[EM] Simple Tournament Proposal

[Forest Simmons]

Time for an example or two.

48 C
28 A>B
24 B

A>B 28 to 24
B>C 52 to 48
C>A 48 to 28

The respective minPS scores are their Top counts 28, 24, &48
So C is the offensive champ with MaxminPS(C)=48.

MaxPO(C)=52 (from B)

A and B are tied for minMaxPO at 48 from C. This tie is broken by looking
at the other PO scores 24 and 28, respectively. So A is the minMaxPO
candidate (defensive champ) with MaxPO(A)=48+24epsilon, ehich is less than
MaxPO(B)=48+28epsilon, and (as already noted) less than MaxPO(C)=52.

Since the Offensive Champ C beats the Defensive Champ A, (48 to 28) and
there is no undefeated candidate, our tournament method elects C.

Now let's look at ...

a A>B(sincere A>C)
b B>C
c C>A

If the A faction is largest its burial of the sincere CW C pays., but the
because A is both the Offensive Champ and the Defensive Champ with
minPS(A)=a.

But suppose C takes the customary sincere CW precaution of bullet voting:

a A>B
b B>C
c C

Then the respective minPS scores are a, b, and c.

Continuing our assumption that enabled the A faction to bury C with
impunity ... namely a>max(b,c), we see that A retains its status as
Offensive Champ with minPS(A)=a>max(b,c).

The MaxPO's are now ...
(b+c)=MaxPO(A), a=MaxPO(B), and (a+b)=MaxPO(C).

The min of these is a=MaxPO(B). So B is the Defensive Champ.

The pairwise contest between the two champs is A vs B ... which still lets
the A faction get away with its burial of the sincere CW.

We see that under this method the sincere CW is at the mercy of the largest
faction.

So despite the nice sports tournament heuristic... we cannot endorse this
method!

It's a good thing we have better methods that are conceptually and
operationally just as simple ... if not as innately appealing to sports
enthusiasts.

Is there a better pair of fall back candidates to choose between when there
is no ballot CW?

How about the implicit approval candidate versus the Max Equal Top
candidate?

The beauty of this class of methods is only one pass through the ballots,
as long as the two contenders are kept simple enough.

-Forest





On Tue, Mar 21, 2023, 9:00 PM Forest Simmons <forest.simmons21 at gmail.com>
wrote:

> Here's my suggestion for choice of tournament champion:
>
> Lacking an undefeated team, elect the pairwise victor of the defensive and
> offensive champs.
>
> In the sports context the natural choices of offensive and defensive
> champs, respectively are the team with the greatest point total, and the
> team with the smallest opposing team points total, both totalled for the
> entire tournament.
>
> In the election context the natural choices of offensive and defensive
> champs, respectively, are the candidate with the greatest vote total, and
> the candidate with the smallest opposing vote total ... noth totalled for
> all of the direct democratic matchups between pairs of candidates.
>
> However, in the context of elections the presence of clone candidates will
> unfairly distort these results. This distortion is easily overcome by the
> following modification:
>
> The Offensive Champ is the candidate that can honestly say "everybody else
> had a matchup where they got fewer votes than I did in my worst matchup."
>
> The Defensive Champ is the candidate that can honestly boast, "Every other
> candidate had a matchup where their opponent got more votes than mine did
> in my worst matchup."
>
> How do we get the matchup votes for candidates X and Y in their direct
> democratic compsrison?
>
> They are simply the transferred votes that they would have if all of the
> other candidates were eliminated.
>
> Candidate X would get one vote from every ballot on which X is ranked
> ahead of Y.
>
> Similarly, Y's vote total in this contest is the number of ballots that
> rank Y ahead of X.
>
> Now, some technical notes for the election technicians ... everybody else
> thanked with heartfelt appreciation for their participation ... and excused:
>
> The Pairwise Support Matrix is the matrix whose entry PS(j, k) in column k
> of row j ... is the number of ballots on which candidate j is ranked ahead
> of candidate k.
>
> In each row j, circle the smallest entry ... m(j) = min over k of PS(j,k).
>
> In each column k, highlight its largest entry M(k) = Max over j of PS(j,
> k).
>
> Let J be argMax m(j), obtained by looking at the row number of the largest
> circled entry in the matrix.
>
> Candidate J is the Offensive Champ.
>
> Let K be argmin M(k), obtained by looking at the column number of the
> smallest highlighted matrix entry.
>
> Candidate K is the Defensive Champ.
>
> If PS(J, K) is larger than
> PO(J, K)=PS(K,J),  then candidate J, the Offensive Champ defeats the
> Defensive Champ K.
>
> If PS(J,K) is less than PO(J,K), then the Offensive Champ J is defeated by
> the Defensive Champ K.
>
> (Otherwise candidates J and K are tied)
>
> -Forest
>
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