Let's suppose that there are three candidates, and that one of them C is preferred over the other two by fifty percent plus majorities (not just by more for than against).
Suppose that candidate X (not equal to C) is the perceived front runner going into the first election. Then strategic voters will approve all candidates that they prefer to X. Since more than half of the voters prefer C to X, more than half of the voters will approve C. This is enough to ensure that C will be one of the two front runners in the next election (since the loser of the frontrunner battle will get less than fifty percent approval). Assume that preferences haven't changed so that C and (say) Y are the two frontrunners in the next election. [If Y=X, as in the likely case that X (the previous top front runner) did not come in last, then C would be the top front runner in the second election. But we are not assuming that here.] Then on every strategic voter ballot precisely one of these two (C or Y)will be approved (whichever the voter prefers) along with others approved by the voter of the ballot. Since C is preferred pairwise over Y by more than fifty percent of the voters, C will get more than fifty percent approval, and Y will get less than fifty percent approval. This is enough to make C the top frontrunner in the third election, being the only candidate to get more than fifty percent approval in both previous elections. In this election all strategic voters will put their approval cutoff next to C on the side of the other frontrunner (say) Z. So C's margin of approval over Z will be the same as Z's margin of pairwise defeat by C. And C will get more than 50% approval (by the same reasoning as for the previous election). The only question is could the other candidate (say) B get more approval than C in this election? In other words, can B be above the approval cutoff (which is right next to C) more than C is? C is above the approval cutoff more than fifty percent of the time. Could B be above the cutoff more than fifty percent of the time? Well, B cannot be above the cutoff without also being above C, no matter which side of C the approval cutoff is placed. So the question becomes, can B be above C more than fifty percent of the time? The answer is no, because C beats B pairwise. So no candidate B can have more approval than C in this election. This makes C the election winner of the third approval election. So once C makes it to top frontrunner status, C will get more approval than either of the other two candidates. This argument shows that approval winner status (once attained) is stable for CW's. In summary, within three elections (and probably sooner than three) the CW will reach and keep approval winner status (until voter preferences change). I'm sure that this crude analysis can be improved upon, including relaxing the fifty percent plus condition and the three candidate condition. Forest ---- Election-methods mailing list - see http://electorama.com/em for list info
