Balinski's new Proportional Representation Method

Revolutionary, but unfortunately we do not think it will eliminate gerrymandering

By Warren D. Smith with some contributions by David P. Langer, Ivan Ryan, S.J.Brams, and Andrew Jennings

Social theorist Michel Balinski, in a 2008 article, introduced a revolutionary new "proportional representation" method for electing (e.g.) congressmen. It's revolutionary because it overcomes the traditional accountability problem that always has bedeviled PR government. However, it unfortunately appears to suffer from other (nontraditional) problems. We shall explain Balinki's ideas and their advantages, but at the end we explain their problems and why it will not cure gerrymandering, and it also will not be politically obtainable in the USA.

Michel Balinski: Fair Majority Voting (or how to eliminate gerrymandering), American Mathematical Monthly 115,2 (Feb. 2008) 97-113.

Before Balinski, there were two main ways to try to accomplish proportional representation:

1. "Party list" methods
2. Schemes involving "multiwinner elections" in each district.

Balinski's new method combines the accountability and simplicity of single-winner elections in districts, with nationwide proportionality properties! Here is his method.

1. Voters vote (using ordinary plurality-style "name one candidate" votes) for congressmen. It might also somehow be possible to use nonplurality-style votes, but Balinski does not consider that; Ivan Ryan suggests that approval voting might work better.
2. We add up (countrywide) the votes for each party to get the total counts of Dem votes, Repub votes, Green votes, etc countrywide.
3. These totals determine how many Dem winners, Repub winners, Green winners, etc there will be, countrywide. Balinski uses Jefferson's rounding-to-integer procedure here to produce those seat-counts.
4. To determine who wins, multiply all the Green vote counts (in each district) counts by a constant scaling factor G. Similarly all the Dem vote counts get rescaled by D, and the Repub vote counts by R, etc.
   
   Also, in addition to those party-based scaling factors, each district has its own scaling factor; all votes in district[k] are weighted by S[k].
   
   The idea is to choose all these magic scaling factors so that, using the rescaled votes, the right total number of congressmen from each party ("right" based on step 3) get elected. Brilliantly, Balinski observes (theorem 2; proof on page 109) that such scaling factors may be found by solving a "linear program" which happens automatically to yield exact-01-integer solutions [i.e. exactly 1 congressman gets elected from each district] if it has any solutions.
   
   Generically, Balinski's linear program's solution is unique (although in the rare event it is not, we must flip some coins to break ties by awarding 0.001 extra votes randomly to one party or another). Also, Balinski claims that if every candidate gets at least one vote (e.g. if they vote for themselves) then a solution always exists [consequence of theorem 1 on page 105]. This solution generically is essentially unique, in the sense that the same winner-sets arise generically for any two solutions so that nonunuqieness is irrelevant. If some candidates can get zero votes, then he gives an example on page 104 where no solution exists. Balinski's linear program has a special form and is much nicer than a random linear program. He examines various forms of it in his paper and recognizes that it has various specially-enjoyable and easy-to-solve forms such as it is (what is called in the literature) a "max-flow min-cut problem," and a "transportation problem," and which is "unimodular."

Balinski's method is a very cute idea, both delivering proportionality and featuring the simplicity and accountability of single-winner elections in districts.

Disadvantages and worries

More complex (but acceptably so, in my view): Although Balinski is very simple from the point of view of a voter, it, since it involves "solving a linear program" it is pretty complicated to count. Still, the counts within each district are conventional and simple (and could be published by each district), and the party-rescaling factors could also be published. Any moron (including somebody who has no idea even what a "linear program" is) could verify that the announced scaling factors do work with the announced district totals to yield the correct total seat counts. And due to the uniqueness theorem we know the scaling factors, once found and verified to work, cannot be altered to give any party any advantage. So this aspect seems acceptable.

What about independents? While Balinski is great for candidates running under the flag of some large party. It might not be so good for those who wish to run independently and unaffiliated. Doesn't Balinski victimize them? (And what about small parties struggling to get off the ground?)

You could establish a "party" consisting of just yourself, then run. But a party running 10 candidates and getting 10% of the vote for each, will win a seat; while you, running alone, will probably not win a seat even if you get, say, 30% of the vote. Further, there might be 10 unaffiliated candidates running, each quite popular and thus each getting 30% of the vote. None win a seat, while meanwhile the Unpopular Party running 10 candidates and getting 10% of the vote for each, does win a seat. That seems unfair.

One possible way to solve this problem is to make a fake party consisting of agglomerated independent/unaffiliated candidates. Perhaps this is the best solution, although it still seems unhappy because they might be very politically opposed to each other.

Andrew Jennings comments: Jefferson's round-to-integer method has the property that a constituency of size X+Y will win at least as many seats as (and possibly one more than) the sum of the number of seats that constituencies of size X and Y could win separately. Balinski apparently specifically chose Jefferson's method to discourage party splintering and encourage parties to join together.

What about turnouts? Suppose it rains heavily in the West while the East is sunny. So we get lower turnout in the West. Therefore the party composition of congress is artificially altered.

Gerrymandering? Balinski seems to hold the view that with his voting method, gerrymandering will vanish because it is no longer useful. From then on, districts would be drawn sensibly, for a refreshing change!

Unfortunately, we believe Balinski is wrong about that. We think it quite likely, unfortunately, that gerrymandering would persist and continue to corrupt everything massively.

Nationwide, we get proportionality, therefore it does not pay to gerrymander. Agreed. But locally, gerrymandering pays – plenty!

Suppose you are Joe Democrat Congressman. You'd like to have a safe seat. So you get your buddies in the statehouse to draw you a safe district full of Democrats.

Result (under Balinski voting): Joe is almost certain to be elected. The only way Joe can fail to be elected is if the entire country decides it hates Democrats hugely, and/or Joe's district stops voting Democratic – both of which are almost impossible. (*- see below about "monkeywrench effect.")

So what I predict would happen (with Balinski voting) is this. The Dems and Repubs will make interparty deals (just as they already do) to draw "safe" districts for all their high-ranking members. As usual, none of them will be defeatable.

But there will also be some districts occupied by lower-ranking Dems and Repubs, who do not have as much influence with the party bosses, which will be "at play". Those will be unsafe seats. So Balinski will improve things, but by no means will eliminate either gerrymandering or "congressmen for life."

Because of proportionality, the Greens will be able to get elected too. There could in principle actually be districts that are safe for Greens (i.e. essentially guaranteed to elect a Green). That is because the Greens will (by proportionality) win, say 5% (or whatever) of congress. The most-Green district in the country will then be essentially guaranteed to elect a Green, even though it (say) contains only 10% Greens.

Ivan Ryan comments: Locals might be pissed at a third-party which sets up its base in their district so it can consistently get 5% of the vote – the result being that 95% of the residents of the district are represented by a wacko-fringe party that they don't support.

Now the Ds and Rs could (and probably will?) conspire to hurt the Greens as follows. Although the Ds and Rs will be unable to prevent 10% Greens from being elected if the voters are 10% Green, they can keep shifting the Green district-boundaries with the goal of preventing any Green from being reelected. Then every Green will be a freshman. Or, they could try to make all the districts safe for everybody, so nobody ever gets defeated. This won't entirely work (if the country shifts from 60-40 R/D to 40-60, say, then there will be a 20% change) but it will work in the sense that all the established party members will be safe everywhere. Further, if, say, the Dems arrange to make their seatholders all have districts packed with Democratic voters, we don't see how that will hurt the Democratic party:

1. Their seatholders will now be safe.
2. The remaining parts of the country, which contain (say) only 1% Democratic voters because the gerrymanderers sucked it dry, still will elect Democrats because of the proportionality guarantee, so the party as a whole is not hurt.

It isn't that easy – the "monkeywrench effect"

Some of the statements we made above are not quite true. The reason is this:

Interestingly, third parties could win relatively easily in any one district, even a district designed to be very Republican.

The Dems and Repubs would likely have scaling factors around 1, as they don't need that much help to win.

Third parties would likely end up with scaling factors of 20-30, as they would need a major boost to win any one district.

If some locals don't like their current rep, they just need to pick a third party and vote for them. Even if only 5% of the voters do that, they would likely swing the district for the 3rd party!

Is this "monkeywrench effect" a good or bad thing?

Dr. Steven J. Brams comments:

I don't think [Balinski's proposal's] biggest problem is gerrymandering. Rather, it's a political nonstarter because I cannot conceive that a politician, at least in the United States, would find it acceptable to win an election in his or her district and then lose this seat. But this can happen, because an underrepresented party nationally can claim a seat in a district it lost, simply because this seat is needed to bring it up to its national proportion. [Therefore something like Germany's MMP system would be better] wherein the district winners all get their seats and underrepresented parties get additional seats, but not necessarily up to their full proportional representation. [Brams cites pp.85-87 of his book Mathematics and Democracy.]

Dr. Douglas Woodall comments:

that Balinski's idea is reminiscent of a method invented by Anton Buhagiar in 1994/95 which uses D'Hondt's rule to produce PR out of an election held by STV in multi-winner constituencies.

Finally, we remark that Balinski would be a complete revamp of the US governmental system, requiring a constitutional amendment to happen. Either major party has both the power and the (already demonstrated) desire to block such an amendment forever. Therefore it cannot happen.

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