When wouldn't SODA elect the CW? SODA meets the Condorcet criterion for voted preferences, and a true [majority/nonmajority] Condorcet winner would be visible in the voted preferences in a [unique/nonunique] strong Nash equilibrium. So in the large majority of cases, it would elect the CW.
Here are (the only) two scenarios where I think it might not. *Scenario 1 (one dimensional pseudo-chicken):* 35: A>B 25: B>A 40: C>>B>A This is almost the classic chicken dilemma scenario, with a little bit more one-dimensional logic. But I am *NOT* saying that SODA would fall prey to the chicken dilemma here. That is to say, unlike Approval and Range, and to a lesser extent MJ and Condorcet, there is no danger that it will elect C. What I am saying is that, if candidate C's preference for B over A is weak, they may choose not to predeclare that preference, in order to be able to campaign more forcefully against B. Since B's natural preference is A, B cannot retaliate in kind. Without C's preference, the Condorcet and SODA winner is A (as with IRV). Who's the correct winner in this scenario? Definitely not C, but you could argue either way between B and A. B is the honest Condorcet winner, but since the preference of the C voters is weak, it may well be that A is the honest utility winner. (If the C voters' preference for B over A is not weak, then at least 11 C voters will explicitly approve B, and B will win.) *Scenario 2 (Weak Condorcet winner):* In this scenario, B is a relative unknown, and throughout most of the campaign, A and C focus their attention on each other. After the votes are counted, the situation is something like the following: 48: A>B 5: B>C 47: C>B B is the Condorcet winner. "Correct" SODA strategy would be: C doesn't approve B, A approves B, B wins. But when it's candidate A's turn to assign delegated votes, they're faced with a concrete choice between B and C. A realizes that, while B gives voice to important centrist concerns in a way that speaks to the median voter, that B just doesn't have the experience or qualifications for the job. So A doesn't approve B. As I've given the scenario, B's preference C happens to be the weaker of the two candidates. This will happen about half the time a scenario like this arises. In this case, B has a choice of whether to give the election to C or leave A as the winner. Perhaps B can use this power to gain some centrist concession from the eventual winner. I think that SODA is getting the right answer in the scenario as portrayed. Of course, you could in principle vary the details to the point where such a result would be arguably wrong. One thing is certain though; whatever is the right winner for the society, both A and C (the major-party candidates, certainly the winners under plurality) would be unhappy with a B win. Thus, plurality incumbents will be happy to know that SODA does not necessarily favor B in this situation. In other systems – Approval, Range, Condorcet, or even MJ – B could in principle win. I actually find it pretty implausible that B would win in any of these systems, but I do think that even the possibility would be scary to plurality incumbents. Thus, I find the fact that SODA does not necessarily elect the CW in this scenario to be a net plus for SODA. Jameson
---- Election-Methods mailing list - see http://electorama.com/em for list info
