Ellsberg paradox

The Ellsberg paradox is a paradox in decision theory in which people's choices violate the postulates of subjective expected utility.[1] 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.[2]

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. For example, 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.[3]

Ellsberg proposed two separate thought experiments, the proposed choices of 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.

Two-urns paradox

There are two urns each containing 100 balls. It is known that urn A contains 50 red and 50 black, but there is no information on the mix of balls in urn B.

Consider offering the following bets to a subject:

Bet 1A: get $1 if red is drawn from urn A, $0 otherwise

Bet 2A: get $1 if black is drawn from urn A, $0 otherwise

Bet 1B: get $1 if red is drawn from urn B, $0 otherwise

Bet 2B: get $1 if black is drawn from urn B, $0 otherwise

Typically, people are indifferent between bet 1A and bet 2A (consistently with the expected utility theory), but strictly prefer Bet 1A to Bet 1B and Bet 2A to bet 2B.

However, the probability distribution of ball colors, i.e. the proportion of red and black balls, for urn B is not known: therefore there is not enough information to actually justify these preferences on the basis of the expected utility theory. The proportions of balls of a given color in urn B may either give better or worse chances to the gambler. A different decision mechanism is at work.

The result is generally interpreted to be a consequence of ambiguity aversion (also known as uncertainty aversion): people intrinsically dislike situations where they cannot attach probabilities to uncertain outcomes, therefore prefer a bet in which they know to have a 50% chance to get 1$ to one where the chances are unknown.

The one-urn paradox

Consider an urn containing 90 balls: 30 balls are red, while the remaining 60 balls are either black or yellow in unknown proportions.The balls are well mixed so that each individual ball is as likely to be drawn as any other. 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 1/3 vs. 2/3.

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 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.

Numerical demonstration

Mathematically, the 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( ) 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

The result holds regardless of your utility function. Indeed, the amount of the payoff is likewise irrelevant. Whichever gamble is selected, 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: receive a specific amount of money, or receive nothing. Therefore, it is sufficient to assume that the preference is to receive some money to nothing (and, 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[4] 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).[5]

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.[6]

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.[7]

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

Decisions under uncertainty aversion

In order to describe how an individual would take decisions in a world where uncertainty aversion exists, modifications of the expected utility framework have been proposed. These include:

Choquet expected utility
Minmax expected utility

Alternative explanations

Other alternative explanations include the competence hypothesis[8] and comparative ignorance hypothesis.[5] These theories attribute the source of the ambiguity aversion to the participant's pre-existing knowledge.

References

Ellsberg, Daniel (1961). "Risk, Ambiguity, and the Savage Axioms" (PDF). Quarterly Journal of Economics. 75 (4): 643–669. doi:10.2307/1884324. JSTOR 1884324.
Keynes 1921, pp. 75–76, paragraph 315, footnote 2.
EconPort discussion of the paradox
Segal, Uzi (1987). "The Ellsberg Paradox and Risk Aversion: An Anticipated Utility Approach". International Economic Review. 28 (1): 175–202. doi:10.2307/2526866. JSTOR 2526866.
Fox, Craig R.; Tversky, Amos (1995). "Ambiguity Aversion and Comparative Ignorance". Quarterly Journal of Economics. 110 (3): 585–603. CiteSeerX 10.1.1.395.8835. doi:10.2307/2946693. JSTOR 2946693.
Ben-Haim, Yakov (2006). Info-gap Decision Theory: Decisions Under Severe Uncertainty (2nd ed.). Academic Press. section 11.1. ISBN 978-0-12-373552-2.
Lima Filho, Roberto IRL (July 2, 2009). "Rationality Intertwined: Classical vs Institutional View": 5–6. doi:10.2139/ssrn.2389751. SSRN 2389751. Cite journal requires |journal= (help)
Heath, Chip; Tversky, Amos (1991). "Preference and Belief: Ambiguity and Competence in Choice under Uncertainty". Journal of Risk and Uncertainty. 4: 5–28. CiteSeerX 10.1.1.138.6159. doi:10.1007/bf00057884.