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 could do 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’t do 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.
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 could do 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’t do 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.