The average number of divisors of the size of legislative chambers is greater than the average number of divisors of simulated data which replicates the mean and dispersion of those sizes. It’s therefore easier to split them up into equally-sized districts.
What Plato wrote
In book V of his Laws – helpfully summarized by the Internet Encyclopedia of Philosophy as “Plato’s last, longest, and, perhaps, most loathed work”, Plato claimed that there was an ideal number of citizens for a polity, and justified this on unusual grounds:
“The number of our citizens shall be 5040 – this will be a convenient number; and these shall be owners of the land and protectors of the allotment… Every legislator ought to know so much arithmetic as to be able to tell what number is most likely to be useful to all cities; and we are going to take that number which contains the greatest and most regular and unbroken series of divisions. The whole of number has every possible division, and the number 5040 can be divided by exactly fifty-nine divisors, and ten of these proceed without interval from one to ten: this will furnish numbers for war and peace, and for all contracts and dealings, including taxes and divisions of the land. These properties of number should be ascertained at leisure by those who are bound by law to know them; for they are true, and should be proclaimed at the foundation of the city, with a view to use” (Jowett translation)
Plato’s claims about the number 5040 are still true today. There are fifty-nine divisors of 5,040 if we exclude the number itself, and 5,040 is divisible by two, three, four, … all the way up to ten. It’s a highly divisible number, and more particularly a highly composite number.
Although Plato’s claims about the number 5,040 are still true, the relevance of these claims is almost nil. Plato’s views about citizenship are abhorrent today; there are no self-governing polities with so few citizens;1 and almost all self-governing polities rely on representation rather than direct participation in the work of government.
If we want to take what is good from Plato, and work out what numbers are “most likely to be useful to all cities”, we can take this property of divisibility and apply it not to the number of citizens, but to the number of legislators. Plato himself did this: he proposes in book VI that any colony have a council of 360, which “will be a convenient number for sub-division” and which, like 5,040, is highly divisible (23 divisors) and highly composite.
Victorian number generator
I was thinking about Plato thanks to a recent trip to Terra Australis Australia, where the Parliament of Victora is reviewing electoral arrangements for its upper chamber, the Legislative Council.
Two parts of the review concern the size of the Legislative Council and the number of electoral districts. At present, the Legislative Council has 40 members, or roughly four-ninths the size of the Legislative Assembly, which has 88 members. The forty members of the Legislative Council are elected in eight districts, each with a district magnitude of five.
Considering first the size of the Legislative Council, the ratio between the Council and the Assembly is very slightly greater than the ratio between most upper chambers and their lower chambers. In general, upper chambers are around 40% the size of the corresponding lower chamber.
You can see that in Figure 1, which uses data from the International Parliamentary Union showing lower chamber size on the horizontal axis against upper chamber size on the vertical axis. The dotted line gives the best-fitting line according to ordinary least squares; the solid line shows the line where the size of the upper chamber is equal to the size of the lower chamber. The Legislative Council is slightly above the trend line, but still well within the confidence interval of the associated linear regression.
Figure 1: A scatter plot of lower chamber size (horizontal axis) against upper chamber size (vertical axis), with the position of the Victorian Assembly highlighted. The sole dot above the solid line is the part-time House of Lords.
How does the size of the legislative council interact with the size of the districts? If you want equal sized districts (and that’s a big if), then the total size of the Legislative Council sets limits on how many regions you can have. If you have 40 members, then you can have eight districts each of five members, or five districts of eight members, but you can’t have six evenly sized districts or seven evenly sized districts. If you had the twin primes (39 or 41 members), you wouldn’t be able to have evenly sized districts at all.
(Having evenly sized districts isn’t a necessary criterion, but the discussion paper which accompanies the review is mostly illustrated with even-sized districts, and the impact of variation in district magnitude upon system properties is in my view still an open question.)
Since you can change districts more easily than you can change the number of members of a legislature, it seems like we should choose the size of the legislature so it’s appropriate (is smaller than the lower chamber by some commonly found ratio) and permits lots of possible divisions.
In the case of the Victorian Legislative Council, a chamber with 36 members would more closely approximate the ratio found amongst national upper chambers (36/88 \(\approx\) 40%), and is highly divisible, allowing evenly-sized districts with 1, 2, 3, 4, 6, 9, 12, or 18 members, or eight different options. Indeed 36 is not only highly divisible in the sense of having a lot of divisors for a number its size, it’s highly composite. This means that a assembly size of 36 allows more flexibility than a legislative council of 40, which allows evenly sized districts with 1, 2, 4 5, 8, 10 and 20 members, or seven different options.
It’s all a bit odd
Perhaps this property of divisibility strikes you as verging into numerology. I’m certainly not aware that anyone has analysed how easily legislative assemblies can be divided. The only remotely similar property I’ve heard discussed is oddness and evenness, where there is an argument for having odd-sized chambers.
The argument goes like this: that democracy is about majority rule, and in an odd-sized chamber there will always be a majority if votes are binary votes and if abstentions are not allowed. By contrast, in an even-sized chamber there might be a tie. There are therefore some circumstances in which even-sized chambers fail to identify a majority either in favour or in opposition to some proposal. If a tie leads to failure of a motion, then this creates an anti-democratic bias towards the status quo, with the size of the bias depending on how big the chamber is and how often controversial issues will be finely balanced.
At least one parliament was designed explicitly to have an odd number of members and thereby prevent a tie. In the 1973 election to the Swedish Riksdag, government and opposition forces both won 175 of 350 seats. The number of seats was changed to 349 for the following 1976 election.2
Unfortunately, it seems that no one is paying attention to the Swedish experience. Most legislative chambers have an even number of members, as Figure 2 shows. This figure uses the same IPU data used to plot Figure 1, except that this time I include all chambers, whether or not the system is unicameral or bicameral, and I drop the Chinese National People’s Congress, because it’s absolutely huge.
Figure 2: Proportions of chambers that are odd- or even-sized, according to chamber type.
The preference for even-sized assemblies is not overwhelming, but it is a bit more pronounced amongst upper chambers, where almost three out of every five chambers have an even number of members. Although having an odd number of members might help prevent ties, having an even number of members does help with Plato’s concern of divisibility. I’ll look at divisibility now.
A house divided
When we looked at whether odd- or even-sized chambers are preferred, we could calculate whether each individual chamber has an odd or even number of members, and then compare that to the expected proportion of even and odd numbers. That proportion is constant, at fifty percent.
When we look at divisibility, it’s not so clear what we’re supposed to compare to. We can calculate the number of divisors for each number, but it’s not clear what benchmark we should use for this.
Here’s an R function, taken from Stack Overflow, which will calculate the divisors for a given assembly size:
Code
divisors <-function(x){# Vector of numbers to test against y <-seq_len(x)# Modulo division. If remainder is 0 that number is a divisor of x so return it y[ x%%y ==0 ]}
We can list the divisors for the current size of the Victorian Legislative Council:
Code
divisors(40)
[1] 1 2 4 5 8 10 20 40
and calculate the number of divisors (including the number itself):
The average legislative chamber therefore has just over seven divisors. We can calculate this, but we’re forced back to two classic questions, “is that a lot or a little?” and “a lot or little compared to what?”
Getting to the kernel of the comparison
Ideally we would like to compare the average divisibility of the chambers that actually exist to the average divisibility of numbers that are similar to the numbers we have in relevant respects (some bigger, some smaller) but which can be dissimilar in terms of their divisibility.
One way of doing this is to estimate a smoothed distribution which matches the numbers we see. Kernel density estimators are a good way of doing this. It’s not necessary to understand how a kernel density is constructed, but it is necessary to understand the output of the procedure.
Figure 3: A histogram of chamber sizes with overlaid kernel density estimate. Vertical axis shows density (count divided by sum of counts of all bins in the histogram).
Figure 3 shows a histogram of chamber sizes. The orange line is the kernel density estimate. It reproduces the shape of the histogram – greater density at lower values, a long tail out towards the seven hundreds – but without reproducing exactly the spikes at certain values. The smoothness of the shape depends on a bandwidth parameter: here I’ve used the ggplot default, which is also the R default.
What we can therefore do is:
estimate a kernel density using the actual sizes of legislative chambers;
randomly draw from that estimated distribution
calculate the average number of divisors of those “fake” chamber sizes
compare the actual average number of divisors to the average on the fake data.
Here’s some code that will do just that. Note that I’ve taken logs so that we don’t get any negative numbers from our kernel density estimate.
Code
set.seed(448)z <-density(log(all_chambers),kernel ="gaussian")rdens <-function(n, z, data) { width <- z$bw rkernel <-function(n) rnorm(n, sd = width)sample(data, n, replace=TRUE) +rkernel(n)}repl_func <-function(x) {### Generate the density on the log scale newx <-rdens(length(x), z, x)### Exponentiate and round to get back on the integer scale newx <-round(exp(newx)) newstat <-mean(n_divisors(newx))return(newstat)}res <-replicate(1000, repl_func(log(all_chambers)))
We now have one thousand numbers giving the average number of divisors of fake chamber sizes in one thousand fake sets of data. We can plot that distribution, and plot the real world figure next to it.
Figure 4: Distribution of the average number of divisors in simulated data (orange density) and in the real world (green line).
The green line, representing data from the real world, is way out to the right of the distribution resulting from our fake data. This means that real chamber sizes are much more divisible than fake data which try to match the overall characteristics of chamber sizes.
A simpler way to the same conclusion
Although kernel density estimates are the most obvious way of mimicking the distribution of a sample, we don’t have to resort to such a complicated technique. Here’s an alternative way of generating something that “looks like” chamber sizes, but can have different divisibility:
take each actual chamber size
draw as many random numbers from [-1, 0, +1] as there are actual chamber sizes
add on these increments to create some fake data
calculate the average number of divisors for the fake data
This code does just that, and stores the average number of divisors.
Code
repl_func2 <-function(x) {### Generate the density on the log scale increments <-sample(c(-1, 0, 1), length(x), replace =TRUE) newx <- x + increments newstat <-mean(n_divisors(newx))return(newstat)}res2 <-replicate(1000, repl_func2(all_chambers))
As before, we can plot the distribution of the fake data and overlay the actual figure.
Figure 5: Distribution of the average number of divisors in simulated data (orange density) and in the real world (green line).
Just as before, the real world legislative chambers we see are more divisible than the fake data we’ve generated by adding or subtracting a single seat. Whether we use a fancy technique (kernel density estimation) or a simple technique (adding or subtracting one), we reach the same conclusion: legislative chamber sizes are more divisible than they might otherwise be.
Why do we get this pattern?
It would be nice to think that constitutional framers are thinking carefully about the divisibility of their legislatures, and choosing that number “most likely to be useful to all [polities]”. Unfortunately, I don’t think that constitutional framers think like that, and that some of these patterns have to do with how much we like particular numbers.
Our like or dislike of certain numbers might seem a pretty slender basis for choosing the size of a legislature. After all, some likes and dislikes are culturally constructed. My house number is 44, which is just about the worst number you can have in Chinese, where 44 sounds a lot like “death death”. Many hotels have no thirteenth floor, but the association between thirteen and bad luck probably only holds in cultures that have some Christian roots.
Some numbers, however, do seem to be liked more than others because they’re easily accessible. Dan King and Chris Janiszewski claim that people like small numbers, and numbers which are the result of common additions and multiplications. Other authors reach similar conclusions.
To return to the numbers used by the Victorian Legislative Council: I might like the number forty more than I like the number thirty-nine or the number forty-one because I spent time as a child learning my times tables, and because forty appears in those times tables as the answer to five times eight and eight times five.
The likeability of numbers is therefore connected to their divisibility, but likeability probabily trumps divisibility. In King and Janiszewski’s data, numbers ending in five are liked more than other numbers. That probably has something to do with the fact that most humans have five fingers on each hand, and use hands to count with. But numbers ending in five aren’t always very divisible. Twenty-five has far fewer divisors (3) than twenty-four (8), but people apparently like twenty-five more than they like twenty-four. Even in the histogram shown in Figure 3 you can see the spikes at 25, 50, 75 and 100. We like the numbers that we like, and divisibility is a happy accident.
This of course is an explanation of why the distribution of chamber sizes is what it is, rather than a justification. For my part, I’ll be writing to suggest that the Victorian Legislative Council have a highly divisible, highly composite number of members (36) rather than the 40 it has now.
Acknowledgements
Thanks to @jameschalmers.bsky.social, @urbaneprofessor.bsky.social and @paulfscott.bsky.social, all of whom sent me copies of Bort’s article “The Numbers Game”.
Footnotes
Palau comes closest, with a population of 10,645, of whom roughly 60% are over the age of 18, giving a total adult population of around 6,500. I don’t consider the Vatican to be self-governing.↩︎
For a time I thought that the size of the Scottish Parliament might have been determined with this in mind. This view was sadly scotched (pun intended) by Eberhard Bort’s article, “The Numbers Game”, where I learned that the number 129 was a compromise between Labour (who wanted a parliament of 112, or the seventy-two existing constituencies plus forty top-up members) and the Liberal Democrats (who wanted a parliament of 145, or seventy-three constituency members plus seventy-two top up MPs). The Liberal Democrats would have had seventy-three constituency members rather than seventy-two because Orkney and Shetland would each have had a constituency member. Not coincidentally, the Westminster constituency of Orkney and Shetland “has elected only Liberal and Liberal Democrat MPs since 1950; the longest run of any British parliamentary constituency”.↩︎
