[EM] Condorcet as compromising DSV, and multiwinner ideas

[Kristofer Munsterhjelm]

Suppose that we have a Plurality election and that A wins, but B is the 
CW. Then the voters who preferred B to A could all compromise for B and 
make B win; and no other faction would be able to change the result 
after the voters compromised.[1]

(If there's a cycle, then repeating this reasoning would make the winner 
sequence follow the Smith set cycle.)

So in some sense, electing the CW executes a majority-compromise on 
behalf of the voters when such an opportunity exists, without the voters 
having to guess ahead of time who they're supposed to compromise for.

Now suppose we replace the simple single-winner method (Plurality) with 
a simple multiwinner method (SNTV). Then we could define a Condorcet 
outcome similarly: as a set of winners that nobody can change by 
compromising. (If there is no such one outcome, there would analogously 
be a cycle.)

Multiwinner compromising is a bit more tricky than single-winner. Say we 
have three candidates A, B, C; and we want to elect two of them; and 
that the SNTV outcome is {A, B} with A having more first preferences 
than B. Then if we want to see if it's possible to change {A,B} to {A,C} 
through compromising, the voters must honestly prefer C to B (or they 
wouldn't be interested in changing the outcome), and they can't be 
ranking C first (or they're already maximally "compromising" for C). So 
the voters who would be interested are A>C>B voters who would vote CA>B 
instead. This is a form of vote management, because if too many A>C>B 
voters switch to C>A>B, that might cost A his victory, which they 
wouldn't want.

If there are more candidates, then plain old unqualified compromising 
might still work - e.g. if the candidates are A...Z and the winner 
outcome is {A,B}, then everybody who prefers C to B and ranked one of 
D...Z first would be interested in compromising for C.

So if we want a multiwinner analog to Condorcet with SNTV as a base 
method, based around single-candidate differences, it could be something 
like:
{A,C} > {A,B} if the honest outcome is {A,B} and voters who prefer C to 
B can turn the outcome into {A,C}.

And how can we determine if this is possible? Linear programming is 
probably the easiest way. With three candidates:

maximize T
subject to:
	fpA - T >= fpB	(1)
	fpC + T >= fpB	(1)

	T <= (A>C>B)	(2)
	
where T is the number of voters changing from A>C>B to C>A>B, 
constraints (1) indicate that the winners will be A and C; and (2) that 
there are enough voters to go around.

If the program is feasible, then {A,C} > {A,B} with strength T; 
otherwise not (strength zero).

This is a pretty rough sketch because it doesn't answer what happens 
when the sets differ by more than one candidate,[2] nor how one may find 
the magnitude of say {X,Y}>{X,Z} if the honest outcome is {A,B}.

As I said there would be a cycle if compromising could always be 
executed, there may be some notion where {X,Y}>{X,Z} is based on some 
strategic election where people compromise for X and Z, and then seeing 
if others can then replace Z with Y, but I have no detailed idea of how 
to formalize that. (Or it's possible to ignore the problem and create a 
method that just depends on successive compromise attempts by different 
factions.)

Still, I thought it'd be an interesting observation :-)

-km

[1] This might even work with equal-rank as long as the voters who are 
indifferent between A and B either stay out of the strategy or lean A or 
B equally often, but I'm not sure if the "no other faction would be able 
to change the result" part would hold.

[2] One option is to just string together a bunch of single-candidate 
changes; this is what Schulze STV does. But it's not easy to understand.