Ellsberg paradox
Encyclopedia : E : EL : ELL : Ellsberg paradox
The Ellsberg paradox is a paradox in decision theory and experimental economics in which people's choices violate the expected utility hypothesis. It is generally taken to be evidence for ambiguity aversion. The paradox was first discovered by Daniel Ellsberg.
The 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 balls 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 0 if you draw a red ball | You receive 0 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 0 if you draw a red or yellow ball | You receive 0 if you draw a black or yellow ball |
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). 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 strongly prefer Gamble A to Gamble B and Gamble D to Gamble C. Therefore, some assumption of expected utility theory is violated.
Mathematical demonstration
Mathematically, your estimated probabilities of each color ball can be represented as: R, Y, and B. Since you are told the value R = 1/3, it is assumed you assign this probability to R. If you prefer Gamble A to Gamble B, by utility theory, it is presumed this preference represents your estimates of expected utility. This is represented in the following inequality (implied by preference):
where [U(\cdot)] is your utility function. Solving for [B] gives:
If you also prefer Gamble D to Gamble C, the following inequality is similarly obtained:
Solving for [B] gives:
Hence, a contradiction.
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 implicitly assumed U($100) > 0, but a contradiction can still be obtained for U($100) < 0 and for U($100) = 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.
See also
- Allais paradox
- Experimental economics
- Utility theory
References
- Ellsberg, D. (1961) "Risk, Ambiguity, and the Savage Axioms" Quarterly Journal of Economics 75: 643-669
From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.