He found that it had already been proved that there is always a Condorcet Winner among lotteries provided that you compare lotteries in a certain way.
These winning lotteries choose from among the Dutta set, a subset of the Smith set, so the method satisfies the generalized Condorcet Criterion.
It seems to me that this idea (of electing lotteries that, in turn, pick candidates) is worth more exploration.
How about an Approval Winner among lotteries?
Here's one way to approach it:
At first let's just restrict ourselves to lotteries that are constant on their support, i.e. lotteries that give all candidates with a positive probability the same probability.
Each such lottery (supported by a proper subset of the candidates) has a natural complementary lottery, a lottery whose support is the complement of the given proper subset. For example, (1/2,0,0,1/2,0) has a complementary lottery of (0,1/3,1/3,0,1/3).
What if we could find out which lottery was preferred over its complement by the greatest number of voters?
If voters submitted sincere utilities (or CR ballots that are affinely related to their utilities) then we could answer this question by assuming that (between a lottery and its complement) the voter would prefer the one that gave her the higher expected utility.
In other words, on each ballot just find the respective CR averages of the candidates in each of the two complementary sets, and the set with the higher average CR will be the preferred of the two sets for the voter of that ballot.
This makes sense if the voter understands that the winning lottery is the one that will be used to pick the winning candidate. If candidates are to be chosen at random from a given set, then her expected utility for (the lottery that chooses at random from) that set of candidates is just the average of their individual utilities.
So each ballot "approves" one lottery of each complementary pair (except when the two expected CR's are the same).
The lottery with the greatest approval total is the winning lottery, i.e. the lottery used to actually pick the winning candidate.
That's the basic idea. Now for some variations:
(1) We could use ordinal ballots instead of cardinal ballots. Instead of using expected CR to infer which of two complementary lotteries was preferred on a given ballot, we would use the relative rank of the median candidate from each of the two complemantary candidate sets being compared on the ballot in question. The set whose median candidate was ranked higher would be the preferred set.
(2) We could allow lotteries other than those in this restricted class of lotteries. For example, in the case of three candidates I would suggest that we consider using the following pairs of lotteries:
(2/3,1/3,0), (0,1/3,2/3), (1/3,0,2/3), (1/3,2/3,0), and (2/3,0,1/3), (0,2/3,1/3),
in addition to the three complementary pairs of the special type mentioned earlier:
(1,0,0), (0,1/2,1/2), (0,1,0), (1/2,0,1/2), and (0,0,1), (1/2,1/2,0).
[Each of these six pairs corresponds to pairs of opposite hour marks on the twelve hour clock face.]
These lotteries can be used as easily as the special complementary lotteries in the case of CR ballots.
In the case of ordinal (i.e. ranked) ballots, the "weighted median" idea used in Chris Benham's WMA method should replace the ordinary median in the first variation suggested above.
As always my special attention to the three candidate case arises from its importance in the spruced up version of the method.
In particular, Spruced Up Approval Lottery (SUAL) trims either one or two branches from each node of the trinary tree representing the beat clone structure of the Dutta set. So the resulting tree is a binary tree.
The winning lottery probability for each candidate (i.e. leaf of the tree) can be computed by multipling the probabilities along the path from the root to the leaf.
Once you have these probabilities you can use the resulting lottery to pick the winner, or (if you only like deterministic methods) you can use them to find approval cutoffs on the ballots, and then find the winner by some deterministic method that makes use of approval cutoffs. [Voters could over-ride with their own cutoffs if they don't trust SUAL.]
A personal note related to a current discussion on another thread: Some folks don't like Approval because they think it is too hard to decide where to draw the line between approved and not. I can understand that, but let's not throw out the baby with the bath water.
If you have a way of getting a fair lottery, i.e. a fair distribution of candidate probabilities based on the true strengths of the candidates (calculated from the actual ballots, not the pre-election polls), then (short of using that lottery itself to pick the candidates) using those probabilities to automate approval cutoff calculations preserves (as far as possible) the good qualities of Approval (except the simple ballot).
Personally, I think those properties are worth trying to preserve. In particular, (1) Approval strives to find the candidate (relatively) acceptable to the greatest number of voters, and (2) Good Approval strategy strives to maximize the likelihood that your vote will make a "positive difference," and (3) Approval satisfies the FBC, and doesn't give more than infinitesimal incentive for order reversal of any kind.
By (2) I mean maximizing the chance that your ballot will be pivotal either in making or breaking a tie. [If it does, then the resulting outcome will be better (in your view) than it would have been without your ballot's contribution, because the only way your ballot can be pivotal is if it raises some candidate approved on your ballot relative to some candidate that you did not approve, presumably someone you liked less.]
What I would I, as a voter like?
(1) I would like to be able to vote my sincere preferences without fear of it back firing.
(2) I would like my ballot to have the max chance of being pivotal.
(3) If my ballot is pivotal, I would like it to be pivotal in a direction I consider to be positive, i.e. in a direction consistent with my sincere ballot.
Can any other method beat SUAL on all three of these points? Probably. That's why I'll keep searching :')
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