Adding a Utility Function to the St. Petersburg Paradox

Logic Level 2

The traditional resolution of the St. Petersburg paradox involves adding a utility function to the problem, taking into consideration diminishing marginal utility.

If the utility of winning $n$ dollars is a logarithmic function, specifically $\log(n)$ , then what is the expected payout, factoring in utility, that a player would get from playing the St. Petersburg paradox game? Put another way, if the value of $n$ diminishes $\log(n)$ , what is the break-even point such that a player should only pay less than this answer to play the game?

2\log n

4\log n

6\log n

8\log n

10\log n

Infinity, a player should pay any amount to play

2 solutions

Christopher Williams
May 25, 2016

If utility is calculated as $\log(n)$ then the expected marginal utility payout from playing the St. Petersburg Paradox can be calculated as:

$\frac{1}{2}\log(2) + \frac{1}{4}\log(4) +\frac{1}{8}\log(8) + \cdots = \sum_{k=1}^{\infty} \frac{\log(2^k)}{2^k} = \$(2\log n)$

Written out, this is the utility from winning $2 ( $\log(2)$ ) times the odds of winning $2 ( $\frac{1}{2}$ ) plus the utility from winning $4 ( $\log(4)$ ) times the odds of winning $4 ( $\frac{1}{4}$ ) and so on. The infinite sum of this series equals $4. Meaning $( $2\log n$ ) is the expected payout, and a player should be willing to pay anything less than that to play.

Something is going wrong here because $\sum_{k=1}^{\infty} \frac{log(2^k)}{2^k} \neq \$4$

Darryl Stein - 3 years, 1 month ago

Indeed, $\sum_{k=1}^{\infty} \frac{\log(2^k)}{2^k} = 2\log(2) \approx 1.386$

Ulquiorra Schiffer - 2 years, 10 months ago

Thanks. I've updated the answer and options to reflect this.

Brilliant Mathematics Staff - 9 months, 3 weeks ago

Can't figure out how to get $4. With a computer program I found ~4$ only when the log base equals to ~1.41421356231...

Delemotte Barthelemy - 2 years, 10 months ago

Using Wolfram Alpha, I get that the answer is log(4).

Nicole Czakon - 2 years, 9 months ago

1/2 log(2) + 1/4 log(4) + 1/8 log(8) + ... = 1/2 log(2) + 1/4 2 log(2) + 1/8 3 log(2) + ... = log(2) * ( 1/2 + 2/4 + 3/8 + 4/16 + ... ) The sum between brackets can be written as S = ( 1/2 + 1/4 +1/8 + ...) + ( 1/4 + 2/8 + 3/16 + ...) = 1 + 1/2 S, hence S = 2 and the correct result equals 2 log(2), where the log base still has to be defined.

Willy Bracke - 2 years, 5 months ago

Joel Lahenius
Apr 26, 2018

The base of the logarithm is not specified. Is it a natural logarithm or 10-based or something else?

log(n) is not well-defined, it should be eg. ln(n) or log_10(n).

The sum calculated in the answer by Christopher also depends on the base of the log.

Adding a Utility Function to the St. Petersburg Paradox

2 solutions

0 pending reports