# [EM] fpA - fpC

Forest Simmons fsimmons at pcc.edu
Tue Mar 16 16:45:47 PDT 2021

```In his book, The Geometry of Voting, Saari approaches the problem of a
resolving a Condorcet cycle in the three candidate case by first canceling
ABC  ballots against CBA ballots, BCA ballots against ACB ballots, and CAB
ballots against BAC ballots. This first step cannot eliminate the Condorcet
cycle, but it reduces the number of factions to three: either ABC, BCA, &
CAB or else
CBA, BAC, & ACB.

Without loss in generality suppose the three factions are

x: ABC
y: BCA &
z: CAB

Now suppose that y = min(x,y,z). Saari now removes y ballots from each
faction, which eliminates the middle faction entirely:

(x-y): ABC
(z-y): CAB

With only two factions there can no longer be a Condorcet cycle.
Majoritarians would say that A or C must win depending on which of the two
remaining factions is larger. A should win if (x-y) is larger than (z-y),
which is true iff x - z is positive.

Saari continues his anaysis to argue in general (no matter which of the
three cyclical factions is smallest) the largest difference among (x -y),
(y -z), (z-x) should determine whether A, B, or C wins, respectively.

[Note x-y is fpA - fpC]

Saari goes on to show that this result is equivalent to the Borda Count,
and rests his case that Borda is the best election method (to make a long
story short).

To be continued ....

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