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A history of the Allais paradox

Published online by Cambridge University Press:  08 August 2014

Institute for Management Research, Radboud University Nijmegen, Holland. Email:


This article documents the history of the Allais paradox, and shows that underneath the many discussions of the various protagonists lay different, irreconcilable epistemological positions. Savage, like his mentor von Neumann and similar to economist Friedman, worked from an epistemology of generalized characterizations. Allais, on the other hand, like economists Samuelson and Baumol, started from an epistemology of exact descriptions in which every axiom was an empirical claim that could be refuted directly by observations. As a result, the two sides failed to find a common ground. Only a few decades later was the now so-called Allais paradox rediscovered as an important precursor when a new behavioural economic subdiscipline started to adopt the epistemology of exact descriptions and its accompanying falsifications of rational choice theory.

Research Article
Copyright © British Society for the History of Science 2014 

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1 Savage took the exchange rate at the time to be 350 francs to the dollar, which makes the 100 million francs equal roughly 285,000 1952 dollars. Similarly, 500 million francs equals about 1.4 million 1952 dollars.

2 Allais, Maurice, ‘Le comportement de l'homme rationnel devant de le risque: Critique des postulats et axioms de l’école americaine’, Econometrica (1953) 21, pp. 503546CrossRefGoogle Scholar, 527.

3 Daston, Lorraine, Classical Probability in the Enlightenment, Princeton: Princeton University Press, 1988Google Scholar; Hacking, Ian, The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference, Cambridge: Cambridge University Press, 1975Google Scholar.

4 As such, this article derives from, and elaborates upon, Chapter 2 of Heukelom, Floris, Behavioral Economics: A History, New York: Cambridge University Press, 2014CrossRefGoogle Scholar.

5 Bernoulli, Daniel, ‘Exposition of a new theory on the measurement of risk’ (1738), Econometrica (1954) 22, pp. 2336CrossRefGoogle Scholar.

6 For histories of Bernoulli's and others' solutions to the St Petersburg paradox see Jorland, Gerard, ‘The Saint Petersburg paradox 1713–1937’, in Kruger, Lorenz, Daston, Lorraine and Heidelberger, Michael, The Probabilistic Revolution 1, Cambridge, MA: MIT Press, 1987, pp. 157190Google Scholar; Teira, David, ‘On the normative dimension of the St. Petersburg Paradox’, Studies in History and Philosophy of Science (2006) 37, pp. 210223CrossRefGoogle Scholar; Jallais, Sophie, Pradier, Pierre-Charles and Teira, David, ‘Facts, norms and expected utility functions’, History of the Human Sciences (2008) 21(2), pp. 4562CrossRefGoogle Scholar; and Basset, Gilbert W., ‘The St. Petersburg paradox and bounded utility’, History of Political Economy (1987) 19, pp. 517523CrossRefGoogle Scholar. As a synonym for moral wealth the original Latin text used the term emolumentum, which in the English translation of 1954 (by, among others, Baumol) was translated as ‘utility’. This translation is somewhat unfortunate as it suggests a pre-Benthamite use of utility as a measurement of pleasure derived from wealth.

7 Edwards, Ward, ‘The theory of decision making’, Psychological Bulletin (1954) 51, pp. 380417CrossRefGoogle ScholarPubMed; Jorland, op. cit. (6).

8 This paragraph briefly indicates a few points in a large literature. Useful overviews include Eriksson, Lina and Hájek, Alan, ‘What are degrees of belief?’, Studia Logica (2007) 86, pp. 185215CrossRefGoogle Scholar; Hájek, Alan, ‘Interpretations of probability’, in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy, 2007, Winter editionGoogle Scholar, at; and von Plato, Jan, Creating Modern Probability, Cambridge: Cambridge University Press, 1994CrossRefGoogle Scholar.

9 de Finetti, Bruno, ‘Le vrai et le probable’, Dialectica (1949) 3, pp. 7893CrossRefGoogle Scholar; Ramsey, Frank, The Foundations of Mathematics and Other Logical Essays, London: Routledge, 1931Google Scholar.

10 Carnap, Rudolf, Logical Foundations of Probability, Chicago: The University of Chicago Press, 1950Google Scholar; Keynes, John Maynard, Treatise on Probability, London: Macmillan, 1921Google Scholar.

11 Wald, Abraham, Statistical Decision Functions, New York: Wiley, 1950, p. 2Google Scholar.

12 The first edition was published in 1944. References here are to the 2004 reprint of the second edition (1947). More strongly, one could argue that von Neumann and Morgenstern started the new field of decision theory which subsequently led to the (re)discovery of some earlier publications and developments that could be reconstructed as the field's long history.

13 Leonard, Robert, Von Neumann, Morgenstern, and the Creation of Game Theory, from Chess to Social Science, 1900–1960, Cambridge: Cambridge University Press, 2010, p. 222CrossRefGoogle Scholar.

14 The axioms are these: ‘We consider a system U of entities u, v, w,… In U a relation is given, u>v, and for any number α, (0<α <1), an operation αu + (1−α)v=w. These concepts satisfy the following axioms:

  • (A) u>v is a complete ordering of U.

    (A:a) For any two u, v one and only one of the three following relations holds: u=v, u>v, u<v.

    (A:b) u>v, v >w imply u>w.

    (B) Ordering and combining.

    (B:a) u<v implies that u<αu+ (1 – α)v.

    (B:b) u>v implies that u>αu+(1 – α)v.

    (B:c) u<w<v implies the existence of an α with αu + (1 – α)v<w.

    (B:d) u >w>v implies the existence of an α with αu + (1 – α)v>w.

    (C) Algebra of combining.

    (C:a) αu+(1 – α)v=(1 – α)v+αu.

    (C:b) α(βu+(1 – β)v)+(1 – α)v=γu + (1 – γ)v where γ=αβ.’

    John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior (1944), Princeton: Princeton University Press, 2004, p. 26, emphasis in the original.

15 Von Neumann and Morgenstern, op. cit. (14), pp. 33, 1.

16 Von Neumann and Morgenstern, op. cit. (15) pp. 33, 5.

17 ‘The mathematician’ was first published in Works of the Mind, vol. 1, Chicago: The University of Chicago Press, 1947, pp. 180–196. References here are to a 1988 reprint: von Neumann, John, ‘The mathematician’, in Newman, James. R. (ed.), The World of Mathematics: A Small Library of the Literature of Mathematics from A'h-mosé the Scribe to Albert Einstein, New York: Simon and Schuster, 1988, pp. 20292039Google Scholar. See also Leonard, op. cit. (13).

18 Von Neumann, op. cit. (17), pp. 2029 and 2030.

19 Savage, Leonard Jimmie, The Foundations of Statistics, New York: John Wiley & Sons, 1954, p. xiGoogle Scholar.

20 Stigler, George J., Memoirs of an Unregulated Economist, New York: Basic Books, 1988, p. 61Google Scholar.

21 Machina, Mark J., ‘Expected utility analysis without the independent axiom’, Econometrica (1988) 50, pp. 277323CrossRefGoogle Scholar. Other visible attempts of these years to incorporate von Neumann and Morgenstern's Theory of Games in existing economic theory include Marschak, Jacob, ‘Neumann's and Morgenstern's new approach to static economics’, Journal of Political Economy (1946) 44, pp. 97115CrossRefGoogle Scholar; Marschak, , ‘Rational behavior, uncertain prospects, and measurable utility’, Econometrica (1950) 18, pp. 111141CrossRefGoogle Scholar; and Arrow, Kenneth, ‘Alternative approaches to the theory of choice in risk-taking situations’, Econometrica (1951) 19, pp. 404437CrossRefGoogle Scholar.

22 Friedman, Milton and Savage, Leonard Jimmie, ‘The utility analysis of choice involving risk’, Journal of Political Economy (1948) 56, pp. 279304, 279CrossRefGoogle Scholar.

23 Friedman and Savage, op. cit. (22), pp. 279, 282, 283. Friedman and Savage did not base their analysis on the von Neumann and Morgenstern axioms directly, but instead argued to have their own analysis ‘self-contained … by the paraphrasing of essential parts of [von Neumann and Morgenstern's] arguments’ (p. 281). It is hence a matter of debate whether Friedman and Savage's 1948 article is indeed entirely in line with the axioms of von Neumann and Morgenstern's Theory of Games. For instance, von Neumann and Morgenstern suggested – it was indeed only a suggestion – that it is ‘not clear at all … what significance [marginal utility] has in determining the behavior of a participant in a social exchange economy’ (p. 31), which seems at odds with the wiggly utility curve of Friedman and Savage. To Friedman and Savage, however, their wiggly utility curve was in line with von Neumann and Morgenstern because its curve is constantly increasing.

24 Friedman and Savage, op. cit. (22), p. 280.

25 Friedman and Savage, op. cit. (22), pp. 281–282; Hicks, John and Allen, Roy G.D., ‘A reconsideration of the theory of value’, Economica (1934) 1, pp. 5276CrossRefGoogle Scholar; and Hicks, and Allen, , ‘A reconsideration of the theory of value. Part II. A mathematical theory of individual demand functions’, Economica (1934) 1, pp. 196219CrossRefGoogle Scholar. No specific references were given for Edgeworth, Pareto and Fisher.

26 Ingrao, Bruno and Israel, Giorgio, The Invisible Hand, Cambridge, MA: MIT Press, 1990Google Scholar; Mandler, Michael, Dilemmas in Economic Theory, New York: Oxford University Press, 1999Google Scholar; Mongin, Philippe, ‘Duhemian themes in expected utility theory’, French Studies in the Philosophy of Science, Berlin: Springer, 2009, pp. 303357CrossRefGoogle Scholar; Moscati, Ivan, ‘History of consumer demand theory, 1871–1971: a neo-Kantian rational reconstruction’, European Journal of the History of Economic Thought (2007) 14, pp. 119156CrossRefGoogle Scholar; Moscati, , ‘Were Jevons, Menger and Walras really cardinalists? On the notion of measurement in utility theory, psychology, mathematics and other disciplines, 1870–1910’, History of Political Economy (2013) 45, pp. 373414CrossRefGoogle Scholar.

27 Samuelson, Paul A., ‘Probability and the attempts to measure utility’, Economic Review (1950) 1, pp. 167173Google Scholar, 169 and 170. Samuelson's remarks are an obvious reference to the epicycle theory that accounts for planets deviating from the perfect circular orbit predicted in the Ptolemaic system of astronomy, until Copernicus, Galileo and Kepler put the Sun instead of the Earth at the center of the astronomical system. In other words, Samuelson accused von Neumann, Morgenstern, Friedman and Savage of building a Ptolemaic system descriptively incongruent with reality.

28 Savage to Samuelson, 19 May 1950, cc Friedman, Leonard Jimmie Savage Papers, Yale University Library (subsequently LJSP), Box C2.

29 Bernoulli, op. cit. (5). Letters not immediately copied to the others were often forwarded. The letter just cited, for instance, was forwarded by Samuelson to Baumol.

30 Baumol, William J., ‘The Neumann–Morgenstern utility index: an ordinalist view’, Journal of Political Economy (1951) 59, pp. 6166CrossRefGoogle Scholar.

31 Allais, op. cit. (2); Allais, , ‘Fondements d'une théorie positive des choix comportant un risque et critique des postulats et axioms de L'Ecole Americaine’, Econometrie (1953) 40, pp. 257332Google Scholar; Ellsberg, Daniel, ‘Ambiguity, and the Savage axioms’, Quarterly Journal of Economics (1961) 75, pp. 643669CrossRefGoogle Scholar; Kahneman, Daniel and Tversky, Amos, ‘Prospect theory: an analysis of decision under risk’, Econometrica (1979) 47, pp. 263292CrossRefGoogle Scholar.

32 Note that a decreasing utility of money, or any other valuation of monetary income, is incorporated in the util. Thus 100 utils by definition is twice as valuable to the individual as 50 utils.

33 Baumol, op. cit. (30), p. 64.

34 Friedman, Milton and Savage, Leonard Jimmie, ‘The expected-utility hypothesis and the measurability of utility’, Journal of Political Economy (1952) 60, pp. 463474CrossRefGoogle Scholar.

35 Friedman, Milton, ‘The methodology of positive economics’, in Friedman, (ed.), Essays in Positive Economics, Chicago: The University of Chicago Press, 1953, pp. 343Google Scholar; Archibald, George C., Simon, Herbert A. and Samuelson, Paul A., ‘Discussion’, American Economic Review (1963) 53, pp. 227236Google Scholar; Fourcade, Marion, Economists and Societies, Discipline and Profession in the United States, Britain, & France, 1890s to 1990s, Princeton: Princeton University Press, 2009Google Scholar; Hausman, Daniel, The Inexact and Separate Science of Economics, Cambridge: Cambridge University Press, 1992CrossRefGoogle Scholar; Nagel, Ernst, ‘Assumptions in economic theory’, American Economic Review (1963) 53, pp. 211219Google Scholar; Maki, Uskali, The Methodology of Positive Economics: Reflections on the Milton Friedman Legacy, Cambridge: Cambridge University Press, 2009CrossRefGoogle Scholar.

36 Maurice Allais to William Baumol, 20 September 1951, William Baumol Papers, Duke University Rare Book, Manuscript, and Special Collections Library, Duke University (subsequently WBP), Box C1. The article that Allais referred to was Allais, op. cit. (31). Its English translation formed the basis for Allais, Maurice and Hagen, Ole (eds.), Expected Utility Hypotheses and the Allais Paradox, London: D. Reidel Publishing Company, 1979CrossRefGoogle Scholar. In their correspondence, Allais wrote in French and Baumol, Savage and Friedman in English, a common practice between French- and English-speaking scientists at the time. Translations are the author's.

37 Bernoulli, op. cit. (5); Friedman and Savage, op. cit. (22). Most likely, Allais and Friedman first met in person during the first meeting of the Mont Pelerin Society in Lausanne, Switzerland, in 1947 – initiated by Friedrich Hayek to advocate (neo)liberal ideals of free societies, free markets and small governments. Allais and Friedman started a correspondence in early 1948 on various economic issues, including the question whether the economic organization of France was such that France could attain the level of welfare of the United States, and regarding the price and revenue elasticities of various goods in the United States. The discussion on the measurement of utility to which Allais referred in his letter to Baumol consisted of Allais sending Friedman the same papers and questions as he was sending Savage. In contrast to Savage, Friedman never answered with more than one or two lines. The exception was the extensive questionnaire that Allais also sent to Savage – to be discussed below.

38 William Baumol to Maurice Allais, 18 October 1951, WBP, Box C1.

39 Next to Samuelson, Baumol and Allais there were others who disagreed with the von Neumann and Morgenstern approach. Robert Solow from the Massachusetts Institute of Technology suggested to Baumol that, like Samuelson, he and many others were sceptical of the von Neumann–Morgenstern utility index. Robert Solow to William Baumol, 14 May 1952, WBP, Box C1.

40 Jallais, Sophie and Pradier, Pierre-Charles, ‘The Allais paradox and its immediate consequences for expected utility theory’, in Fontaine, Philippe and Leonard, Robert (eds.), The Experiment in the History of Economics, New York: Routledge, 2005, pp. 2549Google Scholar.

41 Samuelson, Paul A., ‘Economic theory and mathematics: an appraisal’, American Economic Review (1952) 42, pp. 5666Google Scholar.

42 It is not exactly clear why Baumol did not attend the 1952 Paris symposium. Given Allais's remarks in his letters to Baumol, it seems unlikely that Allais would invite Samuelson, Friedman and Marschak but not Baumol. In view of the fact that in 1954 Baumol had to decline an invitation by Allais to attend a conference on dynamic models in Paris because Princeton would not fund such a visit, a possible explanation is that Baumol did not have the means to attend the symposium in 1952. Alternatively, it may be that Baumol was simply overwhelmed by the different obligations he had at the time, and did not have the time and energy a trip to Europe required. Personal communication (email), William Baumol to author, 18 October 2011.

43 The description of Allais's position in this and the next paragraphs draws on Allais, opera cit. (2) and (31), as well as Allais's position as set out in his letters to Savage.

44 Allais considered ordinal versus cardinal utility to be an established distinction.

45 Harro Maas, William Stanley Jevons and the Making of Modern Economics, Cambridge: Cambridge University Press, 2005.

46 Allais's theory of the ‘psychological mean’ is essentially the same as the reference dependence later introduced by Kahneman and Tversky. See, for example, Kahneman and Tversky, op. cit. (31).

47 Allais, Maurice, ‘The foundations of a positive theory of choice involving risk and a criticism of the postulates and axioms of the American school’, in Allais, Maurice and Hagen, Ole (eds.), Expected Utility Hypotheses and the Allais Paradox, Dordrecht: D. Reidel Publishing Company, 1979, pp. 27148, 34CrossRefGoogle Scholar.

48 Savage's archive at Yale University contains eight letters from Allais to Savage (LJSP, Box 1, Folder 11). The first is dated 21 March 1952, the last 30 December 1952. One of Allais's letters, however, refers back to a letter from Savage to Allais, dated 20 December 1951, the same letter to which Allais referred in his correspondence with Baumol. As said, that letter seems to have been the first between Savage and Allais and the letter in which Savage asked Allais to comment on his manuscript. In addition, the Savage archive contains a lengthy letter from Savage to Allais dated 10 March 1953 (LJSP, Box 1, Folder 11), which more or less concluded the discussion. As said, translations are the author's. Savage was generally interested in German and French and spent a few months in Paris on a Guggenheim Fellowship in 1952, during which he closely befriended Maurice Fréchet, among others. As a result, Savage's reading of French in particular seems to have been fluent. In the mix between serious and tongue-in-cheek way of writing that was his trademark, Savage stated under ‘Foreign languages’ on the Guggenheim application form, ‘Read mathematical German fluently, other German fairly well. Can understand lectures in German and increasing due to current study and practice of spoken French. Expect to be able to lecture in French, and can already converse reasonably well, especially about mathematical subjects'. Savage's Guggenheim application, LJSP, late 1950, Box 2, Folder 11.

49 Allais to Savage, 2 April 1952, LJSP, Box 1, Folder 11.

50 Jallais and Pradier, op. cit. (40).

51 Allais to Savage, 19 May 1952, LJSP, Box 1, Folder 11, emphasis in original.

52 Allais to Savage, 28 May 1952, LJSP, Box 1, Folder 11.

53 Allais to Savage, 28 May 1952, LJSP, Box 1, Folder 11.

54 Allais, Maurice, ‘La psychologie de l'homme rationnel devant le risque: la théorie et l'experience’, Journal de la Société statistique de Paris (1953) 94, pp. 4773Google Scholar, 55.

55 Allais to Savage, 18 June 1952, LJSP, Box 1, Folder 11.

56 Allais to Savage, 24 September 1952, LJSP, Box 1, Folder 11.

57 Savage, graph, 23 December 1952, LJSP, Box 1, Folder 11.

58 Attached at the end of this paper are Savage's initial sketches of his own utility curve, as well as the final version that was sent to Allais.

59 Savage to Allais, 10 March 1953, LJSP, Box 1, Folder 11.

60 Savage to Allais, 10 March 1953, LJSP, Box 1, Folder 11.

61 Draft, LJSP, Box 11, Folder 251.

62 Savage, op. cit. (19), pp. 19–20.

63 Baumol, op. cit. (30); Baumol, William J., ‘The cardinal utility which is ordinal’, Economic Journal (1958) 68, pp. 665672CrossRefGoogle Scholar; Ellsberg, op. cit. (31); Shackle, George L.S., Expectations in Economics, New York: Wiley, 1949Google Scholar; Shackle, , Decision, Order and Time in Human Affairs, Cambridge: Cambridge University Press, 1961Google Scholar.

64 For more details see Heukelom, Floris, Behavioral Economics: A History, Cambridge: Cambridge University Press, 2014CrossRefGoogle Scholar.

65 Heukelom, op. cit. (64)

66 Nobel website, accessed 9 June 2012.

67 Income–utility curve, LJSP, Box 1, Folder 11.