## Thursday, December 16, 2010

### St. Petersburg, Part 1

This is the first post in a planned series on the St. Petersburg paradox. But don't follow the link just yet, unless you want to spoil the excitement. Note, I use the formulation give by Robert Martin, rather than the one in wikipedia.

How much would you pay for a 1/2 chance to earn \$2? (Ignore taxes and transaction costs)
How much would you pay for a 1/4 chance to earn \$4?
How much would you pay for a 1/8 chance to earn \$8?

How much would you pay for a 1/16 chance to earn \$16?
....

How much would you pay for a 1/2^10 chance to earn \$2^10? (2^10 is about 1,000)
How much would you pay for a 1/2^20 chance to earn \$2^20? (2^20 is about 1 million)
How much would you pay for a 1/2^30 chance to earn \$2^30? (2^30 is about 1.1 billion)
How much would you pay for a 1/2^40 chance to earn \$2^40? (2^40 is about 1.1 trillion)

Consider each of the 40 similar bets from 2 to 2^40. Suppose you could take them all at the same time in a mutually exclusive game, such that the chance of winning nothing is 1/2^40. For example, the game could be a sequential coin flip until the coin shows tails, and paying 2^n when the nth flip is the first one to show tails, and paying \$0 if the all of the first 40 flips are heads.

Is the amount you would pay to play this game equal to the sum of the amounts you would pay for each of the individual games listed above? If not, why not?

Why would you not be willing to pay \$40 to play this game?

A variation follows below the fold...

For those of you who protest about diminishing marginal utility of wealth, consider the following variation:

What is the smallest prize that would induce you to pay \$1 for a 1/2 chance to win the prize?

Repeat this question for every probability 1/2^n from n=1 to n=20.

Suppose someone were to offer a portfolio of prizes matching the 20 bets listed above. Would you pay \$20 for the portfolio? If no, why not?