[EM] Non-proportional representation

[Colin Champion]

In the British electoral system MPs are chosen by local single-winner 
elections and a government is constructed from a plurality of MPs, with 
an opposition being made up of whoever's left over. A single party 
usually ends up with a majority of MPs. The standard alternative voting 
systems sacrifice the likelihood of getting a majority, and their 
proponents tend to argue that the shift to coalition government would be 
a benefit. For my part I find it hard to agree, and suspect that people 
who think this way assume that any majority government will be 
unrepresentative and therefore needs to have its wings clipped, or may 
just be pretending that a bug is a feature.

The AV+ system was intended to provide a judicious compromise between 
fairness and decisiveness but it was a dog's breakfast. For this reason, 
if I was asked which voting method I favoured, I'd be embarrassed to 
answer. What follows is a suggestion of my own, stemming from the fact 
that I haven't seen any others which try seriously to solve in a fairer 
way the problems addressed by the British electoral system.

The electorate is divided into constituencies and the ballot papers are 
much like now. Each ballot paper contains a list of candidates and their 
party affiliations (at most one candidate per party). The difference is 
that voters are asked to assign an order of preference to the 
candidates, and that it is the party rather than the individual who 
plays the main role in the system.

The first step is to decide which party forms the government. This is 
done by applying a simple Condorcet method (such as Minimax) to the 
ranked party preferences, lumping together the ballots from all 
constituencies. One party is identified as the winner.

The remainder of the electoral process is determined solely by voters' 
first preferences. The second step is to decide how many seats in 
parliament are assigned to each party. The winning party, which will 
form the government, is treated separately. If its proportion of (first 
preference) votes is p, then its proportion of seats will be (1+p*p)/2. 
The remaining seats will be distributed amongst the remaining parties in 
proportion to their first preference votes.

The final step is to decide which candidates will be included in the 
party quotas thus determined. This is done to maximise the number of 
voters represented by their first-choice candidates, which is a linear 
programming problem solvable by the simplex algorithm (which is 
guaranteed not to return fractional numbers of candidates per constituency).

CJC