Mathematics of Voting: 1. Introduction


This is intended to be the first of a sequence of blog posts about voting; a timely issue given the imminent Australian Federal Election. Although most of these posts will be based on the Australian system, the mathematics involved is of course applicable everywhere.

Voting is something we all do at some time in our lives. Be it a simple raising of hands at a club or committee meeting, or the highly organized and formal secret ballot for electing politicians to office, group decision making affects all of us.

I intend to look at some of the mathematical aspects of voting – how votes are counted, how preferences are distributed, and methods of quantifying the power of a voting body.

Later I will discuss a number of different methods of voting, and different methods of finding the winning candidate in a given voting situation. I will also discuss various methods of electing more than one candidate at a time.

I will also devote a few posts to the topic of “voting power”; to investigate what power is in a voting situation, and the various means of assigning meaningful numerical values to a voting body or political party in such a way as to reflect their power. I’ll also discuss ways in which knowledge of the power of a party can be a very useful tool.

The Australian Parliament

Many of the examples will be drawn from the Australian parliamentary system, so here is a brief introduction to its workings.

The Australian Federal parliament contains two main bodies: the House of Representatives, or “lower house” and the Senate, or “upper house”. (Bill Bryson, in his book on Australia, claims that it’s “interesting, in a very low-grade sort of way, that they use the British term for the institution, and the American terms for the chambers.”) The House of Representatives is designed to represent the Australian people, and the Senate to represent the states, and to act as a sort of watch-dog over the House of Representatives.

Australia is divided (as of 2010) into 150 electoral divisions, or “electorates”. The House of Representatives contains one person from each of these electorates, and that person is supposed to represent his or her electorate.

The main purpose of the House of Representatives is to initiate, amend or reject legislation; that is, possible laws governing the country and in particular appropriating money for various uses (defence, pensions, roads, health, education and so on). Until a law has been passed – that is, agreed on by both houses – it is called a “bill”. Even if a bill is passed by the House of Representatives, it cannot be made law until also passed by the Senate. Other proposals on which yes/no votes are to be made are called “motions”.

Most of the members of the House of Representatives and the Senate are members of one or other of Australia’s major political parties: the Australian Labor Party (and yes, even though Australian English mostly follows British spelling conventions, the Australian Labor Party is so spelled), the Liberal Party, the Greens, the National Party – there are many others. Generally all members of a particular party will vote together on a given issue, and it is sometimes more convenient to consider a vote within either house not as a vote of individual persons, but as vote of several parties, each with a fixed number of votes.

To be in power; to establish a government, a party must have an absolute majority in the House of Representatives. At an election, each of the major parties will field a candidate in each of the electorates. The people in each electorate will vote as to which party they prefer, and by methods to be discussed, the winning candidate is chosen to be the one who has amassed an absolute majority of votes.

An “absolute majority” is 50%+1, so in an electorate of 60,000 voters, an absolute majority is 30,001.

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