[EM] Single Contest

[Andy Jennings]

On Sat, Jul 23, 2011 at 11:28 AM, <fsimmons at pcc.edu> wrote:

> If one of the finalists is chosen by a method that satisfies the majority
> criterion, then you can skip step
> one, and the method becomes smoother.
>
> Here are some possibilities for the method that satisfies the majority
> criterion:  DSC, Bucklin, and the
> following range ballot based method:
>
> Elect the candidate X with the greatest value of p such that more than p/2
> percent of the ballots rate X at
> least p percent of the maxRange value.
>

Forest,

Can you clarify your definition of the majority criterion?  I don't think
this method satisfies it.

As a general example, suppose there are two candidates, A and B.  The voting
range is 0-100 and there are 5 voters:
1 voter: A=10 B=30
1 voter: A=30 B=50
1 voter: A=50 B=70
1 voter: A=70 B=90
1 voter: A=90 B=10

B is strictly preferred to A by 4 out of 5 voters, but these two candidates
have the exact same set of votes.  Any method which forgets which voter gave
which vote must consider them exactly tied.  This includes score (range)
voting, majority judgement, the chiastic median, and any of the other
generalized medians.  Thus, with only some minor perturbation, A can defeat
B in any of these methods.

In the chiastic median (or the majority judgement), both candidates have
societal grades of 50, but if you change A=50 to A=51 for the third voter,
A's societal grade becomes 51 and A defeats B, despite strong majority
opposition.

In the p/2 system, 40% of the voters gave grades of 70 or above and 20% of
the voters gave grades strictly above 70, so both candidates get a societal
grade of 70.  But if you change A=70 to A=71 for the fourth voter, A's
societal grade becomes 71 and A defeats B, again despite a strong majority
opposition.

I think any method which forgets which voter gave which vote will never
satisfy the majority criterion.

- Andy
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