Animal Rights and the Wrongness of Killing
5. Utilitarian Maximizing Principles
5.4 Ranking Nonhomogeneous Populations
The populations that I have considered so far have been homogeneous populations; that is; populations in which each member has an identical utility to all other members of the same population. The task of comparing the preferability of populations in which their constituent members possess nonidentical utilities is considerably more complex. At this early stage in the development of the 'mixed' view maximizing principle, I will content myself with describing two methods for making comparisons between populations with relatively simple utility distributions.
These two methods are:

simple addition
used where all populations in the comparison set share a common subpopulation type 
segmentwhole comparison
used where a segment of one population is morally preferable to the whole of a comparison population
I will illustrate the application of each method in turn. Beginning with the simple addition method, consider these two examples.
Example 1
The diagram below depicts four populations: A, B, C, and D. The average utility for each population is represented in the diagram on the vertical axis, while the running total of the number of beings in each population is represented on the horizontal axis.
Once again, let us stipulate 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.
Populations B, C, and D are identical to A, except for the addition of extra members. All four populations share a subpopulation type comprising 50 members with an average utility value equalling 8. The mixed utility of the subpopulation type that all four populations have in common must also be identical. Let us designate this mixed utility ms. Let us call the extra population segments that populations B, C and D possess, 'b', 'c' and 'd' respectively. These are labelled in Diagram 2 as such. To calculate the mixed utility of population B, C or D, we can simply add the mixed utility of the population's additional segment (mb, mc or md respectively) to the mixed utility of the segment they share, ms.
We can calculate the mixed utility of the common segment and each additional segment using the procedure given in §5.2 above. First, rank the segments in order of average utilities. The result of this ordering is shown is Diagram 3 below.
Using Equation 1 and Equation 2 in §5.2 above, calculate the mixed utilities of the segments mb, ma, mc and md in turn.
mb =  5,000 
ms =  mA = 1,638.4 
mc =  2,950.8 
md =  1,208.6 
The mixed utilities of the segments can now be summed to find the mixed utilities of populations A, B, C, and D, or mA, mB, mC and mD, as follows:
mA =  1,638.4 
mB =  ms + mb 
= 6,638.4 
mC =  mS + mc 
mD =  ms + md 
= 4,589.2 
= 2,847 
So, in order of moral preferability, population B is the most preferable, then C, D and A in descending order. I think this marries with our considered judgements about this scenario.
Example 2
For the second example of using the simple addition method for calculating the preferability of populations with nonuniform utility distribution, consider a more complicated scenario. Populations A, B and C are represented in Diagram 4 below. Once again, average utility for each population is represented on the vertical axis, while the running total of the number of beings in each population is represented on the horizontal axis. This scenario is more complex as all three populations do not share a common complete subpopulation.
None the less, commonalities and segments can be identified. Population C is identical to A, except for the addition of a subpopulation comprising 200 beings, each with a utility of 3 units. Let us designate this extra segment, 'c'. Population C is also identical to B, except for the addition of one being with a utility of 10 units. Let us designate this extra segment comprising one individual, 'd'. Let us also designate the segment that population C has in common with A, 'a', and the remaining segment of B, 'b'. Segment a, then, is the summation of segments b and d. All four segment types are labelled as such in Diagram 4.
As before, we can calculate the mixed utility of population B and C by adding together the mixed utilities of the separate segments comprising each population (ma, mb, and mc) as required. Start by ranking the segments in order of average utilities. The results are shown in Diagram 5 below.
Again, using Equation 1 and Equation 2 in §5.2 above, calculate the mixed utilities of the segments md, mb, ma and mc in turn.
md =  100 
mb =  900 
ma =  1,000 
mc =  100  

= 94•7 
Finally, sum the mixed utilities of the segments to find the mixed utilities of populations A, B and C, or mA, mB, and mC.
mA =  ma = 1,000 
mB =  mb + mc = 994.7 
mC =  ma + mc = 1,094.7 
Using this method reveals that Population C is the most morally preferable, then A, then B. This result is consistent with Postulate 1 to Postulate 4 and so is to be expected. Compared with population C, the lack of one being with a utility of 10 units in population B is less preferable than the lack of 200 beings, each with a utility of 3 units, in population A.
I now want to illustrate the second method for making ethical comparisons between populations; the segmentwhole comparison method. The basic idea here is that for any two given populations, if a homogeneous segment of the first population is morally preferable to the whole of the second population, we can conclude that, all other things being equal, the first population is morally preferable to the second. I will illustrate this with an example.
Consider the three populations depicted in the diagram below; populations A, B, and C. Average utility for each population is represented on the vertical axis, while the running total of the number of beings in each population is represented on the horizontal axis.
Population A comprises two segments, labelled 'a' and 'b'. If it turns out that one segment of population A, considered in isolation, is morally preferable to the whole of population B or C, then the combination of segments a and b in population A is morally preferable to the comparison population. Examining the two segments in population A reveals that segment a satisfies this requirement. This judgement is reached by applying Postulate 3 in §5.1. No member of population B or C has a utility of greater than or equal to 7/10ths that of the average utility of segment a. Using the segmentwhole comparison method, we can conclude that population A is morally preferable to both population B and population C.
One important caveat to applying this method is that the remaining segments of the first population have an average utility of zero or greater. It cannot be the case that a population with miserable beings necessarily outweighs in moral preference another population simply because some of the happy beings the former contains have an average utility greater than or equal to 7/10ths that of the average utility of the beings in the latter population.
In cases of segmentwhole comparison such as this, as yet there is no metric available to measure the moral desirability of each population. So far, we are unable to determine computationally whether population B is morally preferable to population C. However, with this method, we are able to find out that population A is preferable to B and C. What the segmentwhole comparison method does is give us a limited ability to order complex populations. A more powerful calculus must await further interpretation of the postulates and equations.[11]
Now that we have completed this complicated journey through various maximizing principles, what have we achieved? Recall the question that was posed at the beginning of the previous section: Is there some finite number of chickens that can be killed that will outweigh the moral wrongness of the killing of one normal adult human being, all other things being equal? The 'mixed' view maximizing principle developed above allows us to answer this question once we know the relative utilities of the animal being killed and that of the human being.
Footnotes
 [11] The four postulates and four equations advanced in this essay seem to provide the basic structure for an adequate theory of maximization, or what Derek Parfit [1984: part 4] calls 'Theory X'. It appears to satisfy Parfit's four requirements for an adequate theory: it solves the NonIdentity Problem, avoids the Repugnant and Absurd Conclusions and dissolves the Mere Addition Paradox.