Goldfish GMC Example

[Mike Ositoff]

What would Goldfish do in this example?:


100 voters.

A>B 45
B>C 30
C>D 61
D>A 60

A>C 62
B>D 10


***

Though Anderson showed that any set of pairwise preference
vote totals has a set of rankings that will yield those
pairwise preference vote totals, his theorem probably doesn't
say anything about how many voters will be needed in order to
make that happen. For many of our examples the total number
of voters doesn't matter, but of course it matters when we're
talking about majorities. I believe that the percentages voting
the pair-preferences in the cycles in the above example are
mutually consistent.

Requirements for mutual consistency can be calculated for cycles
of various sizes, but they're more demanding for smaller cycles,
and so it's sufficient, it seems to me, to show that the 3-alternative
cycles are consistent.

In a cycle with N alternatives, each voter has to have at least
1 preference in that cycle that he doesn't vote, in order for
his preferences to be transitive.

So, if everyone votes a complete ranking, and, in keeping with
the previous paragraph, each voter doesn't vote one of the 
pair orderings in the cycle, then the sum, over all the
orderings in the cycle, of the percentages of the voters not
voting that ordering, has to add up to 1.

In practice, since there can be truncations, pairwise abstentions,
they must add up to at least 1.

Of course the percentage of voters not voting a particular 
pair-ordering is 1 minus the percentage who _do_ vote that
pair ordering, and so that can be substituted into the statement
in the paragraph before last.

***

Based on those statements, it seems to me that the cycles
in the example in this letter are consistent.

***

Mike