Ellsberg paradox

The Ellsberg paradox is a paradox in decision theory in which people's choices violate the postulates of subjective expected utility. It is generally taken to be evidence for ambiguity aversion. The paradox was popularized by Daniel Ellsberg, although a version of it was noted considerably earlier by John Maynard Keynes.

The basic idea is that people overwhelmingly prefer taking on risk in situations where they know specific odds rather than an alternative risk scenario in which the odds are completely ambiguous—they will always choose a known probability of winning over an unknown probability of winning even if the known probability is low and the unknown probability could be a guarantee of winning. That is, given a choice of risks to take (such as bets), people "prefer the devil they know" rather than assuming a risk where odds are difficult or impossible to calculate.

Ellsberg actually proposed two separate thought experiments, the proposed choices which contradict subjective expected utility. The 2-color problem involves bets on two urns, both of which contain balls of two different colors. The 3-color problem, described below, involves bets on a single urn, which contains balls of three different colors.

The 1 urn paradox

Suppose you have an urn containing 30 red balls and 60 other balls that are either black or yellow. You don't know how many black or how many yellow balls there are, but that the total number of black balls plus the total number of yellow equals 60. The balls are well mixed so that each individual ball is as likely to be drawn as any other. You are now given a choice between two gambles:

Gamble A	Gamble B
You receive $100 if you draw a red ball	You receive $100 if you draw a black ball

Also you are given the choice between these two gambles (about a different draw from the same urn):

Gamble C	Gamble D
You receive $100 if you draw a red or yellow ball	You receive $100 if you draw a black or yellow ball

This situation poses both Knightian uncertainty – how many of the non-red balls are yellow and how many are black, which is not quantified – and probability – whether the ball is red or non-red, which is ⅓ vs. ⅔.

Utility theory interpretation

Utility theory models the choice by assuming that in choosing between these gambles, people assume a probability that the non-red balls are yellow versus black, and then compute the expected utility of the two gambles.

Since the prizes are exactly the same, it follows that you will prefer Gamble A to Gamble B if and only if you believe that drawing a red ball is more likely than drawing a black ball (according to expected utility theory). Also, there would be no clear preference between the choices if you thought that a red ball was as likely as a black ball. Similarly it follows that you will prefer Gamble C to Gamble D if, and only if, you believe that drawing a red or yellow ball is more likely than drawing a black or yellow ball. It might seem intuitive that, if drawing a red ball is more likely than drawing a black ball, then drawing a red or yellow ball is also more likely than drawing a black or yellow ball. So, supposing you prefer Gamble A to Gamble B, it follows that you will also prefer Gamble C to Gamble D. And, supposing instead that you prefer Gamble B to Gamble A, it follows that you will also prefer Gamble D to Gamble C.

When surveyed, however, most people strictly prefer Gamble A to Gamble B and Gamble D to Gamble C. Therefore, some assumptions of the expected utility theory are violated.

Mathematical demonstration

Mathematically, your estimated probabilities of each color ball can be represented as: R, Y, and B. If you strictly prefer Gamble A to Gamble B, by utility theory, it is presumed this preference is reflected by the expected utilities of the two gambles: specifically, it must be the case that

$R \cdot U(\$100) + (1-R) \cdot U(\$0) > B\cdot U(\$100) + (1-B) \cdot U(\$0)$

where $U(\cdot)$ is your utility function. If $U(\$100) > U(\$0)$ (you strictly prefer $100 to nothing), this simplifies to:

$R [U(\$100) - U(\$0)] > B [U(\$100) - U(\$0)]$

$\Longleftrightarrow R > B \;$

If you also strictly prefer Gamble D to Gamble C, the following inequality is similarly obtained:

$B\cdot U(\$100) + Y\cdot U(\$100) + R \cdot U(\$0) > R \cdot U(\$100) + Y\cdot U(\$100) + B \cdot U(\$0)$

This simplifies to:

$B [U(\$100) - U(\$0)] > R [U(\$100) - U(\$0)]$

$\Longleftrightarrow B > R \;$

This contradiction indicates that your preferences are inconsistent with expected-utility theory.

Generality of the paradox

Note that the result holds regardless of your utility function. Indeed, the amount of the payoff is likewise irrelevant. Whichever gamble you choose, the prize for winning it is the same, and the cost of losing it is the same (no cost), so ultimately, there are only two outcomes: you receive a specific amount of money, or you receive nothing. Therefore it is sufficient to assume that you prefer receiving some money to receiving nothing (and in fact, this assumption is not necessary — in the mathematical treatment above, it was assumed U($100) > U($0), but a contradiction can still be obtained for U($100) < U($0) and for U($100) = U($0)).

In addition, the result holds regardless of your risk aversion. All the gambles involve risk. By choosing Gamble D, you have a 1 in 3 chance of receiving nothing, and by choosing Gamble A, you have a 2 in 3 chance of receiving nothing. If Gamble A was less risky than Gamble B, it would follow that Gamble C was less risky than Gamble D (and vice versa), so, risk is not averted in this way.

However, because the exact chances of winning are known for Gambles A and D, and not known for Gambles B and C, 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).

Possible explanations

There have been various attempts to provide decision-theoretic explanations of Ellsberg's observation. Since the probabilistic information available to the decision-maker is incomplete, these attempts sometimes focus on quantifying the non-probabilistic ambiguity which the decision-maker faces – see Knightian uncertainty. That is, these alternative approaches sometimes suppose that the agent formulates a subjective (though not necessarily Bayesian) probability for possible outcomes.

One such attempt is based on info-gap decision theory. The agent is told precise probabilities of some outcomes, though the practical meaning of the probability numbers is not entirely clear. For instance, in the gambles discussed above, the probability of a red ball is 30/90, which is a precise number. Nonetheless, the agent may not distinguish, intuitively, between this and, say, 30/91. No probability information whatsoever is provided regarding other outcomes, so the agent has very unclear subjective impressions of these probabilities.

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 info-gap approach supposes that the agent implicitly formulates info-gap models for the subjectively uncertain probabilities. 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. When faced with the choice between a red ball and a black ball, the probability of 30/90 is compared to the lower part of the 0/90-60/90 range (the probability of getting a black ball). The average person expects there to be fewer black balls than yellow balls because in most real-world situations, it would be to the advantage of the experimenter to put fewer black balls in the urn when offering such a gamble. On the other hand, when offered a choice between red and yellow balls and black and yellow balls, people assume that there must be fewer than 30 yellow balls as would be necessary to deceive them. When making the decision, it is quite possible that people simply forget to consider that the experimenter does not have a chance to modify the contents of the urn in between the draws. In real-life situations, even if the urn is not to be modified, people would be afraid of being deceived on that front as well.

A modification of utility theory to incorporate uncertainty as distinct from risk is Choquet expected utility, which also proposes a solution to the paradox.

Alternative explanations

Other alternative explanations include the competence hypothesis and comparative ignorance hypothesis. These theories attribute the source of the ambiguity aversion to the participant's pre-existing knowledge.

See also

• Allais paradox
• Ambiguity aversion
• Experimental economics
• Subjective expected utility
• Utility theory