Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Pascal's Mugging (2009) [pdf]

nickbostrom.com

Post reply on HN
Print view View original

61–70 of 71 posts

Re: Pascal's Mugging (2009) [pdf]

#61

Post by YeGoblynQueenne »

Dylan16807 wrote: post #60
YeGoblynQueenne wrote: post #59

Earlier quoted context omitted.

But I'm not hallucinating and I'm not dreaming either. Also, I don't know why you're saying I'd round anything below one-in-a-million to zero. I wouldn't. But would assign zero probability to a mugger being an Operator from the Sevent Dimension because that's a patently absurd idea that I see no good reason to grace even with the slightest degree of belief. I mean, if you take what you are saying here at face value I…

> But I'm not hallucinating and I'm not dreaming either. You've never been unsure if something actually happened for a moment? Because even if you only spend a few moments like that per month, we can assign it a probability. > I mean, if you take what you are saying here at face value I actually have to assume that there is a probability that there exists a Seventh Dimension with magikcally powerful Operators inhabit…

>> The point of the thought experiment is just a very likely but very positive act.

(You mean very _un_likely eh?)

The reason I'm objecting specifically to the seventh dimension stuff is that it's just something fanciful that someone came up with, so it's obviously fake and I don't have to believe it even a little bit.

The probability of someone just handing out money (if I read you correctly this time) is very low, but not zero, yes. But I'm contesting the claim that I'm never allowed to assign 0 probability to anything, because then I'm at risk of losing out. Sometimes, you don't risk being wrong by disbelieving something.

Anyway I'm getting more and more confused by this conversation. I don't think it's getting anywhere. Thanks for your patience- you have the floor.

Re: Pascal's Mugging (2009) [pdf]

#62

Post by teambayleaf »

ikeboy wrote: post #48
roenxi wrote: post #46

Earlier quoted context omitted.

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, Alm…

>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 t…

> 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.

Re: Pascal's Mugging (2009) [pdf]

#63

Post by ikeboy »

teambayleaf wrote: post #62
ikeboy wrote: post #48

Earlier quoted context omitted.

>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 t…

> 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.

How are you getting those numbers?

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.

Re: Pascal's Mugging (2009) [pdf]

#64

Post by teambayleaf »

ikeboy wrote: post #63
teambayleaf wrote: post #62

Earlier quoted context omitted.

> 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.

How are you getting those numbers? 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.

Re: Pascal's Mugging (2009) [pdf]

#65

Post by ikeboy »

teambayleaf wrote: post #64
ikeboy wrote: post #63

Earlier quoted context omitted.

How are you getting those numbers? 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 K…

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?

Re: Pascal's Mugging (2009) [pdf]

#66

Post by teambayleaf »

ikeboy wrote: post #65
teambayleaf wrote: post #64

Earlier quoted context omitted.

> 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 K…

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?

> 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.

Re: Pascal's Mugging (2009) [pdf]

#67

Post by ikeboy »

teambayleaf wrote: post #66
ikeboy wrote: post #65

Earlier quoted context omitted.

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?

> 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(…

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.

Re: Pascal's Mugging (2009) [pdf]

#68

Post by teambayleaf »

ikeboy wrote: post #67
teambayleaf wrote: post #66

Earlier quoted context omitted.

> 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(…

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.

> 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.

Re: Pascal's Mugging (2009) [pdf]

#69

Post by ikeboy »

teambayleaf wrote: post #68
ikeboy wrote: post #67

Earlier quoted context omitted.

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.

> 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 comp…

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.

Re: Pascal's Mugging (2009) [pdf]

#70

Post by teambayleaf »

ikeboy wrote: post #69
teambayleaf wrote: post #68

Earlier quoted context omitted.

> 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 comp…

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…

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.

Post reply on HN