It is pretty simple to formalise - if there is a pool of people who routinely accept positive-expected-value gambles where there is a P = 99.999999999% chance of going broke then we expect everyone in that pool will be broke unless the size of the pool is comparable to 1/(1-P). Tighten up the bounds on 'comparable' a bit and that is formalised.
The mistake is accepting uncritically that expected value is the best metric to optimise. Nobody ever proved that expected value is a strategically superior metric. In fact it would be quite hard to prove that since it is not true. It leaves people vulnerable to making very stupid decisions as illustrated in Pascals Mugging.
Optimium strategy involves at a minimum considering your available opportunities and available resources. Opportunity alone is not enough.
You need to maximize expected utility. In order to avoid this, you need utility to be bounded: there must be some multiple of your current utility that's impossible to ever have regardless of what happens.
Agreed that most people could reject the mugging on those grounds but I think that is more than is needed.
Even if someone being mugged had an unbounded utility function the finite resources argument forces them to rationally reject the deal. We've reintroduced infinities; so now our rationalist must accept that there are infinitely many situations with the same properties as the Mugging (positive expected value, Almost Sure to lose the stake). Their strategy would have to be to reject deals like the Mugging and seek out deals that have positive expected return and probably keep the stake/have a tiny stake compared to their reserves.
I'm basically saying a rationalist with finite wealth is probably using the Kelly criterion [0] and would reject the Mugging on that basis.
>We've reintroduced infinities; so now our rationalist must accept that there are infinitely many situations with the same properties as the Mugging (positive expected value, Almost Sure to lose the stake).
Yes, in the presence of infinities the decision function is inconsistent.
>Their strategy would have to be to reject deals like the Mugging and seek out deals that have positive expected return and probably keep the stake/have a tiny stake compared to their reserves.
Try formalizing that.
Kelly criterion doesn't work. Your link says it maximizes log wealth. If potential wealth is unbounded, you will still take bets that are positive E(log utility).
> Kelly criterion doesn't work. Your link says it maximizes log wealth. If potential wealth is unbounded, you will still take bets that are positive E(log utility).
Actually Kelly criterion works well, since it would limit your exposure to the game.
In this situation, Pascal as a Kelly-better would bet 12 deniers out of 10 livres (= 0.05% of his wealth), which is a penny.
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. The amount may need to be slightly more insane than it would be if you were just maximizing wealth, but with massive numbers such as used in Pascal's mugging this is easy enough.
> 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.
> 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.
The mistake is accepting uncritically that expected value is the best metric to optimise. Nobody ever proved that expected value is a strategically superior metric. In fact it would be quite hard to prove that since it is not true. It leaves people vulnerable to making very stupid decisions as illustrated in Pascals Mugging.
Optimium strategy involves at a minimum considering your available opportunities and available resources. Opportunity alone is not enough.