 units. Applying Equation 2, solve for the mixed utility of population p5:
In the case of mixed utility m6, my is the lowest value in the set {m1, m2, m3, m4, m5}. This value accords with the mixed utility of population p4 at 1,894·7 units. Next, apply Equation 2 to population p6. The result is:
So, in order of moral preferability, p3 is the most preferable with a mixed utility of 6,553.6 units, with p2, p1, p5, p4 and p6 next in order of preferability. I think this is in accord with the considered judgements of those who attend carefully to the populations depicted in Diagram 1.
Copyright © 2015 Leslie Allan