FBC Ambiguity?

[Forest Simmons]

Here's an example of an Approval election that could be used to test the
current best efforts at FBC definition (absolute utilities in
parentheses). 

45 A($100) B($60) C($20)
30 B($100) C($60) A($20)
25 C($100) A($60) B($20)

[I attach dollar units to the utilities to emphasize the lack of
altruistic considerations in the strategies considered below.]

You'll recognize this as the example for which Richard recently suggested
the equilibrium Approval of

26 AB
19 A
30 BC
25 CA

Approval totals are 70 for A, 56 for B, and 55 for C.

A is the winner, and there is nothing that B and C can do about it, even
in full cooperation with each other as long as the B faction is loyal to
its favorite.

Suppose that the preferences and utilities are known perfectly by all of
the voters.

Suppose that the first faction is so confident in the strength of their
position (based on their partial understanding of the FBC) that it goes
ahead and votes publically before the other folks go to the polls.. 

Then the B faction volunteers to go next and votes 2 C and 28 BC.

The sure result would be that the C faction votes straight C, giving
the win to C.

This would be the only way (given the A faction's public vote) that the B
faction could avoid their lowest utility outcome without even more
favorite betrayal. 

The result would be

AB 26
A  19
BC 28
C  27

Approval totals: 45 for A, 54 for B, and 55 for C.

So in small groups, where voting is often neither secret nor simultaneous,
there is a possibility for strategically advantageous Favorite Betrayal in
Approval voting.

The same effect could possibly happen in national elections where exit
polls are announced immediately as the voting moves from East to West. 

Forest