A Note on Gerrymandering

During a discussion of Irish election results in August 1999, cross-posted variously between ni.politics, uk.politics.electoral, and soc.culture.irish, both Jim Riley and I felt challenged to prove that to require a rigid ratio between electors and representatives is not a sufficient safeguard against gerrymandering .

I started the bidding with the old chestnut about dividing the mythical county of Tymanagh with a population of 60% Yugoslavs and 40% Belgians into three single-seat constituencies. There are as it were 1.8 seats' worth of Yugoslavs, and 1.2 seats' worth of Belgians.

diagram A 'fair' result is clearly that the Yugoslavs should win two seats and the Belgians one; this picture illustrates just one possible arrangement that wold bring that about.
diagram But since the Yugoslavs are a majority overall, it will usually be possible - in fact, it will usually be easier - to create three seats with Yugoslav majorities.
diagram On the other hand, if the Yugoslavs are sufficiently geographically concentrated, this can be turned against them by creating a homogenous Yugoslav seat, leaving the Belgians a 60/40 advantage in the other two seats

I concluded by stating that:

"Setting strong constraints on the size of electoral districts doesn't make gerrymandering very much more difficult at all. In fact if it's not compensated by other geographical constraints it can make it even easier to incorporate the next little village of Belgians just to make up the numbers."

Gerry Cunningham responded sceptically:

"Nice, if you're dealing with FPTP [ie first-past-the-post, UK or US-style]. Now let's see it with multi-seat STV."

So I came up with these two examples, illustrating how you could engineer different results where you had 80,000 voters, electing eight representatives by STV, and again 60% Yugoslavs (total of 48,000) and 40% Belgians (total of 32,000). Of course the "fair" result is that the Yugoslavs should win five seats and the Belgians three.

diagram But if the Belgians are drawing the boundaries, you might find you have:
1) Tymanagh North-East, a 3-seat constituency, total electorate 30,000, 15,100 Belgians and 14,900 Yugoslavs. The quota is 7,500 so it elects 2 Belgians and 1 Yugoslav.
2) Tymanagh South-West, a 5-seat constituency, total population 50,000, 16,900 Belgians and 33,100 Yugoslavs. The quota is 8,334 so it elects 2 Belgians and 3 Yugoslavs.
Total result: Belgians 4, Yugoslavs 4.
diagram However if the Yugoslavs are in charge (which after all is more likely) you could get:
1) Tymanagh North-West, a 3-seat constituency, total population 30,000, 7,300 Belgians and 22,700 Yugoslavs. The quota is 7,500 so it elects no Belgians and 3 Yugoslavs.
2) Tymanagh South-East, a 5-seat constituency, total population 50,000, 24,700 Belgians and 25,300 Yugoslavs.
The quota is 8,334 so it elects 2 Belgians and 3 Yugoslavs.
Total result: Belgians 2, Yugoslavs 6.

Jim Riley did it much more elegantly with some ascii maps which I have tuned into pretty colours here. He had 65% Yugoslav and 35% Belgians (to be more exact, 144 blocks of which 94 were Yugoslav and 50 Belgian), electing nine representatives by STV.

diagram Using 3-seaters we can have:
Tymanagh North (65%/35%), Tymanagh South (65%/35%), and Tymanagh Mid (67%/33%) each return 2 Yugoslavs and 1 Belgian.
Totals - Yugoslavs 6, Belgians 3 - the "fair" result.
diagram or:
Tymanagh West (100%/0%) returns 3 Yugoslavs, while Tymanagh North (48%/52%) and Tymanagh South (48%/52%) each return 1 Yugoslav and 2 Belgians.
Totals - Yugoslavs 5, Belgians 4.
diagram or switching to a 4-seater and a 5-seater:
Tymanagh West (81%/19%) returns 4 Yugoslavs, while Tymanagh East (53%/47%) returns 3 Yugoslavs and 2 Belgians.
Totals - Yugoslavs 7, Belgians 2.

All 3 of Jim's plans have an identical number of represented per representative in all districts. The districts are compact with regular boundaries. In fact, plans B and C, which produce the most deviation from the popular vote are more compact than plan A. (Thanks to Jim for letting me put his schemes on this site.)

Of course this is all a bit unrealistic in two respects. First, nowhere in the world insists on such a degree of accuracy in the ratio between representatives and population/electorate (and we'll just ignore that last distinction for now). Populations move all the time in the real world, and tolerances of 5% within a given region are usual; 10% is not unreasonable. The more members per constituency, the easier it is to make a difference. But second, which largely cancels out the first point, with margins as tight as these, the gerrymanders are very vulnerable to differential turnout and even more so to people of one tribe voting for candidates from another, or failing to transfer their votes down the line.

It is intrinsically more difficult to gerrymander proportional systems to get the result you want. Usually all you can hope to affect is where the 'last seat' in each constituency goes. The problem with any single-seat system is that every seat is the last seat in that constituency!

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Nicholas Whyte with input from Jim Riley, 4 September 1999; last modified 3 October 2000

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