On Wed, 2002-11-06 at 19:05, Forest Simmons wrote: > Only in those cases where at least one of the subsets (of the CC > hypothesis) has no CW is (the conclusion of) the Consistency Criterion > violated. > > But when one or more subsets is indecisive, i.e. does not give definite > support to any opinion as to which candidate should win, we should not be > surprised if the over all outcome disagrees with a common subset outcome > whose commonality was just a statistical fluke.
I'm going to take advantage of your argument here, without necessarily agreeing with it. Let's imagine that groups can have opinions. Your argument, as I understand it, is that groups have opinions, but not always. Sometimes groups are undecided. Not every election result is a group decision, but a good election result should be consistent with whatever group decisions exist. By declaring a group to be undecided when there is no Condorcet Winner, the strength of the consistency criterion is reduced. Similarly, an IRV supporter might declare that there is only a definite group opinion when there is a first-place majority winner. Various methods could be defended by appropriately weak definitions of when a group decision is made. The main problem with these definitions, including the two above, is that they do not always give enough information to determine a winner (even in non-ties). In general, you can think of the group as expressing opinions on certain pairwise comparisons, and remaining undecided on others. For example, you might declare that if X is the CW, the group has decided that it is superior to each other candidate. You could do the same if X is the majority winner. But as I say, there are probably many ways you could translate a set of ballots into a set of pairwise group-decisions. So, I've come up with these properties to narrow the search. 1. Unanimity. If every single voter orders their ballot X1>X2>X3...>Xn, then for any i and j, Xi must be group-chosen over Xj, iff i>j I hope unanimity will be uncontroversial. This just gets rid of some very silly possibilities. 2. 1-Result. Exactly one ranking should be consistent with the groups pairwise decisions (except for ties breakable by a single vote). Obviously, we don't want groups to be deciding contradictory things, so at least one ranking must be possible based on their decisions. And, as I stated above, my goal is a set of decisions where only one ranking is possible. Or you might loosen this to allow multiple possible rankings, if they give the same winner. It doesn't make any difference for the procedures I considered. 3. Consistency. Group decisions should have the property that if we divide the ballots into two subsets, such that each ballot is in exactly one subset, and the procedure declares that the each subset has a group decision of X>Y, then the procedure should find a group-decision of X>Y in the ballots as a whole. This just declares that pairwise group decisions must be consistent, in much the same kind of way that the consistency criterion does for winners. Let me show an example of how this could work. Consider the following elections 40: A>B>C 35: B>C>A 25: C>A>B Most Condorcet methods pick A, but seeing A as a solid group choice (group-decided over each of B and C) leads to consistency problems according to property 3. So, instead I will say only that the group has declared {A>B,B>C}. The group has not decided on A vs. C. However, in order to give a result consistent with the group-decisions, the election method must give the ranking A>B>C. Now, when I add ballots 51: A>C>B 49: C>A>B Let's say that I declare the group decisions to be {A>C,A>B,C>B}, using a rule yet to be revealed. The election result must be A>C>B. Now, let's say the combined ballots give group decisions of {C>A,C>B,A>B}. Final ranking C>A>B. Although the election method still gives a different result for the combined group than each of the component groups, pairwise choices given as group decisions are consistent. The only group decision common between the subgroups was A>B, and this was in the final result as well. OK, so here's my rule that fits the above properties. A group-decision of X>Y is made iff X pairwise defeats Y with some margin of victory (m). And, there is no way to trace a path from Y to X, by moving along pairwise victories of margin >=m. So, for example if we have pairwise victories: A->B [40] B->C [30] C->A[20], C>A is not a group decision, because here m=20, and I can travel to B via A->B [40], 40>=20 and then back to C via B->C [30], 30>=20. But A>B and B>C are both group decisions because no such path exists. I contend that this rule meets the three properties I give above, and in fact, I will prove it in a later post. It is the only rule I have found that meets all three properties, but anyone can try their hand at finding another. Note that my use of margins is not just habit; property 3 is not obeyed by the winning-votes version of my rule. As property 2 says, except for ties, there is only one ranking consistent with the rule. Not surprisingly, for my rule, this is the Ranked Pairs ranking. But, I have no proof that other rules couldn't be found in favour of other methods. --- Blake Cretney http://condorcet.org ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em