Mark Gritter (markgritter) wrote,
Mark Gritter
markgritter

Ellsberg's Paradox

I'm reading Jordan Ellenberg's "How Not to Be Wrong". He describes a paradox I don't recall seeing before. Here are four bets on drawing a ball from an urn, which has 30 red balls and 60 balls that are a combination of black and yellow (in some unknown proportion):

RED: the next ball is red
BLACK: the next ball is black
NOT-RED: the next ball is black or yellow
NOT-BLACK: the next ball is red or yellow

Survey time--- if you can bet $100 on one of the first two if you are offered a $100 prize based on one of the first two conditions occurring, which one do you prefer? How about your preference among the second pair?

If offered RED or BLACK, I prefer

RED
5(38.5%)
BLACK
1(7.7%)
I am indifferent
7(53.8%)

If offered NOT-RED or NOT-BLACK, I prefer

NOT-RED
6(46.2%)
NOT-BLACK
2(15.4%)
I am indifferent
5(38.5%)


So, most people (even economists and poker players) have strict preference RED > BLACK, and NOT-RED > NOT-BLACK. But under classic expected utility theory, people with these preference should also have a strict preference

RED + NOT-RED > BLACK + NOT-BLACK

but these combinations of bets have exactly equal outcomes (not just equal expectation, but guaranteed $100 on each side!)

Obviously you either need to get rid of expected utility theory, or you need to insist, despite evidence, that people should be indifferent to the choices in the survey.
Tags: economics, gambling, mathematics
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