THE UTILITY ANALYSIS OF CHOICES INVOLVING RISK 28I The shift from the kind of utility ginal utility is unnecessary to explain sis employed by Marshall to the riskless choices, writers have continued ference-curve analysis of F. Y to reject maximization of expected utility worth, Irving Fisher, and Vilfredo Pa- as"unrealistic. This rejection of maxi- reto revealed that to rationalize riskless mization of expected utility has been choices, it is sufficient to suppose that in- challenged by John von Neumann and dividuals can rank baskets of goods by Oskar Morgenstern in their recent book, total utility. It is unnecessary to suppose Theory of Games and Economic behavior that they can compare differences be- They argue that "under the conditions tween utilities. But diminishing, or in- on which the indifference curve analysis creasing, marginal utility implies a com- is based very little extra effort is needed parison of differences between utilities to reach a numerical utility, ' the ex and hence is an entirely gratuitous as- pected value of which is maximized in sumption in interpreting riskless choices. choosing among alternatives involv The idea that choices among alterna- risk. 8 The present paper is based on their ives involving risk can be explained by treatment but has been made self-con the maximization of expected utility is tained by the paraphrasing of essentia ancient, dating back at least to D. Ber- parts of their argument noulli's celebrated analysis of the St. If an individual shows by his market Petersburg paradox. It has been re peatealy referred to since then but almost 6"It has been the assumption in the classical invariably rejected as the correct expla- question will always try to maximize the mathe- ing belief in diminishing marginal utility may appear plausible, but it is certainly not an his nation--commonly because the prevail- matical made it appear that the existence of surmpilointed out tust hold true in all cases. It has t the individual may also be gambling could not be so explained. interested in, and influenced by, the range or the Even since the widespread recognition standard deviation of the different derived or some other measure of that the assumption of diminishing mar- pears pretty evident from the behavior of e In aniel Bernoulli, Versuch einer neuen influenced by the skewness of the probability dis- Theori Wertbestimmmung won Gliicksfaillen (L tribute hard Tintner, "A Contribution to zig, I896), translated by A Pringsheim from"Speci- the Non-Static Theory of Choice, "Quarterly Journe politanae, Vol. V, for the years I730 and I731, ourselves to the mathematical expectation In an interesting note appended to his Marschak, "Money and the Cramer[1704-52 ), a famous mathemat Econometrica, VI [1938], 320 letter in French by Cramer contain what, to us, is consistent with maximization of expected utility is the truly essential point in Bernoullis paper, name- erroneous(see secs. 3 and 4 below). He is led to ly, the idea of using the mathematical expectation consider a formally more general solution because of his failure to appreciate the real generality of the the mathematical on of income to com- kinds of behavior explicable by the maximization of pare alternatives involving risk. Cramer has not expected utility. in general been attributed this much credit apparently because the essential point in Ber- Princeton University Press, Ist ed, 194 ed 1947: pp. I5-3I(both eds ) pp, that the logarithm of income is an appropriate ed, only); succeeding references are to 2d ed s Ibid., P. I7