chalkdust

Plurality (AKA ﬁrst-past-the-post) Plurality is a very common voting system. Each voter chooses their favourite candidate, and whichever candidate receives the most votes is declared the winner. Problems with this system are well known, and I will give an example of how it violates independence of irrelevant alternatives. Consider an election with three candidates x, y, and z where the voters can be split into three groups, and their preferences are shown in Table 1. Under plurality, candidate x would win. But if nine or more voters change their preference from x ≻ z ≻ y to z ≻ x ≻ y, then the result is a victory for y. This is in clear violation of independence of irrelevant alternatives since these voters kept their preference x ≻ y, with the situation being exactly as that described in the definition.

Anthony Burgess

A British ballot paper, using the plurality, or first-past-the-post, voting system.

This is also a violation of the majority loser criterion, which is as follows: if a majority of voters prefers every other candidate over a given candidate, then the given candidate must not win. Once the nine voters switch their support, there is still a majority that prefers both x to y and z to y. In Table 1, candidates x and z are clones, meaning that no Number of voters Preference voter ranks any other candidate to be between (or equal 110 x ≻z ≻y to) x and z. This definition expands to a multi-member 102 y ≻x ≻z clone set, a strict subset of the candidates with the prop8 z ≻x ≻y erty that no voter ranks any candidate outside the clone set to be between (or equal to) any members of the clone Table 1: Voters’ preferences. set. The independence of clones condition requires that removing a clone from a clone set must not improve or harm the ranking of any candidate outside the clone set. Plurality violates this condition because z’s removal would hinder candidate y. There is, therefore, a simple way to aﬀect the outcome of a plurality election in your favour without having to convince anyone else to support you. If you introduce a clone of an opponent then the vote for your opponent may split between your opponent and their clone, meaning that you require fewer votes to win. In practice, this fact is well known and some people in British elections do not vote for their preferred candidate because they do not want to split the vote against the party they dislike.

Alternative vote On 5 May 2011, the UK voted to keep the plurality system for parliamentary elections, rejecting the alternative vote. The alternative vote is defined as follows: voters rank the candidates, and the initial results are based on each voter’s first-preference votes. If a candidate receives more than half of first-preference votes then that candidate wins. Otherwise, the candidate with the fewest 13

spring 2016

Chalkdust, Issue 03

Popular mathematics magazine from UCL

Chalkdust, Issue 03

Popular mathematics magazine from UCL