Allais Paradox

The Allais Paradox refers to a classic hypothetical choice problem in behavioral economics that exposes human irrationality. Daniel Kahneman offered a simplified version of the puzzle in his seminal book, Thinking, Fast and Slow

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Problem A: 61% chance to win $520,000 OR 63% chance to win $500,000

Problem B: 98% chance to win $520,000 OR 100% chance to win $500,000

If you haven’t already, take a moment to consider your own preferences for each problem… The majority of people choose the left-hand option for Problem A and the right-hand option for Problem B. As Kahneman put it: “If these were your preferences, you have just committed a logical sin and violated the rules of rational choice.” The irrationality stems from the willingness to take on an additional 2% risk in exchange for an extra $20,000 in potential winnings for one problem but not the other. Put differently, we are risk-seeking and choose the option with a greater expected value for Problem A but are risk-averse and choose the option with a lower expected value for Problem B, despite the possible outcomes being of equal value for both problems.

Expected Utility Theory: A rational choice theory that rests on the notion that given a choice under uncertainty, individuals will choose the option with the highest expected utility.

Utility Function: A mathematical and often hypothetical representation of an individual’s preferences.

Prospect Theory: A form of decision theory that suggests people perceive value in relation to gains and losses rather than in absolute terms. Derived from experimental results, it assumes that the prospect of a loss looms larger than that of a gain.

Choice Set: A hypothetical scenario often used in experiments where people are typically presented with options and asked which they would choose

In 1952, during a convention on the economics of risk in Paris, a French economist named Maurice Allias presented a few choice sets, logically similar to the problems outlined above, to a reputable audience. It was during a time when expected utility theory was gospel within economics, and when Allias found the audience to unwittingly violate the theory through their preferences in the choice sets, he expected a revelation among them, but his result at the time was seen as nothing more than an anomaly.¹

A year later, Allais published his results in the popular journal, Econometrica. However, the Allais Paradox remained a relatively obscure phenomenon in decision theory through to the 1970s, known only among specialists in the field. It wasn’t until the 1980’s that it grew into its present popularity. According to the behavioral economics historian, Floris Heukelom, this was due to a variety of reasons, such as publications of Allais’ work in English, most of which had been in French prior to 1979, and the fact that the field of economics was becoming more of a behavioral science.²

The Allais Paradox also caught the attention of Amos Tversky and Daniel Kahneman, who cited Allais’ work in their now iconic paper on prospect theory, published in 1979.³ As mentioned earlier, Kahneman would go on to include the Allais Paradox in Thinking, Fast and Slow, which has been dubbed by many as the “bible” of behavioral economics.

The Allais Paradox presents a notable challenge to expected utility theory, and while many economists in the second half of the 20th century ignored its implications, it set the stage for alternative theoretical approaches to decision-making under uncertainty. As we’ve highlighted, prospect theory was one of these alternatives. In addition to Kahneman and Tversky’s work was Graham Loomes and Robert Sugden’s 1982 publication Regret Theory: An Alternative Theory of Rational Choice Under Uncertainty,⁵ which also made mention of the Allais Paradox.

These theories didn’t outright reject expected utility theory, but merely sought to make revisions to them that are more consistent with actual human behavior. This idea of incorporating experimental evidence and empirical understandings from psychological science into economics, which had been and is still sometimes referred to as the “dismal science,” is essentially the crux of behavioral science and economics. Although the Allais Paradox didn’t directly ignite the modern discipline, it certainly plays a role in the early part of the historical narrative.

The Allais Paradox has survived the challenge of replication, with several successful replications in the 1970s providing the phenomenon with an empirically robust foundation.⁶ Therefore, most of the critiques in the finding rest in its interpretation rather than the credibility of the result. As previously mentioned, much of the economics discipline rendered Allais’ choice sets as a novelty rather than a serious challenge to expected utility theory, arguing that the axioms outlined in expected utility theory were not intended to be exact descriptions of human behavior.³ Others felt that not only was the result was not enough to refute the theory, but it could actually be compatible.

For instance, Luc Wathieu of Georgetown University claimed in 1993 that the Allais Paradox is not in fact a paradox as it can be interpreted within the framework of expected utility theory. He pointed out that a given utility function can sometimes be contingent on the situation.⁷ In other words, expected utility theory does not necessarily demand that a utility function must be consistent across choice sets.

Wathieu’s criticism can be further examined in the context of the zero-risk bias, the tendency to have a strong preference for absolute certainty, where an individual’s utility function is actually different for Allais’ two choice sets, as one of them always offers an outcome with 100% probability. This implies that someone’s utility function may be contingent on whether there is an option with zero risk, as this scenario would increase utility. This psychological value of certainty is also Kahneman’s interpretation of the paradox,¹ though he would presumably see this as more aligned with prospect theory rather than expected utility theory. With utility being more of a hypothetical metric than a quantifiable one, the debate over what is and isn’t rational will carry on.

Allais Paradox among traders

In 2005, John List and Michael Haigh of the University of Maryland ran an experiment to explore the susceptibility of traders to the Allais Paradox.⁸ The researchers presented choice sets containing different hypothetical lotteries to futures and options traders from the Chicago Board of Trade (CBOT). The traders exhibited a level of behavior consistent with the paradox, however, they were less prone to the effect than a control group of undergraduate students.

Evolution of Decision Making: The Rational Revolution

This piece builds on the history of behavioral science and economics, further highlighting the contentious subject of rationality in economics.

Evolution of Decision Making: Irrationality’s Revenge

This is the second part to the piece cited above, diving into how experimental findings began to challenge expected utility theory.

1. Kahneman, D. (2011). Thinking, fast and slow. Macmillan.
2. Allais, M. (1953). The behavior of rational man in the face of risk: critique of the postulates and axioms of the American school. Econometrica: Journal of the Econometric Society , 503-546.
3. Heukelom, F. (2015). A history of the Allais paradox. British Journal for the History of Science, 48(1), 147.
4. Kahneman, D., & Tversky, A. (1979). Prospect Theory: An Analysis of Decision under Risk. Econometrica, 47(2), 263.
5. Loomes, G., & Sugden, R. (1982). Regret theory: An alternative theory of rational choice under uncertainty. The economic journal, 92(368), 805-824.
6. Mongin, P. (2018). The Allais paradox: what it became, what it really was, what it now suggests to us. Economics and Philosophy, Forthcoming.
7. Wathieu, L. (1993). A Critique of the Allais Paradox. DOI: 10.13140/RG.2.1.3119.7529
8. List, J. A., & Haigh, M. S. (2005). A simple test of expected utility theory using professional traders. Proceedings of the National Academy of Sciences, 102(3), 945-948.