[teambayleaf]

> If you're trying to maximize log wealth, someone promising an insane amount of wealth for a trivial investment is a positive value for log wealth.

Nah it does not work in such way.

Mathematically speaking, even if the mugger can propose an infinite large return in this situation, a kelly better would not bet more than 0.1% (= p = 1 / 1000) of his wealth.

It's very sound mathematics and nothing mystifying.

Edit: A Kerry better thinks in term of a long running sequence of bets. If you all of your wealth just because the game is favorable, you'll eventually lose all of your capital with probability one (in other words, betting everything minimizes the expected value of your wealth in long term). The natural conclusion here is that you need to bet a fraction of your wealth to maximize your long term wealth. But how much? Kelly criterion answers this queation formally. Read the Wikipedia article (or better, read Kelly 1956. It's a good paper) for how it handles the question.

[ikeboy]

The link says it's trying to maximize log wealth. If it's not actually doing that, then sure, it can limit loss. What exactly is being optimized?

[teambayleaf]

> 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.

[ikeboy]

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.

[teambayleaf]

> 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.

[ikeboy]

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.

[teambayleaf]

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.