Mathematics, democracy, paradox and pitfall

The New Scientist Editorial refers to an article in the magazine, and says that “The mathematics of democracy turns out to be so fraught with pitfalls and paradoxes that complete fairness is probably unattainable.”

The principle of voting in the UK is currently simple. Each of our electoral divisions/constituencies elects just one representative. The candidate who gained the most votes in the election is the winner. In this system votes for anyone other than the winning candidate are disregarded. In Bracknell the winner was Dr Phillip Lee who gained over 50% of the vote.

In the NS article it points out that if more than two parties with substantial support contest a constituency, a candidate does not have to get anything like 50% of the votes to win. In this case a majority of votes may be considered to be "lost".  Take somewhere like Hampstead and Kilburn where Glenda Jackson got just under 33% of the vote. It could be that 67% of the people there are now quite dissatisfied.

From the NS Article an example by mathematician Donald Saari at the University of California:

Suppose 15 people are asked to rank their liking for milk (M), beer (B), or wine (W).

6 rank them M-W-B,
5 rank them B-W-M,
4 rank them W-B-M.

In a plurality system where only first preferences count, the outcome is simple: milk wins with 40%, followed by beer, with wine trailing in last.

Do voters actually prefer milk?

9 voters prefer beer to milk, and
9 prefer wine to milk
Clear majorities in both cases. Meanwhile,
10 people prefer wine to beer.

By pairing off all these preferences, we see the truly preferred order to be W-B-M - the exact reverse of what the voting system produced.

Saari showed that given a set of voter preferences you can design a system that produces any result you desire. You can read more about systems/anomalies in the article.

Preferential voting comes closer to being fair than plurality voting, but it does not eliminate ordering paradoxes.
Given three candidates, A, B and C, and three voters who rank them
A-B-C,
B-C-A and
C-A-B.
Voters prefer A to B by 2 to 1. But B is preferred to C and C preferred to A by the same margin of 2 to 1.
Every one a winner!

In a proportional representation system each party is awarded a number of seats in proportion to the number of people who voted for them. This may be fairer in a mathematical sense than plurality or preferential voting, but implies large, multi-representative constituencies and central lists of people that may be quite remote from the voter. This system also carries with it the possibly of paradoxes occurring.

The NS article points out that there will always be the possibility that one voter, simply by changing their vote, can change the overall preference of the whole electorate.

It seems that in any system we could end up with a hung result. One way to quantify this is the Banzhaf power index. First, list all combinations of parties that could form a majority coalition, and in all of those coalitions count how many times a party is a "swing" partner that could destroy the majority if it dropped out. Dividing this number by the total number of swing partners in all possible majority coalitions gives a party's power index.There are a number of calculators available that one can Google for.

There is a bit in the article about the word gerrymander, which  was coined by a newspaper editor in reaction to a redrawing of boundaries.  Governor Elbridge Gerry. It included one sprawling supposedly salamander-shaped constituency. This leads me on to the Conservative policy to reduce the number of MPs by 10%, and ensure each vote has equal value by reducing the wide discrepancies between constituency electorate sizes. 

The Guardian says that the adoption of the "alternative vote" electoral system would have had only a minimal impact on the outcome of last Thursday's general election with the Liberal Democrats gaining only an additional 22 seats, according to analysis by the Electoral Reform Society. Neither Labour nor the Conservatives would have benefited significantly from transfers based on last Thursday's vote. Significant regional imbalances would remain between the parties.

The newspaper also has a useful bit on the AV and Plus systems. AV Plus was recommend by the Jenkins Commision. PDF of report HERE. So much for the promises made by Labour since 1997?