# [EM] Tournament Finish Order

Forest Simmons forest.simmons21 at gmail.com
Mon Jan 16 19:23:13 PST 2023

```Which is the more onerous strategic burden to be forced onto a voter? 1.
Being required to choose an approval cutoff? Or 2. being required to choose
a break link in their own beat cycle?

Both are unnecessary externalizations .... cop-outs of the voting method at
the expense of the voters ... an expense measured in stressful frustration
and uncertainty.

On Mon, Jan 16, 2023, 6:35 PM Forest Simmons <forest.simmons21 at gmail.com>
wrote:

> Round Robin tournaments require some method of determining the tournament
> finish order based entirely on the information that can be found in a
> pairwise score table T whose entry in the j_th row of its k_th column (for
> off-diagonal entries) is the number of points scored by the j_th team in
> the contest between it and the k_th team.
>
> You can see the analogy with an election method whose finish order among
> the candidates is entirely determined by the pairwise matrix M whose off
> diagonal entry M(j,k) is the number of ballots on which candidate j
> outranks candidate k.
>
> This raises an interesting question: is it always possible when given a
> tournament table T, to find a ranked choice ballot set beta whose pairwise
> matrix M is identical to T?
>
> If not, then it would seem that what Steven J Brams calls "voter
> sovereignty" should allow voters to amend the pairwise matrix m that
> encodes their ballot's contribution to M by explicitly specifying the value
> of any or all  entries m(j,k) (whether zero or one) ... over-riding
> potentially unfaithful ballot-to-matrix conversions.... meaning unfaithful
> to the voter's intent or desire.
>
> For example, suppose your RCV ballot ranks j and k equal top.  The default
> value of m(j,k) could be either 0 or 1, (or even 1/2 ... but let's not go
> there) depending on the default rule.
>
> Voter sovereignty would allow the voter to over-ride the default.
>
> This flexibility would allow any T matrix to be realized as an M matrix,
> thus answering in the affirmative our question about that possibility.
>
> In particular, any monotonic tournament method could be used as a
> monotonic voting method.
>
> For example, consider the tournament method that lists the teams in order
> of their weakest scores. This is a monotonic finish order because ... if
> team j were to get additional points against team k, all else being equal,
> the increase in theT(j,k) entry would be the only change in T. So the only
> possible change in the finish order would be an upward movement of team j
> ... all other teams retaining their previous finish orders relative to each
> other.
>
> By way of contrast, in the voting context raising j from j<k to j>k on a
> ballot will (in general) not only increase M(j,k) but will also decrease
> M(k,j). The sovereign voter can prevent that decrease if she wants to.
> Therefore, when that sovereignty is part of the rules, a decrease in M(j,k)
> should not be counted as part of a mono-raise move ... instead it is a
> mixed raise/drop move that cannot be used to contradict/nullify a method's
> monotonicity compliance (e.g.as part of a counter example).
>
> In sum, the same standard for mono- raise criterion compliance should
> apply in the pairwise election methods context as in the tournament context
> ... especially when the voter sovereignty guarantee is in place ... as it
> should be.
>
> In particular, with voter sovereignty in place ... the Max Min Pairwise
> Support (MMPS) method passes the Mono-raise Criterion... or if that is too
> radical ... we can just say that the MMPS satisfies Tournament Monotonicity
> ... which would be good enough in a real democracy with real voter
> sovereignty!
>
> What say ye?
>
> -Forest
>
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