RangeVoting.org - Bentham, Bayes, Bernoulli, et al.

Jeremy Bentham, Thomas Bayes, Daniel Bernoulli, John Von Neumann, and some of the early history of utility theory

Utilitarianism

The writers who introduced the concept of "utility" were Jeremy Bentham (1748-1832), Thomas Bayes (1702-1761), and Daniel Bernoulli (1700-1782).

Jeremy Bentham, in his 1781 book Principles of Morals and Legislation (still in print), introduced the concept of "utility." He believed that utility in terms of pleasure and pain could be measured numerically on a scale unique up to an affine transformation. It also could be estimated by considering intensity, duration, and degree of certainty of pleasure or pain. One could use it and arithmetic to make interpersonal comparisons and to aggregate individual utilities into a social utility. The philosophy of "utilitarianism" – "the greatest happiness principle" – was invented by Bentham and has been very influential. Bentham wanted to deduce all moral and legal principles from this underlying principle plus logic, arithmetic, and experimental evidence.

How to make social policy decisions: For example (Bentham said), a legislator deciding how to vote on some proposed law should assess (perhaps with the aid of experiments) how much happiness or pain it would lead to for each individual, add these all up to get a social utility, and then vote the way that would maximize social utility.

Important writers often described as Bentham's followers include William Godwin 1756-1836, John Austin 1790-1859, James Mill 1773-1836, John Stuart Mill 1806-1873, Herbert Spencer 1820-1903, Bertrand Russell 1872-1970*, John Von Neumann 1903-1957, John C. Harsanyi 1920-2000*, and Peter Singer 1946- (the *s denote Nobel laureates). Other philosophical or social movements which trace some of their ideas to Bentham and the utilitarians include "social Darwinists," "libertarians," and "animal rights."

In mathematics and probability theory, the concept of "utility" is central to the "Bayesian" school, started by Thomas Bayes and now by far the most important framework in which to view statistics. Daniel Bernoulli also invented the notion of "utility" as a concept in probability theory and gambling distinct from "money." Bernoulli had arguments something like this to prove utility and money were not the same thing:

**Allais "paradox":** Given the choice between A and B, most people choose A. (By means of a large number of experiments of this ilk one could try to determine more precisely the curve expressing the relationship between utility and money.)
Gamble A		Gamble B
Winnings	Chance	Winnings	Chance
$1 million	100%	$1 million	89%
		Nothing	1%
		$5 million	10%

Bernoulli's (more dramatic) example: You are offered two choices.

I give you $1000.
You flip a fair coin. If your first N coin flips all are heads, then I give you $2^N/N. (If the first flip is tails, you get nothing.)

Most people choose (A) even though with (B) you get an infinite expected amount of money. Your expected winnings are ½ ∑_1≤N≤∞ 1/N=∞ because Prob(exactly N heads)=2^-N-1, and winnings(N) = 2^N/N if N≥1.

These examples prove "utility" is different from "money."

This framework was married to economics in Von Neumann and Morgenstern's book Theory of Games and Economic Behavior (1944 & 1947). This book made the important point that – although some authors had tried to replace "cardinal" (numerical) utilities with mere "ordinals" (non-numerical, defined only by preference relations), this was useless if one were trying to make decisions in the presence of random events or play "games of incomplete information." (Also, ordinals run into problems with "cycles.") The 1947 edition also presented the first axiomatic development of utility.

Bentham's decision-making procedure, note, is extremely similar to range voting. But oddly enough (as far as I can tell from, e.g, examining the book Leslie Stephen: The English Utilitarians, vols I, II, III, 1900), Bentham appears never to have actually thought of range voting. (Later note: Professor Claude Hillinger points out that Harsanyi did think of range voting, or something very similar, in the 1950s, and made the point it evaded Arrow's theorem. Online text of yet another book: Utilitarianism by John Stuart Mill 1879.)

By applying his procedure, Bentham deduced:

homosexuality should be legal (previously a crime called "sodomy"),
and so should the practice of loaning money with "interest" (previously a crime called "usury");
women and men should have equal political rights;
democracy was a good idea (and even people who were not rich should be permitted to vote);
slaves should be emancipated (gradually).

In all of these, Bentham's views eventually won out, although for the most part only after (mostly very far after) his death – and in some areas of the world and some religious groups these views are still not accepted, even today. A reform Bentham actually substantially succeeded in getting rolling during his lifetime was getting rid of mandated draconian punishments for minor crimes (such as the death penalty for over 200 crimes – including a child stealing twopence from a house – mandated flogging of women, etc.). Bentham indeed played a substantial part in writing the initial laws of many countries, including Australia and Portugal.

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