[EM] Holy grail: a "condorcet compliant" cardinal method (MCA/Bucklin variant)

Jameson Quinn jameson.quinn at gmail.com
Sun Sep 5 09:20:58 PDT 2010

> 9 (???) I think it even deals with a 3-way Condorcet cycle, if which all
> voters vote the 3 candidates separately (one-per-rank-level), by electing
> the same candidate as Minimax, Schulze, or RP. (!!) (I'll investigate this
> further, and share my results either way.)

Oops, that's not quite true as stated. Consider the prototypical cycle, with
no acyclic or anticyclic ballots. Candidates A, B, and C, numbers of each
vote type a (ABC), b (BCA), and c (CAB). If A is the minimax (and Schulze,
RP, etc.) winner, then b<a and b<c. Thus, A will be the AW. However, C could
be the RW iff c+b>2a. If so, A and C will go into a runoff and, barring
preference or turnout changes, C will win.

With acyclic(bullet) and anticyclic votes, the minimax winner and the AW are
still the same, it's just more complicated to explain.

So, in a 3-way cycle, if the "broadest overall support" (minimax winner/AW)
and the "deepest overall support" (RW) are the same, it's resolved in one
ballot; otherwise, the two will go into a runoff. This is pretty much ideal.

Obviously, if there is a large Smith set, things get more complicated, and,
since the 2+1-ranked ballots don't accommodate the number of distinctions
needed, you can't even be sure that the minimax winner will make it to the
runoff. But if this system does so well up to a 3-way cycle, I am confident
that it will perform reasonably well in the (vanishingly?) rare case of a 4-
or more-way cyclical tie.

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