Utilities: Transitivity, Multiple Dimensions, and the Voting Paradox Robert M. Hayes 2005

Utility Transitivity §In von Neumann and Morgenstern, the starting point for considering utilities is as follows: § §(1) For an individual the system of preferences is complete. That is, for any two alternative events, he is able to tell which he prefers. § §(2) The individual can compare not only events but combinations of events. § §From this starting point, they move to a set of axioms, interpreted as follows: §

Utility Transitivity §(A1) Completeness: for every u, v either u = v, u > v, or v > u and only one of those three applies §(A2) Transitivity: if u > v and v > w then u > w §(A3) Combining 1: if u > v then u > au + (1-a)v and if u < v then u < au + (1-a)v §(A4) Combining 2: if u > w > v then there is an a such that au + (1-a)v > w §(A5) Algebra of combining: if c = ab then a(bu + (1-b)v) + (1-a)v = cu + (1-c)v § §Each of these axioms, rational though they are, can be questioned. For example, it is by no means self-evident that an individual can assess his preferences among all possible alternative events. He simply may not know whether he prefers u to v or vice versa, and that does not mean that he is indifferent (i.e, that the utilities are equal). § §But it is the axiom of transitivity that I want to examine here.

Multiple Dimensions §To do so, I want to suggest that the choice among alternative events may involve consideration of several dimensions or aspects of those events. Indeed, in any real situation it usually will do so. §To illustrate, suppose I am at a restaurant and have the options of choosing a filet mignon steak, a salmon steak, or a pasta for my dinner. Which is preferable for me at that time? Clearly, I will make a choice, if only because I am hungry and the waitress is impatiently waiting for my decision. But I am torn in weighing several factors—what I will call the dimensions of the utility. For example: §(1) Price: pasta > salmon > filet (i.e., I prefer something less expensive, because I am frugal) §(2) Flavor: filet > pasta > salmon (i.e., I prefer something with richer flavor) §(3) Calories: salmon > filet >pasta (i.e., I prefer something with less calories)

Arrow's Voting Paradox §Now, recall Arrow's "voting paradox" theorem: §If there are at least three alternatives which the members of a society are free to order in any way, then every social welfare function (satisfying conditions 2 and 3) and yielding a social ordering satisfying Axioms 1 and 2 must be either imposed or dictatorial. §Condition 2 states, "If an alternative rises and does not fall in an ordering without change in the orderings and if it was preferred to another prior to that rise, it will continue to be preferred." §Condition 3 states, "If two individuals have separate sets of orderings and if, for both individuals and for all choices the relative rankings are the same, then the social choices will be the same." §The two axioms are the completeness and the transitivity axioms of von Neumann and Morgenstern. §If I treat the three dimensions as the voting members of the society of my mind, I face the paradox. §Now, obviously it can be argued that an internal debate within my own mind is hardly comparable to an external one between political alternatives. But nevertheless, the internal debate is real and the paradox holds.

Arrow's Voting Paradox §The two axioms are the completeness and the transitivity axioms of von Neumann and Morgenstern. §If I treat the three dimensions as the voting members of the society of my mind, I face the paradox. §Now, obviously it can be argued that an internal debate within my own mind is hardly comparable to an external one between political alternatives. But nevertheless, the internal debate is real and the paradox holds.

Societal Decisions §In fact, I will suggest that societal decisions imply such individual internal debates. The dimensions of the utility function apply within each individual and not just among the members of the society. Specifically, each individual must and does weigh the relative importance to himself of at least three dimensions: §(1) What does something cost? §(2) What is the balance of benefits and losses to me personally? §(3) What is the balance of benefits and losses to larger groups (to family, to society, to the world)? §Thus, assessment of the utility function involves at least two stages: §(A) Assessment of the relative position of alternative choices on each dimension. §(B) Assessment of the relative importance of the dimensions.