[EM] PAR theory

[Jameson Quinn]

Basically, the TL;DR message of my previous two messages in this thread is:
if there is a voted majority Condorcet winner, then that candidate's
supporters should be sure to reject any rival who will get more
first-choice votes, even if that means bullet voting; and supporters of a
rival who will get a minority of above-bottom votes should top-rank the
VMCW if the VMCW is their second choice. Both of these "strategies" are
relatively obvious, and thus I believe likely on naive ballots, but they
cannot be guaranteed.

2016-11-15 11:59 GMT-05:00 Jameson Quinn <jameson.quinn at gmail.com>:

> 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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