[EM] voting methods

[Raph Frank]

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On Fri, Jun 5, 2009 at 5:58 AM, Árpád Magosányi<magwas at rabic.org> wrote:
> I guess the list might have opininons in this discussion.
>
> 2009/6/4 Warren Smith <warren.wds at gmail.com>
>> In particular, if all voters are tactical then plurality, IRV, and
>> Condorcet systems all
>> are EFFECTIVELY THE SAME (i.e. all elect the same winner) in scenarios
>> where
>> there are 2 major-party candidates plus also some minor party
>> candidates whom the voters believe have little chance to win.

Do you think that people would vote that way under Condorcet?

> And game theory enters here. With some Condorcet dishonest voting gives
> unnoticeable advantage to the voter. With range vote it does much, and
> eventhe  ballot leaves more room for it. Plurality even punishes honest
> voting.

Right.

A good way of looking at it is finding out the Nash equilibrium.

This means  that each voter is allowed to adjust their vote based on
how everyone else votes (say due to opinion polls), but cannot
coordinate with other people.

Assuming that there are 2 major parties (A,B) but one of the minor
candidates (C) is the condorcet winner.

45: A>C>B
10: C>A>B
45: B>C>A

Plurality
A: 45
B: 45
C: 10

A and B voters wouldn't change their vote (as voting for C is
"throwing your vote away"), but based on the polls, C voters would
shift to A.

The effect is that A wins even though C is the condorcet winner.
A: 55
B: 45
C: 0

Approval (and Range)

A: 45
B: 45
C: 10

C voters would approve A, but have no incentive to disapprove C

A: 55
B: 45
C: 10

B voters would now start to approve B+C as they prefer that to the
actual result, and nothing is lost by approving C.

A: 55
B: 45
C: 55

Now that C is one of the top-2, C supporters would stop approving A.

A: 45
B: 45
C: 55

Thus the Nash equilibrium is that C wins.

If can be show that if everyone votes using the strategy "vote for one
of the top 2 and everyone you prefer to the expected winner", the a
condorcet winner will always win.

Ofc, that assumes clear preference strength.  If some of the B voters
were in fact, B>>C>A, then they might not switch.

The same analysis is harder for condorcet methods as you have to
assume strategy and how they handle ties (i.e the exact method).

That might be a good criterion for condorcet methods, are the only
Nash equilibria where members of the Smith set are winners.