[EM] Exploring Friendly Voting

Forest Simmons forest.simmons21 at gmail.com
Sun Feb 12 21:42:15 PST 2023


For simplicity we'll stick with complete rankings for now ... with three
methods that are equivalent in that context, though not in general ... like
MinMaxPaiwiseOpposition and MaxMinPairwiseSupport are equivalent in this
context.

We say candidate X is Friendly to Y if X doesn't defeat Y pairwise ...
otherwise X Vexes  Y.

Also X has Top or Bottom Status on ballot B according to whether no
candidate outranks it or it outranks no candidate on ballot  B.

Method 1:

Elect the candidate argmax S(Y) defined as Sum{Bot(X)-Top(X)| X vexes Y}
where Bot(X) and Top(X), respectively, are the number of ballots on which X
has Top or Bottom status.

Example 1.

fpA: A>B>C
foB: B>C>A
fpC: C>A>B

C is the only candidate that vexes A, so S(A) is merely Bot(C)-Too(C),
which equals fpA-fpC.

Method 2:

Elect argmax S'(Y) defined as
Sum{Top(X)-Bot(X)| X is friendly to Y }.

Method 2 applied to the same ballot profile from example 1 goes as follows:

S'(A)=(Top(A)-Bot(A))+(Too(B)-Bot(B)), which equals fpA-fpB+fpB-fpC, which
simplifies to fpA-fpC, the same exact expression given by method 1.

Method 3. Elect argmax (S(X)+S'(X)).

Note that S(A)+S'(A) can (by rearrangement of terms) be written as
Bot(C)+Top(A)+Top(B) minus
[Top(C) +Bot(A)+Bot(B)].

The three positive terms cinsist in the Top counts of friendly candidates
and the Botoom count of the unfriendly candidate.

The subtrahend consists of the bracketed terms which are the Bottom counts
of the friendly candidates and the Top count of the unfriendly candidate,
as it should be: we want our friends to have high top counts and low bottom
counts ... and vice-versa for those who vex us.

Does this heuristic make sense as a foundation for a Friendly Voting Method?

-Forest
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