Aha. So if a sum of non-negative numbers converges, than any rearrangement of that sum will converge to the same number, but not so for sums of possibly-negative numbers?
Ok, another angle. If you take Christiano’s lottery:
X∞=12X0+14X1+18X2+116X4...
and map outcomes to their utilities, setting the utility of X0 to 1, of X1 to 2, etc., you get:
1/2+1/2+1/2+1/2+...
Looking at how the utility gets rearranged after the “we can write X∞ as a mixture” step, the first “1/2″ term is getting “smeared” across the rest of the terms, giving:
3/4+5/8+9/16+17/32+...
which is a sequence of utilities that are pairwise higher. This is an essential part of the violation of Antisymmetry/Unbounded/Dominance. My intuition says that a strange thing happened when you rearranged the terms of the lottery, and maybe you shouldn’t do that.
Should there be another property, called “Rearrangement”?
Rearrangement: you may apply an infinite number of commutivity (x+y=y+x) and associativity ((x+y)+z=x+(y+z)) rewrites to a lottery.
(In contrast, I’m pretty sure you can’t get an Antisymmetry/Unbounded/Dominance violation by applying only finitely many commutivity and associativity rearrangements.)
I don’t actually have a sense of what “infinite lotteries, considered equivalent up to finite but not infinite rearrangements” look like. Maybe it’s not a sensible thing.
Your argument doesn’t go through if you restrict yourself to infinite weighted averages with nonnegative weights.
Aha. So if a sum of non-negative numbers converges, than any rearrangement of that sum will converge to the same number, but not so for sums of possibly-negative numbers?
Ok, another angle. If you take Christiano’s lottery:
X∞=12X0+14X1+18X2+116X4...
and map outcomes to their utilities, setting the utility of X0 to 1, of X1 to 2, etc., you get:
1/2+1/2+1/2+1/2+...
Looking at how the utility gets rearranged after the “we can write X∞ as a mixture” step, the first “1/2″ term is getting “smeared” across the rest of the terms, giving:
3/4+5/8+9/16+17/32+...
which is a sequence of utilities that are pairwise higher. This is an essential part of the violation of Antisymmetry/Unbounded/Dominance. My intuition says that a strange thing happened when you rearranged the terms of the lottery, and maybe you shouldn’t do that.
Should there be another property, called “Rearrangement”?
Rearrangement: you may apply an infinite number of commutivity (x+y=y+x) and associativity ((x+y)+z=x+(y+z)) rewrites to a lottery.
(In contrast, I’m pretty sure you can’t get an Antisymmetry/Unbounded/Dominance violation by applying only finitely many commutivity and associativity rearrangements.)
I don’t actually have a sense of what “infinite lotteries, considered equivalent up to finite but not infinite rearrangements” look like. Maybe it’s not a sensible thing.