100 Essential Things You Didn't Know You Didn't Know (19 page)

59

Getting a Mandate

Democracy used to be a good thing, but now it has gotten into the wrong hands.

Jesse Helms

Politicians have a habit of presuming that they have a much greater mandate than they really do. If you have a roster of policies on offer to the electorate, the fact that you gained the most voters overall does not mean that a majority of voters favour each one of your policies over your opponent’s alternative. And if you win the election by a narrow margin, what sort of mandate do you have?

For simplicity, assume there are just two candidates (or parties) in the election. Suppose the winner gets W votes and the loser gets L votes, so the total number of valid votes cast was W+L. In any number of events of this size, the random statistical ‘error’ that you expect to occur is given by the square root of W+L, so if W+L = 100 there will be a statistical uncertainty of 10 in either direction. In order for the winner of the election to be confident that they haven’t won because of a significant random variation in the whole voting process – counting, casting and sorting votes – we need to have the winning margin greater than the random variation:

W−L > √(W+L)

If 100 votes were cast, then the winning margin needs to exceed 10 votes in order to be convincing. As an example, in the 2000 US Presidential election
^fn1
Bush received 271 electoral college votes and Gore received 266. The difference was just 5, far less than the square root of 271 + 266, which is about 23.

More amusingly, it is told that Enrico Fermi, the great Italian high-energy physicist who was a key player in the creation of the first atomic bomb – and a very competitive tennis player – once responded to being beaten at tennis by 6 games to 4 by remarking that the defeat was not statistically significant because the margin was less than the square root of the number of games they played!

Let’s suppose you have won the election and have a margin of victory large enough to quell concerns about purely random errors being responsible, how large a majority in your favour do you think you need in order to claim that you have a ‘mandate’ from the electorate for your policies? One interesting suggestion is to require that the fraction of all the votes that the winner receives, W/(W+L), exceeds the ratio of the loser’s votes to the winner’s votes, L/W. This ‘golden’ mandate condition is therefore that

W/(W+L) > L/W

This requires that W/L > (1+√5)/2 = 1.61, which is the famous ‘golden ratio’. This means that you would require a fraction W/(W+L) greater than 8/13 or 61.5% of all the votes cast for the two parties. In the last general election in the UK, Labour won 412 seats and the Conservatives 166, so Labour had 71.2% of these
578
seats, enough for a ‘golden’ mandate. By contrast, in the 2004 US Election, Bush won 286 electoral votes, Kerry won 251, and so Bush received only 53.3% of the total, less than required for a ‘golden’ mandate.

^fn1
Some aspects of this election remain deeply suspicious from a purely statistical point of view. In the crucial Florida vote the result of the recount was very mysterious. Just re-examining ballot papers produced a gain of 2,200 votes for Gore and 700 for Bush. Since one would expect there to be an equal chance of an ambiguous ballot being cast for either candidate, this huge asymmetry in the destination of the new ballots accepted in the recount suggests that something else of a non-random nature was going on in either the first count or the recount.

60

The Two-headed League

But many that are first shall be last; and the last shall be first.

Gospel of St. Matthew

In 1981 the Football Association in England made a radical change to the way its leagues operated in an attempt to reward more attacking play. They proposed that 3 points be awarded for a win rather than the 2 points that had traditionally been the victor’s reward. A draw still received just 1 point. Soon other countries followed suit, and this is now the universal system of point scoring in football league competitions all over the world. It is interesting to look at the effect this has had on the degree of success that a dull, non-winning team can have. In the era of 2 points for a win it was easily possible to win the league with 60 points from 42 games, and so a team that gained 42 points from drawing all its games could finish in the top half of the league – indeed, Chelsea won the old First Division Championship in 1955 with the lowest ever points total of 52. Today, with 3 points for a win, the champion side needs over 90 points from its 38 games and an all-drawing side will find its 42 points will put it three or four from the bottom, fighting against relegation.

With these changes in mind, let’s imagine a league where the football authorities decide to change the scoring system just after the final whistles blow on the last day of the season. Throughout the season they have been playing 2 points for a win and 1 point
for
a draw. There are 13 teams in the league and they play each other once, so every team plays 12 games. The All Stars win 5 of their games and lose 7. Remarkably, every other game played in the league is drawn. The All Stars therefore score a total of 10 points. All the other teams score 11 points from their 11 drawn games, and 7 of them score another 2 points when they beat the All Stars, while 5 of them lose to the All Stars and score no more points. So 7 of the other teams end up with 13 points and 5 of them end up with 12 points. All of them have scored more than the All Stars, who therefore find themselves languishing at the bottom of the league table.

Just as the despondent All Stars have got back to the dressing room after their final game and realise they are bottom of the league, facing certain relegation and probable financial ruin, the news filters through that the league authorities have voted to introduce a new points scoring system and apply it retrospectively to all the matches played in the league that season. In order to reward attacking play they will award 3 points for a win and 1 for a draw. The All Stars quickly do a little recalculating. They now get 15 points from their 5 wins. The other teams get 11 points from their 11 drawn games still. But now the 7 that beat the All Stars only get another 3 points each, while the 5 that lost to them get nothing. Either way, all the other teams score only 11 points or 14 points and the All Stars are now the champions!

61

Creating Something out of Nothing

Mistakes are good. The more mistakes, the better. People who make mistakes get promoted. They can be trusted. Why? They’re not dangerous. They can’t be too serious. People who don’t make mistakes eventually fall off cliffs, a bad thing because everyone in free fall is considered a liability. They might land on you.

James Church,
A Corpse in the Koryo

If you are one of those people who have to give lectures or ‘presentations’ using a computer package like PowerPoint, then you have probably also discovered one of its weaknesses – especially if you are an educator. When your presentation ends it is typically time for questions from the audience about what you have said. One of the facts of life about such questions is that they are very often most easily and effectively answered by drawing or writing something. If you are at a blackboard or have an overhead projector with acetate sheets and a pen in front of you, then any picture you need is easily drawn. But armed just with your standard laptop computer you are a bit stuck. You can’t easily ‘draw’ anything over your presentation unless you have a ‘tablet PC’. Which all goes to show how much we rely on pictures to explain what we mean. They are more direct than words. They are analog, not digital.

Some mathematicians are suspicious of drawings. They like proofs that
make
no reference to a picture you might have drawn in a way that biases what you think might be true. But most mathematicians are quite the opposite. They like pictures and see them as a vital guide to seeing what might be true and how to go about showing it. Since that opinion is in the majority, let’s show something that will make the minority happy. Suppose you have 8 × 8 square metres of expensive flooring that is made up of four pieces – two triangles and two quadrilaterals – as shown in the floor plan here.

It is easy to see that the total area of the square is 8 × 8 = 64 square metres. Now let’s take our four pieces of flooring with the given dimensions and lay them down in a different way. This time to create a rectangle, like this:

Something strange has happened though. What is the area of the new rectangular carpet? It is 13 × 5 = 65 square metres.
¹⁸
We have created one square metre of carpet out of nothing! What has happened? Cut out the pieces and try it for yourself.

62

How to Rig An Election

I will serve as a consultant for your group for your next election. Tell me who you want to win. After talking to the members I will design a ‘democratic procedure’ which ensures the election of your candidate.

Donald Saari
¹⁹

As we have already seen in Chapter 14, elections can be tricky things. There are many ways to count votes, and if you do it unwisely you can find that candidate A beats B who beats C who loses to A! This is undesirable. Sometimes we find ourselves voting several times on a collection of candidates with the weakest being dropped at each stage, so that the votes for that candidate can be transferred to other candidates in the next round of voting.

Even if you are not spending your days casting formal votes for candidates, you will be surprised how often you are engaged in voting. What film shall we go to see? What TV channel shall we watch? Where shall we go on holiday? What’s the best make of fridge to buy? If you enter into discussion with others about questions that different possible answers then you are really engaged in casting a ‘vote’ – your preference – and the successful ‘candidate’ is the choice that is finally made. These decisions are not reached by all parties casting a vote. The process is usually much more haphazard. Someone suggests one film. Then someone suggests another because it is newer. Then someone says the newer one is too violent and we should pick
a
third one. Someone turns out to have seen that one already so it’s back to the first choice. Someone realises it’s no good for children so they suggest another. People are wearying by now and agree on that proposal. What is happening here is interesting. One possibility at a time is being considered against another one, and this process is repeated like rounds in a tournament. You never consider all the attributes of all the possible films together and vote. The outcome of the deliberations therefore depend very strongly on the order in which you consider one film versus another. Change the order in which you consider the films and the attributes you use for comparing them and you can end up with a very different winner.

It’s just the same with elections. Suppose you have 24 people who have to choose a leader from 8 possible candidates (A, B, C, D, E, F, G and H). The ‘voters’ divide up into three groups who decide on the following orderings of their preferences among the candidates:

1st group: A B C D E F G H

2nd group: B C D E F G H A

3rd group: C D E F G H A B

At first glance it looks as if C is the preferred candidate overall, taking 1st, 2nd and 3rd places in the three ranking lists. But H’s mother is very keen for H to get the leader’s job and comes to ask if we can make sure that H wins the vote. It looks hopeless as H is last, second last and third last on the preference lists. There must be no chance of H becoming leader. We make it clear to H’s mother that everything must follow the rules and no dishonesty is allowed. So, the challenge is to find a voting system that makes H the winner.

All we need to do in order to oblige is set up a tournament-style election and pick the winner of each 2-person contest using the preferences of the three groups listed above. First, pit G against
F
, and we see F wins 3–0. F then plays E, and loses 3–0. E then plays D and loses 3–0. D then plays C and loses 3–0. C then plays B and loses 2–1. B then plays A and loses 2–1. That leaves A to play H in the final match up. H beats A 2–1. So H is the winning choice in this ‘tournament’ to decide on the new leader.