# [EM] A Majoritarian, Clone Free, High Efficiency, Sincere Ratings Lottery

fsimmons at pcc.edu fsimmons at pcc.edu
Wed Jan 6 19:51:33 PST 2010

```An improvement is to let S be the set of candidates that cover all of the candidates with greater approval
(not only the approval winner).  This makes S more exclusive, so the method has higher efficiency.

So here it is:

In this method voters are allowed two ballots ...  an ordinal preference ballot with approval cutoffs, and a
cardinal (range style) ratings ballot.

1. The approval winner A is determined from the approval
cutoff on ordinal ballots.

2. The ordinal ballots are also used to determine the set  S  of candidates that
cover all candidates with higher approval.  [Candidate C covers B   iff   every candidate that
pairwise beats C also pairwise beats B.]

3. The same ordinal ballots are also used to determine the winning probabilities for
the lottery L(S) that uses random ordinal ballot to choose from S.

4. The range ballots are used to choose between A and L(S).

5.  If L(S) wins in step 4, then the winner is the member of S that is ranked
the highest on an ordinal ballot picked at random from those submitted.

Note that the approval winner A is always in S, as is the Condorcet Candidate when there is one.

>
> The method I have in mind is inspired by Jobst's T3ASR, which I
> have copied to
> the end of this message for comparison. The only improvement is
> that this new
> method is clone free, while T3ASR fails clone loser because
> (under T3ASR) if the
> candidate with the most approval loses, one of his clones is
> sure to win,
> provided that he has at least two of them.
>
> In this new method voters are allowed three ballots ... an
> approval ballot, an
> ordinal preference ballot, and a cardinal (range style) ratings
> ballot.
> 1. The approval winner A is determined from the approval ballots
> (or approval
> cutoff on ordinal ballots).
>
> 2. The ordinal ballots are used to determine the set S of
> candidates that
> cover the approval winner A. [Candidate C covers A iff
> every candidate that
> pairwise beats C also pairwise beats A.]
>
> 3. The same ordinal ballots are used to determine the winning
> probabilities for
> the lottery L(S) that uses random ordinal ballot to choose from S.
>
> 4. The range ballots are used to choose between A and L(S).
>
> 5. If L(S) wins in step 4, then the winner is the member of S
> that is ranked
> the highest on an ordinal ballot picked at random from those
> submitted.
> Note that this method satisfies Independence from Pareto
> Dominated Alternatives
> (IDPA), is monotone, clone free, and majoritarian in the sense
> that a majority
> can ensure the election of any candidate by bullet voting on the
> first two
> ballot types, i.e. on the strategic ballots. In fact, if the
> approval winner A
> is also the majority winner, then A is the only member of S.
>
> It has high efficiency because it takes a well liked candidate
> to cover the
> approval winner. Geometrically speaking, candidates that cover
> A are between A
> and the "median voter."
>
> And since there is always a chance for the set S to not be
> trivial, it is to the
> rational advantage of voters to fill out honest range ballots.
>
>
> Forest
>
> P.S.
>
> Here's Jobst's T3ASR:
>
> Method "Top-3 approval sincere runoff" (T3ASR)
> ==============================================
>
> 1. Each voter separately supplies
> a "nomination" approval ballot and a "runoff" range ballot.
>
> 2. From all "nomination" ballots, determine
> the options A,B,C with the top-3 approval scores a>b>c.
>
> 3. Let p be the proportion of nomination ballots
> which approve of C but not of B.
>
> 4. If on at least half of all "runoff" ballots we have a rating
> r(A) > p*r(C) + (1-p)*r(B), then option A wins.
>
> 5. Otherwise draw a "nomination" ballot.
> If it approves of C but not of B, C wins, otherwise B wins.
>
> Most of the time, this will elect one of the top-2 approval
> options, and
> only rarely the 3rd placed.
>
> One can then also compute and publish some kind of "index of
> sincerity"by comparing the submitted approval and range ballots.
>
> The method is majoritarian, since any majority can rule by
> bullet voting
> on both ballots.

```