Large Assemblies, Small Districts

Almost twenty years ago, Rein Taagepera and Matthew Shugart published an innovative book, Seats and Votes, which they dared to present as a potential ‘Rosetta Stone’ not only for electoral studies but for other branches of political science as well. Now Taagepera has published a book-form of his further work with still more provocative ambition and potential impact,
Predicting Party Sizes. In this and other works, the physicist-trained, political scientist turned Taagepera campaigns for both scientifically cumulative and practically applicable knowledge in politics, which, in his view, requires making relations between variables precise and quantifiable. He wants to “present scholarly results in a way practitioners of politics could use”.

To show the potential of Taagepera’s approach, let me just draw a little on what I think is his central finding: a formula relating the number of parties in parliament, P, with the average district number of seats, M, and the total number of seats in the assembly, S; in his notation:

P = (MS)^1/4

The title of Taagepera’s book might be a little misleading because it may suggest that the number of parties is always a dependent variable of the basic elements of the electoral systems –a statistical-regression approach that Taagepera characteristically scorns as too limited. Actually his formula accepts two-direction lines of causality. It can indeed be turned the other way around in order to present the electoral system as derived from the number of parties. Specifically:

M = P⁴/S

Since, according to Taagepera, the number of seats of the assembly depends strongly on the country’s population (in a cube root relation), we can deduct from the above formula that, for similar number of parties, P, the larger the country, and hence the larger the assembly, S, the smaller the expected district magnitude, M. Very large countries, precisely because they have large assemblies, should be associated to small (single-member) districts. The institutional designers in India, for example, are likely to choose single-member districts, while the institutional designers in Estonia are likely to choose multimember districts, typically associated to proportional representation rules. Thus we should usually see large assemblies with small districts, and small assemblies with large districts. Which is what we indeed usually see.

The interest of this finding is that it is counter-intuitive, since apparently small countries should have more ‘simple’ party configurations, so that they could work with simple electoral systems with single-member districts and majority rule in acceptable ways (actually this tends to happen in very small and micro-countries with only a few dozen thousand inhabitants in which only one or two significant parties emerge). But now we could have an answer to the very intriguing question of why large countries, including the United States, in spite of the fact that large size is typically associated to high heterogeneity, keep small single-member districts and have not adopted proportional representation. The answer may be that in large countries such as Australia, Canada, France, India, the United Kingdom and the United States, a large assembly can be sufficiently inclusive, even if it is elected in small, single-member districts, due to territorial variety of the representatives. By contrast, in small countries, including Belgium, Denmark, Estonia, Finland, Netherlands, Norway, Switzerland and so many others, the size of the assembly is small and, as a consequence, the development of multiple parties has favored more strongly the adoption of more inclusive, large multimember districts with proportional representation rules.

A relevant implication for political practitioners of institutional design is that if the size of the assembly is rather stable and depends on the country's size, for a small country with a small assembly just a few parties can be sufficient to produce a change of electoral system in favor of proportional representation. In contrast, for a large country and a large assembly, many parties would be necessary to produce such a result --as some of our colleagues in Britain, for instance, know very well.

REFERENCE:

Rein Taagepera, Predicting Party Sizes: The Logic of Simple Electoral Systems, Oxford University Press, 2007. CLICK

COMMENTS

Jan-Erik Lane said...

I will check this model by looking at the UK, Sweden and Russia.
The problem with this approach is that social reality is not CONSTANT like the universe.
How can you be sure that
(1) P = (MS)^1/4
does not change to 1/3 or 1/5 in the future? But of course the speed of light c or strength of gravity G does not change!

jan

Rein Taagepera said...

Thanks for the publicity!

Yes, P = (MS)^1/4
is at the core of the model. An acquaintance dealing in ceramics offered to put one of my equations on a plaque, and I proposed precisely that one.

And yes, M=P⁴/S is as valid, and can be expected to be at work in constituent assemblies.
(Could it be tested?)
To avoid an impression of directionality, MS/P⁴=1 or P⁴/MS=1 might be the safest way to express it.

The range of P is roughly 2 to 10. Hence P⁴ ranges from about 16 to 10,000.
In comparison, S ranges from about 20 to 700 in present democracies.
Hence the impact of S on M is appreciably smaller than that of P.
Even for S=700, 10 parties in the constituent assembly would push toward M=10,000/700=14.
Even for S=20, 2 parties in the constituent assembly would push toward M=16/20=1.

At present, the strongest predictor for M=1, at any S, from St Kitts to India, is British heritage (cf. p. 45 in Predicting).
It may come about through imitation of Westminster and also through imitation of two-party system in the constituent assembly (or its equivalent).

Re Jan-Erik's question: “How can you be sure that 1/4 in P = (MS)^1/4 does not change to 1/3 or 1/5 in the future?”
The exponent 1/4 is not empirical but theoretically derived.
In this respect, it is more akin to the "2" in E=mc² than to the numerical value of c.

Greetings,

Rein

Jan-Erik Lane said...

If you take the root out of big numbers, then you arrive at small numbers for sure.
But can this fomula capture the difference between Norway and Poland, between France and Italy?
There is a GREAT variation in the number of parties WITHIN the PR family.

jan

Rein Taagepera said...

Dear Jan:

I am referring to Chapter 8 in my book.
Please note that:the relationship applies directly only to "simple" electoral systems -- those where all seats are allocated in districts, without thresholds, runoffs and other complications. Even so, it expresses only the average mean trend. Individual electoral systems may be off by a factor of 2 (= multiply or divide by 2), and individual election results are off by even more. Table 8.1 (which is in the attachment) shows that the law N-zero=(MS)^1/4 applies within 20 % to the mean outcomes of all simple within-district formulas. It doesn't apply to the non-simple Two Rounds nor to SNTV.

What is the use of a law that fits only to the means of simple systems, with a large random error range?
It is better than nothing, a first approximation upon which further factors can be grafted.
Recall that Galileo's law of falling bodies accounts very imperfectly for the path of a falling leaf.

Greetings,
Rein

Jan-Erik Lane said...

You can estimate the true functional form for this “invariance” if you plot the line through the data. What they need is information about:

1) number of parties (real or the effective number)

2) average size of constituencies

3) number of parlamentarians

That should not be too difficult to get hold of. For UK, we would have:

1) 5-6 parties

2) 1

3) 650

Taking the fourth root gets us to about 5-6. So the formula works.

Gianfranco Pasquino said...

Italy: Large Assembly, All Kinds of Districts. At this point, it would be quite interesting to know Colomer's and Taagepera's prediction on the features of the next (if any) Italian electoral law. Please, however, refrain from saying that Italy is, as usual, an outlier...

Gianfranco Pasquino

Rein Taagepera said...

I'm loosing track of what the current Italian electoral laws are.

Iff all seats are allocated within districts according to a simple PR of FPTP rule (no thresholds, no second rounds, no parties within party blocs...), give me assembly size and distribution of district magnitudes, and I'll make a prediction for number of seat-winning parties and their size distribution.

If and only if...
Rein

Gianfranco Pasquino said...

House of Deputies: 630 seats; 27 districts (Puglia 44 seats; Emilia-Romagna and Lombardy 2, each 43 seats; Lombardy 1, 40 seats, Tuscany 38... Electoral sytem PR with majority bonus.

Gianfranco Pasquino

Rein Taagepera said...

Thanks.

On the basis of this information, the best quantitative guess would be based on
average M = 630/27 = 23.3; hence MS = 14,700.

It assumes that all seats are distributed in districts, with no thresholds, and parties do not form alliances.
Estimated likely error range (plus or minus) is indicated as +.

No. of seat-winning parties: N-zero = (MS)^1/4 = 11 + 3
Effective no. of parties: N-two = (MS)^1/6 = 4.9 + 2
Minimum measure of no. of p.: N-infinity = (MS)^1/8 = 3.3 + 1
meaning largest seat share s₁ = 1/N-infinity 0.30 + 0.05

Qualitative adjustments:
The many double-the-average districts make it slightly easier for small parties, enhancing N-zero.
Majority bonus reduces all N and enhances s₁; it depends on how strong this bonus is.

I have not looked up the actual results for Italy 2006 --let someone else do it.

Rein

Josep M. Colomer said...

The actual number of parliamentary parties in Italy changes constantly due to splits and migrations. But currently there are 12 parties, basically corresponding to electoral candidacies, plus a small 'mixed' group of independents. Taagepera predicts 11 + 3.

And there may be anticipated elections very soon!

Matthew Shugart said...

Josep, this is very interesting. I wonder about the following, however, from your post:

"in large countries such as Australia, Canada, France, India, the United Kingdom and the United States, a large assembly can be sufficiently inclusive, even if it is elected in small, single-member districts, due to territorial variety of the representatives."

I wonder because that list of countries includes two with significantly under-sized assemblies, according to the cube root (India and the USA). The UK, on the other hand, has one of the world's most "over-sized" lower houses.

So, I can see where the argument works well for the UK: Many more districts than would be the case for an assembly closer to the cube root, and hence a lot of "territorial variety of the representatives" (e.g. Scottish and Welsh nationalists, as well as LibDems). India has a high territorial variation, despite a "small" assembly, due to numerous state-based parties (most of which aggregate into one of two pre-electoral blocs).

The USA, on the other hand, has a lot less room to represent territorial variety, because the districts are so big in population terms due to the small assembly (for the country's population), and because its party system is much too small (just two parties would not be predicted even with the small assembly, according to Rein's models).

Josep M. Colomer said...

Matthew, Thanks a lot for your comment.

I think the point is about absolute size of the assembly, which permits variety of representatives, even if it's small relatively to other variables (country size...). The so-called two parties in the USA are really very varied --these days, with the primaries going on it's crystal clear: the variety of presidential candidates, as well as of representatives and senators, is as great as in most countries in Europe with multiple parties. This may make many people in the U.S. feel proportional representation is not necessary because many different tastes are already represented.

Matthew Shugart said...

As for variety, of course the only kind of variety that can be represented with SSDs [single-seat districts] is that which is regionally concentrated and even then, they are not accountable to any sympathetic voters outside their districts. There is systematic bias against those that are not even able to win a district plurality (which is almost always majority in the US, unlike all other relatively large FPTP countries), and still greater bias against them the more the number of seats is small. So, I agree that the argument you are making "works" on absolute size of the assembly, but the greater the geographic and population extent of SSDs, the less likely minorities are to find a regional concentration sufficient to win a seat. So, the argument you are making must also depend on the size of the assembly relative to total national population.

Thanks for the exchange!

--m

Rein Taagepera said...

S = P^1/3 [S: Size of the assembly; P: Population] visibly accounts for most of variation in S empirically --and even offers a logical process to explain it. Other factors should stand out as one considers the residue r = S/P^1/3, after the effect of S is taken out ("controlled for"). Better use log r, because it can be expected to be symmetrically distributed around 0, if there are no other factors except for random scatter. Actually, it shows a tail, which is largely corrected for by introducing literate adult population.

In short, play around with log r. As long as you stick to S, the huge effect of P keeps confusing you.

Rein