Harsanyi and utilitarianism
John Harsanyi (1920-2000), in two papers (1953, 1955; see also Harsanyi 1976), considers social choice in risky environments. In his 1955 paper, he presents a major result that, in some sense, justifies utilitarianism, more precisely a weighted version of utilitarianism, from a technical perspective.
The set of social states is supposed to be a set of lotteries, that is, the probability distributions over a finite set of prizes. If all probabilities are permitted, this set is infinite and uncountable. Individuals have a preference over this set of lotteries given by a complete preorder and representable by a utility function. In the lotteries setting, rather than using ordinal utility functions, economists generally adopt a framework developed by John von Neumann and Oskar Morgenstern (1953) for dealing with risky situations in game theory, the so-called von Neumann-Morgenstern utility functions. In fact, given appropriate assumptions on the set of lotteries and the set of complete preorders over the set of lotteries (some of these assumptions are topological assumptions), it can be shown that there exists a utility function representing the complete preorder and having the so-called expected utility property. If we assume that a lottery x is given by k prizes x1,..., xk and a probability distribution p1,..., pk - pi being the probability of receiving prize xi - the utility function u is said to satisfy the expected utility property if:
This means that the utility associated with the lottery x is the sum of the utilities associated to the prizes, weighted by the probabilities of receiving these prizes. An important consequence of this property is that the utility function is not only unique up to an increasing transformation, as in the case of ordinal utility functions, but unique up to a specific form of the increasing transformation: a positive affine transformation; if u is a von Neumann-Morgenstern utility function representing a complete preorder ', then r = au + β, where a and β are real numbers and a > 0, is also a von Neumann- Morgenstern utility function representing this same s'. As a major consequence, the differences of utilities can be compared according to the relation ≥, which was not possible for ordinal utility functions.
Such functions are said to be cardinal. In the following presentation which is essentially due to Weymark (1991) three conditions are introduced. Since the prizes are fixed, a lottery will be assimilated to the associated probability distributionp = (p1,..., pk).In (1), it is not said that the weights attached to the individual utility functions are positive. With a negative weight, an increase in individual utility would decrease social welfare. The result in (3) can be considered as the theorem about weighted utilitarianism.
A number of authors have challenged the use of von Neumann-Morgenstern utility functions. Diamond (1967) in particular has criticized the assumption that the social preference could satisfy the assumptions introduced by von Neumann and Morgenstern. Furthermore, some authors have criticized Harsanyi by arguing that his utilitarianism could not be associated with classical utilitarianism, mainly because of the von Neumann-Morgenstern framework (see Sen 1976; Roemer 1996; and contributions in Fleurbaey et al. 2008).