St. Petersburg Paradox
By Arnold Kling
Will Baude brings up the St. Petersburg Paradox, in which a bet with an infinite expected payoff is rejected by the typical individual. Baude points out the problem with trying to resolve the paradox by invoking diminishing marginal utility.
here’s the pet peeve. While this ad hoc (but plausible) assumption solves this particular version of the St. Petersburg Paradox, it does nothing about a modified version of this game with much higher payoffs. Suppose, for example, that I increased the payoffs for this game by exponentiating 5. That is, when I was going to pay you $1 in the first version, I’ll now give you $5. When I was going to pay you 2, now you can have 5^2 = $25. Instead of $4, you can have $625 instead. It’s pretty easy to show that this new set of payoffs will result in a quite infinite payoff, even if you have that diminishing marginal utility of wealth. Indeed, for any proposed diminising marginal utility of wealth function that’s unbounded, I can concoct some crazy version of this game with sky-high payoffs that will still have an “infinite” value.
If you are not familiar with the paradox, a good introduction is here.
A simple way to describe the paradox is this. Suppose that you were offered a bet where if you flip a coin ten times and it comes up heads every time you win $1 billion. Otherwise, you lose $10,000. Would you take the bet? On average, you stand to win by taking the bet, but most people would not take it. Even if somebody raises the payoff for winning to $10 billion or $10 trillion, most people would not take the bet. Why not?
I think that the answer is that people mentally truncate the upper value of what you might win. It is hard to believe that somebody is really going to pay you $1 billion if you flip ten heads in a row. So you act as if they were offering you something lower.
For Discussion. Is it rational or irrational to act as if somebody would stick to the terms of the bet, even if it meant that they had to pay an enormous sum of money?