Would you believe it: The economists (!) have been discussing the question of the optimal size of legislatures and distribution of seats between constituencies. If we look at the real world – e.g. some of the smaller European countries – there are some fascinating variations in the relative sizes of parliaments which aren’t always easy to explain.

- Denmark – population 5.468.120, 175 MPs = 31.250 inhabitants per MP
- Norway – population 4.627.926, 165 MPs = 28.000 inhabitants per MP
- Sweden – population 9.031.088, 349 MPs = 25.900 inhabitants per MP
- Finland – population 5,238,460, 200 MPs = 26.200 inhabitants per MP
- Netherlands – population 16,570,613, 150 MPs = 110.500 inhabitants per MP
- Belgium – population 10.392.226, 150 MPs = 69.300 inhabitants per MP
- Austria – population 8.199.783, 183 MPs = 44.800 inhabitants per MP

And just for fun – if the Nordic countries used the Dutch formula, our parliaments would be rather small:

- Denmark – 50 MPs
- Finland – 47 MPs
- Norway – 42 MPs
- Sweden – 82 MPs

A Tweede Kamer à la Suède, on the other hand, would mean a lot of construction work in the Hague:

- Netherlands – 640 MPs
- Belgium – 401 MPs

Economists Emmanuelle Auriol and Robert J. Gary-Bobo’s take on this is that

…a parliament with too few representatives is not “democratic” enough, possibly leading to an unstable political system, in which various undesirable forms of political expression, including of course violent ones, will develop. In contrast, too many representatives entail substantial direct and indirect social costs, they tend to vote too many acts, interfere too much with the operation of markets, increase red tape and create many opportunities for influence, rent-seeking activities and corruption.

Quoting an analysis of the size of national legislatures compared to population sizes, they argue that parliament size tends to be linked with population size with a factor close to the square root of the population size – there is some not-too-complicated mathematics involved, but I won’t bother you with that here. And yes, the Dutch parliament is an extreme outlier with regard to size. (Technical note: Auriol and Gary-Bobo use the total number of parliamentarians in bicameral systems). Auriol and Gary-Bobo also argue that oversized parliaments are linked with bad policies, cf. the quote above and this statement:

Our preliminary study of the facts shows that a country’s excess number of representatives is significantly correlated with more red tape (a measure of the direct cost of meeting the requirements to open a new business), more state interference (a measure of whether state interference hinders business development) and more perceived corruption (as measured by Transparency International’s well-known corruption index).

Finally, they have this to say about the European Parliament:

Another interesting question is the optimal number of Euro MPs. The total population of the EU is now 490 million; there are currently 785 euro representatives, and according to our computations, the optimal number of seats should now be roughly equal to 890.

Quite an assembly.

The next question is of cause how to distribute all of these seats – and the votes in the Council of Ministers. One solution – which I don’t think we shall see in my lifetime (that’s some forty years) – could be to go Dutch, i.e. elect all MEPs in one single constituency. Another popular solution has been to use some kind of weighed formula, e.g. the square-root formula.

Statistician Andrew Gelman is not convinced:

If your state has N voters and a block vote of B, the probability that your country is tied on any particular issue is approximately proportional to 1/N, and the probability that your country’s block votes are necessary is approximately proportional to B. So the probability that your vote is decisive–your “voting power”–is roughly proportional to B/N, that is, the number of block votes per voter in your state. The allocation is roughly fair if a country’s vote is proportional to its population.

But why is the square root-formula so popular with both academics and politicians. Gelman points to this assumption:

The hitch is that Penrose, Banzhaf, and others computed the probability of your country being tied as being proportional to 1/sqrt(N). This calculation is based (explicitly or implicitly) on a binomial distribution model, and it implies that elections in large states will be much closer (in proportion of the vote) than elections in small states.

And he continues:

The square-root-rule is derived from a game-theoretic argument that also implies that elections in large countries will be much much closer (on average) than elections in small countries. This implication is in fact crucial to the reasoning justifying the square-root rule. But it’s not empirically correct. For example, if a country is 9 times larger, its elections should be approximately 3 times closer to 50/50. This doesn’t happen. Larger elections are slightly closer than small elections, but by very little, enough that perhaps a 0.9 power rule would be appropriate, not a square-root (0.5 power) rule.

Gelman bases much of his argument on a study of U.S. elections and comparisons with European elections so I think there is some room for discussion about the finer impact of electoral systems. Questions to consider could be:

- What would happen in terms of policy if the size of the Swedish Riksdag was reduced to, say, 290 MPs? (This is what you get if you apply the Danish population/MP-ratio) Or conversely, the Danish Folketing expanded to 210 MPs?
- What would happen in terms of policy if the vote in the European Council of Ministers and the seats in the European Parliament were distributed by something approaching a proportional rule (For mathematics nerds: Gelman would suggest 0,9 power)

Political scientists think such games are interesting. I know. I’m one… ðŸ˜›

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