“Now suppose happiness to consist in doing or choosing the greater, and in not doing or in avoiding the less, what would be the saving principle of human life? Would not the art of measuring be the saving principle; or would the power of appearance? Is not the latter that deceiving art which makes us wander up and down and take the things at one time of which we repent at another, both in our actions and in our choice of things great and small?”
- Socrates in Protagoras.
Abstract
We revisit classical Utilitarianism by connecting and generalizing two ideas. The first is that there is a representation theorem possible for hedonic value (pleasure) similar to, but also importantly different from, the one provided by von Neumann and Morgenstern to measure decision utility. The idea is to use objective time, in place of objective chance, to measure hedonic value. This representation for hedonic value delivers a stronger kind of scale than von Neumann–Morgenstern utility, a ratio scale rather than merely an interval scale. The second idea is that measurement on a ratio scale allows the meaningful aggregation of utilities over a group. This is aggregation by product rather than sum. Aggregation by product is known to have interesting Prioritarian consequences. Aggregation becomes complicated when the two approaches are mixed, when hedonic value is mixed with uncertainly. It becomes problematic when pain as well as pleasure is taken into account.
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Notes
Here we follow the terminology of Kahnemann et al. (1997) “Back to Bentham.”
Bentham does not have the notion of the integral, but the idea is clear in chapter II of Jevons’ Theory of Political Economy, in Edgeworth’s (1879) Mind article and in Edgeworth’s appendix III, on the Hedonimeter, to Mathematical Psychics,
Bentham (1789, 2017), An Introduction to the Principles of Morals and Legislation Chapter IV: “Value of a Lot of Pleasure or Pain, How to be Measured”, section V. The setting here is different. Bentham is thinking here of just noticeable differences. But the principle is clear.
von Neumann and Morgenstern (1947).
Together with a few other technical conditions.
We are indebted to a referee for pointing out that continuity, completeness of the order, and the Archimedean property are all psychologically questionable. These idealizations are far from Bentham, who talked of counting particles of pleasure, but closer to Edgeworth, who wanted to apply the integral calculus.
See Aczél and Roberts (1989), for discussion of uniqueness of product representations.
Observe that if we had used the sum, rather than the product, the scale change would have changed the group preference to favor Peter, 502 to 403. The sum is not meaningful; the product is.
Adler uses the von Neumann–Morgenstern representation, and postulates a zero on ethical grounds. His account is thus quite different in both motivation and character from that examined here.
The arithmetic mean of n values is gotten by adding them together and dividing them by n. The geometric mean is gotten by multiplying them together and taking the nth root
Most of those who have questioned the independence axiom would not welcome a deviation in this direction. The social planner here is not risk-averse but risk seeking.
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Research for this article was supported by Grant SMA-1416907 from National Science Foundation
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Skyrms, B., Narens, L. Measuring the hedonimeter. Philos Stud 176, 3199–3210 (2019). https://doi.org/10.1007/s11098-018-1170-z
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DOI: https://doi.org/10.1007/s11098-018-1170-z