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Balls:

There are three balls. One is red. Each of the others is either white or black. Now I give you a choice between two lotteries.

Lottery A: You win a prize if we draw a red ball.
Lottery B: You win a prize if we draw a white ball.

Which lottery do you choose?

Now I give you another choice between two lotteries.

Lottery C: You win a prize if we draw a ball that's not red.
Lottery D: You win a prize if we draw a ball that's not white.

Which lottery do you choose?

Post your answers, plus any reasoning, in the comments. If you're already familiar with the Ellsberg paradox, you can just watch. Explanations to come later.

UPDATE: Glad this is getting so many comments. Just a few comments of my own:

(1) Many people are assuming that each of the two balls is white or black with a 50-50 probability. Maybe, maybe not. Just keep in mind that it's not part of the assumptions.

(2) Just in case you reject the problem because you don't know the probabilities of white vs. black (though you shouldn't), you can answer the question assuming there's a 50-50 probability. Then, just for fun, answer the problem again where white has a 49% chance.

(3) Also, some people are wondering about the motivations of the "house," i.e., whether it wants you to win or lose. Think what you like about the motivations of the house, but keep in mind that the colors of the balls (however determined) are the same in Part 1 and Part 2.

(4) Some of you are wondering what's the "paradox." I'll explain soon (or you can just look it up on Wikipedia). It may not be right to call it a paradox; perhaps it's just an illustration of an interesting aspect of how people make choices.

67 Comments
Ellsberg paradox, take 2:

Looks like my Ellsberg paradox post below was pretty popular — about two dozen comments just in the first hour, between 11 p.m. and midnight (Eastern)! I'll repeat the problem below, then give my explanation. If you haven't done so before, you may want to think about what you would choose before reading the explanation.

There are three balls. One is red. Each of the others is either white or black. Now I give you a choice between two lotteries. Lottery A: You win a prize if we draw a red ball. Lottery B: You win a prize if we draw a white ball.

Which lottery do you choose? (Mini-update: I allow you to be indifferent, if you want.)

Now I give you another choice between two lotteries. Lottery C: You win a prize if we draw a ball that's not red. Lottery D: You win a prize if we draw a ball that's not white.

Now which lottery do you choose?

UPDATE: Just in case you're confused about this — and apparently some people were — we're talking about the SAME THREE BALLS each time. I haven't changed the balls. Nor have I drawn any balls. We haven't conducted any lotteries in the time it took you to read this post. All there is is a single box of balls, and me asking you your preferences over lotteries. (END OF UPDATE)

UPDATE 2: You ask one of these questions, and you find out all sorts of aspects that you weren't expecting people to find important. This will affect how I phrase the problem next time, but for now, let me just clear up one extraneous aspect. I'm not running the lottery. I don't own the balls. I'm not offering a prize. Someone else, who isn't connected with me, is doing all that. I'm just asking questions about which lotteries you prefer. Also, as I mentioned in the first update, we don't draw any balls between your first choice and your second choice. In fact, we're never going to draw any balls. Why? I'm not running the lottery! I'm just asking questions! If you want to draw balls, take it up with the guy actually running the lottery, who is not me.

64 Comments
Ellsberg paradox, take 3:

Back to the Ellsberg paradox (so-called, 'cause it's not really a paradox). Based on a bunch of previous comments, let me summarize where we're at, with a simplified version of the paradox.

There are three balls. One is red. For each of the other balls, someone flipped a fair coin to determine whether they would be white or black.

You can imagine a number of lotteries based on a draw from these balls. For example, consider the following four lotteries:
Lottery A: Win $100 if we draw a red ball.
Lottery B: Win $100 if we draw a white ball.
Lottery C: Win $100 if we draw a ball that isn't red.
Lottery D: Win $100 if we draw a ball that isn't white.

Do you prefer Lottery A or Lottery B? Do you prefer Lottery C or Lottery D?

(This is different than the previous example in the following ways: First, I've given a specific set of probabilities for white vs. black. Second, I've made it clear that I'm not offering any lotteries, just eliciting your opinion. Third, I've made the prize $100, just to be more specific.)

It turns out that most people prefer A to B, and prefer C to D. This is inconsistent with expected utility theory, which says your preferences over lotteries should only depend on what the ultimate probabilities are and the utility of the item. More below the fold, including the answer to the question: "Who cares?"

83 Comments
Ellsberg paradox, take 4:

Glad you all liked the series of Ellsberg paradox posts. If you're interested in these issues, check out the following fairly accessible article: Mark J. Machina, Choice Under Uncertainty: Problems Solved and Unsolved, Journal of Economic Perspectives, Summer 1987, at 121.

5 Comments
A most ingenious paradox:

This is another old paradox, which I'm posting mostly because I like the author's style in presenting it:

In the Hanged-Man Paradox, a man, K, is sentenced on Sunday to be hanged, but the judge, who is evidently French or enamored of the French wit for surprising those sentenced to the guillotine in their last moments, orders that the hanging take place on one of the next five days at noon. Smiling wistfully, he says to K, "You will not know which day until they come to take you to the gallows."

K, who has evidently been condemned for logical perversions, cannot prevent his mind from nevertheless trying to figure out in advance which day will be his last. He quickly realizes it cannot be Friday, because if he has not been hanged by Thursday noon, he will know nearly a full day before they come to get him that he will be hanged on Friday. He is simultaneously pleased at his cleverness and depressed that he has pushed his date with the gallows closer to Sunday.

Soon enough, he realizes that if Friday is logically excluded, then so is Thursday, because if he has not been hanged by noon Wednesday, he will know that, Friday being excluded, his date must be Thursday. In like manner, he can exclude Wednesday, Tuesday, and Monday. As a logician, he smugly concludes that the judge's decree is false. On Thursday noon he is hanged. The paradox is that he is surprised when they come to take him to the gallows.

(One can easily think up less macabre relatives of the Hanged-Man Paradox, such as the Surprise Quiz, a device with which we are all familiar and by which no doubt many of us have illogically been surprised.)

Russell Hardin, Collective Action 147 (1982) (paragraph breaks added). (Of course this isn't a real paradox — just a cautionary tale.) Hardin concludes (p. 148): "His problem was that facing a hangman focused his mind a little too admirably."

P.S. On people named K, see Kozinski & Volokh, The Appeal, 103 Mich. L. Rev. 1391 (2005).

UPDATE: AnonVCfan refers, in the comments, to the "less refined, ugly cousin of this paradox," the famous dialogue from The Princess Bride. I'll reproduce here what I wrote in the comments: "I see the Princess Bride dialogue as illustrating the fact from Game Theory that the game of Matching Pennies has no Nash equilibrium in pure strategies. The Hanged Man's paradox is 'simpler' in a way, because all you need to refute it is elementary logic."

UPDATE 2: Just so no one gets confused here — this paradox is only "simpler" in a way. It's got an intuitive explanation, but in fact it's very hard, and logicians have written like a hundred articles about it. For a good overview, see this paper by Tim Chow. I can follow the gist of it, but the technical aspects are beyond my knowledge of logic.

56 Comments