[EM] CPW = CW in the 3 by 1 by 1 case.
Forest Simmons
fsimmons at pcc.edu
Thu Dec 12 09:30:27 PST 2002
My previous argument can be extended to the case of N candidates, 1 seat,
and 1 dimensional issue space:
Let the letter X represent the candidate that the voter median is nearest
to on the (one dimensional) issue spectrum.
This "Voter Median Candidate," X, is the Condorcet Winner.
To show that X is also the rational Candidate Proxy Winner we need two
facts:
Fact 1. There exists a stable equilibrium configuration of Election
Completion Approval ballots.
Fact 2. Any configuration in which X is not the winner is unstable.
An example of a stable equilibrium is the one in which all proxies approve
down to and including X. No subset of proxies acting together can improve
the outcome for any part of the subset without making it worse for another
part of the subset.
And in any configuration for which some candidate Y other than X is the
Candidate Proxy Approval Winner, there is a subset of candidates
with a majority of the proxy votes, that can strictly improve the outcome
of each member of the subset by approving down to and including X.
This subset consists of X and all candidates that are not on the Y side of
X.
[Since all candidates (including Y) on the Y side of X are ranked strictly
below X within this majority, X is the only candidate with majority
approval, which makes X win, which, in turn, is a strict improvement for
every member of this majority.]
Since all proxies know exactly how many votes each of the other proxies
control, and they know their "reality check" preference orders, the winner
under rational play is the one and only stable equilibrium winner X.
That concludes the proof (except for the negligible case in which the
voter median position is exactly half way between two candidate
positions).
The only potential for manipulation is a candidate lying on the reality
check in an attempt to appear nearer to the voter median. But this seems
like an extremely risky move that would most likely backfire.
Now let's think about multiwinner PR elections with a one dimensional
issue space.
It's obvious in the case of 2 winners that the Voter Median Candidate
should be replaced by the two Voter Quartile Candidates, i.e. the
candidates nearest the 1st and 3rd quartile voter positions on the issue
spectrum.
In the the case of three winners, the winners should be the candidates
closest to the first, third, and fifth sextile voter positions.
In general (for single issue PR elections with K seats to be filled) the
candidates nearest the odd numbered "(2*K)_tile" voter positions should
fill the seats.
Which Election Completion Procedures have this ideal as the one and only
stable outcome?
An answer to that question would complete the N candidate, K seat, 1
dimensional issue space cases, and allow us to start working on the 2
dimensional issue space cases.
PAV might work, but would only be practical for small to medium K.
Another possibility is Repeated Lone Mark Balloting [until each of K
candidates has at least 1/(K+1) of the Election Completion Procedure
votes].
Forest
On Wed, 11 Dec 2002, Forest Simmons wrote:
> 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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