= 27d567bc2d48d9f40e9ccc20ea370b15)
TimFreeman comments on St. Petersburg Mugging Implies You Have Bounded Utility - Less Wrong
You are viewing a comment permalink. View the original post to see all comments and the full post content.
= 783df68a0f980790206b9ea87794c5b6)
Loading…= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (163)
This doesn't work with an unbounded utility function, for standard reasons:
1) The mixed strategy. If there is at least one lottery with infinite expected utility, then any combination of taking that lottery and other actions also has infinite expected utility. For example, in the traditional Pascal's Wager involving taking steps to believe in God, you could instead go around committing Christian sins: since there would be nonzero probability that this would lead to your 'wagering for God' anyway, it would also have infinite expected utility. See Alan Hajek's classic article "Waging War on Pascal's Wager."
Given the mixed strategy, taking and not taking your bet both have infinite expected utility, even if there are no other infinite expected utility lotteries.
2) To get a decision theory that actually would take infinite expected utility lotteries with high probability we would need to use something like the hyperreals, which would allow for differences in the expected utility of different probabilities of infinite payoff. But once we do that, the fact that your offer is so implausible penalizes it. We can instead keep our money and look for better opportunities, e.g. by acquiring info, developing our technology, etc. Conditional on there being any sources of infinite utility, it is far more likely that they will be better obtained by other routes than by succumbing to this trick. If nothing else, I could hold the money in case I encounter a more plausible Mugger (and your version is not the most plausible I have seen). Now if you demonstrated the ability to write your name on the Moon in asteroid craters, turn the Sun into cheese, etc, etc, taking your bet might win for an agent with an unbounded utility function.
Also see Nick Bostrom's infinitarian ethics paper.
As it happens I agree that human behavior and intuitions (as I weight them) in these situations are usually better summed up with a bounded utility function, which may include terms like the probability of attaining infinite welfare, or attaining a large portion of hyperreal expected welfare that one could, etc, than an unbounded utility function. I also agree that St Petersburg lotteries and the like do indicate our bounded preferences. The problem here is technical, in the construction of your example.
That article is paywalled. It was published in 2003. Hajek's entry about Pascal's Wager in the Stanford Encylopedia of Philosophy is free and was substantively revised (hopefully by Hajek) in 2008, so there's a good chance the latter contains all the good ideas in the former and is easier to get to. The latter does mention the idea that utilities should be bounded, and many other things potentially wrong with Pascal's wager. There's no neat list of four items that looks like an obvious match to the title of the paywalled article.
You can find it here though.
Thanks for the pointer to a free version of Hajek's "Waging War on Pascal's Wager" paper. One of his alternative formulations uses surreal numbers for utilities, much to my surprise.
The main thrust is that either the utility of Heaven isn't the best possible thing, or it is the best possible thing and a mixed strategy of betting on heaven with probability p and betting on nothing with probability 1-p also gives infinite utility, for positive p. Thus, if Heaven is the best possible thing, Pascal's Wager doesn't rule out mixed strategies.
If someone could check my math here -- I don't think surreal numbers let you assign a utility to the St. Petersburg paradox. The expected utility received at each step is 1, so the total utility is 1 + 1 + 1 + ... . Suppose that sum is X. Then X + 1 = X. This is not true for any surreal number, right?