[EM] Preferential voting system where a candidate may win multiple seats

Jameson Quinn jameson.quinn at gmail.com
Fri Jun 28 06:16:38 PDT 2013

> Another option is to introduce weights on each party for a given region.
> Say that the Northern Norway region has 6 leveling seats. Then you
> calculate the desired outcome for the NN region as a whole (using
> Sainte-Laguë) and compare this to the current outcome (by adding up all the
> county results). Finally, you weight the votes to bring the latter closer
> to the former, but never so that more than 6 seats change ownership.
> By "weighting" I mean this: say Progress has too many seats. Then you
> divide the number of votes they got in every county by some common x until
> they get fewer seats.
> I think that this can be done simultaneously for every party by using
> linear programming, but I'm not sure of that. The scheme would also produce
> "floating" leveling seats, because the changes of which party gets which
> seat can happen in any of the counties in the region, and would happen
> where the parties in question are close to getting/not getting a seat. Thus
> it avoids the "where are the regional MPs going to be?" issue.

This system was proposed by Balinski as "Fair Majority Voting"; and is
probably the simplest example of a biproportional apportionment system. I
believe that Zurich uses pretty much this system for municipal government.

> As for how the party list method acts when there are few seats, this is
> probably closer to the problem you're seeing (when one disregards leveling
> seats). Here's an old example I often pull in similar cases:
> 46: L > C > R
> 44: R > C > L
> 10: C > R > L
> In the extreme case of there being only one seat, you would want C to win.
> But every divisor method reduces to Plurality when there's only one seat,
> so L gets elected instead. On the other hand, if you have a thousand seats,
> proportional representation pretty much says you should give 46% of them to
> L (were it a party), 44% to R, and 10% to C.
> So we'd like a method that is Condorcetian when there's only one seat, yet
> proportional when there are loads. And such methods exist. Schulze's
> proportional ordering method comes to mind. It's complex, however, and it
> might be possible to make simpler methods since we can elect each
> "candidate" (party) multiple times.
> A note here, though: it's often tempting to add some kind of strategy
> layer. If you have a large number of seats, some voters that vote X > Y >
> Z, and X "very nearly" has the number of votes needed to get another seat,
> at first glance it would seem that you should somehow distribute the
> X-voters' votes to Y instead so that Y can get another seat, because that's
> what the voters wished. But that kind of logic, when taken to its
> conclusion, leads to a quota method. Doing that sort of automatic
> reallocation would turn Sainte-Laguë into something entirely different - a
> quota method. And quota methods all fail population pair monotonicity as
> given by the Rangevoting page I linked to. So unless you want a quota
> method, that path is not the right one to take.
> And finally, there's the threshold. In some countries, the threshold is
> absolute - if the party doesn't get x% support, it gets no candidates. In
> Norway it is only absolute with respect to the leveling seats. But when
> dealing with thresholds, ranking is fairly simple to apply. If we permit
> voters to rank the parties, then the election can be extended with a sort
> of IRV-like system:
> 1. Count up the first preferences on every ballot.
> 2. If no party has less than x% support, we're done. Exit.
> 3. Otherwise, remove the party with the least support from every ballot,
> and go to 1.
> The result is a support list where no party has less than x% support,
> where x is the threshold. However, this method shares IRV's sensitivity to
> initial conditions, and probably also fails monotonicity. One could
> probably fix that by using a system based on DAC/DSC, but that would be
> very complex. So for a leveling seat system, what comes to mind as a
> suggestion is:
> 1. Count first preferences and give county seats as usual.
> 2. Remove every party that has less than x% national support from every
> ballot.
> 3. Count again.
> 4. Use the support levels thus given for the leveling seat calculation.
> That suggestion might eliminate too much: it has an "anti-Woodall free
> riding" problem. If you vote "Communist > Red > Socialist Left", if the
> Communist Party has no hope and the Red party is just short of the
> threshold, then the IRV method above would retain Red, but this method
> wouldn't. But I don't think those effects would be significant. (If you
> disagree, let me know and I'll try to think of a better system!) Votes of
> the type "Red > Socialist Left > Labor", however, would give support to
> Socialist Left even after Red has been eliminated -- unlike the current
> system that doesn't give support to any other party if you vote for a party
> below the threshold.
> ----
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