The government has announced plans to cut the House of Commons down to 600 MPs from the present 650. Should this go ahead, the Commons would be at its smallest since 1796, when a 558-strong House sat without any members from Ireland.
Of course, that parliament was fitted for a much smaller country. The 1801 census showed the population of Great Britain to be 10.5m—less than a sixth of what it is now. Yet despite enormous growth in population, the size of the Commons has scarcely changed. For two centuries, it has ambled along in the 600s, sometimes shrinking, sometimes expanding, but only rarely making dramatic course corrections.
It’s foolish to think that any part of the British constitution corresponds to a rational design. Yet in our decisions (and lack of decisions) about the size of the Commons, we seem to have elevated incrementalism to an art. Is it right to have even 600 representatives for more than 60m people—and if it is, does that mean it was wrong to have just over 600 for 10.5m (1801) or 20m (1861) or 40m (1951) people? Or is there no right answer, such that we can start picking numbers that just feel right?
I won’t claim here that there is a single right answer to the question of how large the Commons should be. But the best answer I’ve seen suggests that the Commons is far larger than it needs to be, and that the government should go far further in cutting the number of MPs—down to about 400—and should slash the Lords to less than a tenth of its current size.
Surprisingly, that answer happens to come from one of the losing candidates in the 1992 Estonian presidential election. Rein Taagepera is an emeritus professor of political science at the University of California, Irvine. He has spent almost all of his academic career there, taking a break in the early 90s to return to Estonia. During that break, he stood in the first (and so far only) direct Estonian presidential election, winning just under a quarter of the vote. In 2008, Taagepera won the prestigious Johan Skytte Prize in political science (other winners include Francis Fukuyama and Robert Putnam).
Aged 82, he continues to publish. Last year he and his co-author Brian Gaines managed to get into a quarrel with one of British political science’s more irascible proponents, Patrick Dunleavy, about how votes are shared between parties in an election. At one point in their response, Taagepera and Gaines plot all the 46,262 ways in which five parties can split the vote. It’s a mystifying, mesmeric plot, with patterns of party competition radiating outwards along the horizontal axis, cutting across one another. I can’t remember seeing anything like it in the political science literature I’ve read. If anything, it resembles some of the figures used to illustrate particle-wave duality in physics. That’s unsurprising: Taagepera’s training is in physics, rather than political science. After leaving Estonia during the Second World War, he moved to Canada, where he studied engineering physics; it was only after completing his PhD in solid-state physics at the University of Delaware that he began studying politics.
Taagepera’s déformation professionnelle has liberated him, allowing him to approach the subject without some of the burdens born by the rest of us political scientists. It’s sometimes said that other social sciences suffer from economics envy, and economics suffers from physics envy. That envy has led academic economists and political scientists to become, essentially, better statisticians. At first glance, that kind of statistics-heavy research can look similar to the kind that Taagepera does. In both, equations feature prominently. But to hear Taagepera tell it, these statisticians are in the grip of a fundamentally wrong-headed approach. They’re in thrall to the wrong type of equation—the linear regression equation.
Understanding the term “linear regression equation” isn’t that hard. Visualise a scatter-plot: a cloud of points, where the two axes might be measuring height and earnings, or gross domestic product (GDP) and defence spending. A linear regression equation is just the equation for the straight line that, in some sense, best fits that cloud of points. Like any equation for a straight line, it’s got an intercept and a slope. The slope tells us how much the line goes up for every one unit along. That’s the same as saying how much defence spending goes up for every unit increase in GDP. The intercept tells us where the line crosses the y-axis. That’s the same as the amount of defence spending when GDP is zero.
In this example, linear regression might not tell us anything that we couldn’t get by “eye-balling” the scatter-plot. In two dimensions, it’s easy for us to visualise the best-fitting line. But we don’t need to restrict our analysis to just two dimensions, or two variables. We can work out the best fitting line through any number of dimensions. I can’t show you the best-fitting line through a 26 dimensional data cloud—but it’s no hardship for a computer to calculate it.
That means that it’s always possible to add in another variable to a regression equation. Think that population explains defence spending? Stick it in the regression equation. Want to know how much it’s also affected by a country’s landmass? Into the equation with it! Once each of these variables is included in the regression equation, they too will get their own number representing their slope. The “slope” in that dimension will tell you how much each extra unit of population affects defence spending, or each extra square kilometre of landmass. It doesn’t matter that you can’t visualise it—all that matters is working out the number that best fits the data. Regression equations like this are infinitely extensible. They’ve been developed to work with different types of variables—variables that only fall between certain limits, variables that fall into discrete categories and so on. They’re the workhorse of the social sciences. So what’s not to like? Some years ago, Taagepera sent some colleagues in the social sciences a spreadsheet with four columns. He didn’t explain where the data had come from, nor what it represented; he just asked them to build a statistical model which would predict the values of the variable in the first column.
Unfortunately for Taagepera’s colleagues, they weren’t being asked to discover a previously unknown relationship. They had been sent data which corresponded exactly to the universal law of gravitation. All of them flunked the test. They didn’t fail for lack of sophistication. Some proposed some advanced statistical techniques to deal with the problem. But generally, their regression models had the same characteristic. Elbert Hubbard once quipped that history was just one damn thing after another. These regression models were the statistical equivalent. They featured one damned variable after another, with each variable having a constant, isolated, additive effect—and not the interdependent, multiplicative, nonlinear effect predicted by the universal law of gravitation.
Taagepera’s test wasn’t unfair. It should have been possible to “recover” the universal law of gravitation using linear regression techniques. There are few things which can’t be shoe-horned into a regression equation given enough work—but they’re not idiomatic. It’s as if Taagepera’s colleagues were trying to speak English using the word order one would use when speaking German—it’s an approximation to English, and might even give passable results, but it’s far from pleasing to listen to. It doesn’t obey the constraints and connections present in the language.
That lack of constraint is common to regression models—and not just when they’re used to “discover” laws of physics. Regression equations will sometimes give answers for situations that don’t make any sense (like a country with zero GDP)—or given situations that make sense they’ll give nonsensical answers (like negative spending for very small values of GDP). They don’t “know” about the logical foundations of the situation.
When Taagepera turned to thinking about the ideal size of a parliament, he started not by performing a regression analysis on real-world examples, but by considering the logical foundations. What are they? To find out, we’ll have to engage in a thought experiment. Imagine that you’ve been asked by your company to sit on a committee and report back to those working in your area. The committee is dealing with a rather fraught issue, so not only will you need to maintain good links with every other person on the committee, you’ll need to keep a beady eye on their links with each other. The committee is small—apart from you, there are only five other individuals (Alice, Bob, Carol, Dave and Eve). You’ll decide on behalf of all 216 employees of the company. The job’s an onerous one. If you just had to speak directly to each person in turn, it’d be simple: five other individuals, five channels of communication. But in addition to keeping track of what you say to Alice (and Bob, and Carol, and Dave, and Eve) and vice-versa, you have to keep track of what Alice says to Bob (and Carol, and Dave, and Eve), and what Bob says to Carol (and Dave, and Eve), and so on.
With six people on the committee, the total number of connections is 15. If you don’t believe me, count the links in the diagram below. There’s a formula which allows us to calculate the number of links between a committee with n members. To be really exact, we can calculate n(n-1)/2. For big numbers, we can get a good approximation by taking n²/2. As the squared term in those equations suggests, the number of links between different members on the committee grows very quickly. If you were to add an extra colleague, the number of connections would jump from 15 to 21. Double the number of committee members, and the connections more than quadruple.
In some respects, it might be easier if the committee were smaller—if it were just you, Alice and Bob. But that’s forgetting the second half of your responsibilities—reporting back to your colleagues. At the moment, all the committee members share the job of reporting back equally: so you speak to one-sixth of the company. That’s 36 people, minus you, which makes for 35 people. If you were to shrink the committee to four, you’d have to go back and report to over 50 people. That would mean that the total number of conversations you had to have would actually go up slightly.
This means that you have to find the right balance: too large a committee, and you spend way too much time trying to eavesdrop on others’ conversations; too small a committee, and you have to glad-hand every minion in the place once you’ve made your decision. But for companies of your size, the committee is always going to be relatively small: the cost of all that extra eavesdropping grows much faster than the cost of reporting back, so growth in the committee has to be kept in check.
This reasoning works for populations and parliaments just as well as it does for companies and committees. The total size of the parliament can’t be too large, otherwise members would spend all of their time trying to monitor what they themselves were saying to each other. Nor can the parliament be too small, otherwise members would have to respond to thousands of constituents. So what is the “Goldilocks number”—not too big and not too small?
Taagepera showed in the 1970s that this number—which means that the number of channels within the parliament is equal to the number of channels to constituents—is equal to the cube root of the population. Miraculously, this elegant model stands up to ugly empirical reality. Tiny countries rarely end up with elephantine assemblies. The model predicts that Kiribati (population just north of 100,000, on a par with Bedford) should have a parliament with 46 members—not far from the actual number of 44. Spain, although it’s 400 times more populous than Kiribati, is predicted to have a parliament of 347 seats, just seven-and-a-half times larger, and just three seats short of the actual number. We can describe how well Taagepera’s theoretical predictions fit actual assembly sizes with a common term of measurement used by social scientists called R-squared. An R-squared of 1 indicates a perfect fit, but values of 0.6 or greater are usually deemed acceptable. A recent paper calculated that the cube root law has an R-squared of almost 0.8—not bad for a model which works from only one piece of information.
What does that mean for the UK? You may already have whipped out your calculator, but with a total population of 64.6m, the UK should—if the cube root law is right—have a House of Commons with just 400 members. That is 250 fewer than we have now, and much fewer than the government is currently proposing. If it’s any consolation, the UK has historically been an outlier to this rule, and is only now gradually coming closer to the pack as population grows.
This doesn’t even begin to take into account the House of Lords—but Taagepera has ideas there too. In a 2003 paper with Steven Recchia, he suggested some limits for the size of upper chambers—at least those supposed to represent the regions. Such chambers couldn’t be as large as the first chamber, or else there would be problems in maintaining the primacy of the first chamber. Nor could they be smaller than the number of regions, or else the chamber’s claim to regional representation would ring hollow. Some kind of average of the two numbers would be helpful—but which average? Taagepera and Recchia suggested we use the geometric mean, which is calculated by multiplying the two numbers together and then taking their square root (instead of adding the two numbers together and then dividing by two, which would give the arithmetic mean).
Suppose that the UK, in a fit of constitutional innovation, decided both to slim down the Commons and have an elected Senate of the Regions. There are nine regions in England. (Primarily statistical entities, they used to be called Government Operating Regions, but now they’re just Regions). Add that to Scotland, Wales, and Northern Ireland, and you have 12 territorial units to represent. The geometric mean of 412 is a fraction under 70. Pleasingly, this would mean that the UK could move from having 1,410 representatives (650 in the Commons and about 760 in the Lords) to exactly one-third of this number: 400 in the Commons, plus 70 in the Senate.
I doubt that the government or the Conservative Party had Rein Taagepera in mind when they proposed a smaller Commons. A smaller Commons would need new constituency boundaries, and the Conservatives have long thought that redrawn boundaries would benefit them electorally. But the taint of possible partisan benefit shouldn’t obscure the fact that the Commons is oversized: compared both to the predictions of Taagepera’s model, and to other countries with similar populations. Germany, with a population of 80m, makes do with only 631 representatives in the Bundestag. France, with a population of 66m, finds 577 deputies to the National Assembly to be quite sufficient. Only Italy (with a population slightly smaller than the UK) has a similarly over-sized parliament.
I don’t suggest that mathematical aesthetics should determine major issues of constitutional reform entirely—but it’s a start. If neither theory nor the practice of other countries can justify the number of MPs, then do we really need so many?
