Maybe some of the problem is coming from trying to extend dominance to infinite combinations of lotteries. If we’re saying that the utility function is the thing that witnesses some coherence in the choices we make between lotteries, maybe it makes sense to ask for choices between finite combinations of lotteries but not infinite ones? Any choice we actually make, we end up making by doing some “finitary” sort of computation (not sure what this really means, if anything), and perhaps in particular it’s always understandable as a choice between finite lotteries. Like, if we choose between two lotteries, at that time we only had a finite number of concepts in mind, which implies a finite set of possible descriptions of lotteries, so that all lotteries we can distinguish are finite combinations of those finitely many lotteries.
Can the paradox be extended to someone who rejects Weak Dominance but accepts finitary dominance? E.g. by offering an infinite sequence of choices between finite combinations of lotteries, or something?
I think you avoid any contradiction if you reject Weak Dominance but accept a finite version of Dominance. For example, in that case you can simply declare all lotteries with infinite support to be incomparable to each other or to any finite lottery.
If you furthermore require your preferences to be complete, even when asking about infinite lotteries, such that either A>B or A>B or A=B, then I suspect you are back in trouble.
But if you just restrict preferences to finite lotteries you are fine and can compare them with expected value.
Yeah, maybe just truncating off finitely many summands in an infinite lottery induces constraints that force your examples to have infinite value? Maybe you can have complete hyperreal-valued preferences and finite dominance...?
Maybe some of the problem is coming from trying to extend dominance to infinite combinations of lotteries. If we’re saying that the utility function is the thing that witnesses some coherence in the choices we make between lotteries, maybe it makes sense to ask for choices between finite combinations of lotteries but not infinite ones? Any choice we actually make, we end up making by doing some “finitary” sort of computation (not sure what this really means, if anything), and perhaps in particular it’s always understandable as a choice between finite lotteries. Like, if we choose between two lotteries, at that time we only had a finite number of concepts in mind, which implies a finite set of possible descriptions of lotteries, so that all lotteries we can distinguish are finite combinations of those finitely many lotteries.
Can the paradox be extended to someone who rejects Weak Dominance but accepts finitary dominance? E.g. by offering an infinite sequence of choices between finite combinations of lotteries, or something?
I think you avoid any contradiction if you reject Weak Dominance but accept a finite version of Dominance. For example, in that case you can simply declare all lotteries with infinite support to be incomparable to each other or to any finite lottery.
If you furthermore require your preferences to be complete, even when asking about infinite lotteries, such that either A>B or A>B or A=B, then I suspect you are back in trouble.
But if you just restrict preferences to finite lotteries you are fine and can compare them with expected value.
Yeah, maybe just truncating off finitely many summands in an infinite lottery induces constraints that force your examples to have infinite value? Maybe you can have complete hyperreal-valued preferences and finite dominance...?