I'm coming around to the conclusion that reverse symmetry is not such a good idea in multiwinner elections if one's aim is proportional representation.
Consider the following examples: Example 1: 25 ABC 25 BAC 25 CAB 25 CBA In this example if we reverse the preferences, we get the same number of each preference as before, so the candidate subsets stay in the same order. If the method is such that it reverses the order, then the only possibility is that all subsets are tied. In particular, {A,C} is tied with {A,B}. This means that even with 50% of the first place vote, C isn't guaranteed a seat in a two winner election. Example 2. 50 A>B=C=D=E=F 50 B=C=D=E=F>A Again, reversal of all preferences yields exactly the same number of each type of ballot, so the winning order of the subsets must be unchanged, and also reversed if the method is symmetrical. So again, all of the subsets are tied. In particular, {B,C,D,E} is tied with {A,C,D,E} even though A is the only approved choice of half of the electorate. The only way to fair proportional representation seems to be through abandoning the symmetrical method that I have been playing with recently. The lower median m is out, along with the j1 and j2. The upper median M, and the k1 and k2 are adequate by themselves. Doubling the k values gets rid of the daunting integral. The margin is just 1+1/2+...+1/(2*k1) - 1+1/2+...+1/(2*k2) . to be continued ... Forest ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em