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asr comments on Open thread, Aug. 03 - Aug. 09, 2015 - Less Wrong
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It's a tempting thought. But I think it's hard to make the math work that way.
I have a lovely laptop here that I am going to give you. Suppose you assign some utility U to it. Now instead of giving you the laptop, I give you a lottery ticket or the like. With probability P I give you the laptop, and with probability 1 - P you get nothing. (The lottery drawing will happen immediately, so there's no time-preference aspect here.) What utility do you attach to the lottery ticket? The natural answer is P * U, and if you accept some reasonable assumptions about preferences, you are in fact forced to that answer. (This is the basic intuition behind the von Neumann-Morgenstern Expected Utility Theorem.)
Given that probabilities are real numbers, it's hard to avoid utilities being real numbers too.
If we are going into VNM utility, it is defined as the output of the utility function and the utility function is defined as returning real numbers.
I could try to rescue the idea by throwing in units, the way multiplying distance units by time units gives you speed units... but I'd just be trying to technobabble my way out of the corner.
I think the most that I can try to rescue from this failed hunch is that some offbeat and unexpected part of mathematics might be able to be used to generate useful, non-obvious conclusions for utilitarian-style reasoning, in parallel with math based on gambling turning out to be useful for measuring confidence-strengths more generally. Anybody have any suggestions for such a subfield which won't make any actual mathematicians wince, should they read my story?