# [EM] Tournament Finish Order

Forest Simmons forest.simmons21 at gmail.com
Mon Jan 16 18:35:15 PST 2023

```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,

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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