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Ellsberg paradox
From Wikipedia, the free encyclopedia
The Ellsberg paradox is a paradox in decision theory and experimental economics in which people's choices
violate the expected utility hypothesis.[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]
Ellsberg raised two problems: 1 urn problem and 2 urn problem. Here, 1 urn problem is described, which is the
better known one.
Contents
1 The 1 urn paradox
1.1 Utility theory interpretation
1.2 Mathematical demonstration
1.3 Generality of the paradox
1.4 Possible explanations
2 See also
3 References
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 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 – whether the non-red balls are all yellow or all 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).
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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. 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 D to Gamble C, it follows that you will also prefer
Gamble B to Gamble A.
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
where
is your utility function. If
simplifies to:
(you strictly prefer $100 to nothing), this
If you also strictly prefer Gamble D to Gamble C, the following inequality is similarly obtained:
This simplifies to:
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
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comparison of the ambiguous proposition with a less vague proposition (but not when ambiguous propositions
are evaluated in isolation)[3].
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.[4]
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.
See also
Allais paradox
Experimental economics
Subjective expected utility
Utility theory
Ambiguity aversion
References
1. ^ Ellsberg, Daniel (1961). "Risk, Ambiguity, and the Savage Axioms" (http://jstor.org/stable/1884324) .
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Quarterly Journal of Economics 75 (4): 643–669. doi:10.2307/1884324
(http://dx.doi.org/10.2307%2F1884324) . http://jstor.org/stable/1884324.
2. ^ (Keynes 1921, pp. 75–76, paragraph 315, footnote 2)
3. ^ Fox, Craig R.; Tversky, Amos (1995). "Ambiguity Aversion and Comparative Ignorance"
(http://jstor.org/stable/2946693) . Quarterly Journal of Economics 110 (3): 585–603. doi:10.2307/2946693
(http://dx.doi.org/10.2307%2F2946693) . http://jstor.org/stable/2946693.
4. ^ Ben-Haim, Yakov (2006). Info-gap Decision Theory: Decisions Under Severe Uncertainty (2nd ed.).
Academic Press. section 11.1. ISBN 0123735521.
Anand, Paul (1993). Foundations of Rational Choice Under Risk. Oxford University Press. ISBN 0198233035.
Keynes, John Maynard (1921). A Treatise on Probability. London: Macmillan.
Schmeidler, D. (1989). "Subjective Probability and Expected Utility without Additivity"
(http://jstor.org/stable/1911053) . Econometrica 57 (3): 571–587. doi:10.2307/1911053
(http://dx.doi.org/10.2307%2F1911053) . http://jstor.org/stable/1911053.
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Categories: Economics paradoxes | Economics of uncertainty | Decision theory | Utility | Statistical paradoxes |
Paradoxes
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