The ways we vote

The recent elections in Britain led to the usual cries about the ways in which the first-past-the-post system that Britain uses makes it almost impossible for a new, or small, political group to gain any representation.

I last thought about this when Finland had its local elections last month. At that point I looked up the system that Finland uses: the D’Hondt method, “also called the Jefferson method or the greatest divisors method”. According to Wikipedia, “the method was first described in 1792 by American Secretary of State and later President of the United States Thomas Jefferson. It was re-invented independently in 1878 by Belgian mathematician Victor D’Hondt, which is the reason for its two different names”.

Wikipedia explains the motivation for the D’Hondt method as follows:

Proportional representation systems aim to allocate seats to parties approximately in proportion to the number of votes received. For example, if a party wins one-third of the votes then it should gain about one-third of the seats. In general, exact proportionality is not possible because these divisions produce fractional numbers of seats. As a result, several methods, of which the D’Hondt method is one, have been devised which ensure that the parties’ seat allocations, which are of whole numbers, are as proportional as possible. Although all of these methods approximate proportionality, they do so by minimizing different kinds of disproportionality. The D’Hondt method minimizes the largest seats-to-votes ratio. Empirical studies based on other, more popular concepts of disproportionality show that the D’Hondt method is one of the least proportional among the proportional representation methods. The D’Hondt favours large parties and coalitions over small parties due to strategic voting. In comparison, the Sainte-Laguë method reduces the disproportional bias towards large parties and it generally has a more equal seats-to-votes ratio for different sized parties.

The axiomatic properties of the D’Hondt method were studied and they proved that the D’Hondt method is a consistent and monotone method that reduces political fragmentation by encouraging coalitions. A method is consistent if it treats parties that received tied votes equally. Monotonicity means that the number of seats provided to any state or party will not decrease if the house size increases.

The article then goes on to explain the mathematics: how the method actually works. You might well want to read it.

Finally I will just note that

The D’Hondt method is used to elect the legislatures in Åland, Albania, Angola, Argentina, Armenia, Aruba, Austria, Belgium, Bolivia, Brazil, Burundi, Cambodia, Cape Verde, Chile, Colombia, Croatia, the Dominican Republic, East Timor, Estonia, Fiji, Finland, Greenland, Guatemala, Hungary (in a mixed system), Iceland, Israel, Italy (in a mixed system), Japan, Luxembourg, Moldova, Monaco, Montenegro, Mozambique, Netherlands, Nicaragua, North Macedonia, Paraguay, Peru, Poland, Portugal, Romania, San Marino, Serbia, Slovenia, Spain, Switzerland, Turkey, Uruguay and Venezuela”.

Denmark, the Scottish Assembly, the Welsh Parliament, and the London Assembly also make use of it for aspects of their elections.