[EM] ironclad pro-Condorcet argument?
James Green-Armytage
jarmyta at antioch-college.edu
Sun Aug 22 21:36:42 PDT 2004
Has anyone clearly advanced this pro-Condorcet argument? I think that it
is devastating to methods which are not Condorcet efficient. If someone
else has made this same argument already, please let me know, so that I
can cite it in the paper I’m trying to write on weighted pairwise.
BASIC STATEMENT:
If there is a Condorcet winner with regard to the sincere preference
rankings of voters, and the voting method is plurality, then the Vote is
only at equilibrium when the Condorcet winner is selected.
FULL CLARIFICATION AND PROOFS:
By a Vote, I mean the sum of the way each voter casts their vote. Hence,
I’m not just talking about the winner, but in who votes how.
By an equilibrium Vote, I mean a Vote where no group of voters can
immediately get a result which they mutually prefer by changing their own
votes to a different candidate.
Point 1: There is no possibility of a Vote where the CW is not the
plurality winner, and yet the Vote is at equilibrium.
Point 2A: There can be a non-equilibrium Vote where the CW is the
plurality winner,
2B: but for every (plurality) election where there is a sincere CW, there
is at least one equilibrium Vote which selects the CW.
It would make sense to prove Point 1, Point 2A, and Point 2B separately.
Point 1: When someone other than the sincere CW wins by plurality, there
is a number of voters who would mutually prefer the CW, such that this
number exceeds the number who prefer the winner. Hence the Vote is not at
equilibrium.
Point 2A: A single example can show the possibility of this occurrence.
Sincere preferences, and votes
46: A>B>C; vote for B
10: B>C>A; vote for B
44: C>B>A; vote for C
B is the winner, but this is not an equilibrium Vote, because the 46 A>B>C
voters can gain a preferable result by voting for A instead.
Point 2B: For any plurality election where there is a sincere CW, there is
at least one equilibrium Vote wherein every voter votes for the CW. Since
there is no other candidate who is preferred over the CW by a majority,
there is no group of voters who can gain a mutually preferable result by
changing their vote.
Of course, there are likely to be several different equilibrium Votes
where the CW is selected, but the reasoning above shows that there is at
least one, which suffices to prove the point.
By the way, the number of possible Votes is generally very large. If the
number of voters is V, and the number of candidates is C, then the total
number of possible Votes is C to the Vth power.
Here is another example of an equilibrium Vote for the CW, using the same
preferences as above:
46: A>B>C; vote for A
10: B>C>A; vote for B
44: C>B>A; vote for B
You should be able to see that there is no group of voters who can get a
mutually preferable result by changing their votes.
WHY THIS IS IMPORTANT:
This strikes me as one of the most ironclad pro-Condorcet arguments that
I can think of. It implies that if you give each person one vote, and you
hold an extended series of successive ballotings, the outcome cannot truly
come to rest until the CW is selected.
This suggests to me that the election of anyone other than the CW can be
considered an ERROR on the part of the voters. That is, if they had better
information, and enough time to work it out, they would have settled on
the CW... but as it was, due to the protracted process and uncertain
information, the majority allowed itself to be split, leading to the
accidental election of someone else.
It seems to follow that if a group of voters with a certain set of
preferences, perfect information, ample time, and one vote each will
eventually settle on the CW, then the election of the CW is the most valid
expression of these preferences.
Sincerely,
James Green-Armytage
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