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?

How maths lessons have changed over the years

I found this on a Proboards forum

I just had to record it.

TEACHING MATHS IN 1970
A logger sells a truckload of timber for £100. His cost of production is 4/5 of the price. What is his profit?

TEACHING MATHS IN 1980
A logger sells a truckload of timber for £100. His cost of production is 80% of the price. What is his profit?

TEACHING MATHS IN 1990
A logger sells a truckload of timber for £100. His cost of production is £80. How much was his profit?

TEACHING MATHS IN 2000
A logger sells a truckload of timber for £100. His cost of production is £80 and his profit is £20. Your assignment: Underline the number 20.

TEACHING MATHS IN 2005
A logger cuts down a beautiful forest because he is selfish and inconsiderate and cares nothing for the habitat of animals or the preservation of our woodlands. Your assignment: Discuss how the birds and squirrels might feel as the logger cut down their homes just for a measly profit of £20.

TEACHING MATHS IN 2009
A logger is arrested for trying to cut down a tree in case it may be offensive to Muslims or other religious groups not consulted in the felling license. He is also fined a £100 as his chainsaw is in breach of Health and Safety legislation as it deemed too dangerous and could cut something. He has used the chainsaw for over 20 years without incident however he does not have the correct certificate of competence and is therefore considered to be a recidivist and habitual criminal. His DNA is sampled and his details circulated throughout all government agencies. He protests and is taken to court and fined another £100 because he is such an easy target. When he is released he returns to find Gypsies have cut down half his wood to build a camp on his land. He tries to throw them off but is arrested, prosecuted for harassing an ethnic minority, imprisoned and fined a further £100. While he is in jail the Gypsies cut down the rest of his wood and sell it on the black market for £100 cash. They also have a leaving barbeque of squirrel and pheasant and depart leaving behind several tonnes of rubbish and asbestos sheeting. The forester on release is warned that failure to clear the fly tipped rubbish immediately at his own cost is an offence. He complains and is arrested for environmental pollution, breach of the peace and invoiced £12,000 plus VAT for safe disposal costs by a regulated government contractor.
Your assignment: How many times is the logger going to have to be arrested and fined before he realises that he is never going to make £20 profit by hard work, give up, sign onto the dole and live off the state for the rest of his life?

TEACHING MATHS IN 2010
A logger doesn't sell a lorry load of timber because he can't get a loan to buy a new lorry because his bank has spent all his and their money on a derivative of securitised debt related to sub-prime mortgages in Iceland and lost the lot with only some government money left to pay a few million pound bonuses to their senior directors and the traders who made the biggest losses. The logger struggles to pay the £1,200 road tax on his old lorry however, as it was built in the 1970s it no longer meets the emissions regulations and he is forced to scrap it. Some Bulgarian loggers buy the lorry from the scrap merchant and put it back on the road. They undercut everyone on price for haulage and send their cash back home, while claiming unemployment for themselves and their relatives. If questioned they speak no English and it is easier to deport them at the governments expense. Following their holiday back home they return to the UK with different names and fresh girls and start again. The logger protests, is accused of being a bigoted racist and as his name is on the side of his old lorry he is forced to pay £1,500 registration fees as a gang master. The Government borrows more money to pay more to the bankers as bonuses are not cheap. The parliamentarians feel they are missing out and claim the difference on expenses and allowances.
Your assignment: You do the maths.