[EM] Simple Tournament Proposal

[Forest Simmons]

All of this groping for a halfway decent Quick & Clean tournament method
has brought me to this:

Lacking an undefeated candidate, elect the candidate who defeats the
Defensive Champ with the most winning votes.

To define the Defensive Champ, we need to know for each pair of candidates
X and Y the number of ballots on which Y was ranked ahead of X ... this is
the Pairwise Opposition of Y against X, denoted PO(Y,X)

The MaxPO of X or MPO(X) is the greatest value of PO(Y,X) as Y ranges over
all of X's opponents. This MPO(X) is X's defensive score ... the smaller
the better ... it is the most "points" (votes) any opponent of X has made
against X in any pairwise matchup of this election.

The Defensive Champ is argminMPO(X), the candidate whose MPO value is
smaller than any other candidates MPO value.

So the Defensive Champ is the one who "lets slip" the fewest votes to any
of her opponents ... almost as if she had some psychic power to keep votes
against her from reaching the ballot box.

So this Defensive Champ is a strong candidate ... but the candidate that
overcomes that strong resistance and gets the most votes past her defense
is even stronger ... and that's the one our method chooses!

Example:

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

The MPO's of the respective candidates are MPO(A)=(n-a), MPO(B)=(n-b), and
MPO(C)=(n-c). So the Defensive Champ is A, assuming the A faction is
largest (emboldening the burial).

The winner is supposed to be the candidate with the strongest defeat
against the Defensive Champ A. So C is the only candidate that qualifies,
since no other candidate defeats A.

So we see that the sincere CW is restored.

Example 2.

48 C
28 A>B
24 B(Sincere B>A)

In a previous message we found that the Defensive Champ was A, who is
defeated only by C ... therefore the winner.

So the B faction's defection backfired.

So far, so good ... not at all surprising ... because I had these examples
in mind when I designed this method;-)

We need somebody to run some tests.

-Forest

On Wed, Mar 22, 2023, 7:31 AM Hahn, Paul <manynote at wustl.edu> wrote:

> "In sports, what strategies could exist? I'd imagine something more
> like... team B tells team X to play badly against team C, because the
> tiebreaker won't make X win anyway. Thus if say, the Smith set is ABCX,
> then it's possible that X losing more heavily against C could make B win
> instead of A. That's more like compromising, but it's not quite the same
> thing."
>
> AFAIK the majority of deliberate losing (or not winning as handily as one
> is capable) in sports are to take advantage of side bets.  I can imagine
> that in a double elimination tournament one might deliberately go over to
> the loser's bracket to avoid a team one is particularly bad against, in the
> hope that they'll be eliminated before you have to face them.  But that
> means you have to fight your way through the loser's bracket, which means
> more matches; I don't know that it would be worth it most of the time.
>
> The other scenario I am aware of is that in chess and some other sports,
> one can lose or not win as big to avoid having your rating increased, so
> that (again) you get to face lesser opposition.  This definitely happens.
>
> I'm not sure how much of this carries over to an election situation,
> though.
>
> --pH
>
> -----Original Message-----
> From: Election-Methods <election-methods-bounces at lists.electorama.com> On
> Behalf Of Kristofer Munsterhjelm
> Sent: Wednesday, March 22, 2023 8:03 AM
> To: Forest Simmons <forest.simmons21 at gmail.com>; EM <
> Election-methods at lists.electorama.com>; Kevin Venzke <stepjak at yahoo.fr>;
> Andy Jennings <elections at jenningsstory.com>; Colin Champion <
> colin.champion at routemaster.app>; Andy Dienes <andydienes at gmail.com>
> Subject: Re: [EM] Simple Tournament Proposal
>
> On 3/22/23 05:00, Forest Simmons wrote:
> > Here's my suggestion for choice of tournament champion:
> >
> > Lacking an undefeated team, elect the pairwise victor of the defensive
> > and offensive champs.
>
> I'll have to investigate further, but my impression from working with
> burial-resistant methods is that it's impossible to make a method that's
> burial resistant (in the DMTCBR sense) without using positional data.
>
> However, another important property to note is that the modes of strategy
> very much depend on how the data is gathered. In an election situation,
> burial is fairly easy: just change A>X>B>C>D>E>F into
> A>B>C>D>E>F>X. But in sports, the analog would be that A decides to tell
> B to "strategically defeat X", e.g. to score more goals against X (or
> similar) to push X further down the ranking. Presumably any team B would
> be doing its best to defeat X already, so "burial" doesn't really seem to
> be a strategy in sports.
>
> Thus it's not a problem that we don't have positional data, because we
> don't need to defend against that particular strategy.
>
> In sports, what strategies could exist? I'd imagine something more like...
> team B tells team X to play badly against team C, because the tiebreaker
> won't make X win anyway. Thus if say, the Smith set is ABCX, then it's
> possible that X losing more heavily against C could make B win instead of
> A. That's more like compromising, but it's not quite the same thing.
>
> -km
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