# [EM] Even more thoughts on the three-candidate cardinal method

Kristofer Munsterhjelm km_elmet at t-online.de
Sun Oct 24 13:15:52 PDT 2021

```The (currently nameless) cardinal method automatically normalizes the
three-candidate elections resulting from every possible way to choose
three candidates out of the candidates running, and then produces a
"potential winners set" that gets fed to a Condorcet method to determine
the ultimate winner.

To preserve monotonicity, it would probably be better to do something
like PWS,Ranked Pairs; not PWS//Ranked Pairs.

If we ask for indifference lotteries, we'll have to get the preference
orders separately. Ratings, on the other hand, give them directly, but
I'm not sure if the indifference lottery interpretation is sound in that
case. In any case, the method needs rankings to normalize properly, and
these rankings could be used for the Condorcet method.

I think that the method would be monotone if the Condorcet method is; it
would also be reversal symmetric. However, I'm not sure if it would be
cloneproof. Suppose the voters are an epsilon away from indifferent
between the clones[1], and candidate A is cloned into A1 and A2.

Then every contest of the type {Ax, B, C} will have the same loser as
the before-cloning {A, B, C} contest, so the structure of the PW set
inferred by these will be the same. If the Condorcet method is
cloneproof, then we'd be done since the PWS wouldn't change.

However, it's also possible to get contests of the type {A1, A2, B}. For
these, if A beats B pairwise, the loser is B; otherwise it is one of the
As. This might produce a clone problem if B is either booted from the
PWS where he wouldn't be otherwise, or is retained where he wouldn't be
otherwise; particularly in a Condorcet cycle. The addition of a clone
could make a Condorcet cycle enlarge the PWS. (Or perhaps not. I would
have to think about it further.)

Since the method works for arbitrary internal rated methods, and since
there are only three candidates in each mini-election, it might be
possible to do a true DSV of some sort. If there are three candidates,
there are six possible rankings (more with truncation and equal-rank,
but disregard that). Then perhaps we could find a game theoretical
equilibrium of some sort for the six-player simultaneous continuous move
game and use that?

The game is: each rank corresponds to a player, and each player P
provides a rating r_P to serve as the intermediate rating in a (1, r_P,
0) Range ballot. Then the method determines the loser and each player
receives utility according to the players left. Complications include:
we don't know the actual utilities: we only know that an honest A>B>C
voter who rates B at p considers A to give utility a+b, B ap+b, and C b.
And these values of a and b vary for the different voters.

Maybe I should just go through the marginal calculations for
Lp-cumulative voting and convince myself that Euclidean indeed makes
honest lottery rating the optimal zero-info strategy. I've convinced
myself that for 3-candidate Range, the optimum is to rate the in-between
candidate at round(p*) where p* is the true lottery value[2], or any
value at all if you don't care about the outcome or value the in-between
candidate equal to a 50-50 chance at either the best or worst candidate.

[1] This seems to be the convention for rated ballots, although it's a
bit more restrictive than I like -- i.e. it lets rated methods pass what
would be clone violations for ranked methods.

[2] The true lottery value for B is the probability p so that a voter
with preference A>B>C is indifferent between B winning with certainty,
and A winning with probability p, C with probability 1-p.
```