I was wrong

[Richard Moore]

Alex wrote:

 >Given:
 >
 >25 A>B>C
 >49 B>A>C
 >26 C>A>B
 >
 >A is Condorcet winner. Say all members of faction vote same way, so
there
 >are 3 players.

Having three blocks of voters, each block acting as a single player,
doesn't
necessarily mean all members of each block will vote the same way. A
block
can split its votes up so as to prevent its most hated candidate from
winning, but without giving excessive support to its compromise, as in
my
solution to the example Forest gave a few months back.

In that case, Forest agreed with my solution being the more stable one,
so I think we can distinguish between stable and unstable equilibria.

Perhaps Alex's solution is a Nash equilibrium, but I think I can find
an equilibrium that is more stable:

 >Say 25 vote A and B, 49 vote B, 26 vote C. B wins.
 >
 >If the A>B>C faction withdraws its support for B then B still wins.
 >
 >If the C faction also votes for A candidate B still wins.
 >
 >This is a Nash equilibrium that did not elect the Condorcet candidate.

In this case, C is a guaranteed loser. So, the C voters will all approve
A as well as C, no matter what the other two blocks do. They can never
make the result worse for themselves by doing this (given the size of
the
other factions), but they can make the result better if the A faction
cooperates.

With the knowledge that the C voters are motivated to compromise, the A
voters can be assured of an A victory as long as they don't blow it and
give it to B. There's nothing the B voters can do to improve their lot.
However, they should vote B only, knowing C can't win and there is a
remote possibility of the A voters screwing up.

So my solution is

25 A
49 B
26 CA

 >Nash equilibria and the existence of Condorcet winners have no
definitive
 >link, although I suspect that if you enumerated all possible Nash
 >equilibria for any situation (treating all factions as single players) 
most
 >Nash equilibria would elect the CW. Still, everything I said 24 hours
ago
 >was bull.

Maybe not. I hope not, since I've always had an intuitive notion that
there
is some quantifiable advantage to Approval, I just never figured out
what
it was. At any rate it would explain why Approval has a strong tendency
to find the CW in simulations.

  -- Richard