Marginalist Expected Utility and the Frequentist Dead End
Cournot and De Morgan laid the basis for modern choice under uncertainty, and both underlined the conceptual problems related to the use of utility and probability for this purpose.
Schlee (1992) documented that during the so-called marginalist revolution, the expected utility was used to analyse a variety of decisions made under uncertainty, a result ascribed to the definition of a sound notion of marginal (decreasing) utility. From the point of view adopted here this explanation is only partial. A more correct one requires a distinction between scholars that in the last 30 years of the nineteenth century discussed these questions in reference to treatises on probability, ethics, psychology and philosophy in general, and those who considered the problem as exclusively belonging, so to speak, to the economic profession. William S. Jevons and Francis Y. Edgeworth are examples of the first kind of approach, Alfred Marshall and Arthur Cecil Pigou of the second.Bernoulli’s hypothesis was “self-evident” for Jevons who took its application not only to gambling but also to commerce for granted (Jevons 1879: 173-4). The basic structure of human reasoning “in selecting a course of action which depends on uncertain events, as, in fact, does everything in life”, consists in multiplying
the quantity of feeling attaching to every future event by the fraction denoting its probability. A great casualty, which is very unlikely to happen, may not be so important as a slight casualty which is nearly sure to happen. Almost unconsciously we make calculations of this kind more or less accurately in all the ordinary affairs of life; and in systems of life, fire, marine, or other insurance, we carry out the calculations to great perfection. In all industry directed to future purposes, we must take similar account of our want of knowledge of what is to be.
(Jevons 1879: 36)Behind all this there was not only a new idea of utility, but also a logical theory of probability developed in The Principles of Science (Jevons 1874). According to Jevons, probability is the “noblest creation of the intellect” (Jevons 1874: 200), in that it “deals with quantity of knowledge, an expression of which a precise explanation and measure can presently be given. An event is only probable when our knowledge of it is diluted with ignorance, and exact calculation is needed to discriminate how much we do and do not know” (Jevons 1874: 199). According to Jevons, the idea that probability “belongs wholly to the mind” (Jevons 1874: 198) does not signify that it is personalistic, as in De Morgan. On the contrary, it is the basis of “rational expectation” obtained “by measuring the comparative amounts of knowledge and ignorance” (Jevons 1874: 200). The principal value of the theory is normative, consisting “in correcting and guiding our belief, and rendering our states of mind and consequent actions harmonious with our knowledge of exterior conditions” (Jevons 1874: 199).
Marshall’s Principles of Economics (Marshall 1890 [1961]) professionalized the question of choice under uncertainty; it was treated in a footnote in chapter 6 of book 3 and in note 9 of the mathematical appendix. Marshall cited the introductory chapter of Jevons as his main reference. He used expected utility to solve truly economic problems. In particular:
to measure numerically the present value of a future pleasure, on the supposition that we know, (i) its amount, (ii) the date at which it will come, if it comes at all, (iii) the chance that it will come, and (iv) the rate at which the person in question discounts future pleasures. (Marshall 1890 [1961]: 135)
Its present value is the value of future expected utility (ibid.: 840). He then discussed the problem that the maximization of the expected utility of wealth is inconsistent with gambling at fair odds, introducing the idea that gambling may be explained if the pleasure of gambling is separately considered (Schlee 1992).
Marshall’s treatment paved the way to the (moderate) diffusion of expected utility hypothesis in economics - almost always associated with the assumption of diminishing marginal utility. Scholars applied it in order to explain advantages of fair insurance or the necessity of a reward for risk bearing (Schlee 1992: 737). Pigou (1924) treated risk or “uncertainty bearing” as a factor of production. On the whole, uncertainty in the post- Marshallian tradition was treated routinely and in a soft manner, probably following Marshall’s statement that the measurement of expected utility “belongs to Hedonics, and not properly to Economics” (Marshall 1890 [1961]: 840).
Edgeworth adopted a radically divergent view since he was completely engaged in the traditional vision according to which probability theory was the main problem (Baccini 2009: 2011). His reference point was the frequentist approach to probability, presented in the epoch-making work of John Venn The Logic of Chance (1888). Both Edgeworth and Venn agreed that a coherent frequentist theory prevented the cogent use of probability for choice under uncertainty for logical and philosophical reasons. From a logical point of view, frequentist probability is objective, it is defined as relative frequency, and may be applied properly only to a particular class of things: series. Probability serves to construct assertions such as, “The relative frequency of individuals with property R in population P is x”. If probability is not a property of individual events constituting the series, then it is invalid to infer from the preceding one an assertion such as “The probability that individual i, who belongs to population P, enjoys property R is x”. This has relevant effects for decision theory: if choices are exercised essentially around single events, probability can be of no help in the choice of actions, being unable to say anything at all about those events. As a consequence, probability cannot be used as a decision-making instrument other than in extremely rare cases where some very peculiar conditions hold (Baccini 2001).
The second point in the argumentation was related to the nineteenth-century debate on the theory of human action. According to the philosopher Alexander Bain (1859), belief is preparedness to act, a necessary condition for human action. In order to apply probability to decision theory, the relations between belief and probability had to be defined properly. At the end of different strategies of reasoning, Venn and Edgeworth concluded that probability is not a direct measurement of belief, that it does not necessarily give rise to belief, and, as a consequence, that it is not useful for a theory of human choice (Baccini 1997, 2001, 2007).At the turn of the nineteenth century two roads are clearly delineated for uncertainty in economics. The first shuts out uncertainty and expected utility from the professional toolbox of economists, limiting its use to the narrow problems of gambling and insurance. The second road, inherited by the secular tradition of probabilists and philosophers, faces the frequentist dead end.