Their coin flip thought experiment is extremely misleading and it took me a while to understand why but now I think I can explain it.

Intuitively, it seems like everyone is making a fair bet because you're equally likely to win or lose. If you have an initial net worth of $1000 and flip a coin you're equally like to gain or lose $200 and your expected net worth after the flip is still $1000 (50% chance of $1200 or $800) so it's a wash, right? However their simulation kept having me end up poor which confused me, so I ran the same simulation in a Python script. What I found was as the number of flips increases your net worth approaches zero! I found this surprising because if the expected net worth after a single flip is unchanged, I would expect this to stay true for multiple flips. But based on simulations, against my intuition, it seems like this is actually a bad bet in the long term and you'll always lose money. This is still true even if you start to skew the odds and give them a 51% chance to win the coin flip.

So after some googling I found something called the Kelly Criterion which calculates whether a bet is good or bad based on the gains and losses and chance of each and decided to plug in these numbers: https://en.wikipedia.org/wiki/Kelly_criterion#Proof

For the game in the article, the rules are that the poorer person bets 20% of their net worth on a coin flip, so these are the variables:

    f=20%
    p=50%
    q=50%
    a=20%
    b=20%

    r = (1 + f*b) ^ p * (1 – f*a) ^ q
      = (1 + 0.2 * 0.2) ^ 0.5 * (1 – 0.2 * 0.2) ^ 0.5
      = 0.99919967974

So the long term geometric return of playing this game is 0.999, and since it's slightly below 1 you will lose money in the long term. And the really misleading part is it seems like everyone is playing the same game, but what's really happening is the POORER person is playing this game (because the net worth value comes from them) and the rich person is just taking the inverse bet against them. In other words, this thought experiment is "force a poor person to play this gambling game with a geometric return below 1 (so on average they lose money), and pair them up with a rich person who gains money equal to the poor person's losses", which is obviously going to result in rich people being favored and gaining money.

If you forced a rich person to play this same game of repeatedly betting 20% of their money on a coin flip they would also end up losing all their money! When you frame it like this it's obvious that having a poor person play an unfavored gambling game and deposit their losses to a rich person is going to favor the rich people. This doesn't seem like a critique on capitalism or inequality, it's more analogous gambling at a casino.

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However I am still confused how you can have a game where the expected gain after a single match is 0%, but when playing multiple rounds your expected gain is negative (this is what plugging numbers into the expected value formula in the Kelly Criterion wiki seems to prove). I find this counterintuitive and hoping someone can explain this.

user:	adamerose
created:	March 24, 2022
karma:	2