Ellsberg's paradox: why do the common preferences between bets on an urn of known and an urn of unknown...
decision/ellsberg New, unsolved
New entry, edited from material collected for the archive and not yet in a released version.
Problem
Ellsberg's paradox: why do the common preferences between bets on an urn of known and an urn of unknown composition violate the independence axiom of expected utility theory? Suppose that: Urn I contains 100 red and black balls, in a ratio entirely unknown to you. In Urn II, you confirm that there are exactly 50 red and 50 black balls. An observer who is ignorant of the state of your information about the urns wants to measure your subjective probabilities by asking you questions. 1. Which would you rather bet on, Red[I] or Black[I]? (That is, drawing a ball from Urn I, do you prefer a bet that wins if the ball is Red, or one that wins if the ball is Black?) 2. Which would you prefer to bet on, Red[II] or Black[II]? 3. Which would you prefer to bet on, Red[I] or Black[II]? 4. Which would you prefer to bet on, Black[I] or Black[II]? Suppose you say that in (1) and (2) you are indifferent. Most people say so. And suppose you also prefer Red[II] to Red[I], in (3), and Black[II] to Black[I], in (4). Again, most people have those preferences. The 'justification' (I'm not saying that there is any real justification!) is supposed to be that betting on the draw from the urn of unknown composition is 'riskier'. Now you are in trouble with the independence axiom. (ObPuzzle: demonstrate.) Daniel Ellsberg, _Quarterly Journal of Economics_ 75 (1961) 643-669. -- Jamie Dreier <pl436000@brownvm.brown.edu> [ Editor: the first paragraph is the editor's summary. See also decision/allais and decision/zeckhauser. ]
Show solution
No solution was found in the collected postings. [ Editor: the problem was posted, with Zeckhauser's problem, as one of "two relatives of the Allais paradox" (see decision/allais). ] [ Editor: further material on this puzzle, edited separately from other collected postings (batch 19); it may repeat some of the above. ] [ Editor: problem as stated there: ] Urn I contains 100 red and black balls in an unknown ratio; urn II contains exactly 50 red and 50 black. Most people are indifferent between betting on Red[I] or Black[I], and between Red[II] or Black[II], yet prefer Red[II] to Red[I] and Black[II] to Black[I]. Show that these preferences violate the independence axiom of expected utility theory (Ellsberg's paradox). [ Editor: problem statement reconstructed from the posting. ] ELLSBERG'S PARADOX Suppose that: Urn I contains 100 red and black balls, in a ratio entirely unknown to you. In Urn II, you confirm that there are exactly 50 red and 50 black balls. An observer who is ignorant of the state of your information about the urns wants to measure your subjective probabilities by asking you questions. 1. Which would you rather bet on, Red[I] or Black[I]? (That is, drawing a ball from Urn I, do you prefer a bet that wins if the ball is Red, or one that wins if the ball is Black?) 2. Which would you prefer to bet on, Red[II] or Black[II]? 3. Which would you prefer to bet on, Red[I] or Black[II]? 4. Which would you prefer to bet on, Black[I] or Black[II]? Suppose you say that in (1) and (2) you are indifferent. Most people say so. And suppose you also prefer Red[II] to Red[I], in (3), and Black[II] to Black[I], in (4). Again, most people have those preferences. [ Editor: question 3 reads "Red[I] or Black[II]" in the original, but the answer compares Red[II] with Red[I]; the intended question is probably "Red[I] or Red[II]". ] The 'justification' (I'm not saying that there is any real justification!) is supposed to be that betting on the draw from the urn of unknown composition is 'riskier'. Now you are in trouble with the independence axiom. (ObPuzzle: demonstrate.) Daniel Ellsberg, _Quarterly Journal of Economics_ 75 (1961) 643-669. -- Jamie Dreier <pl436000@brownvm.brown.edu> No solution was found in the collected postings.