# [EM] Problem solved (for pure ranked ballot)

Forest W Simmons fsimmons at pcc.edu
Mon Jan 29 18:44:14 PST 2007

```As Kevin pointed out in one of his posts I have been using a overly
difficult standard of Favorite Betrayal.

Here's a simpler proof based on the following definition of FBC:

Raising favorite to top rank must not decrease expected utility.

Given three voters with utilities consistent with the ranked preferences

1 A>B>>C
1 B>>C>A
1 C>>A>B

Sincere pure ordinal ballots would be

1 A>B>C
1 B>C>A
1 C>A>B

Neutrality and anonimity require that A, B, and C win with equal
probability, so the expected utility for the A voter is below the
utility of B.

Swapping B and A on that ballot yields

1 B>A>C
1 B>C>A
1 C>A>B

Clone immunity together with Majority Rule in the two candidate case
implies that B wins with this ballot set.

Moving favorite back up to top rank displaces B down a rank because we
are still assuming pure rankings.  This puts us back where we started
with less utility than B.

That's it.

```