I’m upvoting this post because it caused me to rethink some of my thoughts about utility and Pascal’s mugging, something that very few of these posts [on Pascal’s mugging] so far have done.
However, I am now somewhat confused. Doesn’t the Archimedean property for utility (axiom P3′ here) automatically imply that utility should be bounded? My argument for this would be as follows. I’m using the terminology from the Wikipedia page, but in case you don’t want to click the link, u is utility, Eu is expected utility.
If Eu(N) was ever infinite, then there would be no epsilon > 0 such that (1-epsilon)L+epsilon N < M (here we start with some fixed L,M with L<M).
von Neumann utility is a statement about all lotteries, not just the ones induced by our beliefs. So in particular we have to consider the lottery A(E) where a fixed event E happens with probability 1. Since Eu(A(E)) is finite, and u(E) = Eu(A(E)), u(E) must be finite.
Finally, suppose that there was a sequence of events E1,E2,… whose utilities grew without bound. We can always pick out a subsequence F1,F2,… such that u(F(n+1)) >= 2u(F(n)) for all n. Then consider the lottery where Fn occurs with probability 2^(-n). This has infinite expected utility since its expected utility is sum(n=1 to infty) 2^(-n)u(F(n)) > sum(n=1 to infty) u(F(1)) = infty
Is there a flaw in this argument? Are the VNM utility axioms too strong (should preferences only need to be defined over lotteries that can actually be induced by updating our priors based on some amount of evidence; or alternately, is the Archimedean property too strong)?
Or is utility bounded? While I personally think that human preferences should lead to bounded utility functions, the conclusion that all rational agents must have bounded utility functions seems possibly too strong.
The vNM argument only uses preference over lotteries with finitely many possible outcomes to assign utilities to pure outcomes. One can use measure theory arguments to pin down the expected utilities of other lotteries. That some lotteries have infinite value does not seem to me to be a big problem.
I suppose what it comes down to is that I don’t believe a conditionally convergent expected utility is possible, and thus does not count as a possible lottery.
I’d say that even this idea is too strong, and I’m only willing to go with it because if I break expected utility, I have no idea what to do.
Also, doesn’t the set of “possible lotteries” have to be the space of lotteries about which VNM can be proved? Probably there are multiple such spaces, but I’m not sure what conditions they have to satisfy. Also, since utilities don’t come until after we’ve chosen a space of lotteries, we can’t define the space of lotteries to be “those for which expected utility converges”, so I’m not quite sure what you are looking for.
Using priors that allow for unconditionally convergent expected utility.
Also, doesn’t the set of “possible lotteries” have to be the space of lotteries about which VNM can be proved?
I don’t understand what you mean. The set I’m using is the largest possible set for which all of those axioms are true. If I use one with utility that increases/decreases without limit, axiom 3 (along with 3′) is false. If I use one with utility that doesn’t converge unconditionally, axiom 2 is false.
I’m upvoting this post because it caused me to rethink some of my thoughts about utility and Pascal’s mugging, something that very few of these posts [on Pascal’s mugging] so far have done.
However, I am now somewhat confused. Doesn’t the Archimedean property for utility (axiom P3′ here) automatically imply that utility should be bounded? My argument for this would be as follows. I’m using the terminology from the Wikipedia page, but in case you don’t want to click the link, u is utility, Eu is expected utility.
If Eu(N) was ever infinite, then there would be no epsilon > 0 such that (1-epsilon)L+epsilon N < M (here we start with some fixed L,M with L<M).
von Neumann utility is a statement about all lotteries, not just the ones induced by our beliefs. So in particular we have to consider the lottery A(E) where a fixed event E happens with probability 1. Since Eu(A(E)) is finite, and u(E) = Eu(A(E)), u(E) must be finite.
Finally, suppose that there was a sequence of events E1,E2,… whose utilities grew without bound. We can always pick out a subsequence F1,F2,… such that u(F(n+1)) >= 2u(F(n)) for all n. Then consider the lottery where Fn occurs with probability 2^(-n). This has infinite expected utility since its expected utility is sum(n=1 to infty) 2^(-n)u(F(n)) > sum(n=1 to infty) u(F(1)) = infty
Is there a flaw in this argument? Are the VNM utility axioms too strong (should preferences only need to be defined over lotteries that can actually be induced by updating our priors based on some amount of evidence; or alternately, is the Archimedean property too strong)?
Or is utility bounded? While I personally think that human preferences should lead to bounded utility functions, the conclusion that all rational agents must have bounded utility functions seems possibly too strong.
The vNM argument only uses preference over lotteries with finitely many possible outcomes to assign utilities to pure outcomes. One can use measure theory arguments to pin down the expected utilities of other lotteries. That some lotteries have infinite value does not seem to me to be a big problem.
I suppose what it comes down to is that I don’t believe a conditionally convergent expected utility is possible, and thus does not count as a possible lottery.
I’d say that even this idea is too strong, and I’m only willing to go with it because if I break expected utility, I have no idea what to do.
Can you clarify what you mean by “this idea”?
Also, doesn’t the set of “possible lotteries” have to be the space of lotteries about which VNM can be proved? Probably there are multiple such spaces, but I’m not sure what conditions they have to satisfy. Also, since utilities don’t come until after we’ve chosen a space of lotteries, we can’t define the space of lotteries to be “those for which expected utility converges”, so I’m not quite sure what you are looking for.
Using priors that allow for unconditionally convergent expected utility.
I don’t understand what you mean. The set I’m using is the largest possible set for which all of those axioms are true. If I use one with utility that increases/decreases without limit, axiom 3 (along with 3′) is false. If I use one with utility that doesn’t converge unconditionally, axiom 2 is false.