However, how I assign value to divergent sums is subjective—it cannot be determined precisely from how I assign value to each of the elements of the sum, because I’m not trying to assume anything like countable additivity.
This implies that you believe in the existence of countably infinite bets but not countably infinite dutch booking processes. Thats seems like a strange/unphysical position to be in—if that were the best treatment of infinity possible, I think infinity is better abandoned. Im not even sure the framework in your linked post can really be said to contain infinte bets: the only way a bet ever gets evaluated is in a bookie strategy, and no single bookie strategy can be guaranteed to fully evaluate an infinte bet. Is there a single bookie strategy that differentiates the St. Petersburg bet from any finite bet? Because if no, then the agent at least cant distinguish them, which is very close to not existing at all here.
In a case like the St Petersburg Lottery, I believe I’m required to have some infinite expectation.
Why? I haven’t found any finite dutch books against not doing so.
Perhaps you can try to problematize this example for me given what I’ve written above—not sure if I’ve already addressed your essential worry here or not.
I dont think you have. That example doesn’t involve any uncertainty or infinite sums. The problem is that for any finite n, waiting n+1 is better than waiting n, but waiting indefinitely is worse than any. Formally, the problem is that I have a complete and transitive preference between actions, but no unique best action, just a series that keeps getting better.
Note that you talk about something related in your linked post:
I’m representing preferences on sets only so that I can argue that this reduces to binary preference.
But the proof for that reduction only goes one way: for any preference relation on sets, theres a binary one. My problem is that the inverse does not hold.
This implies that you believe in the existence of countably infinite bets but not countably infinite dutch booking processes. Thats seems like a strange/unphysical position to be in—if that were the best treatment of infinity possible, I think infinity is better abandoned. Im not even sure the framework in your linked post can really be said to contain infinte bets: the only way a bet ever gets evaluated is in a bookie strategy, and no single bookie strategy can be guaranteed to fully evaluate an infinte bet. Is there a single bookie strategy that differentiates the St. Petersburg bet from any finite bet? Because if no, then the agent at least cant distinguish them, which is very close to not existing at all here.
Why? I haven’t found any finite dutch books against not doing so.
I dont think you have. That example doesn’t involve any uncertainty or infinite sums. The problem is that for any finite n, waiting n+1 is better than waiting n, but waiting indefinitely is worse than any. Formally, the problem is that I have a complete and transitive preference between actions, but no unique best action, just a series that keeps getting better.
Note that you talk about something related in your linked post:
But the proof for that reduction only goes one way: for any preference relation on sets, theres a binary one. My problem is that the inverse does not hold.