Paul Del Piero, a student at Pomona College, did a study of four redistricting plans in California. He uses uniform partisan swing (which can be viewed as an approximation to the method used by the Judgeit program developed by Gary King and myself) to estimate seats-votes curves. Here’s his paper, here are the appendices for the paper, and here’s the abstract.

I also have one comment of my own, which I’ll give after Paul’s abstract:

From Paul Del Piero:

The process of legislative and congressional redistricting after each decennial census has been a heated issue in the past, especially within the State of California. Moreover, control over the California redistricting process has come to the forefront in recent months with the qualification of Proposition 77 for the November California Special Election ballot. Over the past thirty years, California has experienced three distinct redistricting plans: two court-ordered plans (1974-1980 and 1992-2000), a partisan gerrymander (1982-1990), and a bipartisan gerrymander (2002-present). The purpose of this project was to determine through statistical analysis whether the common arguments against a perceived gerrymander were, in fact, true. This project evaluates the four redistricting plans in each legislative category (the State Assembly, the State Senate, and the California Congressional Delegation) using hypothetical seats-votes curves derived from the “uniform partisan swing” theory. These curves are then compared with an ideal proportional representation curve to determine the plan that most accurately represents the political preferences of the California electorate. The results demonstrate that the 1992 court-ordered district plans reflect the partisan preferences of the electorate far more accurately than both the current bipartisan gerrymander and the partisan gerrymander of the 1980’s. Thus, partisan and bipartisan gerrymanders increase electoral responsiveness to the point that legislative seats are overly-responsive and grossly reward the party with the most statewide votes. In stark contrast, court-ordered plans that closely resemble the standards set forth in Proposition 77 produce a more proportional response and accurate reflection of the electorate’s partisan preferences in California.

My comment:

I would just be careful about describing the legislature as “overly-responsive.” This implies a normative standard, but people disagree on these things. To quote from my article, Voting, Fairness, and Political Representation (published in Chance in 2002):

Most of Europe uses proportional representation: if a party gets, say, 20% of the vote, they are given 20% of the seats in the legislature. In some countries, a party that gets only 2% of the votes still receives 2% of the seats. Other countries have a threshold of 5% before a party gets representation–they don’t want every group with 1 percent of the vote to have a representative in the legislature, screaming, putting chewing gum in the elevators, and generally causing problems.

The United States does not have proportional representation. There’s a separate election in each Congressional district, and whoever wins that election goes to Congress. In theory, a party could get 49% of the vote and still get zero representation–if they got 49% in every district in the country. More realistically, a party could get 45% of the vote and only 40% of the seats, or 20% of the vote and no seats. Suppose some unfortunate third party gets 20% of the vote in every district. In one district, the Democrat might receive 42% and the Republican 38%. In another, the Democrat might win 60% while the Republican ties with the third party at 20%. Regardless, the third party loses out.

Some people, especially Europeans, consider the lack of proportionality to be a defect in our system, with the 20% of votes that went to that third party as wasted, since they do not lead to any representation. Of course, you could also consider the 38% for that Republican candidate as wasted, or that all votes for a losing candidate are wasted. For that matter, you can consider extra votes for a winning candidate as wasted, too–did that guy really need 60% of the vote–but it seems particularly tough on that third party, since all their votes are wasted.

But . . . proportional representation has a problem too: small changes in the vote give you only small changes in seats for the political parties. This is a problem because swings in votes between national elections are typically only about 5%. In the American (or British) system, a 5% swing in votes can easily produce a 10% swing in seats–enough to possibly change which party controls the legislature. Look at what happened with Newt Gingrich and the Republicans in 1994. You might not have been happy with that particular outcome, but it’s reassuring that a change in votes has the power to change who ruled Congress. Under proportional representation, this can happen too, but in a more subtle, less voter-controlled way: perhaps some minor party increases its vote share from 10% to 15%, and then it can make a deal with another party, ultimately changing the government. The change in votes has an input, but not so directly–the proportional representation system with multiple parties is more like a pinball game where the voters shoot the ball, and then the parties keep it bouncing all by themselves.