1
A distribution is a list
An outcome is described by one number per person, the welfare of that person’s life. The numbers are stipulated by the example; whether welfare can be measured so is a real question that the lab does not answer.
Lesson · Ethics and the Arithmetic of Welfare · Consequentialism
Utilitarianism and the Sum of Welfare
Utilitarianism is stated here as a rule of aggregation: add up each person’s welfare and prefer the larger total. Rank two distributions by that rule and see that the ranking ignores who gets what, including a case where two of three people are worse off.
an outcome: one welfare number per person
total = x₁ + x₂ + … + xₙ, largest wins
10, 10, 10 totals 30; 1, 30, 2 totals 33
the total cannot see who has which number
Utilitarianism as a rule of aggregation
Core ideas
What this lesson establishes
To make a moral theory computable you need two things: what it counts, and how it combines the counts. Utilitarianism supplies both in one sentence.
2
The rule is a sum
Utilitarianism ranks distributions by total welfare. Add the list, compare the sums, and the larger total is the better outcome. Ties are ties.
3
A sum forgets order and owner
Two lists with the same numbers in a different order have the same total. So the rule is indifferent to who has which welfare, and to how the welfare is spread.
The material
Welfare as a column of numbers
Two distributions, one rule, and what the rule does not ask.
Definition · Utilitarianism as aggregation
The theory ranks outcomes by total welfare, the sum of every person’s welfare, and holds that the outcome with the larger total is better. It is a rule that turns a distribution into one number.
Compare A = 10, 10, 10 with B = 1, 30, 2, each list giving the welfare of three people. Under A the total is 30. Under B it is 33. The lab prints that B is ranked above A, because the larger total wins.
Look at what that ranking does. Compared with A, the three people change as follows.
person under A under B change
first 10 1 −9
second 10 30 +20
third 10 2 −8
net +3
Two of the three people are much worse off under B, and the rule prefers it. Bentham’s slogan, the greatest good of the greatest number, makes that sound impossible, because it sounds like two aims. As a rule of aggregation there is one: the sum. When “the greatest good” and “the greatest number” pull apart, as they do here, the number of people helped does not count, and Bentham himself came to drop the second clause.
Example · A transfer
Compare A = 5, 15 with B = 15, 5. The two people have swapped welfare. Each list totals 20, so the rule calls them equal. The same indifference covers any redistribution that leaves the sum alone, whether it takes from the badly off to give to the well off or the reverse.
Example · A sacrifice
Compare A = 10, 10, 10 with B = 0, 15, 16. One person is left with nothing, and the total rises from 30 to 31. The rule prefers B.
These are the cases critics press, and the replies should be heard at full strength. A utilitarian can say that in practice resources have diminishing returns in welfare, so sacrifices like this are rare. That is a claim about how often the sum favours them, and it leaves the rule’s verdict on the stated numbers where it was. Another reply accepts the verdict and holds that intuition against it should be revised.
Two of the lab’s tiles are for later. The population at ε is read in “Total, Average and the Repugnant Conclusion” and the Gini coefficient in “Priority, Equality and Levelling Down”; here the verdict and the two scores are the whole of the rule.
The next lessons keep the rule’s structure and change the part that bothers: what is summed, how people are counted, and how heavily the worse off are weighted. First comes the plain version, because the plain version is the one the others are defined against.
Interactive
Two distributions, six rules
the same people ranked by total, average, priority, the worst off, and sufficiency
Method
Ranking two outcomes by total
Five steps. The last one is the one that connects the arithmetic to the objection.
1
Write each outcome as a list
One welfare number per person. Keep the people in the same order in both lists so changes can be read off.
2
Add each list
The total is the score. Do not average, weight or sort.
3
Compare the totals
The larger total ranks higher; equal totals tie. That is the whole of the rule.
4
Write the change for each person
Subtract the first list from the second, person by person. The changes add up to the difference of the totals, and their signs show who gains and who loses.
5
Ask what the ranking ignored
Check whether swapping two people’s numbers would change the verdict. It would not. Then ask whether the example is one where that matters.
What the total sees
Score 0 / 0Choose the answer you can justify.
Ranking A and B
Add each list, compare, then look at the individuals.
A = 10, 10, 10 total 30
B = 1, 30, 2 total 33
total ranks B first 33 against 30
changes from A to B: −9 +20 −8 net +3
worst-off person: 10 under A, 1 under B
The rule did what it says. The ranking is correct, and the third line is where a critic starts: the person left with 1 is the price of the person with 30.
Common mistakes
1
Reading the slogan as two separate aims“The greatest good for the greatest number” sounds like two aims, but the rule has one: the sum. In the skewed example two of three people lose and the rule still chooses B, so the number of people helped did not count.
2
Expecting the total to favour fairnessThe transfer preset swaps two people’s welfare and the total does not move. A sum is the same for every ordering, so it is the same whether welfare is shared or concentrated.
3
Reading the arithmetic as an argument for or against the theoryThe numbers are stipulated. Nothing here shows that anyone’s welfare is the number written, or that welfare can be added across people. The argument lives in those stipulations, not in the sum.
Check your understanding
✓
Finish when you can rank by total and say what the ranking ignored.
Given two distributions, you should be able to compute each total, state the ranking, write the change for each person, and say whether the ranking would survive swapping two people’s welfare.
Note: Everything in this course assumes welfare can be put on one scale and added across people. That is a substantive assumption, and one the labs cannot check.