# One of the challenges of decision theory is figuring out exactly what the utility function is.

One of the challenges of decision theory is figuring out exactly what the utility function is. The utility of money, for example, is notoriously nonlinear for most people. In fact, the behavior of many people cannot be captured by a coherent utility function, as illustrated by the Allais paradox, which runs as follows.

Which of these choices do you find most attractive?

A. \$1 million guaranteed.

B. 89% chance of \$1 million;

10% chance of \$2.5 million;

1% chance of nothing.

Now consider these choices:

C. 89% chance of nothing;

11% chance of \$1 million.

D. 90% chance of nothing;

10% chance of \$2.5 million.

Many people prefer A to B, and, at the same time, D to C. Prove that these preferences are inconsistent with any utility function U(x)for money.

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