[EM] My Favorite Majoritarian Ordinal Ballot Method
fsimmons at pcc.edu
fsimmons at pcc.edu
Thu Jul 8 17:40:02 PDT 2010
Let's consider the prospects for burial in three candidate Condorcet when cycles
are resolved by random ballot. In particular, let's consider the electorate
profile given by
40 A>C>B
30 C>A>B
30 B>C>A
There is a potential for burial strategy here: if the 40 A>C>B voters bury the
Condorcet candidate C, they create an artificial A beats B beats C beats A
cycle, in which "C beats A" is the weakest defeat, so unless C takes counter
measures, most deterministic Condorcet methods reward A for going ahead with the
burial of C.
But when cycles are resolved by random ballot, the respective probabilities for
A, B, and C are 40, 30, and 30 percent. If the 40 voter faction has a high
regard for C relative to B, which we can denote by
40 A>C>>B,
then the burial doesn't pay.
Note that the sincere ballots show that A and C are clones, so the most likely
case is indeed that A and C are closer to each other than to B.
However it is possible that the true feelings of the faction of 40 voters are
more like
40 A>>C>B.
If so, there are still two sub cases:
Subcase I: The 30 B>C>A voters have relative preference strengths of
30 B>C>>A.
In this case 70 percent of the voters are of the opinion that C is closer to B
than to A. But this contradicts the 30 sincere C>A>B voters. So subcase I is
not very reasonable or likely.
Subcase II: The 30 B>C>A voters have strengths of 30 B>>C>A.
In this subcase we have
40 A>>C>B
30 C>A>B
30 B>>C>A.
In this case we see that C is a very low utility Condorcet candidate. Sincere
approval votes would make A the winner. So it shouldn't be considered a bad
thing that A gets 40 percent of the probability and C only gets 30 percent of
the probability in the case of an artificial cycle created by burial of C.
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