## Tuesday, 29 March 2011

### To AV or not to AV

Never having voted for anyone who then got elected, I have always been attracted to the alternative vote (AV) system. At least with AV, people like me would have the chance to vote for our second choice, so there is a greater possibility of us actually having a vote that counts.

So, surely more people would feel happier with the result than in the first-past-the-post (FPP) system we have at present.

Well, very soon we shall have the chance to vote on it, in a referendum. So, I’ve been doing some investigations into the question: would AV increase the amount of happiness with the result? I’ve been surprised at what has emerged from my mathematical modelling.

My model attempts to measure how much happiness there is amongst the voting electorate with the result of an election. Keeping it simple, we score 1 for happiness if our first choice candidate is elected and 0.5 if our second choice is elected. So, 1 = ‘happy’, and 0.5 = ‘quite happy’.

I apply this model to a straightforward situation, in which there are only three candidates (A, B and C) and everyone indicates their second choice candidate under the AV system.

Now, clearly, in the majority of cases where AV produces the same result as FPP, the degree of happiness with the result is not going to be any different.

So, we need to look at examples where AV produces a different result. In the examples that follow, I assume in each case that A would win under FPP. But B wins under AV. Numbers are percentages of the number of votes cast.

EXAMPLE 1

 Candidate Total 1st choice votes 2nd choice A 2nd choice B 2nd choice C A 40 n/a 19 21 B 38 19 n/a 19 C 22 9 13 n/a

Note that under AV, C gets eliminated and votes for C get transferred to the second choice.

A then has a total of 49%, and B 51%. So, B gets elected.

But is there greater happiness with the result?

If A is elected (under FPP) the total happiness score is 40 + (19+ 9) × 0.5 = 54.

If B is elected (under AV) the total happiness score is 38 + (19 + 32) × 0.5 = 54.

So the two systems produce different results, but the same amount of happiness. All that we achieve by going for AV is that different people are happy!

EXAMPLE 2

Here’s an example where all three candidates do well for first choice votes, but the majority of the votes for C when eliminated go to B.

 Candidate Total 1st choice votes 2nd choice A 2nd choice B 2nd choice C A 37 n/a 17 20 B 32 14 n/a 18 C 31 11 20 n/a

Under AV, C gets eliminated and votes for C get transferred to the second choice. A then has a total of 48%, and B 52%. So, B gets elected.

Is there greater happiness with the result?

If A is elected (under FPP) the total happiness score is 37 + (14+ 11) × 0.5 = 49.5.

If B is elected (under AV) the total happiness score is 32 + (17 + 20) × 0.5 = 50.5.

So in this case AV does produce a little more happiness with the result.

But now look at example 3.

EXAMPLE 3

In this example, supporters of A generally do not want B elected, and vice versa. This is therefore a very realistic possibility.

 Candidate Total 1st choice votes 2nd choice A 2nd choice B 2nd choice C A 40 n/a 3 37 B 38 5 n/a 33 C 22 9 13 n/a

Under AV, C gets eliminated and votes for C get transferred to the second choice. A again has a total of 49%, and B 51%. So, B gets elected.

But is there greater happiness with the result?

If A is elected (under FPP) the total happiness score is 40 + (5+ 9) × 0.5 = 47.

If B is elected (under AV) the total happiness score is 38 + (3+ 13) × 0.5 = 44.

So, this time AV actually decreases the amount of happiness with the result!

My assumption was that AV is bound to produce greater happiness with the result of an election. This is not supported by the results of my mathematical model. What we forget is that under FPP there are still a lot of people who are ‘quite happy’ with the result, because their second choice candidate (if they had a second choice) is actually the one who gets elected.

Often all that AV achieves is to make different people happy or quite happy with the result.

The maths has convinced me. To my surprise, I shall vote ‘no to AV’ in the referendum. It's not worth the hassle of changing the system.

#### 1 comment:

1. Good analysis - but I think there are 2 good reasons to vote for AV:
- Many constituencies are "safe" for one party, as no other party individually can get enough votes. So the other voters always feel they have no say. With AV, there is an opportunity for the others to work together to unseat an incumbent. I think that this will mean that voters in these seats feel their votes are actually worth something at last.
- Your analysis says that the total sum of happiness stays about the same in both systems. But with FPP more voters have full happiness, and fewer score half. With AV, more have half, and fewer have full. I think that we should give a higher score to a result with more happy people, even if they did not get their first preference.
I think that the change is worth the hassle.