Wednesday, May 18, 2016

The Ellsberg Paradox and Ambiguity Aversion

• OK, this is not really an outline of an article, but at least there is an urn involved. The urn has a total of 90 balls inside of it. Thirty of the balls are black, and the other 60 balls are either red or blue. (That is, anywhere between 0 and 60 of those balls are red, and the remainder of the non-black, non-red balls are blue.) A single ball will be pulled at random from the urn. 

• Situation A: You can choose Bet 1A, which pays $100 if the ball that is chosen is black. Alternatively, you can choose Bet 2A, which pays $100 if the chosen ball is red. Which bet do you prefer? [Spoiler alert: most folks prefer Bet 1A.]

• Situation B: You can choose Bet 1B, which pays $100 if the ball that is chosen is either black or blue. Alternatively, you can choose Bet 2B, which pays $100 if the chosen ball is either red or blue. Which bet do you prefer? [Spoiler alert: most folks prefer Bet 2B.]

• The modal choices in these hypothetical urn-related decision problems, already spoiled for you, are to choose Bet 1A and Bet 2B. 

• These modal choices are inconsistent with expected utility maximization. A person who (strictly) prefers Bet 1A to Bet 2A, and is an expected utility maximizer, must believe that the probability of choosing a black ball (here, precisely one-third) exceeds the probability of choosing a red ball. A person who (strictly) prefers Bet 2B to Bet 2A, and is an EU maximizer, must believe that the probability of choosing a black ball is smaller than the probability of choosing a red ball (because the probability of winning via the blue ball is the same in either alternative, 2A or 2B).

• The disposition that (presumably) leads to these modal choices is termed ambiguity aversion. In Situation A, the subject knows precisely the probability of winning Bet 1A, but is unsure of the probability of winning Bet 2A. In Situation B, the situation is reversed, with Bet 2B being the option with the known probability (precisely 2/3) of winning.

• The modal choices, inconsistent with expected utility maximization, are an example of what has become known as the Ellsberg Paradox, after the analysis given by Daniel Ellsberg in "Risk, Ambiguity, and the Savage Axioms," Quarterly Journal of Economics 75(4): 643-669, 1961 [pdf here]; Ellsberg's version is on pages 654-655. The version in this post follows closely the presentation in the Introduction (pages 3-4) by Adam Oliver in Behavioural Public Policy, Adam Oliver, ed., Cambridge University Press, 2013.

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