Here’s a concrete example. Start with a sum that converges to 0 (in fact every partial sum is 0):
0 + 0 + …
Regroup the terms a bit:
= (1 + −1) + (1 + −1) + …
= 1 + (-1 + 1) + (-1 + 1) + …
= 1 + 0 + 0 + …
and you get a sum that converges to 1 (in fact every partial sum is 1). I realize that the things you’re summing are probability distributions over outcomes and not real numbers, but do you have reason to believe that they’re better behaved than real numbers in infinite sums? I’m not immediately seeing how countable additivity helps. Sorry if that should be obvious.
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.
Here’s a concrete example. Start with a sum that converges to 0 (in fact every partial sum is 0):
0 + 0 + …
Regroup the terms a bit:
= (1 + −1) + (1 + −1) + …
= 1 + (-1 + 1) + (-1 + 1) + …
= 1 + 0 + 0 + …
and you get a sum that converges to 1 (in fact every partial sum is 1). I realize that the things you’re summing are probability distributions over outcomes and not real numbers, but do you have reason to believe that they’re better behaved than real numbers in infinite sums? I’m not immediately seeing how countable additivity helps. Sorry if that should be obvious.
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.