You've profoundly misunderstood McGee's argument, Eliezer. The reason you need the expectation of the sum of an infinite number of random variables to equal the sum of the expectations of those random variable is exactly to ensure that choosing an action based on the expected value actually yields an optimal course of action.

McGee observed that if you have an infinite event space and unbounded utilities, there are a collection of random utility functions U1, U2, ... such that E(U1 + U2 + ...) != E(U1) + E(U2) + .... McGee then observes that if you restrict utilities to a bounded range, then in fact E(U1 + U2 + ...) == E(U1) + E(U2) + ..., which ensures that a series of choices based on the expected value always give the correct result. In contrast, the other paper -- which you apparently approve of -- happily accepts that when E(U1 + U2 + ...) != E(U1) + E(U2) + ..., an agent can be Dutch booked and defends this as still rational behavior.

Right now, you're a "Bayesian decision theorist" who a) doesn't believe in making choices based on expected utility, and b) accepts Dutch Books as rational. This is goofy.

Eliezer:

Never mind having the expectation of a sum of an infinite number of variables not equalling the sum of the expectations; here we have the expectation of the sum of two bets not equalling the sum of the expectations.If you have an alternating series which is conditionally but not absolutely convergent, the Riemann series theorem says that reordering its terms can change the result, or force divergence. So you can't pull a series of bets apart into two series, and expect their sums to equal the sum of the original. But the fact that you assumed you could is a perfect illustration of the point; if you had a collection of bets in which you

coulddo this, then no limit-based Dutch book is possible.Ensuring that this property holds necessarily restricts the possible shapes of a utility function. We need to bound the utility function to avoid St. Petersburg-style problems, but the addition of time adds another infinite dimension to the event space, so we need to ensure that expectations of infinite sums of random variables indexed by time are also equal to the sum of the expectations. For example, one familiar way of doing this is to assume a time-separable, discounted utility function. Then you can't force an agent into infinitely delayed gratification, because there's a bounded utility and a minimum, nonzero payment to delay the reward -- at some point, you run out of the space you need to force delay.

If you're thinking that the requirement that expected utility actually works puts very stringent limits on what forms of utility functions you can use -- you're probably right. If you think that philosophical analyses of rationality can justify a wider selection utility functions or preference relations -- you're probably still right. But the one thing you

can'tdo is to pick a function of the second kind and still insist that ordinary decision-theoretic methods are valid with it. Decision theoretic methods require utility to form a proper random variable. If your utility function can't satisfy this need, you can't use decision theoretic methods with it.