[EM] CPW = CW in the 3 by 1 by 1 case.

[Forest Simmons]

Candidate Proxy with 3 candidates, 1 seat to be filled, and a 1
dimensional issue space:

The following conditions taken together are sufficient to ensure that the
Candidate Proxy Winner and the Condorcet Winner will be one and the same
in a three candidate, single winner election.

(1) The issue space is one dimensional.

(2) The Election Completion Procedure is (strategically equivalent to)
Approval.

(3) The candidates and the other voters act rationally, i.e. in their own
best interest as measured by their position along the one dimensional
issue spectrum.

(4) The candidates know which candidate is between the other two along the
issue line.


Proof:

Label the three candidates A, B, and C.

Use vertical bars to represent the position in issue space half way
between adjacent candidates, and without loss of generality take the order
to be A|B|C .

Since the method satisfies the Majority Criterion, the only interesting
case is when there is no single candidate which is the favorite of more
than half of the voters.

In this case the median voter M must be found (with B) somewhere between
the vertical bars: we must have either  A|MB|C  or  A|BM|C .

[Let's not worry about the borderline/tie cases when the median falls
precisely on one of the bars.]

If the voters (including the candidates) are rational, then all of the
voters to the left of the right bar (including A and all of those for whom
A is proxy) will prefer B to C, while all of the voters to the right of
the left bar (including C and all of those for whom C is proxy) will
prefer B to A.

That makes B the Condorcet Winner.

The remainder of our argument will show that B is also the Candidate Proxy
winner.

When the Election Completion Convention is convened (and well before
candidates A, B, and C cast their Election Completion Procedure ballots),
the proxies (i.e. candidates) all know the exact number of voters that
each proxy represents, i.e. the exact number in each of the following
three factions, none of which has a majority:

x% A>B
y% B
z% C>A

The middle faction splits into two factions B>C and B>A, but it turns out
that, no matter the relative size of these two subfactions, candidate B is
the one and only Nash equilibrium winner.

We break this claim down into the following two facts:

(Fact 1) Candidate B is the winner of at least one Nash equilibrium.

(Fact 2) There is no Nash equilibrium in which B is not a winner.

To see the truth of Fact 1 consider the configuration in which all proxies
approve down to and including candidate B.  This is a Nash equilibrium
because no player (i.e. candidate) could improve the outcome for himself
without cooperation from another candidate (who would be made to suffer a
loss thereby).

To see the truth of Fact 2, imagine a configuration in which B is not the
winner.  Then the other loser (the weaker of A and C) could improve his
own outcome (from last choice to second choice) by approving B.

Since the players are rational and have perfect information about each
others' preferences, the one and only Nash equilibrium winner is the
certain winner of the game, i.e. winner of the Candidate Proxy Election.

If we combine this result with Candidate Proxy's immunity to fake poll
manipulation, we can see that in this one dimensional, three candidate
case, Candidate Proxy is as good as any Condorcet method.

I doubt that any other method of comparable simplicity (besides Approval
itself) can make such a claim.

Forest

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