[EM] Condorcet and Reversal Symmetry are incompatible with DMTBR
km_elmet at t-online.de
Tue Apr 3 14:16:58 PDT 2018
Here's a proof that we can't have all of Condorcet, reversal symmetry,
and dominant mutual third burial resistance.
It's a proof by exhaustion, and I see what Wikipedia means when it says
"A proof with a large number of cases leaves an impression that the
theorem is only true by coincidence, and not because of some underlying
principle or connection. "
I handle ties by saying that "X wins with certainty" if there is no tie
and "X wins with positive probability" if the probability of X being
winning after whatever tiebreaker is greater than zero.
DMTBR is incompatible with Reversal Symmetry and Condorcet:
Start with (A):
A is the CW and by Condorcet, wins with certainty.
Bury A on the 4 BACs to get (B):
If B wins with positive probability, we're done: (A)->(B) is a DMTBR
failure. Otherwise, either C wins with certainty or A wins with positive
probability. See the end for the case where C wins with certainty.
Suppose A wins with positive probability on (B). Reverse the ballots to
By reversal symmetry, at least one of B and C must win with positive
probability on (D).
Suppose C wins with positive probability. Then flipping 4 CBA to CAB
where A is the CW, and so (E)->(D) is a DMTBR failure.
On the other hand, if B wins, then flipping 1 BAC to BCA gives us (F)
where C is the CW, so (F)->(D) is a DMTBR failure.
The case where C wins in (B):
Suppose that C wins with certainty. Consider the reversed ballots (D) again:
By reversal symmetry and our assumption that C won on (B), C can't win.
First suppose A wins with positive probability.
Relabel the candidates according to ACB -> abc. We get:
By relabeling and assumption that A won, a must win with positive
But this is just (B), where c must win with certainty by our initial
assumption, so we have a contradiction.
Hence B must win with certainty, and (F)->(D) gives us a DMTBR failure
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