> The link says it's trying to maximize log wealth
Yes it is maximizing log wealth.
Think this way: If you give all of your fortune to the mugger, your capital will end up being 1) 0 livres(99.9%) or 2) 20000 livres (0.1%). So the expected log wealth is:
log(0) * 0.999 + log(20000) * 0.001 = -inf
On the other hand, if you bet 10% of your wealth, you will end up having 1) 9 livres (99.9%) or 2) 2009 livres (0.1%)
log(9) * 0.999 + log(2009) * 0.001 = 2.3
So you will prefer to bet 10% over 100%. The math does not bring you to "bet all of your fortune!" even if the odds is 1:100000000.
> Yes, maximizing the log implies you shouldn't bet everything. But you should still bet 99.99%, if the odds are a googol to 1.
Not quite. If the odds are infinitely favorable, the log wealth is maximized when you bet 0.1% of your wealth. Any fractions other than that produce inferior results. This might be counter-intuitive but actually can be easily proven by basic calculus.
If you still doubt it, you can just compute it to be sure! For example, if odds is indeed 1:googol (1:10^100), the log wealth for betting 99.99% is -6.67, less than betting 0.1% which produces 2.52.
I figured out the issue. You're assuming you can bet any fraction you want, while Pascal's mugging requires a specific bet, take it or leave it.
Maximizing log wealth will require taking such bets at less than 100% of your current wealth, provided the payout is high enough. That's easily proven with simple algebra: if you start with X, taking a bet requiring 99.99% of X and paying Y:1 and a probability of Z of paying out, then expected log value of taking it is log(X/10,000)(1-Z)+log((YX*9999/10000)+X/10,000)(Z). This goes to infinity as Y goes to infinity.
You are right in a narrow sense. Yes, you can construct a situation that makes "betting 99.99% of my wealth" profitable. But:
* You're assuming the winning probability p does not decrease as the odds Y goes up. This is a silly assumption. Do you really believe that the probability is all the same when "Hay I will pay you 2000 livres tomorrow" and "Hay I think I can pay you a infinite amount of livres tomorrow"?
* For example, if p = 1 / (odds), then E(log) never goes to infinity.
If you really assume that there is 1/10000 chance that the mugger can pay you an infinite amount of money, then ...... why not? You can just start a hedge fund on that. You'll gather 100000 people and make them bet independently with muggers, then there is 99.99% of chance that someone actually gets an infinite amount of returns. Now everyone is happy receiving an infinite amount of money.
>You're assuming the winning probability p does not decrease as the odds Y goes up. This is a silly assumption.
No, the only assumption required is that the probability decreases much slower than the odds goes up. Which is self-evidently true; the complexity of the claim doesn't go up nearly as fast as the odds being offered.
If the probability is 1/googol (in reality it's much higher than that, 1/googol epistemic probabilities never show up) but the odds being offered is 3^^^3, then you should take the bet, whether you're trying to maximize wealth or log wealth.
Yes it is maximizing log wealth.
Think this way: If you give all of your fortune to the mugger, your capital will end up being 1) 0 livres(99.9%) or 2) 20000 livres (0.1%). So the expected log wealth is:
On the other hand, if you bet 10% of your wealth, you will end up having 1) 9 livres (99.9%) or 2) 2009 livres (0.1%) So you will prefer to bet 10% over 100%. The math does not bring you to "bet all of your fortune!" even if the odds is 1:100000000.