Equating money with value is a simple trap as well. Who cares if you can win millions when a single loss wipes out all your savings? Since anything below a certain level of money leaves you trapped with no way out it could be argued that the value of being destitute is not 0 but -infinity which makes any risk of losing all money unacceptable. This is especially true in a world where people are willing to offer strange bets with arbitrarily high expected value as long as you have some money.
Literally anyone, you included. Every day, there is a chance that the roof collapse on you while you're sleeping; that's -Inf of value! Yet you don't put infinity resources into preventing that since you consider the probability low enough to not obsess over it.
that's just loss aversion codified.
The missing context is that in case A, you need in 10 minutes to repay $50 debt to the Sicilian mafia, or else they'll kill you to make an example for others, and you have no other assets or other ways to make money in this short time. In case B, the situation is the same, but you owe $100 instead of $50.
Putting them together suggests it's flagrantly irrational to apply naive toy models to the real world. Even if they do have a nice mathy sheen.
Engineers (mostly) know this, but for some reason gamblers and economists (mostly) act as if they don't.
There are standard arguments (e.g. the Von Neumann–Morgenstern utility theorem) that an agent with rational preferences, with remarkably weak definitions of the word “rational”, must have an utility function and a subjective probability function such that their behaviour is always governed by the EV of that utility with respect to that probability.
I’m glad to now know there’s a common example of that weakness.
Yes! The St. Petersburg "Paradox" shows that we intuitively know that. I put "paradox" in quotes because I don't think it's a paradox, it's just a sane reaction.
(Sam Bankman-Fried was a big fan of EV and famously declared that he would toss a coin where heads would double the "value" (?) of the world but tails would destroy it.)
In short, the St. Petersburg paradox goes as follows: a fair coin is tossed until heads come up, and the player wins $2^n, where n is the number of times the coin was flipped. So for example if heads come up on the first flip the player gets $2, if it comes up on the second they get $4, on the third, $8, on the tenth $1024 (2^10), etc. It's easy to show that the expected value of the game is infinite (approaches infinity).
Therefore, someone perfectly rational (?) should be willing to pay virtually any amount of money to play the game, because any finite amount of money is less than an infinite amount of money, and therefore the expected gain is always positive.
Yet you will probably not find many people (except SBF?) willing to pay millions of dollars to play that game.
It's only a paradox if we think it shows that people are not "rational". But I think it simply shows EV is not a good measure of risk, and everyone knows it.
Very complete and fascinating article about the St. Petersburg Paradox here:
https://plato.stanford.edu/entries/paradox-stpetersburg/