[EM] Ranked Robin

[Forest Simmons]

El mié., 19 de ene. de 2022 4:52 p. m., Daniel Carrera <dcarrera at gmail.com>
escribió:

> On Wed, Jan 19, 2022 at 4:32 PM Forest Simmons <forest.simmons21 at gmail.com>
> wrote:
>
>> Better yet use de-cloned Copeland, which has a statistically negligible
>> chance of ties.
>>
>> The offensive score of candidate X is the sum of last place votes of the
>> candidates pairwise defeated by X.
>>
>> The defensive score of candidate X is the sum of first place votes of the
>> candidates that pairwise defeat .
>>
>> In the unlikely case that the offensive and defensive champions are are
>> not the same, elect the pairwise winner of the two.
>>
> I'm not familiar with de-cloned Copeland. I think I'm seriously
> misunderstanding how it works... So... X's offensive score is the sum of
> the total number of ballots that favor the OTHER candidate?
>
"Defeated by X..."

>
> 6 votes: A>B
> 5 votes: B>A
>
> A beats B, 6 vs 5. So... the last place votes is 5... and B is defeated by
> A... so A's offensive score is 5? ???  I must have misunderstood.
>
A's offensive score is the number of last place votes of the candidate
defeated by A, namely the six last place votes of B.

Here's another way to say it in general: A's offensive score is the number
of candidates pairwise defeated by A weighted by their average number of
last place votes.

So like St Patrick, candidate A gets more credit for  driving out the
snakes and dragons than for controlling the mole and cricket populations.

Similarly, A's defensive score is the  number of candidates that defeat A
weighted by their average number of first place votes.

The smaller this score, the better it speaks for A,  .. this score could be
small from A suffering few defeats or from the candidates defeating A not
being strong enough to get very many first place votes.

If there is a Condorcet winner, she will be both best offensive and
defensive candidate simultaneously. Otherwise, the head-to-head winner
between the two is the natural choice.

Those are the heuristic justifications for the two scores, but beyond that
it is obvious that if B covers A, then B's offensive score will be higher
than A's, and B's defensive score will be lower than A's, so the winner
must be in the Landau Set.

The weights serve to de-clone standard Copeland.

And using weights from opposite ends of the rankings for the offensive and
defensive scores allows the right kind of decoupling of mono-raising and
lowering to preserve Copeland's monotonicity, which proved to be impossible
when we only used first place votes for de-cloning.

Thanks for taking an interest!


--
> Dr. Daniel Carrera
> Postdoctoral Research Associate
> Iowa State University
>
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