Von Neumann and Morgenstern’s Utility
According to Ellsberg, “it was the feeling that the emphasis on mathematical expectation was arbitrary and unrealistic which led to the decline of the concept even before doubt arose that a measurable utility could be discovered to make it meaningful” (1954: 537).
The decline of the Bernoulli-Marshallian tradition (Marschak 1938; Tintner 1942; Friedman and Savage 1948; Arrow 1951) was stopped by the appearance of von Neumann and Morgenstern’s book (1944), containing a complete and, to professional economists, satisfactory axiomatic treatment of choice under uncertainty. It represented a dramatic break in continuity and opened the way to a central field in twentieth-century economic analysis.Von Neumann and Morgenstern’s discussion of the question of utility was “mainly opportunistic” (von Neumann and Morgenstern 1953: 8). They needed a cardinal measure of utility to be considered “identical” to “money or a single monetary commodity” (unrestrictedly divisible, substitutable and transferable) (von Neumann and Morgenstern 1953: 8). Their goal was to define a cardinal notion of utility without following the unsatisfactory patterns of their predecessors. In particular they were interested in a cardinality not based on a more or less introspective measure of pleasure or satisfaction derived from goods. They tried also to divorce their approach from cardinality based on the comparability of preference differences (Fishburn 1989: 131). The solution consisted in the definition of numerical utility “as being that thing for which the calculus of mathematical expectation is legitimate” (von Neumann and Morgenstern 1953: 28). The basic operation in deriving a cardinal utility index is the analysis of a situation where an agent is choosing between a sure outcome and two possible outcomes with given probabilities. In order to fix the origin and unit of the utility index, arbitrary numbers are assigned to the two outcomes A and B, with an order respecting the order of preference of the agent.
For the sake of simplicity, set the utility of the worst outcome A to Ua = 0, and of the best B to Ub = 1. Consider now a third outcome C which the agent ranks between the first two. The utility index of C is the probability p at which the agent is indifferent between having C with certainty or participating in a lottery with a prob- abilityp of winning B, and (1 -p) of winning A. Thus Uc = pUB + (1 -p)UA = p.The same reasoning may be applied to define other utility numbers reflecting other choices consistent with this last one for other intermediate outcomes. If, for example, an alternative outcome D which the agent ranks between C and B is considered, the utility index is Ud = q where q is the probability for which the agent is indifferent between having D with certainty or participating in a lottery with a probability q of winning B and (1 - q) of winning C. The only consistency requirement is that Ub > Ud > Uc. It is therefore possible to elicit the entire utility function of the agent (for modern expositions, see Binmore 2009; Gilboa 2009); the utility index so defined is unique up to a linear transformation that is a cardinal measure of utility. It is defined directly through the observation of choices of agents in risk situations, and it allows us to describe agents with different patterns of behaviours in risk situations. Von Neumann and Morgenstern derived thisoperational result from an “axiomatic treatment of numerical utilities”. A “controversial” axiom is also introduced, according to which the agent is indifferent between two possible outcomes which are derivable from each other according to the rules of probability. On this basis it is possible to reduce systematically to a simple lottery lotteries in which prizes are other lotteries. The controversial nature of this axiom rests on the fact that it constrains the agent to be indifferent about the number of steps of the gamble, since the interest lies only in the final possible outcome, and not in intermediate possible winnings (Ellsberg 1954: 543).
Now it is finally possible to calculate the expected utility of different possible outcomes as EU = ∑piU(xi), where U(xi) are the utility values constructed with the operations described above regarding possible outcomes (xi), and not the utility values of riskless outcome, as in the Bernoulli-Marshallian tradition. Consider two even lotteries with prizes respectively A and D, and C and D, as defined above. The expected utility of the first one is EU1 = 0.5 ? Ua + 0.5 ? Ud = 0.5p, and of the second one EU2 = 0.5 ? Uc + 0.5 ? Ud = 0.5p + 0.5g; the two lotteries can be ordered in relation to their expected utilities EU2 > EU1; the rational agents choose respecting this order. The choice of agents can be described as they are maximizing their expected utility.From our point of view, the nature of probability must still be discussed. Von Neumann and Morgenstern’s construction used probability as an individual numerical estimate of utility. They therefore stated clearly that a “subjective concept” of probability “would not serve” their purpose. They insisted “upon the alternative, perfectly well founded interpretation of probability as frequency in long runs. This gives directly the necessary numerical foothold” (von Neumann and Morgenstern 1953: 19). Moreover they did not cite any authors on this point. It is therefore possible to conjecture that they were unaware of, or considered irrelevant, the discussion about the proper domain and applicability of frequentist probability. They evidently considered the discussion on the nature of probability involved in their theory as a minor point. This interpretation is supported also by a footnote in which they suggested that it is possible to make a joint axiomatization of probability and utility (von Neumann and Morgenstern 1953: 19, fn 2), without resorting to statistical probability.
Von Neumann and Morgenstern’s book opened a lively discussion among economists in the course of which emerged the complete professionalization of the theme. The first question was that of cardinal utility: Von Neumann and Morgenstern were accused of taking economics back to the pre-Pareto and Hicksian era (Baumol 1951). The second concerned the kind of probability assumed. This probability, as underlined by Savage, “can apply fruitfully only to repetitive events” and cannot be used to elicit “which of several actions is the most promising” because probability is not assigned to the truth of propositions (Savage 1954 [1972]: 4). It was through searching for a solution to this last problem that Savage generalized the structure of the von Neumann and Morgenstern utility.