### Expected utility

In the 1930s and 1940s, economists reformulated economic analysis in terms of preferences, eliminating, seemingly once and for all, the troublesome notion of utility and the link between classical economics and utilitarianism. Almost immediately, however, the concept of cardinal utility theory was revived by von Neumann and Morgenstern in their analysis of behavior under uncertainty, and its application to game theory, based on the idea of expected utility maximisation. When faced with an uncertain prospect, under which any of a set of outcomes could occur with known probability, von Neumann and Morgenstern suggested attaching a numerical utility to each outcome and evaluating the prospect by calculating the mean value of the utilities. This procedure is feasible only for cardinal measures of utility.

Von Neumann and Morgenstern denied that the cardinal nature of the utility function they used had any normative significance, and most advocates of expected utility agreed. Savage (1954) warned against confusing the von Neumann–Morgenstern utility function with “the now almost obsolete notion of utility in riskless situations.” Arrow (1951) described cardinal utility under certainty as “a meaningless concept”. However, as Wakker (1991a, p. 10) observes

The same cardinal function that provides an expectation representing individuals’ preferences over randomized outcomes is also used to provide the unit of exchange between players. The applicability of risky utility functions as a means of exchange between players is as disputable as their applicability to welfare theory, or to any other case of decision making under certainty.

This view was adopted by Allais (1953), the most prominent critic of the expected utility model. Allais argued that a proper analysis of choice under risk required both a cardinal specification of utility as a function of wealth under certainty and a separate specification of attitudes towards uncertainty. Allais’ position has been strengthened by the development of the rank-dependent family of generalisations of the expected utility model (Quiggin 1982), in which there is a clear separation between diminishing marginal utility of wealth and risk attitudes derived from concerns about the probability of good and bad outcomes. These models have been combined iwth other generalisations such as the prospect theory of Kahneman and Tversky (1979).

The use of cardinal utility models of social choice has been encouraged by the popularity of contractarian models such as that of Rawls (1971). Rawls introduces the device of a ‘veil of ignorance’ behind which individuals choose social arrangements without knowing what place they will occupy in those arrangements. Rawls argues, largely on the basis of intuition about choices under uncertainty, that rational individuals will adopt a ‘maximin’ criterion, focusing on the worst possible outcome. This is an extreme form of the decision-weighting process represented in rank-dependent expected utility. From the maximin criterion of choice under uncertainty, Rawls derives his theory of justice based on concern for the worst-off members of the community. The approach used by Harsanyi (1953) may be interpreted in similar terms. Unlike Rawls, Harsanyi assumes that rational individuals seek to maximise expected utility. He therefore derives the conclusion that they will prefer utilitarian social arrangements.

Bibliography

Allais, M., (1987), The general theory of random choices in relation to the invariant cardinal utility function and the specific probability function: The (U, q) model - A general overview,, Centre National de la Recherche Scientifique Paris.

Arrow, K. (1951), ‘Alternative approaches to the theory of choice in risk-taking situations’, Econometrica 19, 404–437.

Harsanyi, J. (1953), ‘Cardinal utility in welfare economics and in the theory of risk taking’, Journal of Political Economy 61, 434–435.

Quiggin, J. (1982), ‘A theory of anticipated utility’, Journal of Economic Behavior and Organization 3(4), 323–43.

Rawls, J. (1971), A Theory of Justice, Clarendon, Oxford.

Savage, L. J. (1954), Foundations of statistics, Wiley, N.Y.

von Neumann, J. and Morgenstern, O. (1944), Theory of Games and Economic Behavior, Princeton University Press.

Wakker, P. (1991), ‘Separating marginal utility and probabilistic risk aversion’, paper presented at University of Nijmegen, Nijmegen.

Von Neumann and Morgenstern denied that the cardinal nature of the utility function they used had any normative significance, and most advocates of expected utility agreed. Savage (1954) warned against confusing the von Neumann–Morgenstern utility function with “the now almost obsolete notion of utility in riskless situations.” Arrow (1951) described cardinal utility under certainty as “a meaningless concept”. However, as Wakker (1991a, p. 10) observes

The same cardinal function that provides an expectation representing individuals’ preferences over randomized outcomes is also used to provide the unit of exchange between players. The applicability of risky utility functions as a means of exchange between players is as disputable as their applicability to welfare theory, or to any other case of decision making under certainty.

This view was adopted by Allais (1953), the most prominent critic of the expected utility model. Allais argued that a proper analysis of choice under risk required both a cardinal specification of utility as a function of wealth under certainty and a separate specification of attitudes towards uncertainty. Allais’ position has been strengthened by the development of the rank-dependent family of generalisations of the expected utility model (Quiggin 1982), in which there is a clear separation between diminishing marginal utility of wealth and risk attitudes derived from concerns about the probability of good and bad outcomes. These models have been combined iwth other generalisations such as the prospect theory of Kahneman and Tversky (1979).

The use of cardinal utility models of social choice has been encouraged by the popularity of contractarian models such as that of Rawls (1971). Rawls introduces the device of a ‘veil of ignorance’ behind which individuals choose social arrangements without knowing what place they will occupy in those arrangements. Rawls argues, largely on the basis of intuition about choices under uncertainty, that rational individuals will adopt a ‘maximin’ criterion, focusing on the worst possible outcome. This is an extreme form of the decision-weighting process represented in rank-dependent expected utility. From the maximin criterion of choice under uncertainty, Rawls derives his theory of justice based on concern for the worst-off members of the community. The approach used by Harsanyi (1953) may be interpreted in similar terms. Unlike Rawls, Harsanyi assumes that rational individuals seek to maximise expected utility. He therefore derives the conclusion that they will prefer utilitarian social arrangements.

Bibliography

Allais, M., (1987), The general theory of random choices in relation to the invariant cardinal utility function and the specific probability function: The (U, q) model - A general overview,, Centre National de la Recherche Scientifique Paris.

Arrow, K. (1951), ‘Alternative approaches to the theory of choice in risk-taking situations’, Econometrica 19, 404–437.

Harsanyi, J. (1953), ‘Cardinal utility in welfare economics and in the theory of risk taking’, Journal of Political Economy 61, 434–435.

Quiggin, J. (1982), ‘A theory of anticipated utility’, Journal of Economic Behavior and Organization 3(4), 323–43.

Rawls, J. (1971), A Theory of Justice, Clarendon, Oxford.

Savage, L. J. (1954), Foundations of statistics, Wiley, N.Y.

von Neumann, J. and Morgenstern, O. (1944), Theory of Games and Economic Behavior, Princeton University Press.

Wakker, P. (1991), ‘Separating marginal utility and probabilistic risk aversion’, paper presented at University of Nijmegen, Nijmegen.