History of Utility Theory

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 1789 book Principles of Morals and Legislation, introduced the concept of "utility:"

The principle of utility is the foundation of the present work: it will be proper therefore at the outset to give an explicit and determinate account of what is meant by it. By the principle of utility is meant that principle which approves or disapproves of every action whatsoever. according to the tendency it appears to have to augment or diminish the happiness of the party whose interest is in question: or, what is the same thing in other words to promote or to oppose that happiness. I say of every action whatsoever, and therefore not only of every action of a private individual, but of every measure of government. By utility is meant that property in any object, whereby it tends to produce benefit, advantage, pleasure, good, or happiness, (all this in the present case comes to the same thing) or (what comes again to the same thing) to prevent the happening of mischief, pain, evil, or unhappiness to the party whose interest is considered: if that party be the community in general, then the happiness of the community: if a particular individual, then the happiness of that individual.

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. This is also called hedonistic utility since a hedonist pursues the increase of pleasure and the diminution of 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.

Bentham's decision-making procedure, note, is extremely similar to range voting. But oddly enough, Bentham and his utilitarian followers appear never to have actually thought of range voting.

For more on utilitarianism, check this book by Leslie Stephen: The English Utilitarians, 3 vols, 1900, (also available online here)

By applying his procedure, Bentham deduced:

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.

A refinement to Benthamite utility would be basing one’s utility not solely on pleasure vs pain but on enlightened self-interest. For instance, I might find pleasure in the consumption of a Big Mac, fries and an ice cream sundae, in other words in consuming sugar, salt and fat. However, it might be healthier for me in the long run to eat tofu, organic vegetables and free range chicken. Basing one’s utility on enlightened self-interest might be preferable to just basing it on pain and pleasure.

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."

John Stuart Mill refined the definition of utility by arguing that intellectual and moral pleasures are superior to more physical forms of pleasure. More on Millian utility here.

Up to this point utility theory had mainly to do with economics, in particular, normative economics. Normative economics is concerned with what should be, not what is. This later became welfare economics. The notion of utility as applied to voting methods came later. In 1938 Abram Bergson introduced the concept of a social welfare function. His conception was strictly an economic one. Kenneth Arrow1 in his book “Social Choice and Individual Values” (1951) created the field of social choice which applied the concepts of utility to voting systems as well as economic systems. It was assumed that each individual had a preference ordering over all alternatives. Then the social welfare function aggregated these individual preference orderings in such a way as to maximize social utility which was defined as the sum of the individual utilities. Arrow based his work on the utilitarian philosophy of Jeremy Bentham.2 “The utilitarian philosophy of Jeremy Bentham and his followers sought instead to ground the social good on the good of individuals. The hedonist psychology associated with utilitarian philosophy was further used to imply that each individual’s good was identical with his desires. Hence, the social good was in some sense to be a composite of the desires of individuals. A viewpoint of this type serves as a justification of both political democracy and laissez-faire economics or at least an economic system incolving free choice of goods by consumers and of occupations by workers.” So it was Arrow who tied the notion of utility to both economic and voting systems. However, concepts of utility which might be well suited to economics may not be well suited to voting systems, or, indeed, may have to be appropriately modified. Arrow based his notions of utility on preference orderings alone and this conception of utility is called ordinal utility. Instead of preference orderings it is possible to have utilities specified on the real number axis. This is called cardinal utility and is more in line with Bentham’s notion. Cardinal utility implies interpersonal comparisons of utility which means that one person’s utility can be compared to another’s. Arrow rejected such notions, but they are assumed in welfare economics. They are also assumed in voting systems in which all votes have equal power.

There is a distinction between “desire-based” utility and “choice-based” utility. A concept of utility which is based on desire satisfaction or is basically hedonistic is close to the Benthamite conception. On the other hand, “choice-based” utility assumes that one does not need to take into account the underlying pleasure or pain principle. All that is important in evaluating utility are the preference orderings (whether ordinal or cardinal) themselves. For instance, an individual who ranks or rates candidates in an election is said to have maximized his utility if his first choice wins irregardless of how much actual pleasure this state of affairs brings him. There is an implied interpersonal comparison of utility in the sense that anyone’s first choice would be considered to have maximized (to the same degree) that person’s utility regardless of how much pleasure that state of affairs actually brought to that individual. This is also called preference utility.

Arrow further distinguises between utility based on tastes and that based on values. Taste-based utility is similar to Benthamite or hedonistic utility. It only takes the individual’s selfish interests into account. Values-based utility takes the individual’s3 “general standards of equity (or perhaps his standards of pecuniary emulation)” into account. Arrow assumes that each individual orders social states not only in terms of his own rewards and reponsibilities but in terms of everyone else’s as well. Thus one would vote not only in terms of one’s own pleasures, but to increase or decrease others’ pleasures. One could vote to help others, thereby accruing perhaps a diminution of one’s one pleasure, or to hinder others thereby accruing supposedly some additional pleasure for oneself. Or one could vote the schadenfreude vote: to decrease someone else’s pleasure while not even necessarily increasing one’s own. According to Jonathan Riley4 “In effect, Arrow’s conditions lend support to his characterization of his approach as ‘ordinal utilitarian.’ There is no essential difference between Arrow’s normative premises and those of Benthamite utilitarianism. The essential difference is rather in respective factual assumptions, i.e., Arrow assumes that only purely ordinal utility information is feasible, whereas the classical utilitarian assumes that richer cardinally comparable utility information is feasible. Arrow’s famous ‘impossibility theorem’ effectively shows that his informational assumptions provide inadequate factual support for his normative premises to yield rational social choices. In other words, ordinal utilitarianism fails for want of adequate utility information, and not because of some inherent conflict among ‘seemingly mild’ utilitarian norms.”

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.

  1. I give you $1000.
  2. You flip a fair coin. If your first N coin flips all are heads, then I give you $2N/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) = 2N/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." The 1947 edition also presented the first axiomatic development of utility.

Moral and Ethical Considerations

There is a distinction between ‘rule utilitarianism’ and ‘act utilitarianism.’ According to John Harsanyi5: “'act utilitarianism' is the view that each individual act must be judged directly in terms of the utilitarian criterion. Thus a morally right act is one that, in the situation the actor is actually in, will maximize social utility. In contrast, 'rule utilitarianism' is the view that the utilitarian criterion must be applied, in the first instance, not to individual acts but rather to the general rules governing these acts. Thus a morally right act is one that conforms to the correct moral rule applicable to this sort of situation, whereas a correct moral rule is that particular behavioural rule that would maximize social utility if it were followed by everybody in all social situations of this particular type.”

John Harsanyi has proposed a dual structure taking into account one's selfish and altruistic motives6:

…John Harsanyi's important distinction between a person's 'ethical' preferences and his 'subjective' preferences: 'the former must express what this individual prefers (or, rather would prefer), on the basis of impersonal social considerations alone, and the latter must express what he actually prefers, whether on the basis of his personal interests or on any other basis'. This dual structure permits us to distinguish between what a person thinks is good from the social point of view and what he regards as good from his own personal point of view.

Here he is talking about an individual’s judgement regarding another individual. In a voting system it seems clear that the system should accept an individual’s vote at face value and not try to judge the underlying experiential utilities involved. Preference autonomy or preference utility should be the kind of utility measured and compared if possible.

Harsanyi has formulated the concept of preference utilitarianism which is7 "the principle that, in deciding what is good and what is bad for a given individual, the ultimate criterion can only be his own wants and his own preferences."

Thomas Scanlon makes this comment on Harsanyi's definition of Preference Autonomy:8

Now consider Harsanyi's claim that "in deciding what is good and what is bad for a given individual, the ultimate criterion can only be his own wants and his own preferences." If Harsanyi were claiming that it is a person's wants and preferences that make things good or bad for that person, then he would be saying something that appears on reflection to be false. There is, however, a more plausible reading of his principle of Preference Autonomy, which takes it to be making the moral claim that when we must decide what is good or bad for others, for the purposes, for example, of making decisions about the use of public resources, we should take those people's wants and preferences as our ultimate standard even though these are not what makes things good and perhaps … even though these wants and preferences may sometimes be mistaken.

Revealed Preference

Revealed preference is preference that an observer can infer from watching the choice an individual makes. For instance, if an individual is observed to choose A over B, he is said to have revealed a preference for A over B. According to Amartya Sen9,

…there are several related but different statements about a person's interests, actions, etc., that need to be distinguished, even though they are often identified in the literature: (1) the person gets more satisfaction in state x than in state y (statement about satisfaction or pleasure); (2) the person thinks that he or she is better off with x than with y (statement about introspective welfare); (3) the person is better off with x than with y (statement about individual welfare which may or may not be introspective); (4) the person prefers that x rather than y occurs (statement about the mental condition of preference, or desire, regarding states); (5) the person would like to so choose that x rather than y occurs (statement about desired choice); (6) the person believes that it would be right to so choose that x rather than y occurs (statement about normative judgement regarding choice); (7) the person believes that it would be better if x were to occur rather than y (statement about normative judgement regarding states of affairs); (8) the person so chooses that x rather than y occurs (statement about actual choice).

Sen does not think that using cardinal information rather than ordinal allows the escape from Arrow’s impossibility theorem, but the prescence of interpersonal comparability does.10

The Arrow informational format does not permit the use of cardinal information regarding individual utilities. This, however, proves to be not really a serious limitation, since the Arrow impossibility result can be generalized to cover cases in which the individual preferences are expressed as cardinal utility functions (see Theorem 8*2 in Collective Choice and Social Welfare). However, the absence of interpersonal comparability of individual utilities is a binding limitation, and its removal does indeed permit many rules satisfying all of Arrow's conditions, redefined for such a broader framework (see chapters 7 and 9 in Collective Choice and Social Welfare). It can be argued that it is the imposed poverty of the utility information that dooms Arrow's aggregation exercise to failure.

According to Paul Samuelson11,

The individual guinea-pig, by his market behaviour, reveals his preference pattern - if there is such a consistent pattern. If a collection of goods y could have been bought by a certain individual within his budget when he in fact was observed to buy another collection x, it is to be presumed that he has revealed a preference for x over y. The outside observer notices that this person chose x when y was available and infers that he preferred x to y. From the point of view of introspection of the person in question, the process runs from his preference to his choice, but from the point of view of the scientific observer the arrow runs in the opposite direction: choices are observed first and preferences are then presumed from these observations.

But what about indifference? How does one “reveal” that he is indifferent between two alternatives?12

The point can be illustrated with a variation af the classic story af Buridan's ass. This ass, as we all know, could not make up its mind between two haystacks; it liked both very much but could not decide which one was better. Being unable to choose, this dithering animal died ultimately of starvatian. The dilemma of the ass is easy to understand, but it is possible that the animal would have agreed that it would have been better off by choosing either of the haystacks rather than nothing at all. Not choosing anything is also a choice, and in this case this really meant the choice of starvation. On the other hand, had it chosen either of the haystacks, it would have been presumed that the ass regarded that haystack to be at least as good as the other, which in this version of the story was not the case. The ass was in a real dilemma vis-a-vis the revealed preference approach.

The traditional interpretation of the story is that the ass was indifferent between the two haystacks. That indifference may be a cause for dithering has often been stated. For example, Ian Little (I.M.D. Little, ‘A Reformulation of the Theory of Consumer’s Behaviour,’ Oxford Economic Papers, 1 (1949), P. 90) prefaced his closely reasoned attack on the concept of indifference by posing the rather thoughtful question: 'How long must a person dither befare one is really indifferent, since the loss from choosing one alternative rather than another is exactly zero. The person can choose either alternative and regret nothing in either case. This, however, is not the case if the preference relation is unconnected over this pair, i.e. if the chooser can neither say that he prefers x to y, nor y to x, nor that he is indifferent between the two.

If Buridan's ass was indifferent, choosing either haystack would have been quite legitimate and would not have misled the observer armed with revealed preference theory provided the observer chose a version of the theory that permitted indifference. The real dilemma would arise if the ass had an unconnected preference. Choosing either haystack would have appeared to reveal a view that that haystack was no worse than the other, but this view the ass was unable to subscribe to since it could not decide what its preference should be. By choosing either haystack it would have given a wrong signal to the revealed preference theorist since this would have implied that he regarded the chosen haystack to be at least as good as the other. There is very little doubt that Buridan's ass died for the cause of revealed preference, though -alas- he was not entirely successful since non-choice leading to starvation would have looked like the chosen alternative, at any rate from the point of view of mechanical use of the fundamental assumption of revealed preference. There was no way the ass could have rescued that assumption given its unconnected preference.

While traditional economists assume that man operates according to egotistical and selfish preferences, utilitarians are concerned that people operate according to a mixture of egoistic and altruistic considerations. In a voting system distinct from an economic system, it seems not to be necessary to engage in this debate. An individual’s submitted vote is a revealed preference. It can either be totally selfish, totally altruistic or somewhere in between. A voting system can accommodate a wide range of possible votes. On the other hand, there is no way of knowing what the true utilities of the voter for each alternative are. Because of the basic premise of any democratic voting system that all votes should have equal weight or power, all votes are made to be interpersonally comparable ipso facto. Therefore, a consideration of the “true” utilities for each voter is meaningless. In addition extended sympathy is of little value when the voter cannot possibly know what is best for anyone other than himself. To presume otherwise is paternalistic.

Some history is contributed by Kahneman and Varey13:

In the essay that initiated the modern analysis of decision making, Daniel Bernoulli (1738) proposed that people evaluate financial options by weighting the utilities of possible outcomes by their probabilities. His argument and his references to earlier writings by Gabriel Cramer identify utility as satisfaction - a subjective state or experience. Jeremy Bentham and John Stuart Mill also used the term utility to refer to the hedonic quality of experience. Bentham spoke of the two sovereign masters that govern mankind - pleasure and pain - and developed the notion of a 'hedonic calculus.' Indeed, the basic tenet of utilitarianism is variously referred to as 'the principle of utility' and as 'the greatest happiness principle.' Sen (1986) finds a similar interpretation of utility in more recent writings by many economists, notably Pigou. However, the modern view of utility has abandoned any explicit reference to hedonic experience or happiness. The positivistic movement that swept the social sciences between the two world wars gave us behaviorism, strict operational definitions, and a suspicious attitude to mentalist notions. In this spirit it was natural to seek a definition of utility in terms of observable choices - revealed preferences. The definition of utility in terms of choices still rules the sciences of decision, although operationalism and behaviorism have largely lost their hold on psychology.

It should be clear that the original and the modern notions of utility are different concepts, and that the use of the same term for both is likely to produce confusion. In referring to Bentham's concept we shall speak of experience utility: the hedonic quality of experience, broadly construed to include satisfaction as well as pleasure. The value associated with a particular consequence in a decision context is its preference utility. ...

The [two] notions of utility are separable in measurement as well as in principle, and how they relate to each other is an empirical issue. As will be shown, there is reason to believe that the correspondence between the measures is not always close. The empirical dissociation of experience and decision utility has significant implications: Unless it can be shown that experience utility and preference utility correspond very closely, any utilitarian ethics, and indeed any attempt to perform interpersonal comparisons of utility or intrapersonal comparisons of utility across time, must be explicit about the notion of utility that is involved. The task of interpersonal comparison of utility is best stated as a comparison of strengths of preference only if people know what will be good for them and if this knowledge is appropriately reflected in their preferences. However, if people do not know their future experience utilities, or if their preferences of the moment do not accurately reflect what they do know, a case can be made for using experience utility rather than preference as the unit of account in utilitarian calculations. The task of evaluating and comparing subjective experiences of utility will be central to such an approach. The task is not an easy one, but perhaps not as hopelessly impossible as the positivist dogma holds it to be. ...

Objectivist versus psychological approaches

There are characteristic differences between the approaches of various disciplines to the measurement and interpretation of utility. From the perspective of psychology, the methodological and conceptual presuppositions of the analysis of utility and choice in economics and in the formal sciences of decision share an objectivist core. In this section we sketch - or perhaps caricature - the objectivist position on utility and contrast it to the presuppositions that guide a psychological analysis of that topic.

We have already mentioned that the objectivist position favors a notion of utility based on publicly (objectively) observable choices. Several arguments bolster the implicit rejection of subjective experience as the criterion of welfare analysis. The methodological argument is the standard behaviorist one - that subjective experience is irrelevant to scientific analysis because it is not publicly observable. Sen (1986, p. 18) notes that "The popularity of this view in economics may be due to a mixture of an obsessive concern with observability and a peculiar belief that choice (in particular, market choice) is the only human aspect that can be observed." The substantive argument for measuring utility from choices is that people know what is good for them - or at least that they know it better than any observer, certainly better than any agency of society.

Interpersonal Comparison of Utility (ICU)

Many writers have chosen to let individuals deal with ICU through the notion of ‘extended sympathy.’ Individuals put themselves in the other person’s shoes when deciding how to vote and vote in such a way as to take the other person’s interests into account. Here is Scanlon's version: “It is just as good (no better and no worse) to be an individual with i's preferences in circumstances A as to be an individual with j's preferences in circumstances B." There are many problems with this approach, not the least of which is knowing how other people would want you to vote in order to best represent their interests. It seems much more feasible for each voter to vote in such a way as to represent their own interests instead of voting to promote 'the good of society'. However, in any democratic voting system, an individual is free to vote in any way he or she chooses from totally selfish to totally altruistic. A voter is free to vote completely hedonistically or completely masochistically. Because of this, it’s hard to see how utility, insofar as voting is concerned, has any other meaning than choice or preference based utility. Is the masochist’s utility based on how he votes or some notion of utility which would be based on the inverse of his vote because he’s a masochist? No, a masochist wants his first preference candidate to win as much as a hedonist does and his utility should be based on his choice not on his underlying psychology.

In Peter J Hammond’s survey paper14, we find the following quote from Robbins:

... I still believe that it is helpful to speak as if interpersonal comparisons of utility rest upon scientific foundations - that is, upon observation or introspection .... I still think, when I make interpersonal comparisons ... that my judgments are more like judgments of value than judgments of verifiable fact. Nevertheless, to those of my friends who think differently, I would urge that, in practice, our difference is not very important. They think that propositions based upon the assumption of equality are essentially part of economic science. I think that the assumption of equality comes from outside, and that its justification is more ethical than scientific. But we all agree that it is fitting that such assumptions should be made and their implications explored with the aid of the economist's technique. Robbins (1938, pp. 640-641)

His phrase “the assumption of equality comes from the outside,” while a dilemma for welfare economists, should not be a problem for voting theorists who ipso facto assume the equality of all votes. Thus the ethical assumption of equality is built in to all democratic voting systems and makes interpersonal comparisons as well as escape from Arrow’s impossibility theorem possible.

Harsanyi and Rawls have proposed a method of social choice in which each participant operates from a “veil of ignorance” so that his choice will be fair to all participants since he won’t be submitting only his own personal, selfish preferences. This is not appropriate to a voting system in which a person’s vote could be anything from totally selfish to totally altruistic. It is a restriction of a voter’s freedom of expression.

Hammond continues15:

"Impersonality" is the term used by Harsanyi (1953b, 1955) to describe the idea that, in order to free oneself from an unduly selfish perspective in weighing moral issues, an ethical observer should pretend to be completely uncertain which individual he will become after the issue is decided. In this formulation of utilitarianism, therefore, individuals are meant to choose as though behind a "veil of ignorance" - to borrow Rawls's (1971) felicitous term - uncertain what positions they will eventually occupy in the society being affected by the decisions under consideration. This Kantian idea is similar to Hare's (1951, 1963) principle of "universalizability," under which any person should only prescribe what he would still be willing to prescribe even if he were somebody else completely. And it is perhaps even close in spirit to what Rawls describes as "the original position" - see Rawls (1959,1971). Harsanyi and Rawls both use this concept of impersonality or the original position in order to arrive at alternative specific forms of social ordering. Harsanyi assumes that a person who acts as though he does not yet know who he is will be "Bayesian rational" and maximize the expected utility of a von Neumann-Morgenstern utility function, giving equal probability to becoming each possible individual in the society. Rawls, on the other hand, hypothesizes a much less orthodox view of behavior under uncertainty in the original position, which focuses upon the person who one would least like to be. This leads to his "difference principle." If one restricts oneself to a utilitarian framework (which Rawls does not), this would suggest maximizing the minimum utility level - that is, maximin.
John Von Neumann and Oskar Morgenstern introduced a probabilistic concept16:

3.3.2. Let us for the moment accept the picture of an individual whose system of preferences is all-embracing and complete, i.e. who, for any two objects or rather for any two imagined events, possesses a clear intuition of preference. More precisely we expect him, for any two alternative events which are put before him as possibilities, to be able to tell which of the two he prefers. It is a very natural extension of this picture to permit such an inclividual to compare not only events, but even combinations of events with stated probabilities. By a combination of two events we mean this: Let the two events be denoted by B and C and use, for the sake of simplicity, the probability 50% - 50%. Then the "combination" is the prospect of seeing B occur with a probability of 50%. We stress that the two alternatives are mutually exclusive, so that no possibility of complementarity and the like exists. Also, that an absolute certainty of the occurrence of either B or C exists. To restate our position. We expect the individual under consideration to possess a clear intuition whether he prefers the event A to the 50-50 combination of B or C, or conversely. It is clear that if he prefers A to B and also to C, then he will prefer it to the above combination as well; similarly, if he prefers B as well as C to A, then he will prefer the combination too. But if he should prefer A to, say B, but at the same time C to A, then any assertion about his preference of A against the combination contains fundamentally new information. Specifically: If he now prefers A to the 50-50 combination of B and C, this provides a plausible base for the numerical estimate that his preference of A over B is in excess of his preference of C over A. If this standpoint is accepted, then there is a criterion with which to compare the preference of C over A with the preference of A over B. It is well known that thereby utilitie - or rather differences of utilities become numerically measurable. That the possibility of comparison between A, B, and C only to this extent is already sufficient for a numerical measurement of "distances" was first observed in economics by Pareto. Exactly the same argument has been made, however, by Euclid for the position of points on a line - in fact it is the very basis of his classical derivation of numerical distances.

Probabilistic considerations were thought to be useful in helping an individual come up with his set of utilities based on his preference intensities so that he could translate these, for instance, into range votes. However, there are other methods for doing this and probabilistic considerations don’t seem particularly relevant for voting theorists although they may have their place in welfare economics. D. Ellsburg has been critical of the claim that von Neumann and Morgenstern had succeeded in synthesizing “measurable utility.”17

It is ten years since von Neumann and Morgenstern, in their famous aside to the economic profession, announced they had succeeded in synthesizing "measurable utility." That feat split their audience along old party lines. It appeared that a mathematician had performed some elegant sleight-of-hand and produced, instead of a rabbit, a dead horse. The most common reaction was dismay. To "literary" economists who had freshly amputated their intuitive feelings of cardinal utility at the bidding of some other mathematicians, it seemed wanton of von Neumann and Morgenstern so soon to sprinkle salt in their wounds with the statement: "It can be shown that under the conditions on which the indifference curve analysis is based very little extra effort is needed to reach a numerical utility." To others, who had said all along that surgery was unnecessary, the verdict was no surprise but still welcome, coming as it did from an unexpected (non-Cambridge) source. But before long both these groups had joined in expressing doubts that von Neumann and Morgenstern had succeeded in doing what (these readers believed) they had set out to do. The spokesman for the "cardinalists" interpreting their cause as his own, was forced to conclude that they "seem to me to have done as much harm as good to the cause to which they have lent their distinguished aid."

… What von Neumann and Morgenstern asserted, in their famous digression, was the possibility that the notion of maximising the mathematical expectation of utility might (a) be made meaningful, and (b) describe a wider range of risk-behavior than in its old usage, if "utility" were measured (defined) in a special way. Since they were concerned only with risk-behavior, the operation they proposed was the observation of choices in risk-situations. If a person's preferences among prospects — described merely in ordinal terms — should satisfy certain, apparently weak, axiomatic restrictions, then von Neumann has proved that it would be possible to find a set of numbers which could express these preferences in a particularly convenient way. This set of numbers would be a utility index, because it would be one among all the sets of numbers (related by monotonic transformations) expressing the person's preferences (not "intensities of preference" or "quantities of feeling") among sure outcomes. The novelty would be that this same set of numbers, applying explicitly only to sure outcomes, could also summarise the person's preferences among prospects. In a complete description of the individual's entire preference-structure, it would be unnecessary to list prospects separately or to record explicitly his preferences among prospects; these preferences would be known, through observation, but they could be expressed implicitly by the numbers attached to sure outcomes.

The Measurability of Cardinal Utility

Edgeworth proposed a mathematical approach to utilitarianism he called hedonimetry18:

Marrying utilitarianism and psychophysics, Edgeworth synthesized the works of Bentham and Fechner. Following Fechner, he postulated the unit of utility to be a just perceivable increment of pleasure. But he went further. In order to be able to compare the welfare of different persons, he also chose a just perceivable increment to be an interpersonal unit of utility. Edgeworth was fully aware that these two postulates were value judgements, or - in his own terminology - "incapable of proof", but he felt strongly that comparison of happiness between persons and groups of persons was an important scientific task: “Such comparison can no longer be shirked, if there is to be any systematic morality at all. It is postulated by distributive justice. It is postulated by the population question.” EY. Edgeworth (I881, pp. 7-8)

One may easily attack Edgeworth for being arrogant and male chauvinist, but nevertheless, in this passage he goes right to the heart of utilitarianism. Allocating a pound between two persons, utilitarianism prescribes giving it to the person who would "light up" most, rather than, for example, the person initially less happy. The Italian economist Vilfredo Pareto denied that interpersonal comparisons of utility was a meaningful operation (Pareto, 1971, p. 192). Still his construction of a cardinal utility index resembles that of Edgeworth: “a man can tell fairly well whether in passing from combination I to combination II, he experiences a greater pleasure than in passing from combination II to another combination III. If his judgement could be of sufficient precision, he would be able, at the limit, to know whether in passing from I to II this man experiences a pleasure equal to that which he experiences in passing from II to III. And consequently, in passing from I to III he would experience a pleasure double that which he gets in passing from I to II. That would be enough to permit us to consider the pleasure or ophelmity as a quantity.” On the same page Pareto argues that such perfect precision is not attainable. Pleasure is only imperfectly measurable. Note that Edgeworth's and Pareto's ways of defining measurable utility/ophelmity are not identical.

The concept of measurability of cardinal utility was also taken up by Erik Gronn19:

Suppose, one day, the temperature in Oslo is 15°C, the temperature in Rome is 30°C, and the temperature in Cairo is 40°C. Then everybody will agree that it is hotter in Rome than in Oslo, and still hotter in Cairo. This conclusion will be reached regardless of the way different people have learned to measure temperature. Will we further say that the temperature-difference: between Oslo and Rome is greater than the temperature-difference between Rome and Cairo? We probably will. But how many will insist that it is twice as hot in Rome as in Oslo? Probably very few.

The usual way to measure a quantity is to attach a numerical value (a real number) to the quantity, with the tactical understanding that we intend to employ the ordering-property ("a is less than b") of the system of the real numbers to the measures.

But now we must be careful: The difference and the ratio between two numbers are also numbers, but this does not mean that we are automatically willing to apply the concepts of the ordering-relation of the real numbers to the differences between (the measures of) two quantities or the ratios between (the measures of) two quantities. We will in this paper try to delineate exactly what "we can do" with the measures of measurable quantities. Our first task is then to settle what we shall mean by the term "measurable quantities". An absolute requirement for measurements to be meaningful is that the ordering between (the measures of) two quantities should be independent of every (acceptable!) way of measuring the two quantities. The temperature at a certain place, at a certain time is measurable, because at a fixed point in time the temperature at place A, measured in degrees Celsius, is greater than the temperature at place B, measured in degrees Celsius if and only if the temperature at place A is greater than the temperature at place B measured in any other acceptable way of measuring temperature.

We can characterize measurable quantities further according to whether or not the ordering properties of the real numbers can be applied to certain combinations of the measures of measurable quantities. If the differences between the measures of two quantities is ranked independent of every acceptable measure, we say that the quantities are cardinally measurable or just cardinal. If not, the measurable quantities are ordinally measurable or just ordinal. (These definitions will be refined in the next section.) Thus, temperature should be termed a cardinal variable. This is in the final analysis more a convention, a reflection of how people have become accustomed to think about these matters, than a statement about how the world "really is". The convention is based upon a compromise between what is relevant and what is objectively attainable. In many cases we are interested in what the temperature means to us. Then two temperature-differences of equal Celcius-size do not necessarily mean the same thing: On a normal winter day in Oslo a change in temperature from -2°C to -6°C could be far less important than a change from, say, -20°C to -24°C. We can accordingly, for a single individual, imagine a temperature-scale based on "intensities of dislike-like", "willingness to pay", etc., that cannot be transformed linearly into the Celsius-scale. The resort to the usual ways of measuring temperature represents an improvement in scientific objectivity and facilitates interpersonal comparisons, but something may have been lost in essentially.

If also the ratio between the measures of two quantities is always greater than the ratio between the measures of two other quantities, independent of the methods of measurements, the quantities will be called purely cardinal. Temperature is not a purely cardinal quantity; the weight of a sack of potatoes is. Suppose we measure sacks of potatoes in kilograms. If sack A weighs 50 kg and sack B weighs 25 kg, the weight of sack B is just half the weight of sack A, irrespective of the method of measurement. Accordingly, the ratio between the weights of sacks A and B is fixed, so is the ratio between the weights of sacks C and D, and the result, that sacks of potatoes are purely cardinally measurable, follows.

Is there a place for these distinctions in economic theory, and are they important in that theory? The answer to the first question is an unequivocally yes; the answer to the second question is more debatable. The technical term for a person's well-being, satisfaction, happiness etc. from consuming a certain commodity-bundle is called the person's utility. An old debate is whether utility is measurable as a cardinal quality or only as an ordinal quantity. … It was further questioned whether it was possible to make meaningful in an empirical sense the evaluation of the intensities of the preferences which seemed necessary for a cardinalistic utility measure. Accordingly, cardinalistic concepts were gradually expelled from economic theory, particularly after the "Hicks-Allan revolution" in the 1930s. This is, however, not an accurate description of the practice of economic theorists: Cardinal utility is used freely, with the apology, when pressed, that it is only used for "illustrative purposes", the emphasize more effectually a point which is also valid with only ordinal utility, etc. Anyhow, as the saying goes, the distinction between cardinal and ordinal utility is not an important distinction.

But it can be argued that this distinction is not necessarily entirely innocuous. Is it true that the ban on cardinal utility really represented a scientific progress? As pointed out by Cooter and Rappoport (1984), economic theory did not provide better answers to old questions when cardinal utility went out of fashion; instead economic theory posed new questions and deemed some of the old questions meaningless, not worthy of scientific inquiry. For instance, many economists around the turn of the century were passionate about income distribution and economic welfare. They wanted, among other things, to find out what better housing really "was worth" and sought justifications for redistributing income. If differences in utility levels could be compared, within a person and between persons, we can find a scientific basis for an activist welfare policy. But this seems to require a cardinalistic utility concept; ordinalists will judge these endeavours as being "unscientific". Cooter and Rappoport (1984) listed some of the questions the cardinalists wanted to answer and the ordinalists claimed were unanswerable by economics: Is a dollar more valuable to the average poor person than to the average rich person? Should economists give different weight to additional income for the rich and poor when doing cost-benefit analysis? Does a hungry person need food more than a bored person needs theater tickets? If income is redistributed to the poor, with no change in total income, does national welfare go up or down? Is there an economic justification for progressive income tax schemes?

Our concept of meaningfulness of cardinal utility is the following: Cardinal utility is meaningful if there is a canonical way of picking out a subset, closed under only positive linear transformations, of representations of the preference field. Well known examples are the different forms of separability properties, mentioned for instance by Debreu (1960) and Koopmans (1960). Their results are briefly commented upon in Section 4, where we also refer to results by Fishburn (1978), which succeeds in generating a cardinalistic utility concept from certain axiomatizations governing the individual's behaviour when the individual's lifetime is uncertain.

Again the concepts of utility are more complicated for welfare economists than they are for voting theorists. Welfare economists seem to have a built-in bias for using utilitarianism to help the less well off. They seem to assume that a rational person would place himself in the place of a less well off person and through “extended sympathy” choose a social state which benefitted the less well off person more than himself. This, in general, is not true of human nature. More often we have witnessed the opposite: the advantaged person uses his or her power to act in his or her own best interests and not in the interests of the less advantaged.

Ole Hagen20 claims to have come up with a way of assigning meaningful cardinal utilities without using any probabilistic methods such as those of von Neumann and Morgenstern:

I have (Hagen, 1984) extensively argued that the normal consumer can distinguish between pairs of goods that are complements and those that are substitutes. The main point can be summarized in one of the "Bartender's Games"; By the pitch of a coin you will get either whisky or gin, also soda or tonic. Do you prefer soda with whisky and tonic with gin, or soda with gin and tonic with whisky? Tastes differ. I was intrigued watching an Indian lady pour salt over a slice of orange. My horror at seeing her actually eat it with great pleasure can only be compared to that of the English bartender receiving an order for gin and Coca-cola (In the same glass, Sir?!). The diversity in tastes does not diminish the clarity and strength of individual feelings, at least in extreme cases, of what goods go together and what are substitutes or, even stronger, should be kept apart for physiological reasons (alcohol and antabus, apple vinegar and sugar). Let us revisit the friendly bartender. This time he offers for sure: whisky on odd dates and gin on even dates, but tonic with whisky and soda with gin. We can, however, have soda instead of tonic on m days or tonic instead of soda on n days. A person's choice will tell that not only does he know the sign of his second order derivatives, he also has a pretty good idea of the ratios between them, which implies cardinality. (Of course an ordinal preference function can describe his preferences by adding to the original 4 arguments another 4: the two "normal" and the two "perverse" combinations.)
Finally, Claude Hillinger in his paper, “Voting and the Cardinal Aggregation of Judgments,” (online here: ) has some interesting comments:

The paradoxes of social choice arise from the fact that the ordinal scales that are taken as the starting point are also context dependent. On such a scale, the distance between two alternatives is given by the number of intermediate alternatives and changes as these are added or subtracted. This is the reason for the violation of Arrow’s axiom of independence of irrelevant alternatives. Both Fleming (1952) and Harsanyi 1955, 1977) have shown that social choice procedures that satisfy reasonable conditions are possible, as long as preferences are expressed by means of an independent cardinal scale. For a given normalization, each alternative has a unique value on this scale and the problem of irrelevant alternatives does not arise. The present paper may be interpreted as an operationalization, in the electoral context, of their abstract theories.

‘Utility’ and ‘preference’ are the most fundamental concepts of economics and, coming from there, have assumed a similar importance in political science, particularly in voting theory. The meaning of these concepts has, in spite of their centrality, by no means been satisfactorily clarified. On the contrary, I will argue that there are deep seated confusions both regarding the relevance of alternative concepts and about how to operationalizes them.

Both terms have a wide range of meanings, with considerable overlap. In economics and social choice theory ‘preference’ is linked directly to choice; the preferred alternative, if available will be chosen. The associated mathematical concept is that of a preference ordering, either of the weak form, ‘a is not worse than b’, or of the strict form, ‘a greater than b’. ‘Utility’, in a theoretical context, is usually defined as a numerical value given by a function u = f(a), where a may be a discrete alternative, as in voting theory, or a vector of continuous variables as in consumer demand theory. Choice is then determined indirectly by assuming that the alternative chosen from the available set is the one that maximizes u. If the choice is invariant under a monotone transformation of f(.) , as is the case in the static theory of consumer choice, utility is said to be ordinal. If choice remains invariant only under a positive linear transformation of f(.), we speak of cardinal utility. A monotone transformation leaves the ordering of the alternatives unchanged. A linear transformation leaves in addition the ordering of the utility differences of the alternatives unchanged. These utility differences are usually interpreted as preference intensities. Cardinal utility has been central to the theory of choice under uncertainty. According to Harsanyi, it is also central to the theory of social choice.


1. K. Arrow, Social Choice and Individual Values (New Haven: Yale University Press, 1951, 1963 Second Edition)
2. ibid p. 22
3. ibid p. 18
4. J. Riley, Liberal Utilitarianism (Cambridge: Cambridge University Press, 1988) p. 22
5. J. Harsanyi, Morality and the Theory of Rational Behavior in "Utilitarianism and Beyond" edited by Amartya Sen and Bernard Williams(Cambridge: Cambridge University Press, 1982) p. 39
6. A. Sen, Choice, Welfare and Measurement(Boston: MIT Press, 1982) p. 99
7. Harsanyi, op. cit. p. 55
8. T. Scanlon, Moral Basis of Interpersonal Comparisons in "Interpersonal Comparisons of Well-Being" edited by Jon Elster and John Roemer (Cambridge: Cambridge University Press, 1991) p. 26
9. A. Sen, Choice, Welfare and Measurement(Boston: MIT Press, 1982) pp. 8-9
10. ibid p. 18
11. P. Samuelson, 'A Note on the Pure Theory of Consumer's Behaviour', Economica, 5 1938
12. Sen, op. cit. pp. 61-62
13. D. Kahneman and C. Varey, Notes on the Psychology of Utility in "Interpersonal Comparisons of Well-Being" edited by Jon Elster and John Roemer (Cambridge: Cambridge University Press, 1991) pp. 127-129
14. P. Hammond, Interpersonal Comparisons of Utility: Why and how they are and should be made in "Interpersonal Comparisons of Well-Being" edited by Jon Elster and John Roemer (Cambridge: Cambridge University Press, 1991) pp. 200-254
15. ibid p. 212
16. J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior(Princeton: Princeton University Press, 1944)
17. D. Ellsburg, Classic and Current Notions of "Measurable Utility" in "Interpersonal Comparisons of Well-Being" edited by Jon Elster and John Roemer (Cambridge: Cambridge University Press, 1991) p. 269
18.T. Ellingsen, History of Hedonimetry in "Cardinalism" edited by Maurice Allais and Ole Hagen (Dordrecht: Kluwer Academic Publishers, 1994) pp. 111-112
19. E. Gronn, Cardinalism and Dynamic Analysis in Economic Theory in "Cardinalism" edited by Maurice Allais and Ole Hagen (Dordrecht: Kluwer Academic Publishers, 1994) p. 167
20. O. Hagen, From Ordinal to Cardinal Utility in "Cardinalism" edited by Maurice Allais and Ole Hagen (Dordrecht: Kluwer Academic Publishers, 1994) p. 212

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