[EM] Re: Lottery methods. wv between all possible lotteries? New methods.

[Forest Simmons]

I consider Mike's recent posting under the above subject heading to be 
very thoughtful and a good summary of some of our common interests and 
where we are currently in our quest to find methods that are in line with 
those interests.

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.

Forest