## Is the British voting system fair?

At the last general election, the percentages of votes and numbers of seats in parliament for the three main parties in Britain were as follows: Labour, 35.3% of votes, 356 seats; Conservatives, 32.3% of votes, 198 seats; Liberal Democrats, 22.1% of votes, 62 seats. In the election coming up on the 6th May, there is a distinct possibility of some quite bizarre outcomes. For example, if some recent polls give a true picture of how people will vote (which is of course far from certain), then there is a good chance that the Liberal Democrats will get more votes than Labour, but well under half the number of seats. It is also a commonplace that the Conservatives will need a higher percentage of votes than Labour to become the party with the largest number of seats. In the past there have been occasions where the party with the largest number of votes has lost the election. (Much of what I am saying applies equally to the system for electing a US president, but I shall stick with the British system in this post.)

Supporters of the first-past-the-post system argue, correctly, that it makes it much more likely that one party will have an absolute majority. They also argue, much more controversially, that this is a good idea. However, regardless of outcome of that argument, there can be no doubt that it has the potential to lead to anomalous results, and this potential has been thrown into sharp focus in the last week or two because it has a good chance of being realized. Here I would like to discuss whether it is correct to describe these anomalies as unfair.

In case I have just given the impression that I am about to dazzle you by arguing convincingly that they are not in fact unfair, let me come straight out and say that there is no such surprise in store: our voting system is as unfair as it looks. My purpose is to discuss in more detail what “fair” means in this context. I should also add that I do not plan to discuss a whole array of different voting systems, give an account of Arrow’s paradox, etc. etc. That has been done to death in my opinion. (The rest of what I shall say is hardly new either, but I care about it enough to want to say it anyway.)

Why bother to vote, even if you care about the result?

A question that many people have asked themselves is this: what is the point of voting? After all, my constituency is huge, so it’s inconceivable that the result here will be decided by just one vote. Therefore, my vote will make no difference.

An obvious answer would be, “If lots of people did that, then those people between them would make a difference.” But that argument won’t do. The fact is that you know that lots of people are going to vote, and given that information it really is true that your vote is almost certain to make no difference.

Note, however, that I said “almost certain” rather than “certain”. It is certainly possible that by changing your vote you could change the outcome in your constituency. It’s even possible that changing that outcome would tip the balance of power and thus have a significant impact on future policy decisions.

The justification usually given for voting is that your expected reward for doing so is just about right: there is an incredibly small chance that your vote will have a very big influence, so your expected influence is small but non-zero, just as it should be given that you are just one voter amongst millions. Let us examine this argument in more detail.

What is the expected impact of my vote?

First let us consider the probability that your vote will change the result in your constituency. For simplicity, let us suppose that there are only two candidates who have a realistic chance of being elected, and that if there is a tie then the outcome will be decided on the toss of a coin. Let us call these two candidates A and B, and let us suppose that you vote for candidate A. Then a necessary condition for your vote to make a difference to the outcome is that, when all the other votes are counted, either A has one less vote than B (note for pedants — my opinion is that because “vote” is singular there, “less” is the right word rather than “fewer”, but not everyone agrees with this) or they have the same number of votes. And in both those two cases the chance that your vote actually does make a difference is 50%, because in the first case you need A to win the coin toss and in the second case your vote will not have helped A if A would have won the coin toss.

How likely is it that A and B will get almost the same number of votes? The obvious way of modelling the situation is to assume that there are $N$ voters who vote for either A or B and that each such voter has a probability $p$ of voting for A and $q$ of voting for B, where $p+q=1.$ Thus, the expected number of votes for A is $pN$. For A and B to have almost exactly the same number of votes, A needs to have almost exactly $N/2$ votes. Therefore, if $p\ne 1/2$, the total number of A’s votes must differ from its expectation by approximately $|p-1/2|N$.

Now a commonplace in probabilistic combinatorics is that the binomial distribution is highly concentrated about its mean. In particular, the probability that it differs from its mean by $\alpha N$ is at most $e^{-c\alpha^2 N}$, where $c$ is some constant. (I think one can take $c=1/2$.) In qualitative terms, unless $\alpha$ is very small (and it has to be smaller and smaller the larger $N$ is), the probability that it differs from its mean by $\alpha N$ is vanishingly small.

Let me put that another way. The binomial distribution approximates a normal distribution with standard deviation $\sqrt{Npq}$. Now the probability that a normally distributed random variable differs from its mean by more than a few standard deviations is fabulously small. How does $|p-1/2|N$ compare with $\sqrt{Npq}$? Let’s suppose $p=0.45$, $q=0.55$ and $N=10,000$. (In reality, the typical size of a British constituency is four or five times bigger than this.) Then $|p-1/2|N=500$, and $\sqrt{Npq}$ is roughly $50$. So A requires the result to be 10 standard deviations away from the mean to win. The probability of this is around $e^{-50}$, which is absurdly small. It follows that the probability that your vote will make a difference is also absurdly small.

Incidentally, this model shows up a rather bizarre feature of the first-past-the-post system. Suppose that people’s voting intentions did not correlate with where they lived. Then if the most popular party had a lead of two or three percent over the next most popular party, then with high probability they would win every single seat! The only reason we do not see this phenomenon is that votes do correlate significantly with where people live. In the UK, for example, Labour tends to do better in the north and in inner cities and the Conservatives do better in the south and in the countryside. The Liberal Democrats suffer from a vote that is more evenly spread around the country and so they have much more difficulty winning seats. (So in fact we do see the phenomenon — but not quite in such an extreme form.) Indeed, the only reason the Liberal Democrats have the seats they do have is that there are pockets of the country, such as the extreme south west, the Scottish islands, and much more recently university towns such as Oxford, Cambridge and Bristol, where they traditionally do well.

Now consider what happens if $p=1/2.$ Now the expected number of votes for A is $N/2.$ Let’s assume for simplicity that $N$ is even. What is the probability that A gets precisely $N/2$ votes (from the $N$ votes that go to either A or B)? The answer is on the order of $1/\sqrt{N}$. This can either be calculated by estimating the size of the binomial coefficient $\binom N{N/2}$ or by using the following more hand-wavy argument: we know that the number of votes for A will be $N/2$ plus or minus a small number of standard deviations. The standard deviation is $\sqrt{N}$ so that gives us about $\sqrt{N}$ possibilities, not all equally likely but mostly of comparable likeliness, of which we want one. If $N=10,000$, then in this case our probability of influencing the result in the constituency is on the order of 1/100, which is quite good going.

One can pursue such thoughts and reach the following not very surprising conclusion: if the two top candidates in your constituency are neck and neck, then you have a small but significant chance of influencing the result, whereas if one has a noticeable lead over the other, then your chances of influencing the result (whether the candidate with the lead is the one you support or the other one) are so small as to be effectively zero. Thus, the first-past-the-post system effectively disenfranchises anybody who belongs to a non-marginal constituency.

This might seem a rather curious conclusion. Surely you need people to go out and vote, in order to make the constituency safe for one party or another. So surely the people who vote in these constituencies are not disenfranchised after all.

I find this argument rather hard to counter — maybe others have some idea about it. The best I can think of is two remarks. The first is that other people behave as they behave, and given that behaviour it really is the case that if you are in a safe seat then the probability that your vote will make a difference to the outcome in that seat is negligible. The second is that the apparent paradox is closely related to the prisoner’s dilemma: in a safe seat, any individual voter is better off staying at home (because they lose nothing as far as the outcome is concerned and save themselves the trouble of voting) but if all the people who support the more popular party cooperate and actually vote for that party, then between them they may gain considerably more (because their collective action may effect the outcome) than they lose in the small effort needed to vote.

Is the model a good one?

So far, I have assumed that each voter who votes for A or B has a probability $p$ of voting for A. Is this a reasonable model? Surely there are lots of people who are certain to vote for A and lots who are certain to vote for B and only a few waverers.

This is not a big problem with the model. To see why, consider an extreme situation where everybody who might vote for A or B will definitely vote for A or definitely vote for B. That is, there is nobody who is undecided between A and B. Nevertheless, there is a certain amount of random fluctuation, because some people who are intending to vote will not in fact get round to it. So the outcome is not completely determined, and if the variance in the number of voters is reasonably large and (contrary to what actually happens) A-supporters and B-supporters are equally likely to be non-voters, then treating each person who arrives to vote as if they were making a purely random choice with a certain probability of going for A is not too bad a model.

However, it has a more serious defect if you are considering not voting, which is that you do not know the value of $p$. Even if an opinion poll is taken in your constituency — which it almost never is — you will not necessarily get a reliable value of $p$, since there can be biases in sampling, and people can change their minds in ways that are very far from independent. For example, the surge in support for the Liberal Democrats in Britain a couple of weeks ago was not the result of millions of people independently changing their minds but of millions of people having watched the TV debates and been impressed by the leader Nick Clegg. The best way to model this is to imagine that the value of $p$ has changed.

Now if the value of $p$ can move about, and if you don’t know what it is, then how does that affect the probability, given the evidence you have available, that your vote will make a difference to the outcome? It will usually increase the probability substantially. Why? Well, let us again take a rather extreme case and suppose that what happens is this: first the probability $p$ is chosen uniformly from the interval $[0,1]$ and then people vote for A with probability $p$, their votes being independent once the choice of $p$ has been made. (Those who followed the discussion of DHJ will recognise their old friend equal-slices measure here.) Then, roughly speaking, your vote will have a reasonable chance of making a difference if $p$ is within about $N^{-1/2}$ of $1/2$, and when this is the case the chance that it will make a difference is itself around $N^{-1/2}$. So we end up with a probability of about $N^{-1}$, which sounds rather fair when you come to think of it.

More realistically, you will know $p$ to within some error. For example, you may be pretty confident that it lies between $45\%$ and $55\%$. Then your vote has a reasonable chance of making a difference. If the interval of possible probabilities does not contain 1/2 (even approximately), then you might as well stay at home.

Conclusion.

The conclusion is a fact that all politicians know: only the votes in the marginal constituencies count. This is the big unfairness in the first-past-the-post system. It means that the interests of those who live in the marginal constituencies are more likely to be taken into account by those in power than the interests of people in safe seats.

Does that matter? One might argue that the composition of people in the marginal constituencies is likely to be quite mixed, and therefore a reasonably representative sample of the country as a whole. And to some extent that does seem to be the case — a political party usually needs to court the votes in the centre and can ignore both its core supporters and the core supporters of the other party, who are likely to be more extreme. And perhaps this argument is a reasonable one as long as two parties dominate the political scene, as is the case in the US and as has been the case for a long time in the UK. It is even possible to argue that the interests of voters for minority parties are looked after, since if a major party sees that a third party is getting a lot of votes, it may decide to steal some of that party’s policies. For example, in Britain the Labour party was out of power for 18 years, during which time a new party, the Social Democratic Party, was formed by people who broke away from Labour. In the 1984 election the anti-Conservative vote was evenly split between the Labour Party and an alliance of the SDP with the old Liberal party, with the result that the Conservatives won by a landslide (though with under 50% of the vote — that is how things work in Britain). The Labour Party was forced to move to the centre, and by the time they won power in 1997 their policies were of a kind that the founders of the SDP would have had no objection to whatsoever.

Nevertheless, this kind of indirect influence feels less like true representation than the direct influence to be gained if all votes count equally.

In the interests of fairness, I should mention the criticism that is often levelled at more proportional systems, which is that smaller parties can hold larger parties to ransom. Again, let us consider an extreme case: there are two parties with 48% of the seats each and one party with 4% of the seats. Let’s suppose that the small party represents a small geographical region. Then it could simply auction itself: whichever of the larger parties offers more money to that region gets the support of the small party. It is easy to see that this will probably lead to that region getting more than its fair share of the country’s resources.

This is another game-theoretic situation. It reminds me of a very nice problem that has been studied by cognitive scientists. Imagine that you and a friend are walking down the street and a man appears out of nowhere with a huge wad of notes in his hand. It turns out that he has $100,000. He then says that he will give the money to you and your friend if and only if you agree on how to split it. You say, “Great, let’s split it 50-50,” and to your astonishment your friend says, “No, I’m not interested unless I get$90,000.” And it becomes clear that your friend is being serious. Your choice is either to accept the deal and gain $10,000 or to refuse it and not gain$10,000. What do you do? The opportunity is not likely to come round again …

Actually, the usual presentation of this problem is different. In the usual presentation, it is just your friend who is offered the money, but on condition that your friend shares it (in some proportion) with somebody else. And you are the chosen person. Most people say that they would accept the deal as long as they get about a third of the money — it seems that two thirds is the “reward” that your friend is considered to deserve for having been the one who met the rich man.

The relevance of this is that there is sometimes an interval of outcomes that will suit two parties who are trying to make a deal, and then it is far from clear where in that interval the deal will actually be struck.

I’d like to end with a defence of the alternative vote system. (I’m not saying it’s the best possible system, but just that it is a distinct improvement on what we have now, and not as bad as some people think.) This is the system where if nobody has an absolute majority, then the second-choice votes of the least popular candidate are distributed amongst the other candidates, and this process is iterated until some candidate does have an absolute majority.

Although this system is not proportional, and still leads to marginal constituencies mattering more than safe seats, it avoids one of the big abuses of the usual first-past-the-post system, namely the need for tactical voting. If your first choice is party A but only party B has a chance of beating party C, then under the current system you have a good reason for voting B. But under AV you have nothing to lose by voting for A as your first choice and B as your second. Since many people say that they do not vote for smaller parties because their vote would be “wasted” (blissfully unaware that it will almost certainly be wasted anyway), the alternative vote system could help people vote for the party whose policies they most supported. It doesn’t sound like much to ask, and even this would be a huge improvement on the current system, which, now that there is a third party that is not tiny, is monstrously unfair.

### 44 Responses to “Is the British voting system fair?”

1. Peter de Blanc Says:

Scott Aaronson and Aaron Edlin, Andrew Gelman, and Noah Kaplan have pointed out that the social utility of the election outcome scales with the size of the population, which is proportional to N in a democracy.

So if your model says that your probability of changing the election result is proportional to 1/N, then the expected social utility of voting is independent of the population size.

2. Mugizi Robert Rwebangira Says:

Seems what you Brits call “Marginal constituencies” are what we call “Swing Districts” over here in the States.

Also what you call “Alternative Voting” is what we call “Instant Runoff Voting.”

I am very much interested in this issue, and in general how the setup of the political system can affect policies even if the voters have the same preferences. In general its hard to say a particular system will automatically lead to better outcomes – e.g. Germany and Italy both have proportional representation but one is a model of stability and the other one is … Italy.

A book I liked a lot on this topic is “Gaming The Vote” by William Poundstone.

Actually I am not aware of any empirical study comparing different voting systems that tries to really determine how they shape voter behavior and policies, surely someone must have done that?

But perhaps the voting system is just noise and gets lost in all the other factors which affect policy.

3. William Nelson Says:

These are interesting questions. Voting seems so common-sense, yet these confusing paradoxes appear…

As an American, I definitely know the feeling of pointlessness in voting in “safe districts”. I really like the two-party system, which produces clean policy divisions and generally clean election results, but sometimes I wish that each district could send two representatives, with their votes in congress counting proportionate to the votes they received.

But this opens up a big can of worms, and in the end one would probably have to have a full multi-party, proportionate system, which I don’t want at all, because then half the time the election doesn’t decide anything (case in point: Iraq).

What we definitely do need, and could implement within the current Constitution, are instant runoffs so that third-party candidates can’t produce drastically unfair outcomes.

• Klas Markström Says:

“a full multi-party, proportionate system, which I don’t want at all, because then half the time the election doesn’t decide anything (case in point: Iraq).”

The Swedish voting system is basically a full multi-party, proportionate system, but in order to avoid deadlocks there are two tweaks, first an odd number of seats in the parliament, and a 4% bar for getting a seat in the parliament.

The 4% bar means that in order to get a seat in the parliament a party must get at least 4% of the votes. This rule stabilises the parliment and keeps most of the small fringe parties out ( revolutionary communisst/nazis/ultra religious). The parliament has been lead by different coalitions for many decades now and while there are often negotiations needed to form the coalition that is typically done in a few days when only larger parties are involved.

• William Nelson Says:

Interesting; do people not find it distressing to not know in advance who their party will be joining with in a coalition?

• Klas Markström Says:

Not usually, thanks to the 4% bar there is not a large rotation in terms of which parties are actually in parliment, only their strength. Although there are 1 or 2 parties which oscillate around the bar, and the occasional one seat/ one election party.

For a long time there have been two larger parties, the Social-democrats, right now around 35% and the Moderats(conservative/liberal), right now around 30%, who normally gets to form a ruling coalition with two or three supporting parties. How much influence, and minister posts, the supporting parties get will depend on their number of seats in the parliament.

So for the voters the question is normally first which bloc to support and then which policies to emphasize by choosing a particular party within that bloc.

• Stefan Says:

Yes, to some extent. But you know this in advance, and you’d still rather have your party inside the government than in the opposition.

Moreover, certain coalitions are unlikely (though not impossible – the Netherlands had a period in which the left-wing and right-wing party joined forces with a central party in the government). As long as your political climate doesn’t splinter into too many small parties (Italy comes to mind), and the country is reasonably homogeneous (the Netherlands are – you won’t see things like the Limburgian Party), the proportionate system seems to work reasonably well.

Apart from our current prime minister having run the previous 4 governments into the ground…

• Tim van Beek Says:

But this opens up a big can of worms, and in the end one would probably have to have a full multi-party, proportionate system, which I don’t want at all, because then half the time the election doesn’t decide anything (case in point: Iraq).

The Swedish system has already been mentioned, I would like to add that most European, working democracies have a similar system.
Germany’s system is a little more complicated than full multi-party, proportionate, but as a first order approximation that will do. Germany has a 5% barrier for the parliament (a party has to get at least 5% of all votes to get into the parliament at all).
The situation that a small party held two parties hostage was basically the case during the 1960ies and 1970ies, when there were two big parties (social democrats, SPD and conservatives, CDU) and one small (liberals, FDP). Actually the small party overturned the government once by changing sides (that’s how Helmut Kohl became chancellor).
But all in all the system worked quite well.

My point being: If this system works or does not work depends much more on how the people participating handle it, than game theoretic discussions.

4. William Nelson Says:

Regarding the actual analysis of the blog post, I do have a couple of observations.

One is that not everything has to revolve around parties. Representatives come from a district and have to answer to the voters in that place, not anywhere else. This means that if even if your district representative is from the “wrong” party, they still should be representing your interests at least to some degree. And the different districts produce quite different flavors of representative, even within the same party.

Also, I am not sure you are correct in your summary statement that marginal constituencies get all the representation. I would say that they get most of the publicity, but I don’t see any particular excess of influence in actual policymaking, at least in the U.S. On the contrary, the seniority system gives greater influence to the longstanding members from the “safe seats”.

• gowers Says:

I’d just make the comment that there’s an important distinction between the influence that the member from a safe seat has and the influence that a voter in that safe seat has.

• William Nelson Says:

And that, I guess, returns us to the starting point of the whole debate 😉

In theory, the representatives of different parties in a competitive district would come to resemble each other, thereby reducing the effective “influence” of votes in those districts. In the U.S. this clearly happens to some extent – “safe” conservatives are more conservative that unsafe ones, likewise for liberals. I wish that it occurred to a greater extent, and I am not sure why it does not. Apparently the system of primary elections is partially to blame for the irrational polarization in close districts.

5. Gil Kalai Says:

That’s a very interesting post and, of course, there is a vast literature about related questions. The probabilities that your vore counts in the two possibilities of p=1/2 and of p uniformly distributed in [0,1] are referred to as the Banzhaf power index and the Shapley power index respectively.

One can question if the probability that your vote counts is really the right measure of your impact and whether your impact however it is measured, is the reason why you vote and do not stay home .

In the realistic situation when there is no reason to assume the voters vote independently there are other measures of voter’s effect that can be relevant. (One such measure was studied by Haggstrom, Mossell and me and by Graham and Grimmett. It is a normalized correlation between the outcome of the election and your vote.)

In the British system the impact (any way you measure it) of voters depends on where they live. But I am not sure if equal impact in terms of the measures described in the post is the only criteria for fairness.
From the outside the British system (not just the abstract voting system but the political system as a whole) seems sufficiently fair and rather stable.

The British system is probably more sensitive to rather smalll changes in the votes compared to simple majority method.

• gowers Says:

For some of the reasons I gave above, I think one can argue that at least while the Conservatives and Labour were by far the two largest parties, the first-past-the-post system was reasonable and did sort of reflect the will of the electorate (but if anyone attacks that assertion I will be able to offer only a weak defence). But in this election there are three parties all with roughly a third of the vote — though it’s not clear that will last until polling day — and the number of seats they are set to get bears an extremely arbitrary relation to the number of votes.

It has to be said that Israel is often held up by opponents of proportional representation as an example of the disproportionate power of small parties. I think the question of what political system gives the right amount of influence to various groupings probably has an answer that depends a great deal on how tightly defined those groupings are and the extent to which their interests conflict. For example, suppose you have a country that divides cleanly into two racial groupings, one of 55% and the other of 45%, and suppose that these groups have radically different needs. Then the people in the smaller group will be in a bad way. Devising a good political system that protects large minorities seems to me to be a major challenge, not least because the best such system is likely to be too complicated to seem fair (to non-mathematicians) and the perception of fairness is almost more important than fairness itself.

6. Tom Ellis Says:

If you permit mixed strategies (some probability of voting or not) then presumably the equilibrium exists where some people vote and some people do not.

Admittedly, the theoretical equilibrium is probably a lot lower than the turnout in practice, i.e. voters systematically over-value their votes.

• Mark Bennet Says:

I do think the paradox is closely related to the Prisoners’ Dilemma. In a two-person Prisoners’ Dilemma take a simple model, and assume that both prisoners are equally intelligent, so that they use the same one. We know they don’t because the furious arguments about PD lead people to different conclusions by different means, so this is a definite huge simplification.

But with classical PD if we model each prisoner selecting co-operate with probability p and defect with probability (1-p) it is easy to check that the optimal outcome is with p=1.

In voting, if each person votes with probability p and doesn’t vote with probability (1-p) it is obvious that the best outcome for the candidate that the voter supports is achieved by that person voting.

The probability of voting can be added to the model too, and this would no doubt lead to some interesting observations.

7. polyanon Says:

“Actually I am not aware of any empirical study comparing different voting systems that tries to really determine how they shape voter behavior and policies, surely someone must have done that?”

Together with the dictatorship / democracy (still in the agenda due to economic success of some dictatorships), presidentialism / parlamentarism and multipartidism / bipartidism issues the majority /proportional issue might be one of the most researched issues by contemporary political scientists (of course all this issues are related and can not be considered independently). Some suggested readings:

A good start for the theory could be “classics” such as Duverger´s “L´influence des systèmes eléctoraux sur la vie politique” where he states Duverger´s law, Rae “The political consequences of electoral laws” or Sartori “Comparative Constitutional Engineering”

Regarding more recent empirical studies, Lijphart is a must: “Electoral systems and party systems: a study on twenty seven democracies 1945-1990” 1994; “Patterns of democraties: Government forms and performance in thirty six countries” 1999. The most recent and complete i´m aware of is from Josep M. Colomer, “Cómo Votamos. los sistemas electorales del mundo: pasado, presente y futuro”. 2004.

In the latter it is interesting to see how as societies changed from corporativism (where the individual only counted as a member of a larger group) to individualism, prefered and used electoral systems changed from unanimity and loteries to majority and proportionality.

Regarding the point of the post it must be said that proportionality in itself does not guarantee fairness, it depends also on the size of the assembly, size of districts, rule for allocate votes to seats – remainders (Hamilton-Hare, Jefferson-D´Hondt, Webster-Sainte Lague or Gergonne-Gilpin), political geography etc…

IMO the most interesting issues are how electoral systems (and in general political institutions) relates to economic performance (which implies political stability). But democracy as a form of governement is probably still too young as to asses this issue.

• Mugizi Robert Rwebangira Says:

Thanks for the references!

So it seems the bottom line is that its not too clear what the impact is and other factors come into play.

I will try and check out some of those references.

8. Elsinor Says:

Gowers: “For example, suppose you have a country that divides cleanly into two racial groupings, one of 55% and the other of 45%, and suppose that these groups have radically different needs. Then the people in the smaller group will be in a bad way. Devising a good political system that protects large minorities seems to me to be a major challenge, not least because the best such system is likely to be too complicated to seem fair (to non-mathematicians) and the perception of fairness is almost more important than fairness itself.”

Change the word “racial” to “national” and “needs” to “priorities” and you are describing Northern Ireland.

9. Ori Says:

In the situation you described just before the conclusions (p is distributed uniformly in [0,1], etc.) the probability that your vote will decide the election is *exactly* 1/N. This model is exactly equivalent to Polya’s Urn.

• gowers Says:

Indeed. One of the things that Ryan O’Donnell spotted during the work on the density Hales-Jewett theorem was that the equal-slices distribution (which I referred to above) was the same as Polya’s urn.

10. A HANGING PARLIAMENT « DUCKPOND Says:

[…] Gower’s Weblog has a mathematics related review of FPTP and PR. […]

11. Mark Bennet Says:

The posts on the behaviour of the electorate and how this is influenced by the voting system take us back into game theory. But one factor in this election which is more acute than in previous elections is that the outcome seems more uncertain, and therefore individual voters cannot be so certain as before how their choices will affect outcomes.

For example, someone told me that they were uncertain because LibDem was no longer a safe protest vote – they might get elected.

Someone else quoted a survey which had apparently suggested that if people believed the LibDems could win, they’d get 49% of the vote (unchecked) – so are we in the realm of catastrophe theory? Does it take a “catastrophe” to change a government, or does the ideal voting system give a smooth transition (modelling votes/opinion over time as a dynamic system of some kind)?

12. Anonymous Says:

The main source of unfairness (and inefficiency) in practice is that despite nominal majority rule many democracies are in fact run as oligarchies that benefit a tiny elite at the expense of the vast majority. This seems in part to be because politicians can successfully pretend to represent their constituents while actually representing someone else.To what do the usual models of voting systems depend on assumptions of perfect information, and are there realistic models of voting with imperfect information?

13. polyanon Says:

1. “The main source of unfairness (and inefficiency) in practice is that despite nominal majority rule many democracies are in fact run as oligarchies that benefit a tiny elite at the expense of the vast majority”.

Together with Duverger´s law, Michels “iron law of oligarchy” is one of the most celebrated of classical sociology. One of the fourth steps in the Clegg liberal democrat manifesto is to try to change this (“A fair deal, by cleaning up politics”).

2. For those interested in the empirical side of elections and the “evolution of electoral systems across countries i forgot to cite another good source:

Elections and electoral systems by Dieter Nohlen.

Here he uses more case analysis than statistical analysis. There is an interesting discussion about the real difference within majority-proportional systems and how to evaluate them.

3. It seems that british system evolved to a pure relative majority uninominal system gradualy. Until the 30´s Liberals were the second political force and were supplanted by Labour due to some bad moves regarding the irish issue and probably also due to sociological changes and international environment (cold war). Then in the 50´s the system was closed (relative majority uninominal) and as a result it was impossible for liberals to govern anymore. Now there are new sociological changes (globalisation, green thinking) and i think the liberals has good chance to become the second force again. If the sociological change is robust, this will happen in one election or the other, independently of the electoral system. By reading the Liberal manifesto it is clear that they are trying to gain territory against the Labour while increasing abstention from conservatives.

4. IMO, the British system might be unfair but taking into account the multinational character of UK, it is the best possible. I would not change to a proportional one; i´m not sure about the alternative system.

14. Richard Watson Says:

I think the small party holding two larger parties to ransom is a red herring.

Let’s assume in the UK you had Labour on 49%, Conservatives on 48% and BNP supposedly “holding the balance”. To suggest that the BNP holds the others to ransom means that Conservative and Labour would much rather work with the BNP than each other. I don’t believe this to be true.

I think what would happen is that you would get a minority Labour Government who would find a way to engage support from the only slightly weaker Conservatives.

This might result in a lot of compromises, which could well be a Good Thing. Some things wouldn’t change at all, some things wouldn’t happen. In order to put through legislation people would have to try to come to an agreement. This is also a Good Thing.

• gowers Says:

I was about to make a similar remark before reading yours. Consider the abstract question of why it is that a third party, with fewer seats than the first two, can have an apparently disproportionate influence. At first this seems almost like a contradiction: trivially it cannot harm a party to have more seats. So what is it that gives the Lib Dems an advantage when it comes to forming coalitions? It’s the simple fact that they are more ready to form coalitions because (to oversimplify) they are in the middle. If Labour and the Conservatives were prepared to cooperate, then they could shut out the Lib Dems, so in a sense they are being punished for being extremist and/or uncooperative.

Having said that, I’m not sure that this argument is entirely borne out by experience. For instance, in Israel, some small and very hard-line religious parties appear to have a lot of power.

• Richard Watson Says:

I think you are right (gowers) that the position of a party relative to the others matters a lot as to whether they can be in a position to deal effectively. Also we must bear in mind that there are quite a lot of “others” and there would be more in a proportional system, therefore there would be more potential entities to be allied with, thereby making it perhaps less likely that a single party could hold the balance and other people to ransom.

Consider the current situation, according to the BBC’s poll of polls at the present time, it could be that the vote is split: Con: 35%, Lab: 29%, LD: 26%, Others: 10%. With such a spread it would be unlikely for a single party to hold an outright majority of seats on the current system. The LDs may well have very few seats (much fewer than than their 26% share would suggest) but their large popular vote makes it seem extremely fair to me that they could form a government with Lab, together representing 55% of the voting public.

I think in many ways this is the most interesting election of my adult life.

• Mark Bennet Says:

Richard – your latest highlights for me the fact that in a three-way contest it would be possible for the parties to combine forces in such a way that the most popular party was always in opposition.

• Richard Watson Says:

Mark, I completely agree, but this can happen in any system, it could happen as a result of today’s election. It doesn’t seem to make it fairer if this should happen as a result of the skewed system we have than if it happens as a result of a fairer system.

In any case all we are doing is electing local mps. Coalition makes it more likely that a voter has an mp whose party is governing, its more important that people are represented than some arbitrary thing we call a political party.

• Frederik Bloggs Says:

Austria has been smitten with two major parties (Peoples Party – conservative; Socialist Party) and neither has had an absolute majority in more than 20 years. The result is that they inevitably form a coalition that holds a monstrous majority. This, together with proportional representation results in a system that is far from democratic, and the establishment – with negligible opposition – of extremely inefficient systems. e.g. there are over 13 separate state health systems, each with highly paid executives and a bureaucracy.

The country – with some 8 million population – is divided into regions each with its government and severely under-occupied bureaucracy It has an air force with modern fighter planes.

It is financially convenient to the political class to maintain this incredible situation.

Votes are essentially purchased by offering ever more goodies to the lower income groups and the unemployed, paid for by a highly regressive income tax and other, that reach a maximum of 40% at a low level of income.

There is no escape other than FPTP.

15. From Calculus to Columbia Says:

There is nothing new about criticising the British electoral system and its unfairness…

• From Calculus to Columbia Says:

Sorry about that, got a little ahead of myself there. I had meant to add a “ignore the polotics (much of which is clearly heavily outdated anyway)” caveat. Afterall, this humerous little video does only describe one possible altenative among the many possibilities.

Nonetheless, the graphs near the begining to illustrate how stark the distorting and unfair effects of having a FPTP system are. This was really the main point I was trying to highlight.

16. Anonymous Says:

It remains to be seen whether the Lib Dems are in the end any more willing and able to defy the “iron law of oligarchy” than their rivals.

The robustness of abstract models with respect to modifications of their assumptions is an interesting theoretical question. It might also be helpful to construct realistic models of electoral systems. The failure to have, or at least to apply, realistic models seems to have caused certain difficulties in economics lately. It is far from clear to what extent the public policies of, say, Israel, are an artifact of its electoral system, rather than a product of its history, culture and location. More generally, perhaps the number of essentially distinct voting systems is very small under sufficiently realistic assumptions.

17. Anonymous Says:

The voting system interacts with the other elements of the political system and with the underlying society in very complicated ways. The Weimar Repubic had proportional representation, but its hard to see how this promoted fairness. At the other extreme, Norway has proportional representation, and seems a model of equity, efficiency and stability. The average case scenario seems closer to Italy, which is probably no fairer than Britain, but less efficient and stable. Intuitively,at least, the Italian scenario becomes more likely as a society becomes more fragmented. Thus I’m skeptical that proportional representation will benefit Britain much.

18. polyanon Says:

1. Yet another reference for the empirical, again from Colomer.
“Political Institutions” 2001. According to this book, as for year 2000 all democratic political systems can be clasified in four classes:
–majority parlamentarism. It is the system prominent in the Commonwealth (UK, Canada, Australia, India…). It includes also Botswana, which is one of the most prosperous African countries, by per capita GDP PPP.
–proportional parlamentarism. It is the system of continental western europe, except France. Outside Europe it can be found in Japan, South Africa and New Zeeland.
–presidentialism. It is the system of the Americas. Since the three powers are elected independently (executive, legislative, judicial) it is the most purest democratic model (separation of powers). Outside America it can be found in Namibia, Mali or Philippines and Taiwan.
–semipresidentialism. The difference with Presidentialism is that the assembly nominates the cabinet. It originates in the french system and has been adopted in the last wave of demcratization by Poland, Rumanie and Bulgaria.
Parlamentarism was clearly an historical compromise within dinastic monarchies and democracy. IMO Semi presidentialism is quite weird (not only because his french origine…).

2. “The voting system interacts with the other elements of the political system and with the underlying society in very complicated ways”.
I see more complex the relations of the political system with social structure (which includes economy).
Social structure–>Social problems–>Ideologies–>Political Parties–> Party Manifestos–>Elections / Electoral Systems–>Assembly / Government–>Law/public policies–>Desired Changes in social structures.
IMO the social structure is the key point of the whole cycle.

19. Carnival of Mathematics #65 « Maxwell’s Demon Says:

[…] To finish on a topical note, I was writing this carnival as I watched the British election. With the hung parilament and the discussions now going on the electoral system itself is coming under question. What does a mathematician think of this? Tim Gowers has some interesting comments. […]

20. ogerard Says:

You are of course not alone in wondering about ameliorations of voting systems. Among the mathematically oriented mind that studied this in earnest I stumbled recently on the work of Lewis Carroll in this book “A Mathematical Approach to Proportional Representation: Duncan Black on Lewis Carroll”, Kluwer, 1996.

21. Would an Alternative Vote make a difference? « Maxwell’s Demon Says:

[…] perfect system does not mean that different options are not better or worse. Tim Gowers has given a mathematicians thoughts on how the current system is unfair. In a similar vein the Fabian society pointed out a couple of […]

22. 【数学都知道】2010年5月3日 « 科学博客 Says:

[…] 数学和选举 Steven Brams 写了一本书“Mathematics and Democracy”，就是“数学和民主”。如果能用数学的方法来解释一个选举系统，那它应该是公正的吧。2008年美国数学推广月的主题就是“数学与选举”(Math and Voting)，美国人弄出了一堆学术论文。英国人也有：“Is the British voting system fair?”。在台湾和香港也都有学术研究。在中国这方面似乎根本没有开展研究。中国的选举不需要数学上正确。 […]

23. Voting and the Harmonic series « From Calculus to Columbia Says:

[…] analysis of which uses the harmonic series. Now, I’m certainly not the first person to write a blog post applying a simple mathematical model to the British electoral system, but this still seems worth bring to the attention of anyone who cares to read this. We define […]

24. Is AV better than FPTP? « Gowers's Weblog Says:

[…] to suppose that AV gives “more” votes to supporters of unsuccessful parties. In another post I explained in detail why FPTP does not give equal votes to everybody. To put that slightly better, […]

25. Carnival of Mathematics #101: Prime Numbered Special Edition | The Aperiodical Says:

[…] 2010: Is the British Voting System Fair, at Tim Gowers’s Weblog; featured in Carnival […]

26. Anonymous Says:

The political system in this country is not fair… I don’t need to write a paper on it because its very obvious, seats are almost guaranteed. Cultural differences make a difference. I personally think we should make it simple you vote for the party leader wherever you are and then the amounts of votes count not the seats. If successful then the party can decide /who should win… and I love the Australian system cut the BS and get to the point. As you can see UKIP got a thousands of votes and no seats (good example) yet there is no reflection on their gains. Its not good and not reflective – a change in voting is needed

27. Willowandy Says:

> And perhaps this argument is a reasonable one as long as two
> parties dominate the political scene, as is the case in the US and
> as has been the case for a long time in the UK.

It isn’t reasonable, as the FPTP system locks tends to lock in the two-party dominance – ie. it is a circular argument.