[EM] Two election method observations

[Kristofer Munsterhjelm]

First: Let's say that method B is a more decisive version of method A if 
you can get B by breaking ties in the outcomes produced by A. (I.e. B's 
outcome never ranks X ahead of Y whn A ranks Y ahead of X, but sometimes 
ranks X ahead of Y when A ranks X equal to Y.)

Then it's possible for B to be summable even when A isn't. So 
tiebreaking can make matters *easier*. (This unfortunately makes 
election method design harder.)

Second: Suppose you have some black box method that only does "n-1 out 
of n" multiwinner elections, i.e. where the number of seats is one less 
than the number of candidates. Suppose this method is Droop 
proportional. Then the following multiwinner method is Droop 
proportional in general and extends the method to any-seat elections:

	1. Do an n-1 out of n election.
	2. If n-1 is equal to the number of seats, you're done and that's the 
outcome.
	3. Otherwise, eliminate whoever was not elected and go to 1.

My proof idea is like this: Suppose that we're trying to elect k seats 
and there are n candidates, and |V| voters. Then the Droop quota for k 
seats is |V|/(k+1). So suppose that the Droop proportionality criterion 
states that q candidates from some coalition must be elected because 
q|V|/(k+1) voters ranked them first. Then since the Droop quota for n-1 
seats is lower than that for k seats, at least q candidates from the 
coalition must also be elected in the (n-1) seat election. Hence the nth 
candidate (the one who's not elected when the (n-1) seat election is 
performed) can't be any of the q. So eliminating him can't violate any 
future Droop constraints.

The rest follows pretty straightforwardly from induction.

So in a sense, the hard part of a proportional multiwinner method is the 
(n-1) out of n election. Solve that and you can get Droop 
proportionality in general -- although it would probably be neither 
monotone nor summable.

The funny/interesting part is that no actual multiwinner election method 
I know of is constructed this way. Perhaps something notable could be 
found in the space of election methods that look like this?

-km