Redistricting: A Nasty Political Problem with a Nice Mathematical Solution

Stephen H. Unger
February 12, 2007

(The sections entitled "Multi-Member Districts" and "A Neat Solution" were Revised February 20, 2007)

A classic problem in a representative democracy such as ours, is that of partitioning a state (or some other geographical unit) into subunits, each to be represented in a legislative body. How should such partitioning be done? What would constitute a good partitioning? Even in a world of fair-minded people, this would not be an easy problem. Where we live it's a lot tougher. I will start by explaining why redistricting is a serious problem, and then point out what seems to be as good a solution as we are likely to find.

Let's assume we are talking about defining assembly districts in a state. One obvious goal that is hard to dispute (I can almost see certain people preparing to accept this challenge) is to make the populations in the districts equal. That's about it for "obvious". In most states, it is the state legislature that does the partitioning (referred to as redistricting). This is normally done every decade, following a national census. Population shifts generally necessitate redistricting to maintain district equality. Hard nosed politicians tend to take one of two approaches.

More Power to the Most Powerful

The first use of redistricting is by the powerful, to amplify and perpetuate their power. Assume that there are just two parties represented in the legislature. If one of these parties has sufficient power to control the redistricting process, it may take as its goal the maximization of the number of districts in which it is likely to get majority votes. Suppose party A expects to get 51% of the vote and party B 49%. If the districts can be defined so that the adherents of both parties are evenly divided among all the districts, then A would have a majority of 51% in every district and expect to win all the seats in the legislature, even tho it has a lead over party B of only 2%. Once a party has attained control of the legislature, and therefore of the redistricting process, it may be able to maintain dominance, even if it loses much of its constituency. Some examples illustrating the problem (in Ohio, Florida, and Michigan) can be found in Hebert.

To illustrate this, suppose that, in the above example, voter sentiment undergoes a major shift, so that A becomes the minority, party, expecting support from only 36% of the voters as opposed to 64% for B. Then if the district lines are drawn (under A's direction) so that 30% of the districts are filled completely with B voters, the remaining B-voters will constitute only 34% of the electorate. If these are now distributed evenly among the remaining 70% of the districts, along with the A voters (36% of the electorate), then A would be expected to win all these seats, or 70% of the total, despite being the minority party! It is not hard to show that a gerrymandering party whose identifiable supporters constitute proportion r of the total vote can arrange the districts so that they win all the seats if r>0.5 and 2r of the seats otherwise. An interesting, more detailed, discussion of other factors showing how gerrrymandering can lead to permanent domination by one party is in "One Party Domination".

Quite apart from objections based on such "goody-goody" notions as "fairness", there are a few "practical" problems with this approach. One is that drawing district lines to produce such a result might require some really imaginative geometry. (We assume here that party A, while powerful, is not powerful enough to force people to relocate so as to facilitate this arrangement, altho, incredibly, such an effort was actually made once in England. ) But this is not an insuperable obstacle. The process of redistricting with such objectives has a long history that has resulted in many wonderfully shaped districts. Modern computer technology has led to refinements that have transformed the art of gerrymandering into a science.

A more basic difficulty is that there is significant uncertainty as to how people will actually vote. Because a certain set of people voted for party A in the last election does not mean that they all will do the same next time. So, in the above example, a swing toward B of a few percent could result in a complete reversal, with A losing all or most of the seats. Slightly less greedy gerrymandering would have allowed B to have a few sure seats in order to give A the remaining seats with much greater certainty. For example, with the same 51% to 49% advantage, 10% of the districts could have been filled with B voters, leaving A with a 51% to 39% advantage outside these districts. So evenly distributing the A's and B's among the remaining 90% of the districts, would have given A a virtually certain 9 to 1 advantage in seats, which is pretty good for a 2 % edge in votes. Obviously, this kind of precision is not achievable in the real world, but good approximations are often feasible. Clearly tho, even unscrupulous gerrymanderers with complete control over the redistricting process still have to make trade-offs between the number of seats targeted and risks of suffering great losses if there should be a large enough shift in the vote.

Making Club Membership Permanent

A second application of gerrymandering is in situations where no party is firmly in control. A narrower goal is pursued. Rather than try to load the dice in favor of one party, a cozy bipartisan arrangement is made to ensure that the districts of favored (or perhaps all) incumbents from both parties are shaped so as to make them "safe", i.e., so that, in each such district, the incumbent's party has a safe majority. This kind of gerrymandering is one important reason why incumbent state and federal legislators in the US win re-election with probability 98%.

Helping--or Hurting Subgroups

Another possible goal of redistricting might involve the effect on one or more subgroups of voters. Suppose that, in the example-state of a million people, with ten congressional seats, 15% of the population are members of farm families. Then it might seem reasonable that one of the seats go to a person strongly tied to the farming community. This would likely occur if, in at least one district, the farm families were in the majority. So care might be taken to draw the boundaries so as to produce this result. Depending on the extent to which farmers are scattered over the state, this might or might not require a peculiar shaped district. On the other hand, the goal of those dominating the redistricting process might be just the opposite. They might partition the state in such a manner that the farmers are a small minority in every district, so that their influence in elections is minimized.

Actually, it is not obvious that scattering a group thinly over many districts is necessarily detrimental to their political influence. If elections are generally close in many districts, the farmers (in our example) might thereby become important factors in many elections and so gain significant leverage over many legislators instead of having just one who is completely beholden to them. It might also be the case that the gerrymandering necessary to put farmers in control of one district might have the effect of delivering a number of other districts to a party that is generally hostile to farmer interests. So once again, trade-offs are involved. At different times and places gerrymandering has been used with respect to various groups, both to pack them into one or a few districts, or to scatter them among many districts. The groups so treated are often racial minorities.

Neutral Commissions

Gerrymandering has become a major tactic in the US, mainly for the purpose of increasing the power of a particular party or providing job security for legislators. An important reason for this is that, with few exceptions, redistricting in the US is done by the state legislatures, with minimal effort to rule out partisanship. In many other countries, including Canada, for example, redistricting is carried out by commissions that are set up so as to insulate them from political pressure as much as possible.

But, even if the job were to be turned over to such a commission, it is not clear what marching orders that commission should be given. Following are some possible objectives:

  1. Each district should contain the same proportions of adherents to each party.
  2. The number of districts in which each party dominates should be proportional to the number of that party's followers in the state.
  3. Certain groups should be allocated districts in which they constitute a majority, so that at least one member of the legislature is likely to represent them properly.

The first of these amounts to a winner-take-all approach that would clearly benefit the majority party, possibly resulting in its winning all seats. The second is a form of proportional representation that seems more equitable. Which groups should be the subject of item-3, and are we sure that such treatment would benefit them? Each of these objectives would, in many cases be hard to achieve, and even approximations might require some bizarre partitioning. Obviously, they are also pairwise incompatible. Imagine how difficult it would be to obtain a consensus with regard to these objectives (and the above list is certainly not comprehensive).

Multi-Member Districts

In many countries, the redistricting problem is minimized (or eliminated altogether) by assigning not one, but many representatives for each district. At one extreme, the entire state (or province or nation) is treated as a single district. The election system is designed (and there are various ways to do this) so that the number of seats allocated to each party is proportional to the number of votes cast for that party. Every party that gets some minimum percentage (e.g., 5%) of the votes is entitled to at least one seat in the legislature. This arrangement neatly achieves objective-2 on the above list, and does so, not by assuming voter preferences, but by using the actual votes cast. It also can achieve objective-3, if the group involved organizes a political party.

Any state in the US could change its laws or constitution so as to elect its legislative body, and/or to allocate its electoral votes, in this manner. Doing the same for its delegation to the House of Representatives (if it has more than one representative) would require a change in federal law.

A Neat Solution

Now consider how we could do a better job with our current redistricting problem. It should be clear from the above that it is unrealistic to expect agreement on and implementation of any procedure with objectives of the complexity of those listed above. Experience to date suggests that any procedural leeway will be exploited to benefit the party in power, or to minimize the chances that incumbents will be defeated. The best we can reasonably hope for is a system that rigidly produces a unique solution in which districts are compact without regard for the distribution of party strength in each. (Of course population should be equal in all districts.)

Warren D. Smith, a Temple University mathematician, has developed an elegant technical solution to this problem, called the shortest-splitline algorithm. It generates a unique, compact partition with straight-line internal boundaries. It works as follows:

Suppose we need to partition a state into 5 districts (with equal population). For the first step, we split it into two segments with population ratio 3:2. This is done by finding the shortest straight line that accomplishes this. (If there is more than one such line, choose the one whose orientation is closest to North-South. If there is more than one of these, choose the one furthest West.). We now proceed recursively by splitting the first of these sections into 2 subsections with population ratio 2:1, etc.

The beauty of this scheme is that it is totally objective; anyone can execute the algorithm, which is not a secret. Because the solution is unique, the politicians have no wriggle room at all to bias the outcome in any way. There is no need for a state to hire experts since the partitioning program can be run by state civil service employees. The various political parties can also run their own copies to ensure that there is no trickery. We do have to assume that a suitable population data base is available and that it has not been manipulated. Note tho, that, unlike what is needed for gerrymandering programs, the necessary data base need not (perhaps should not) have information about party affiliations. The major drawback of the Smith algorithm is that it ignores precinct and other political boundary lines. It might be necessary to redraw precinct boundaries every decade in conjunction with the redistricting operation.

For any particular redistricting operation, some people might be happy with the outcome of the Smith algorithm and others not. This is inevitable, since the various measures of goodness are, in general, incompatible. The procedure achieves our goal, which is to prevent those in power from solidifying their positions by biasing the redistricting process. Whatever biases do result from this process will be of a random nature. Alternative redistricting schemes with the same objectives can be found at Westmiller and at Olson. These schemes respect precinct and other boundaries. However, they are much more complex, and take a lot longer to compute.

Implementing the Solution

How do we get redistricting reform adopted by the various states? The normal process by which such improvements in government operation are adopted would be via the passage of laws. But the people with the power to enact this reform, i.e., state legislature members, are the ones who would thereby lose power. Not much chance that the foxes are going to agree to make the hen houses more secure! Furthermore, there now exist redistricting experts. How likely are they to testify on behalf of reform? Not very, since they would thereby be helping to close the market for their skills.

Thus, the only way this is going to happen is via the referendum route. We need to enlist such good government groups as the League of Women Voters and Common Cause to get this reform on state ballots and then to campaign for passage.

Return to "Ends and Means" Blog