Political scientists are used to the idea that the number of legislators elected in a district shapes voting behaviour.
If there are only \(m\) seats available, then voters have reason to switch towards the top \(m\) + 1 parties rather than vote for a party which is unlikely to win.
This is a good first order heuristic, but it’s not a complete account of voter responses to institutional features. A district with a low magnitude works very differently in a system with many much larger districts compared to a system with other similarly small districts.
The best example I’ve seen of this comes from the Department of Madre de Dios in Peru. Madre de Dios is the smallest department in Peru by population, but one of the largest by area. If you follow the river after which the department is named, you eventually run into the Amazon and from there out into the Atlantic after a short 6,000 kilometres.
Because Madre de Dios is so small, it only elects one member of the Congress. That might lead us to expect that only a small number of parties would be competitive in Madre de Dios. That expectation turns out to be flat wrong. In 2006, fourteen parties ran for the single seat in Madre de Dios, and the largest party (Restauracion Nacional) won just over 20% of the vote. The effective number of vote-winning parties is 6.86, only slightly smaller than the effective number for Peru as a whole in that year (7.2).
I was so shocked by this figure that I first assumed that there was an error in the data source I was using, and so I confirmed it with official results. The result must have been shocking to some in Madre de Dios, because the effective number of parties fell substantially in the following (2011) election, to just 3.44.
This is an extreme example, but it shows that we can’t work out what happens at district level just with reference to local features. As they don’t say: “all politics is national”.