# Interesting Decision Theory question

I was reading “Young Money” by Kevin Roose. In it, he cites the following finance entry-level interview question (paraphrased):

Let’s consider the following gamble: I pay you a fixed amount up-front. Then, I flip a fair coin repeatedly until it comes up heads. You pay me 2^N dollars where N is the number of tails – so if heads came up right away, then you pay $1; if the sequence was Tails-Tails-Heads, $4; and so on.

At what price are you willing to play this game?

Take a few minutes to consider this question. You could even write a simple program to simulate it, if you’re feeling so inclined.

I’m going to leave a little space here, but scroll down for my thoughts.

OK, so if you’re like me, you immediately broke out pen and paper and computed the expected value of the gamble. And, perhaps, a surprising thing occurred: you realized that the gamble is valued at negative infinity. (The probability of paying $1 is 1/2; $2, 1/4; $4, 1/8; multiplying this sequence of tiny probabilities by large payments gets you the sum of infinitely many $1/2.)

OK, so now that you’ve gotten that far, at what price do you accept the gamble? Take some more time to consider it.

I wrote a program to simulate the gamble 10,000 times. The average per-play cost came out to $13. I ran it again, and it was $7. I kept running it – it was usually between about $6 and $18, except for the time it was $50. Whoops.

My analysis: it’s probably correct for most people to accept such gambles at a high enough price such that the take-home money would be life-changing. I would obviously accept it for a billion dollars, for instance. I’d probably accept it at a million. It’s really hard for our brains to understand the low chance of getting wiped out, though. As Dave Baxter points out, the subjective value to you of an extra million dollars depends strongly on how much money you already have.

In the real world, you won’t go into debt bringing your net worth too far below zero, because at some point you would just declare bankruptcy. So that also sets a bound on how bad the outcome can be, and makes the gamble (at certain prices) economically rational.

Aha! Jess Riedel points out that Wikipedia answers the question here: St. Petersburg Paradox.

I guess an interesting takeaway for me was that expected value calculations don’t (& shouldn’t) always dominate decisionmaking.