[EM] Condorcet and the Muller-Satterthwaite Theorem
Forest Simmons
fsimmons at pcc.edu
Tue Oct 22 14:39:26 PDT 2002
Good question. I wonder if "Pareto Efficient" means the same as
satisfying the Pareto Condition that we are all familiar with and that you
use in your proof sketch.
I know that the name Pareto is associated with various related but
distinct concepts in game theory because the referee of a paper that I
co-authored on the subject of "consensus halving" and related fair
division problems asked if we could show that the method was Pareto
Efficient, by which he meant that no participant in the consensus
division procedure would have an incentive to lie about any of her
preferences.
Forest
On Mon, 21 Oct 2002, Alex Small wrote:
> I recently learned of the Muller-Satterthwaite Theorem: pareto efficient
> and monotonic social choice functions are dictatorial. I was always under
> the impression that Condorcet, at least in some implementations, was
> monotonic and pareto efficient. Let's first consider Pareto:
>
> If A and B are in the innermost unbeaten set, and if every voter prefers A
> to B, then B's defeat will be stronger than any other defeat (barring a
> tie with another candidate who loses a contest 100% to zero). That defeat
> will never be dropped and hence B will never become "undefeated" (after
> disregarding weak defeats). Hence B will lose.
>
> As far as monotonicity: Suppose A wins all pairwise contests. If a voter
> moves candidate A up in his rankings, without changing the relative
> rankings of the other candidates, A will still win all pairwise contests.
> Hence improving the CC's ranking cannot cause him to lose.
>
> If A is not the Condorcet Candidate, but one such candidate exists, then a
> voter moving A down in his rankings (without changing the relative
> rankings of any other candidates) will not knock the Condorcet Candidate
> off of his pedestal. Hence demoting a non-Condorcet candidate cannot
> cause him to win (if a CC exists).
>
> Say there is a cyclic ambiguity. Let's use winning-votes to keep life
> simple, and assume no truncation (truncation might invalidate some of the
> assumptions of the Muller-Satterthwaite Theorem). Say A is the winner.
> Upgrading your ranking of A, leaving all other relative rankings
> unchanged, will only increase other candidates' magnitudes of defeat. It
> may even cause A to win ALL pairwise contests. So promoting A does not
> hurt him.
>
> Finally, say there is a cyclic ambiguity and A is not the winner.
> Demoting A in your rankings, all other relative rankings unchanged, will
> not lessen the strength of any of his defeats. It may increase the
> strength of the defeat, and it may even cause another candidate to win all
> pairwise contests. Hence demoting a candidate will not help him win if
> there's an ambiguity.
>
>
>
> Where did I go wrong?
>
>
>
>
> Alex
>
>
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