It suffices to show that the Smith lotteries that the above result establishes are the only lotteries that can be part of maximal lottery-lotteries are also subject to the partition-of-unity condition.
I fail to understand this sentence. Here are some questions about this sentence:
what are Smith lotteries? Ctrl+f only finds lottery-Smith lottery-lotteries, do you mean these? Or do you mean lotteries that are smith?
which result do you mean by “above result”?
What does it mean for a lottery to be part of maximal lottery-lotteries?
does “also subject to the partition-of-unity” refer to the smith lotteries or to the lotteries that are part of maximal lottery-lotteries? (it also feels like there is a word missing somewhere)
Why would this suffice?
Is this part also supposed to imply the existence of maximal lottery-lotteries? If so, why?
To avoid confusion: this post and my reply to it were also on a past version of this post; that version lacked any investigation of dominance criterion desiderata for lottery-lotteries.
Lotteries over the Smith set. That definitely wasn’t clear—I’ll fix that.
which result do you mean by “above result”?
Proposition: (Lottery-lotteries are strongly characterized by their selectivity of partitions of unity)
This one. “You can tell whether a lottery-lottery is maximal or not by how good the partitions of unity it admits are.” Sorry, didn’t really know a good way to link to myself internally and I forgot to number the various statements.
What does it mean for a lottery to be part of maximal lottery-lotteries?
Just that some maximal lottery-lottery gives it nonzero probability.
does “also subject to the partition-of-unity” refer to the smith lotteries or to the lotteries that are part of maximal lottery-lotteries? (it also feels like there is a word missing somewhere)
Oh no! I thought I caught all the typos! That should be “also subject to the partition-of-unity condition”, that is, you look at all the lotteries (which we know are over the Smith set, btw) that some arbitrary maximal lottery-lottery gives any nonzero probability to, and you should expect to be able to sort them into groups by what final probability over candidates they induce; those final probabilities over candidates should themselves result in identical geometric-expected utility for the voterbase.
Why would this suffice?
Consider: at this point we know that a maximal lottery-lottery would not just have to be comprised of lottery-Smith lotteries, i.e., lotteries that are in the lottery-Smith set - but also that they have to be comprised entirely of lotteries over the Smith set of the candidate set. Then on top of that, we know that you can tell which lottery-lotteries are maximal by which partitions of unity they admit (that’s the “above result”). Note that by “admit” we mean “some subset of the lotteries this lottery-lottery has support over corresponds to it” (this partition of unity).
This is the slightly complicated part. The game I described has a mixed strategy equilibrium; this will take the form of some probability distribution over ΔC. In fact it won’t just have one, it’ll likely have whole families of them. Much of the time, the lotteries randomized over won’t be disjoint—they’ll both assign positive probability to some candidate. The key is, the voter doesn’t care. As far as a voter’s expected utility is concerned, the only thing that matters is the final probability of each candidate.
That’s where your collapse of different possible maximal lottery-lotteries to the same partition of unity comes in. Because we know that equivalent candidate-lotteries give voters the same expected utility, the only two ways you get a voter who’s indifferent between two candidate-lotteries are 1) they’re the same lottery or 2) the voter’s utility function is just right to have two very different lotteries tie. Likewise, the only two ways you get a voterbase to be indifferent between two lottery-lotteries is 1) they reduce to the same lottery or 2) the geometric expectation of a voter’s utility over candidates sampled from the samples of the lottery-lottery Just Plain Ties.
So: if we can show that whatever equilibrium set of candidate-lotteries Alice and Bob pick always collapses to some convex combination of the Best partitions of unity...? Yeah, I don’t think that the second half of the proof holds up as is.
I think I’ve slightly messed up the definition of lottery-Smith, though not in a fatal way nor (thankfully) in a way that looks to threaten the existence result. The set condition is too strong, in requiring that a lottery-Smith lottery contain all lotteries which correspond to any of the admissible partitions. I’m just going to cut it; it’s not actually necessary.
Is this part also supposed to imply the existence of maximal lottery-lotteries? If so, why?
Yes.
Yes, and in particular, it implies the existence of maximal lottery-lotteries before it even tries to prove that they’re also lottery-Smith in the sense I define.
Scott’s proof doesn’t quite work (as he says there) - it almost works, except for the part where the reward functions for Alice and Bob can quite reasonably be discontinuous. My proof is intended as a patch—the reward functions for Alice and Bob should now be extremely continuous in a way that also corresponds well to “how much better did Alice do at picking a candidate-lottery that V will like than Bob did?”.
Hopefully this helped? Reading this is confusing even for me sometimes—the word “lottery/lotteries”, which appears 59 times in this comment alone, no longer looks like a real word to me and hasn’t since late Wednesday. Your comment was really helpful—I have some editing to do! (update—editing is done.)
I fail to understand this sentence. Here are some questions about this sentence:
what are Smith lotteries? Ctrl+f only finds lottery-Smith lottery-lotteries, do you mean these? Or do you mean lotteries that are smith?
which result do you mean by “above result”?
What does it mean for a lottery to be part of maximal lottery-lotteries?
does “also subject to the partition-of-unity” refer to the smith lotteries or to the lotteries that are part of maximal lottery-lotteries? (it also feels like there is a word missing somewhere)
Why would this suffice?
Is this part also supposed to imply the existence of maximal lottery-lotteries? If so, why?
To avoid confusion: this post and my reply to it were also on a past version of this post; that version lacked any investigation of dominance criterion desiderata for lottery-lotteries.
Lotteries over the Smith set. That definitely wasn’t clear—I’ll fix that.
Proposition: (Lottery-lotteries are strongly characterized by their selectivity of partitions of unity)
This one. “You can tell whether a lottery-lottery is maximal or not by how good the partitions of unity it admits are.” Sorry, didn’t really know a good way to link to myself internally and I forgot to number the various statements.
Just that some maximal lottery-lottery gives it nonzero probability.
Oh no! I thought I caught all the typos! That should be “also subject to the partition-of-unity condition”, that is, you look at all the lotteries (which we know are over the Smith set, btw) that some arbitrary maximal lottery-lottery gives any nonzero probability to, and you should expect to be able to sort them into groups by what final probability over candidates they induce; those final probabilities over candidates should themselves result in identical geometric-expected utility for the voterbase.
Consider: at this point we know that a maximal lottery-lottery would not just have to be comprised of lottery-Smith lotteries, i.e., lotteries that are in the lottery-Smith set - but also that they have to be comprised entirely of lotteries over the Smith set of the candidate set. Then on top of that, we know that you can tell which lottery-lotteries are maximal by which partitions of unity they admit (that’s the “above result”). Note that by “admit” we mean “some subset of the lotteries this lottery-lottery has support over corresponds to it” (this partition of unity).
This is the slightly complicated part. The game I described has a mixed strategy equilibrium; this will take the form of some probability distribution over ΔC. In fact it won’t just have one, it’ll likely have whole families of them. Much of the time, the lotteries randomized over won’t be disjoint—they’ll both assign positive probability to some candidate. The key is, the voter doesn’t care. As far as a voter’s expected utility is concerned, the only thing that matters is the final probability of each candidate.
That’s where your collapse of different possible maximal lottery-lotteries to the same partition of unity comes in. Because we know that equivalent candidate-lotteries give voters the same expected utility, the only two ways you get a voter who’s indifferent between two candidate-lotteries are 1) they’re the same lottery or 2) the voter’s utility function is just right to have two very different lotteries tie. Likewise, the only two ways you get a voterbase to be indifferent between two lottery-lotteries is 1) they reduce to the same lottery or 2) the geometric expectation of a voter’s utility over candidates sampled from the samples of the lottery-lottery Just Plain Ties.
So: if we can show that whatever equilibrium set of candidate-lotteries Alice and Bob pick always collapses to some convex combination of the Best partitions of unity...? Yeah, I don’t think that the second half of the proof holds up as is.
I think I’ve slightly messed up the definition of lottery-Smith, though not in a fatal way nor (thankfully) in a way that looks to threaten the existence result. The set condition is too strong, in requiring that a lottery-Smith lottery contain all lotteries which correspond to any of the admissible partitions. I’m just going to cut it; it’s not actually necessary.
Yes.
Yes, and in particular, it implies the existence of maximal lottery-lotteries before it even tries to prove that they’re also lottery-Smith in the sense I define.
Scott’s proof doesn’t quite work (as he says there) - it almost works, except for the part where the reward functions for Alice and Bob can quite reasonably be discontinuous. My proof is intended as a patch—the reward functions for Alice and Bob should now be extremely continuous in a way that also corresponds well to “how much better did Alice do at picking a candidate-lottery that V will like than Bob did?”.
Hopefully this helped? Reading this is confusing even for me sometimes—the word “lottery/lotteries”, which appears 59 times in this comment alone, no longer looks like a real word to me and hasn’t since late Wednesday. Your comment was really helpful—I have some editing to do! (update—editing is done.)