Vann McGee has proven that if you have an agent with an unbounded utility function and who thinks there are infinitely many possible states of the world (ie, assigns them probability greater than 0), then you can construct a Dutch book against that agent. Next, observe that anyone who wants to use Solomonoff induction as a guide has committed to infinitely many possible states of the world. So if you also want to admit unbounded utility functions, you have to accept rational agents who will buy a Dutch book.
And if you do that, then the subjectivist justification of probability theory collapses, taking Bayesianism with it, since that’s based on non-Dutch-book-ability.
I think the cleanest option is to drop unbounded utility functions, since they buy you zero additional expressive power. Suppose you have an event space S, a preference relation P, and a utility function f from events to nonnegative real numbers such that if s1 P s2, then f(s1) < f(s2). Then, you can easily turn this into a bounded utility function g(s) = f(s)/(f(s) + 1). It’s easily seen that g respects the preference relation P in exactly the same way as f did, but is now bounded to the interval [0, 1).
Vann McGee has proven that if you have an agent with an unbounded utility function and who thinks there are infinitely many possible states of the world (ie, assigns them probability greater than 0), then you can construct a Dutch book against that agent. Next, observe that anyone who wants to use Solomonoff induction as a guide has committed to infinitely many possible states of the world. So if you also want to admit unbounded utility functions, you have to accept rational agents who will buy a Dutch book.
And if you do that, then the subjectivist justification of probability theory collapses, taking Bayesianism with it, since that’s based on non-Dutch-book-ability.
I think the cleanest option is to drop unbounded utility functions, since they buy you zero additional expressive power. Suppose you have an event space S, a preference relation P, and a utility function f from events to nonnegative real numbers such that if s1 P s2, then f(s1) < f(s2). Then, you can easily turn this into a bounded utility function g(s) = f(s)/(f(s) + 1). It’s easily seen that g respects the preference relation P in exactly the same way as f did, but is now bounded to the interval [0, 1).