[EM] PAR theory

[Jameson Quinn]

I said earlier:


> This method will always elect a voted majority Condorcet winner C, because
> such a candidate will always either meet the three criteria, or win by the
> fallback rule.
> - For any Z, C's tally when considering C as a frontrunner will include
> (more than) all ballots with C>Z, by assumption a majority; but Z's tally
> will include only ballots with Z>C, by assumption a minority.
> - C will not be rejected by a majority.
> - If Z has more top-ranks than C, and Z is not rejected by a majority,
> then Z will not have the highest tally when considering Z as a frontrunner,
> and C will have fewer rejections than Z.
>

This last statement is not true.

Say there is a Z who has more top-ranks than C and is not rejected by a
majority. Without loss of generality, assume Z is initially selected as the
frontrunner. Now Z's tally will be their number of above-bottom votes,
while C's tally will be the C>= Z votes. The latter number is, by
assumption, a majority; but it's not necessarily larger than the former
number.

For instance, take the following scenario:
49: Z
48: C>Z
3: >C

C beats Z 51-49, but Z wins PAR 97-3. There are two different semi-honest
strategies available for C to win — either 2 of the >C voters can switch to
C, or 47 of the C>Z voters can switch to C. But C clearly does not win the
election as presented.

What if you add the requirement that every voter will prefer at least one
candidate? That changes the above to:

49: Z
48: C>Z
3: Y>C

But wait a minute; C is no longer the majority Condorcet winner! In order
to restore that property, we must go to:

46: Z
3: Z>C
48: C>Z
3: Y>C

Z still wins. So my claimed property is totally shot. But I have a hard
time imagining that this would really happen. Essentially, 54% of the
voters are "wasting" either the top or bottom of their ballot on Y, when Z
versus C is clearly the contest that matters.

What if we try to make Y more relevant?


23: Z
26: Z>C
25: C>Z
26: Y>C

Yes, Y is more relevant here; they actually have more first-choice support
than C. But still, we have 51% of the voters "wasting" the bottom of their
ballot on Y, even though Y has just 26% above any other candidate. And
furthermore, the Y>C is not at all reciprocated by any C>Y; in that sense,
preferences have some "cyclical tendency".

So I still believe that, in real-world elections, any voted majority
Condorcet winner that exists will win; even though this is definitely not
true as a criterion compliance.
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