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Paradox From an Unexpected Source

You probably recognize the name: Daniel Ellsberg. As a former United States military analyst and RAND Corporation strategist, Ellsberg gained fame in 1971 when he created a national political controversy by releasing the top-secret Pentagon Papers to The New York Times.

But, Daniel Ellsberg has some fame that stretches into the mathematics arena as well. That is, the eponym Ellsberg Paradox is his creation, even though economist John Maynard Keynes considered it much earlier.

The Ellsberg Paradox occurs in decision theory when "people's choices violate the expected utility hypothesis....and is generally taken to be evidence for ambiguity aversion." Right!..let me explain.

Suppose Joe is to draw a ball at random from Box A or Box B, both of which contain black balls and red balls. If Joe draws a red ball, he gets $100. Before Joe draws, he is told that the balls in Box A are half black and half red. Joe knows nothing about Box B. Which box should Joe draw a ball from?

If Joe opts for Box A, he perhaps does so because he associates Box B with a greater chance of picking a black ball. Now the Paradox: if Joe were now told that he would get $100 if he withdrew a black ball...Joe would still "overwhelmingly" pick Box A again. Why? Because definite information (i.e. knowledge of Box A's contents) always trumps ambiguity (i.e. nothing known about Box B).

This type of game "triggers a deceit aversion mechanism." Joe assumes that because he is not told anything about the color ratios in Box B, this was done to deceive him. Originally, Joe expects there to be more black balls than red balls in Box B because it would be to the advantage of the "experimenter" offering the gambling game. Even after making this conclusion (i.e. less chance of getting a red ball in Box B), the revision of the $100 offer has little impact on Joe, as his former conclusion is ignored (or revised) to accomodate the ambiguity he associates with Box B.

And, Joe igores or simply forgets to consider the fact that the "experimenter" does not have a chance to modify the contents of Box B in between changing the offer of $100 from red to black.

Source: Agresti & Franklin's Statistics: The Art and Science of Learning From Data, 2007, p. 590