Regarding "wv between all possible lotteries" (a small part of his posting). True, there are an infinite number or lotteries to deal with, but that fact by itself doesn't preclude a finite procedure for finding the wv winner.
However, until such a procedure is discovered, we could limit ourselves to lotteries that assign probabilities only from some finite set of numbers say 0, .1, .2, .3, .4, .5, .6, .7, .8, .9, and 1.
Even this reduction might be too large to handle if the number of candidates is large.
Another possibility is to limit consideration to lotteries that assign positive probabilities to each set of three or fewer candidates, and that assign those probabilites uniformly.
Another possibility is to limit consideration to lotteries that assign positive probabilities to each set of two or fewer candidates, and (in the case of two candidates A and B with positive probabilities) to assign those probabilities on the basis of the following odds:
the number of ballots on which A beats B
to
the number of ballots on which B beats A.
There is a lot of unexplored territory in the field of lotteries.
Using ballot preferences to set the odds should exert a slight pressure toward sincere order, perhaps enough to push Cardinal Pairwise over the top, since it is already so close.
But suppose that we have straight Cardinal Ratings style ballots with no AERLO, ATLO, nor Approval cutoffs indicated, and that the sincere ratings are
45 A(100), C(25), B(0) 30 B(100), C(50), A(0) 25 C(100), A(50), B(0)
Then C is the sincere CW, and any method like Cardinal Pairwise that satisfies the Condorcet Criterion would give C the win in the case of sincere ballots.
But what about the temptation for the first faction to induce a cycle by rating B above C on their ballots:
45 A(100), B(1), C(0) ?
Surely any reasonable deterministic method would give A the win, given these (insincere) ballots.
So how can we counteract this incentive without resorting to the introduction of voter supplied cutoffs of various kinds (approval, aerlo, atlo, etc.)?
Random ballot would get rid of the incentive for order reversal, but (as Mike duly noted) that would be too extreme, since that would give B (with sincere rating of zero by seventy percent of the voters) a 30 percent chance of winning.
Is there some intermediate, acceptable level of randomness that would do the job?
What if A and C were given one to one odds in the first (sincere ballot) case, while A and B were given one to one odds in the second (insincere ballot) case?
Then by changing from sincere to insincere ballots the voters of the first faction would lower their expected payoff from (100+25)/2 to 100/2.
On what basis could these (or similar) probabilities be assigned without knowing whether the ballots were sincere or insincere?
We note that in both cases the two candidates with the highest average CR scores were the ones given positive probabilities.
But this by itself cannot be the rule: surely there are some cases that should assign all the probability to the CW. If the first faction changed their sincere rating of C from 25 to 99 (due to a wonderful campaign promise introduced at the last minute), most folks here would agree that C should be the unique winner with positive probability (100%).
Furthermore, always assigning the top two CR candidates positive probability might encourage distortion of ratings wherein strong candidate supporters would downplay their second choice, and weak candidate supporters would raise their second choice artificially. The voters would be faced with strategy decisions as in Approval.
What if we said that the top two CR candidates should share the probability unless the top CR candidate turned out to be the CW?
Would there still be incentive to distort ratings as in ordinary CR?
In the case of sharing probability, should the two candidates share equally? Or should the one that beats the other pairwise have a greater share of the probability?
If we gave the one with greater CR greater probability, then there definitely would be an incentive to use approval style strategy.
What about doing Cardinal Pairwise on the seven lotteries
(1,0,0), (0,1,0), (0,0,1), (0,.5,.5), (.5,0,.5), (.5, .5, 0), and (1/3, 1/3, 1/3) ?
One could use the respective expected payoffs for each of these seven lotteries as their cardinal ratings, and then apply James' Cardinal Pairwise method, perhaps the River version, (not to the original ballot set, but rather) to the resulting set of CR ballots for the seven lotteries.
I once tried doing this by hand, but got too bogged down. I intend to try again when I get the time. In the mean time if somebody else were to do it, I would be eternally grateful.
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