The Allais Paradox Revisited

Here is an original version of the “Allais paradox.” Take a moment to think about your own choices, and maybe also what you think other people would be likely to choose.

Discussion will follow! This formulation is from “On the Experimental Robustness of the Allais Paradox” by Pavlo Blavatskyy, Andreas Ortmann, and Valentyn Panchenko, in the February 2021 issue of the American Economic Journal: Microeconomics. (Full disclosure: AEJ: Micro is published by the American Economic Association, which also publishes the Journal of Economic Perspectives, where I work as Managing Editor, but the articles in AEJ: Micro are not freely available online.) The authors write: 

The first Allais (1953) example consisted of two related decision problems, which we call Allais questions.

In the first question, a decision-maker chooses between two options A and B:
• Option A: ₣100 million for certain
• Option B: ₣500 million with probability 0.1, ₣100 million with probability
0.89, nothing with probability 0.01

In the second question, a decision-maker chooses between another two options C and D:
• Option C: ₣100 million with probability 0.11, nothing with probability 0.89
• Option D: ₣500 million with probability 0.1, nothing with probability 0.9

Notice that you need to make two choices here: between A and B, and between C and D. Choice A is 100 million for sure. (The symbol ₣ stands for the French currency of the time, the franc.) Choice B is a 10% chance of 500 million, an 89% chance of 100 million, and a 1% chance of nothing. In short, this choice tells something about your risk preferences. Would you take 100 million for sure? Or would you prefer a small chance of a much larger prize, even at the risk of an even smaller chance of nothing at all?

Now think about the choice between C and D, which on the surface looks very different. Would you prefer an 11% chance of 100 million and an 89% chance of nothing? Or would you prefer a slightly smaller chance of winning but a larger prize if you do win, with 10% chance of 500 million and a 90 percent chance of nothing?

A few years after Maurice Allais won the Nobel price in economics in 1988 (“for his pioneering contributions to the theory of markets and efficient utilization of resources”–the Allais paradox was just one small slice of his body of work), Bertrand Munier wrote an article about his accomplishments in the Spring 1991 issue of the Journal of Economic Perspectives (“Nobel Laureate: The Many Other Allais Paradoxes,” pp. 179-199). With regard to the Allais paradox, Munier wrote:

Allais and Darmois organized a conference in Paris in 1952 on mathematical economics and risk. The above questions were put to several participants by Allais, and in particular to authors of the Neumann-Savage expected utility theory—which Allais calls the “neo-Bernoullian” theory—to researchers supporting the theory, like B. de Finetti, as well as to other persons. Most of these individuals, including Savage himself, preferred a1 to a2 [in the notation above, choices A and B] in the first pair and a4 to a3 [that is, choice D and C] in the second one. Later on, Allais submitted the questionnaire containing, among others, the above questions to a number of colleagues and students. About 65 percent of them made similar choices.

The key insight here is that, from a certain perspective, A and B is actually the same choice as C and D. If you choose A over B, it is logically inconsistent to choose D over C. To understand why, there are theoretical and algebraic explanations of the Allais paradox in the articles by Blavatskyy, Ortmann, ad Panchenko, and the article by Munier. Here, I’ll try a more verbal explanation.

If you look back at the choice between A and B, 89% of the time the outcome is the same: that is, it’s 100 in either option. The difference between A and B lies in the other 11% of what can happen. In A, you get 100 all 11% of the time. In B, you get 500 10% of the time, and zero 1% of the time.

Now look at the choice between C and D. Again, 89% of the time the outcome is the same, but instead of being 100, as it was in the choice between A and B, now the outcome is zero for both C and D, 89% of the time. Again, the difference between C and D thus resides in what happens in the other 11% of outcomes. In choice C, you get 100 all the remaining 11% of the time. In choice D, you get 500 10% of the time and 0 the other 1% of the time.

Notice that once you strip out the fact that the same outcome happens 89% of the time in both sets of choices, the remaining choice over what happens in the remaining 11% of time becomes identical! If you prefer A over B, you thus prefer the more certain outcome over the outcome with more risk in that remaining 11% slice of outcomes. But if you prefer D over C, you prefer the outcome with more risk. In other words, choosing A over B, and then also choosing D over C, is an inconsistent choice according to the expected utility theory. This is the Allais paradox.

You might object that this example–that is, making two choices and the comparing consistency across those choices–is contrived and hard to understand. Fair enough. It definitely is contrived to help Allais make his point. But notice that many prominent supporters of expected utility theory, including Leonard Savage, fell right into the Allais paradox.

You might further object that a choice that offers a life-changing amount with complete certainty, like choice A, has an unfair advantage over any choice with a degree of uncertainty. Or you might say that the fact that the 89% outcome is 100 in the first choice and 0 in the second choice is not something that can be ignored: the reference point that you start from will affect choices, too. Now we’re getting into the heart of the discussion. Is the Allais paradox a fragile outcome that depends on a certain phrasing of the choices? Does it depend on whether a choice is certain or not? Does it depend on large amounts or small probabilities being involved? Would people make these same choices in real-world settings?

In broad terms, what is the applied psychology of making choices like these? The Allais paradox helped to open up these kinds of questions for exploration, and is thus thought of as one of the building blocks of what has now developed into behavioral economics. As Munier put it:

Allais had put the finger on what is now termed the “certainty effect” in experimental decision science. But he suspected, without being able to prove it immediately, that the lesson was more general: attitudes towards risk change not only from an individual to another, but also from a given individual between different patterns of risk.

The recent paper by Blavatskyy, Ortmann, and Panchenko looks at research on the Allais paradox over time. They find:

The Allais Paradox, or the common consequence effect, is a well-known behavioral regularity in individual decision-making under risk. Data from 81 experiments reported in 29 studies reveal that the Allais Paradox is a fragile empirical finding. The Allais Paradox is likely to be observed in experiments with high hypothetical payoffs, the medium outcome being close to the highest outcome and when lotteries are presented as a probability distribution … The Allais Paradox is likely to be reversed in experiments when the probability mass is equally split between the lowest and highest outcomes in risky lotteries.

The specific example of the Allais Paradox, and the many somewhat examples that have been explored in research since then, teach a general lesson: people’s attitudes toward risk can be inconsistent, depending on how the choices are framed, and in particular by valuations people place on certainty and on reference points. Allais died in 2010 at the age of 99. My sense is that his work and legacy is not well-known among many younger economists, but he was a giant among the economists of his time. Munier summed up his philosophy in the 1991 JEP essay:

[Allais] later made clear to his students the few principles on which his economic methodology rests: 1) make reference to original thinkers only; 2) never accept any theory if it has not been successfully checked on empirical data; 3) look for invariants in space and time as much as possible; 4) make use of mathematics only as a way of expressing a theory rigorously (and particularly of explicitly stating the hypotheses on which it rests), but never admit that the mathematical content of a paper is a significant index of quality; and 5) aim at developing synthetic views.