Okay, but is avoiding St. Petersburging risk-aversive or loss-aversive? In my impression, many similar cases just contain equivalent of “you just die” (for example, you lose all your money) which is very low utility, so you can sort-of recover avoiding St. Petersburg via setting utility in log of size of bankroll, or something like that.
Good point! St. Petersburg requires utility being monotonic (ideally linear) in something other than probability (and optionally something like unbounded or at least increasing for a while).
This doesn’t have to be the case for all utility functions. (Especially since unbounded utilities are bad). Probabilities are strictly bounded, so having utility being linear in them is not a huge problem. Thanks for changing my mind!
Yep, but my honest position towards St. Petersburg lotteries is that they do not exist in “natural units”, i.e., counts of objects in physical world.
Reasoning: if you predict with probability p that you encounter St. Petersburg lottery which creates infinite number of happy people on expectation (version of St. Petersburg lottery for total utilitarians), then you should put expectation of number of happy people to infinity now, because E[number of happy people] = p * E[number of happy people due to St. Petersburg lottery] + (1 - p) * E[number of happy people for all other reasons] = p * inf + (1 - p) * E[number of happy people for all other reasons] = inf.
Therefore, if you don’t think right now that expected number of future happy people is infinity, then you shouldn’t expect St. Petersburg lottery to happen in any point of the future.
Therefore, you should set your utility either in “natural units” or in some “nice” function of “natural units”.
Okay, but is avoiding St. Petersburging risk-aversive or loss-aversive? In my impression, many similar cases just contain equivalent of “you just die” (for example, you lose all your money) which is very low utility, so you can sort-of recover avoiding St. Petersburg via setting utility in log of size of bankroll, or something like that.
Good point! St. Petersburg requires utility being monotonic (ideally linear) in something other than probability (and optionally something like unbounded or at least increasing for a while).
This doesn’t have to be the case for all utility functions. (Especially since unbounded utilities are bad). Probabilities are strictly bounded, so having utility being linear in them is not a huge problem. Thanks for changing my mind!
My general reasoning about unbounded utilities see here
This doesn’t work if the lottery is in utils rather than dollars/money/whatever instrumental resource.
Yep, but my honest position towards St. Petersburg lotteries is that they do not exist in “natural units”, i.e., counts of objects in physical world.
Reasoning: if you predict with probability p that you encounter St. Petersburg lottery which creates infinite number of happy people on expectation (version of St. Petersburg lottery for total utilitarians), then you should put expectation of number of happy people to infinity now, because E[number of happy people] = p * E[number of happy people due to St. Petersburg lottery] + (1 - p) * E[number of happy people for all other reasons] = p * inf + (1 - p) * E[number of happy people for all other reasons] = inf.
Therefore, if you don’t think right now that expected number of future happy people is infinity, then you shouldn’t expect St. Petersburg lottery to happen in any point of the future.
Therefore, you should set your utility either in “natural units” or in some “nice” function of “natural units”.