I wondered if someone would bring this up! Yes, some people take this as a strong argument that utilities simply have to be bounded in order to be well-defined at all. AFAIK this is just called a “bounded utility function”. Many of the standard representation theorems also imply that utility is bounded; this simply isn’t mentioned as often as other properties of utility.
However, I am not one of the people who takes this view. It’s perfectly consistent to define preferences which must be treated as unbounded utility. In doing so, we also have to specify our preferences about infinite lotteries. The divergent sum doesn’t make this impossible; instead, what it does is allow us to take many different possible values (within some consistency constraints). So for example, a lottery with 50% probability of +1 util, 25% of −3, 1/8th chance of +9, 1⁄16 −27, etc can be assigned any expected value whatsoever. Its evaluation is subjective! So in this framework, preferences encode more information than just a utility for each possible world; we can’t calculate all the expected values just from that. We also have to know how the agent subjectively values infinite lotteries. But this is fine!
How that works is a bit technical and I don’t want to get into it atm. From a mathematical perspective, it’s pretty “standard/obvious” stuff (for a graduate-level mathematician, anyway). But I don’t think many professional philosophers have picked up on this? The literature on Infinite Ethics seems mostly ignorant of it?
Sorry to comment on a two year old post but: I’m actually quite intrigued by the arbitrary valuation of infinite lotteries thing here. Can you explain this a bit further or point me to somewhere I can learn about it? (P-adic numbers come to mind, but probably have nothing to do with it.)
I wondered if someone would bring this up! Yes, some people take this as a strong argument that utilities simply have to be bounded in order to be well-defined at all. AFAIK this is just called a “bounded utility function”. Many of the standard representation theorems also imply that utility is bounded; this simply isn’t mentioned as often as other properties of utility.
However, I am not one of the people who takes this view. It’s perfectly consistent to define preferences which must be treated as unbounded utility. In doing so, we also have to specify our preferences about infinite lotteries. The divergent sum doesn’t make this impossible; instead, what it does is allow us to take many different possible values (within some consistency constraints). So for example, a lottery with 50% probability of +1 util, 25% of −3, 1/8th chance of +9, 1⁄16 −27, etc can be assigned any expected value whatsoever. Its evaluation is subjective! So in this framework, preferences encode more information than just a utility for each possible world; we can’t calculate all the expected values just from that. We also have to know how the agent subjectively values infinite lotteries. But this is fine!
How that works is a bit technical and I don’t want to get into it atm. From a mathematical perspective, it’s pretty “standard/obvious” stuff (for a graduate-level mathematician, anyway). But I don’t think many professional philosophers have picked up on this? The literature on Infinite Ethics seems mostly ignorant of it?
Sorry to comment on a two year old post but: I’m actually quite intrigued by the arbitrary valuation of infinite lotteries thing here. Can you explain this a bit further or point me to somewhere I can learn about it? (P-adic numbers come to mind, but probably have nothing to do with it.)
Commenting on old posts is encouraged :)