Tuesday, 19 April 2011

The Alternative Vote #5: Do the math

There are certain properties that you might expect a voting system to have. For example, you might think that a good voting system should choose the candidate with the majority of votes, or make it easier for candidates with more votes to win. These ideas have been formalised using mathematics in voting theory and social choice theory, so that comparisons can be made between different voting systems. In this post, Alisdair Cairns explores how AV performs as a system for collecting votes in comparison to First-past-the-post. 

Voting Theory is, predictably, the study of voting systems. One specific endeavour of voting theorists is to come up with rules that 'good' systems should adhere to. There are a multitude of potential criteria that have been suggested, and all can be seen as important depending on what you want from your voting system. I will attempt to explain a few of them and show examples of when FPTP and AV don't conform. It is however sometimes quite complicated to prove when they DO follow these rules, so you'll just have to take my word in those places!

1) Majority - "If more than 50 percent of voters put x as their most preferred candidate, then x should be elected."

This is maybe the most basic property and it is probably obvious that both FPTP and AV satisfy it. Not all voting systems do though. In fact it is not always obvious that the majority winner is the most appropriate. Consider this example:

Let candidates a, b and c be candidates in a vote and let xy represent a ballot where x is first preference and y is second preference.

ballot round 1
ab 9
bc 7
cb 1

a wins with a clear majority but b is at least the second choice for every single voter in the electorate. It might be that b and c voters REALLY hate candidate a, in which case it might be better overall if b were to win. Points-based voting systems, like Borda Count, cope better with this sort of situation.

2) Monotonicity - "A winning candidate should not be harmed if they are placed higher on some ballots." (Monotonicity is actually a whole system of inter-related statements but this formulation is simple and gives a good idea of what it's all about.)

On first glance it might be a confusingly obvious rule that a fair voting system should follow. But in fact AV actually fails this condition, as shown in the following election: 

ballot round 1 round 2
ab 8 14
bc 7 7
ca 6

Here c is eliminated first and its votes pass to a who claims a majority. However if two b voters had been convinced to switch to a then the situation would have looked like this:

ab 10 10
bc 5
ca 6 11

a has gained two extra votes but does worse because of it. Indeed it is because of this problem that tactical voting could still be prevalent under AV. The two voters who switched actually would have done better by voting for a candidate they didn't want!

FPTP has no such problems with monotonicity. If you get more votes, you stand a better chance of winning.

3) Participation - "The inclusion of additional ballots that have x above y should not change the winner from x to y."

This is closely related to monotonicity and seems equally obvious, but again AV fails miserably. Here's an example:

ab 7 7
ba 6 11
cb        5

Here c is eliminated first and its votes go to b who wins in the second round of voting. However if a few more voters had turned out in favour of c (but preferring b to a) then the situation might have looked like this:

ab 8 14
ba        6
cb 7 7

Those extra voters would have been better off not voting at all! This could feasibly be used as an argument that AV actually encourages voter apathy. 

Again, FPTP passes participation. Here a would win both elections.

4) Condorcet Criteria - "If there is a condorcet winner, they should win overall. If there is a condorcet loser, they should not win."

A condorcet winner, if one exists, is a candidate who beats every other candidate in pairwise comparisons. That is to say in an election between a,b,c…etc. if a and x were the only candidates, a would beat x for all other candidates x. Equally a condorcet loser is a candidate who would lose to every other candidate in pairwise comparisons. 

Instinctively these seem like pretty reasonable criteria. If a is preferable to every other candidate, on an individual basis, then surely they are the best candidate? In fact FPTP and AV both fail this criterion.

ac 5 7
bc 4 4
ca 2

In this election under AV, c is eliminated in the first round and a wins in the second. Clearly a would also win under FPTP. But in fact c is the condorcet winner! If either a or b were not running, then their votes would have been cast for c and so c would have won 7 to 4 or 6 to 5 respectively. 

Of course one could easily be suspicious of a situation where a candidate that is under 20 percent of the electorate's first choice could be the winner. Indeed some have suggested that voting systems should NOT satisfy the condorcet winner criterion!

Condorcet loser on the other hand is slightly weaker and possibly more appealing. Indeed AV passes where FPTP still fails:

ab 4
bc 3
cb 3

Here a is the condorcet loser as it would lose by 6 to 4 in pairwise comparisons to both b and c

5) Convexity - "If the electorate is split into two distinct, non-overlapping subsets and the winner in both is the same, then that winner should be the overall winner." 

Also known as consistency. Again this seems logical. Lets say we are placed in a constituency where the population is half urban and half rural. If a candidate wins among both the city and country voters then they should win the whole constituency, right? Not under AV:

Set 1 (urban)
ab 6 6
ba 4 7
cb 3

Set 2 (rural)
ab 3
ba 4 7
cb 6 6

ab 9 17
ba 8
cb 9 9

Clearly b wins in both subsets but when the electorate is considered as a whole she is eliminated in the first round. In fact in single-winner, preferential (candidates are ranked in order of preference) voting systems it has been proved that convexity and majority are mutually exclusive. If we want our voting system to satisfy both then we have to go with a system where only one choice is placed on the ballot, FPTP being the prime example.

Of course the votes in constituencies are never separated in this way, so it would be difficult to tell how often a problem occurs de to failure of convexity, but it is troubling to know that it could.

6) Clone Independence - "The addition of a candidate identical to one already present in an election will not cause the winner of the election to change."

Under FPTP, two very similar candidates suffer because they split the vote for that political stance, leaving the other parties to profit. Because of this, voting systems that don't satisfy clone independence encourage tactical voting. Voters you feel want to vote for a' may be encouraged to vote for a just so that b doesn't win. Here's an example where x' represents a candidate similar to x:

aa' 4 7
a'a 3
b 6 6

Under FPTP b wins very easily however under AV a wins. One could argue that if a and a' are so similar, it wouldn't matter if one were to simply stand down. But subtle differences can be important and a position where voters feel like their vote could be 'wasted' should be avoided at all costs. 

There are cases to be made for placing more or less importance on each of these rules (and the many, many more I haven't gone into) so I leave you to draw your own conclusions. Hopefully I have at least shown that there is very little in it from a purely mathematical point of view. It is perhaps obvious why Nick Clegg once described AV as a "miserable little compromise". 

1 comment:

  1. Thanks for that post. It was very interesting and informative. It is indeed a question of which rule we consider more important, and pragmatically one of which system is more likely to lead to the people we support being elected, or at least to our voice being heard by politicians.