Animal Rights and the Wrongness of Killing
5. Utilitarian Maximizing Principles
5.3 Ranking Populations with Negative Utilities
So far, I have considered homogeneous populations with both positive total and average utilities. I now want to consider populations with values of total and average utility below zero. First, applying our previous definition, any individual being that has a utility below zero leads a life that is not worth living. Also, note that for such populations, average utility a and total utility t will have as their values negative real numbers. However, because I am here speaking of amounts of suffering, whenever I say that one level of negative utility, either average or total, is above another, what I mean is that the former is a greater amount of suffering than the latter.
I think we can now state a corollary to the third postulate, dealing with the comparison between different populations of such miserable beings.
 Postulate 4:

For any given variable population with variable average negative utility (average level of suffering) and variable total negative utility (total amount of suffering), such that t.a = −k, where k is a constant (or, alternatively, a^{2}.n = −k), there is a level of average negative utility above which no further decrease in negative total utility will compensate for the increase in average negative utility.
How should we calculate the moral desirability, or mixed utility, of a set of populations comprised of miserable beings? Consider the simplest case in which the populations are homogeneous, with every member of each population having the same utility as all other members of the same population. First, rank the populations under consideration in order of average negative utility, with the population with the lowest average negative utility (least miserable) ranked first (designated p1) and that with the highest (most miserable) ranked last. If there is more than one population in which the average negative utility is not higher than the average negative utility of any other population (that is, if there is more than one contender for first position), then rank these populations in order of their values of n, with the population with the highest value of n ranked first. For populations other than these that also have identical average negative utilities, their ordering with respect to each other is immaterial.
If one of the populations under consideration has no members (that is, if one option is a state of affairs in which no sentient beings exist), then designate this population p0 and designate the population with the next lowest average negative utility p1, subject to the above condition dealing with multiple contenders for p1. Apply the same rule to those populations under consideration that contain beings with zero utility (that is, beings whose lives are neither worth living nor not worth living). All such populations designated p0 have a mixed utility of zero units and are, therefore, morally preferable to any other population with a mixed utility of less than zero. If only two populations are under consideration, one of which is p0, then the mixed utility of the remaining population m1, is given by the equation:
m1 = a1^{2}.n1
If the set of homogeneous populations under consideration contains at least two members that have an average negative utility of less than zero units, then calculate the moral desirability mx for each homogeneous population px using one of two equations. Apply Equation 3 where the average utility of the first ranked population is greater than or equal to 0.7 of the average utility for that population; otherwise, apply Equation 4.
Equation 3: where  a1  ≥ 0.7, 
ax 
−mx = ax^{2}.nx(  ax  )  ^{3} 
a1 
Equation 4: where  a1  < 0.7, 
ax 
mx = my(  ax^{2}.nx  )  +1 
ay^{2}.ny 
where  'mx' = 'moral desirability of population px'  
'ax' = 'average negative utility of population px'  
'nx' = 'number of individuals in population px'  
'a1' = 'average negative utility of population p1'  
'n1' = 'number of individuals in population p1'  
'ay' = 'average negative utility of population py, where population py is population with highest average negative utility in the set of populations that have average negative utilities < 0.7 ax' 

'ny' = 'number of individuals in population py'  
'my' = 'mixed utility of population py' 