What is the relationship between district magnitude and a party’s advantage in seats, relative to votes?

Using the same district-level dataset that Rein Taagepera and I use for our forthcoming Votes from Seats (and which has its original source in CLEA), we can answer this question. The sample I am using here is simple PR systems–those in which the districts are the sole locus of seat allocation (i.e., leaving out two-tier systems).

The advantage ratio (A) is the best way to examine this; it is just a party’s percentage of seats, divided by its percentage of votes.The table below shows the average A for magnitudes (M) ranging from 2 to 7. The larger the A, the more a given party is over-represented. The table shows mean A values for the first, second, and third largest parties (by vote share in the district), as well as how many districts of a given magnitude are in the dataset .

M A, 1st A, 2nd A, 3rd Num. obs. 2 1.29 1.50 0.00 172 3 1.34 1.10 0.50 98 4 1.25 1.15 0.50 103 5 1.26 1.14 0.64 112 6 1.19 1.17 0.73 72 7 1.19 1.09 0.81 79

We can see that M=2 is the only case where the second largest party gains more than the largest does, on average. This result is well known from the experience in Chile. I had thought we might see a similar pattern at M=4. However, we do not. As with all other magnitudes except 2, the largest party at M=4 tends to have a bigger advantage ratio than the second, although the two largest parties’ A values are closer here than in odd M‘s just above and below.

Also of interest fro the same query to the dataset, we find that the lowest magnitude at which A<1.10 for the largest party is M=17. The average for the largest party never falls below 1.00. The second party first falls to A<1.10 at M=13 and then stays right around 1.00 through all higher values.

As for the third largest party, it stays, on average below A=1.00 till M=13, but falls below 1.00 again at several higher M values. The fourth largest party has to wait till M=20 (and likewise has some higher magnitudes where it falls below 1.00).

At higher magnitudes, these average values tend to bounce around a bit, mainly because the sample at any given magnitude is small and thus subject to vagaries of country-level (or district-level) factors (including allocation rules, although the vast majority of these are D’Hondt).

I have long “known” that 4-seat districts tended to under-represent the largest party, relative to the second largest. Well, apparently one should check what one “knows”. Thanks to Jack (on Twitter) for the prompt to investigate.