In 1955, John Harsanyi published a paper demonstrating that anyone who follows certain reasonable assumptions must be a total utilitarian. The paper is somewhat technical, but the result is relatively easy to understand. I’ve been unable to find a non-technical summary of this result and so, because it is one of the more compelling arguments for utilitarianism, I decided to write one up.
Background
Suppose a group of friends are deciding where to eat. Each individual person has some preference (say, one person most prefers Chinese, then Italian, then Japanese; another prefers Italian, then Chinese, then Japanese) but there is no clear restaurant which everyone thinks is best. How should they choose a place?
One solution is to have each person attach a numeric score to how much they would enjoy a given restaurant. If you really like Chinese food, then maybe you give it 10 points; if you’re lukewarm then you give it 2, and if you really hate Chinese then maybe it’s -5.
Once each person has voted, you simply add up all the scores, and then the group goes to whichever restaurant had the highest total score.
This method is (a simplified form of) “total” utilitarianism, and Harsanyi demonstrated that it is the only “reasonable” way that groups can make a decision.
Theorem
Concretely, assume:
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Each individual in the group is rational (for a commonly used but technical definition of “rational”, hereafter referred to as “VNM-rational”)
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The group as a whole is VNM-rational
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If every individual in the group is indifferent between two options, then the group as a whole is indifferent between those two options
The theorem proves that total utilitarianism is the only method which satisfies these three assumptions.
Note that this theorem just demonstrates that, if there is some way of saying that certain things are better or worse for individuals, then the way to determine whether those things are better or worse for groups is to add up how good it is for the individuals in those groups. It doesn't say anything about the way in which things can be better or worse for individuals. I.e. you could be adding up each individual's happiness (hedonistic utilitarianism), something related to their preferences (preference utilitarianism), or something more exotic.
Example
The above is somewhat abstract, so here is a concrete example demonstrating why anything other than total utilitarianism fails these axioms. (This is my best attempt at creating a simple example; perhaps others in the comments can create even simpler ones.)
Consider a population consisting of 2 people. Because they are VNM-rational, they have utility functions, and therefore we can represent states of the world as a vector of numbers. E.g. the vector (5,7) is a world in which the first person has utility 5 and the second has utility 7.
Let’s prove that the world (2,0) must be as good as the world (1,1).
Consider a lottery in which there is a one-half chance we end up with the world (2,0) and a one-half chance that we end up with the world (0,2). Because we are indifferent between who has the 2 and who has the 0, and the group is an expected utility maximizer, these are equally valuable:
12(2,0)+12(0,2)=(2,0)
We can we write this from the perspective of each individual in society:
(12⋅2+12⋅0,12⋅0+12⋅2)=(2,0)
Because VNM-rational agents are expected utility maximizers we can just multiply the probabilities through:
(1,1)=(2,0)
QED.
The key insight here is that each individual is indifferent between the “50% chance of 2, 50% chance of 0” and “guaranteed chance of 1” lotteries (on account of being VNM-rational). Because each individual is indifferent, the group is also forced to be indifferent (on account of the third assumption).
Conclusion
Total utilitarianism is a fairly controversial position. The above example where (1,1)=(2,0) can be extended to show that utilitarianism is extremely demanding, potentially requiring extreme sacrifices and inequality.
It is therefore interesting that it is the only decision procedure which does not violate one of these seemingly reasonable assumptions.
While not conclusive, this theorem provides a compelling argument for total utilitarianism.
Appendix on Equality
Harsanyi’s original theorem allowed for weighted total utilitarianism. (I.e. everyone gets a vote, but some people’s votes count more than others.)
It’s easy enough to add an assumption like “also everyone is equal” to force true total utilitarianism, but interestingly Harsanyi didn’t think that was necessary:
This implies, however, without any additional ethical postulates that an individual’s impersonal preferences, if they are rational, must satisfy Marschak’s axioms [equivalent to VNM-rationality] and consequently must define a cardinal social welfare function equal to the arithmetical mean of the utilities of all individuals in the society (the arithmetical mean of all individual utilities gives the actuarial value of his uncertain prospects, defined by an equal probability of being put in the place of any individual in the situation chosen). [Emphasis added]
In other words, he thinks it would be irrational to weight people unevenly, because equal weighting is the expected utility-maximizing choice if you don’t know which person in society you will become.
This idea of making decisions behind a veil of ignorance where you don’t know which person in society you will become was later popularized by John Rawls, who used it to argue for his Minimax decision rule.
It is, in my humble opinion, unfortunate that the veil of ignorance has become associated with Rawls, when Harsanyi’s utilitarian formulation has a much more rigorous mathematical grounding. (And was also published earlier.)
Credits
I would like to thank Aaron Gertler, Sam Deere, Caitlin Elizondo and the CEA UK office staff for comments on drafts of this post and discussions about related ideas.
Thanks for writing this up!
For those interested in more info:
Harsanyi had two different theorems like this (his aggregation theorem and his impartial observer theorem) which rely on slightly different assumptions.
The main arguments against Harsanyi's theorems were made by prominent economist Amartya Sen in what has become known as the "Harsanyi-Sen debate" or "Harsanyi-Sen-Weymark debate" (searchable terms). The gist of the counterargument is that "while Harsanyi has perhaps shown that overall good is a linear sum of individuals’ von Neumann–Morgenstern utilities, he has done nothing to establish any connection between the notion of von Neumann–Morgenstern utility and that of well-being, and hence that utilitarianism does not follow.".
I think this last point essentially denies the third axiom above, which is what connects individual vNM utility and social/ethical preferences. (The original statement of the second axiom is just vNM rationality for social/ethical preferences, and has no relationship with the individuals' preferences.)
VNM Utility is the thing that people actually pursue and care about. If wellbeing is distinct from that, then wellbeing is the wrong thing for society to be optimizing. I think this actually is the case. Harsanyi, and myself, are preference utilitarians. Singer and Parfit seem to be something else. I believe they were wrong about something quite foundational. Writing about this properly is extremely difficult and I can understand why no one has done it and I don't know when I'll ever get around to it.