Allais and Ellsberg Paradoxes

allaisindexThis post looks at a little history and a little psychology that make clear that humans do not even try to maximize expected utility in certain circumstances.  If this post were to fit in logically it should have been before prospect theory.  Prospect theory manages to internalize these little foibles.  The bottom line is that in certain circumstances we have plenty of brain power, but we still do not want to maximize expected utility (at least as normally measured).   I have taken the information that follows largely from the materials prepared by John Miyamoto for his class Psychology 466 at the University of Washington.

Allais Paradox

This objection to expected utility theory was raised by Maurice Allais, Nobel laureate economist, in 1953. In my own case, with regard to investments, I call it the “you only live once and don’t get greedy” paradox.

Choice 1:

Option A: Receive 1 million for sure.
Option B: Receive 2.5 million, 10% chance, Receive 1 million 89% chance, Receive 0, 1% chance.

Choice 2:

Option A’: Receive 1 million 11% chance, otherwise $0.
Option B’: Receive 2.5 million 10% chance, otherwise $0.
•Typical choices: Choose A from Choice 1 and choose B’ from Choice 2.

There are a couple of explanations for this typical behavior,  The first is based on regret.

•If you choose option B in choice 1 and get $0, you will feel intense regret. Choosing option A avoids the possibility of regret.
•If you choose option B’ in choice 2 and get $0, you will not feel regret for your decision because you could have gotten $0 with option A’ as well.

If one experiences a negative outcome, one feels greater regret if it is easy to imagine an alternative action that would have produced a better outcome.


The other explanation of the Allais paradox is in terms of the nonlinear perception of probability.



The main point Allais wished to make is the fact that your choice in one part of a gamble may depend on the possible outcome in the other part of the gamble. In the above Choice 1, Option B, there is a 1% chance of getting nothing. However, this 1% chance of getting nothing also carries with it a great sense of regret if you were to pick that gamble and lose, knowing you could have won with 100% certainty if you had chosen Option A. This feeling of regret, however, is contingent on the outcome in the other portion of the gamble (i.e. the feeling of certainty). Hence, Allais argues that it is not possible to evaluate portions of gambles or choices independently of the other choices presented, We don’t act irrationally when choosing A in Choice 1 and B’ in Choice 2; rather expected utility theory is not robust enough to capture such “bounded rationality” choices.

ellsEllsberg Paradox

This objection was raised by decision theorist Daniel Ellsberg (Pentagon Papers) in 1961.

We are going to draw a ball from an urn.  The urn contains 30 red balls and 60 balls that are either blue or yellow, but you do not know the relative proportion of blue and yellow balls. Payoffs are based on the following payoff matrix.


The explanation of this typical choice behavior is that people tend to avoid ambiguous probabilities in the domain of gains while they tend to seek ambiguous probabilities in the domain of losses.


Because the exact chances of winning are known for Options A and B’, and not known for Options B and A’, this can be taken as evidence for some sort of ambiguity aversion which cannot be accounted for in expected utility theory. It has been demonstrated that this phenomenon occurs only when the choice set permits comparison of the ambiguous proposition with a less vague proposition (but not when ambiguous propositions are evaluated in isolation).

There have been various attempts to provide  explanations of Ellsberg’s observation. Some of these alternative approaches  suppose that the agent formulates a subjective (though not necessarily Bayesian) probability for possible outcomes.

In light of the ambiguity in the probabilities of the outcomes, the agent is unable to evaluate a precise expected utility. Consequently, a choice based on maximizing the expected utility is also impossible.  The agent then tries to satisfice the expected utility and to maximize the robustness against uncertainty in the imprecise probabilities. This robust-satisficing approach can be developed explicitly to show that the choices of decision-makers should display precisely the preference reversal which Ellsberg observed.

Another possible explanation is that this type of game triggers a deceit aversion mechanism. Many humans naturally assume in real-world situations that if they are not told the probability of a certain event, it is to deceive them. People make the same decisions in the experiment that they would about related but not identical real-life problems where the experimenter would be likely to be a deceiver acting against the subject’s interests.


Strong evidence that expected utility (EU) theory is not descriptively adequate.
•Hypotheses that explain the Allais Paradox:
People’s decisions are influenced by anticipated regret.
People’s perception of probability is nonlinear.
•Hypothesis that explain the Ellsberg Paradox:
People tend to avoid ambiguous probabilities in the domain of gains.
People tend to seek ambiguous probabilities in the domain of losses



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