Before I start, let me get one thing over and done with: I fully admit that professional mathematicians are as capable as anyone else of making stupid collective decisions.

But I don’t want to imagine what the world would be like if it were run by mathematical researchers. I just wonder how much difference it would make if politicians understood enough mathematics to be able to understand an argument of more than one sentence. Or to put it more accurately, what would it be like if the following rules of political life were no longer accepted?

1. An argument that is slightly complicated but correct is trumped by an argument that is punchy, amusing, and wrong.

2. If option A is better than B in some respects and worse in others, then instead of weighing up the pros and cons, you decide which side you are on and then just mention the pros of the option you prefer and the cons of the other option.

3. If option A is better than B in every respect, but your party supports B, then you support B.

4. If one of your political opponents points out a flaw in your argument, then count to ten and repeat the flawed argument.

If that were the case, then one consequence would be that one could advocate new ways of doing politics and have them discussed seriously. In this post, I would like to mention a few ideas that would be dismissed as utter lunacy by any politician. But perhaps people who read this blog would be prepared to engage with them properly and weigh up the pros and cons. I’m sure there are cons — but I don’t think the ideas are utter lunacy.

I am not talking here of electoral reform, though I very much support some kind of change to the British voting system. Rather, I am talking about reform of the way that business is conducted in between elections. But before I suggest any ideas, let me discuss what I think is wrong about the current system (as it is in Britain, but I think the remarks can be generalized to many other democracies). Let us suppose that we have an ideal voting system: to please everybody, let’s suppose that it’s a first-past-the-post system that just happens to have delivered a wonderfully proportional result; and it has even delivered a good strong government, since one party has received 55% of the vote, and approximately 55% of the seats, giving it a comfortable majority.

What could possibly be wrong with that? In my view, at least two things. The way things work in Britain, the members of each political party agree that they will form a bloc and vote the same way on every issue. If you belong to a party and you think that one of your party’s policies is wrong, then you are faced with a choice. Either you vote against the party line, and become known as a rebel, jeopardizing your chances of advancement to higher office (if that is what you hope for), or you toe the line and support measures that you do not believe in.

There are of course good reasons for behaving this way. You join a party because you believe in its general principles, and the idea is that you compromise on some issues because that is the price to be paid for ensuring that the party has the political strength to make other decisions that you do believe in.

But this system can in principle lead to decisions being made that are not supported by anything like a majority of members of parliament. If the party with 55% of the vote puts forward a policy that is supported by 70% of its members (perhaps there is some committee that reflects perfectly the views of the party, and the policy is voted on in that committee) and opposed by everyone else, then it is supported by under 40% of MPs. But it is still implemented.

The second problem is one that is particularly serious in a country that has a large minority with very different interests from the majority. (This is often the case for ethnic or religious reasons, and has led to many of the worst and most persistent conflicts round the world.) Suppose that 30% of the country belongs to group A and 70% to group B. And suppose that there is a political party that represents people in group A and another political party that represents people in group B. And finally, suppose that the two groups dislike each other intensely. Then if the number of seats is roughly proportional to the number of votes, the party representing group B will have a large majority in government, which will allow it to advance the interests of group B at the expense of group A. For example, it could give all the powerful jobs to people from group B, pay for good infrastructure in regions where people in group B tend to live, and so on. This situation is sometimes referred to as the tyranny of the majority.

So far so good. Now comes the nutty bit. I would like to suggest two systems for parliamentary votes, one that would weaken the party system but without killing it off entirely, and one that would protect large minorities. Neither has the slightest chance of being adopted, because they are both too complicated to be taken seriously. But mathematicians wouldn’t find them complicated at all — hence the title of this post.

An obvious way to weaken the party system is to have secret ballots for every single parliamentary vote. That way, MPs could simply vote on every issue according to their judgment about that issue. I myself would like to see that. But it would kill off the party system almost completely. It would be criticized, with some justification, for making government virtually impossible: how could you plan ahead if every measure you proposed was in danger of being voted down? And what if somebody were to say one thing to get elected and then to vote in an entirely different way once they were elected? (Of course, entire political parties do that with their manifestos, but that’s another matter.) That would make a nonsense of representative democracy.

To meet that objection, I propose the following system. Votes are made electronically and then counted. After they are counted, the way people voted is made public. However, before that happens, each vote is changed, independently, with a certain probability such as 10% (but the precise value could be argued about, and might even vary from vote to vote, being lower for especially important votes). If you feel strongly that your party is wrong on a certain issue, then you can vote against it, and if that annoys the party whips, you can tell them that you voted for it but your vote was flipped. However, you cannot play this game too much, or the number of times your vote appears to be against the party line will be so far above 10% that it will be clear that you are not a loyal party member.

It is this last part that almost no politician would understand, since non-mathematicians have a strong aversion to the probabilistic method.

Now for a political system that would protect minorities and give power to political parties in rough proportion to the number of seats in those parties (and in particular not give 100% of the power to a party with over 50% of the seats). In this system, I shall assume that there is total loyalty within parties, so on each issue, each party can appoint a representative, who will know exactly what the party wants and will act on that knowledge. At the beginning of each year, all parties are given “credit” in proportion to their sizes. They also know how many votes there are going to be and have a good idea of how many will be very important, how many extremely minor, and so on. And then, instead of votes on different issues, there are auctions. If party A wants income tax to be raised, and party B does not, then whichever party offers to give up the most credit wins the auction and gets its way. If the issue is particularly important to both parties, the bidding may well go quite high, but if a party bids too much, then it is significantly weakened later in the year. (There may be better ways of implementing the basic idea, such as starting with a particular level of credit and continuously replenishing it at a certain rate.) And a minor party that cares deeply about one issue can save all its credit up for that issue. Perhaps it would also be permissible for two parties to get together and make a joint bid.

Note that under such a system, it would be difficult to push through a lot of controversial legislation, since your opponents would mind about it enough to force the bidding up to high levels. But if you could find compromises, then they would be cheaper, since it would not be worth your opponents’ while to spend much political capital opposing them. So the system would naturally encourage consensual politics.

To make the system more theatrical, each MP could be given a certain number of cards that was less than the number of votes to be held. If, say, there were 200 votes, then each MP could have 50 cards, each of which could be used as a vote. Then for each vote MPs would take turns: one would vote for, another against, another next for, and so on, until one side gave up. On average, a quarter of the members of each party would participate in any given vote, but they would not have to stick to the average for every single vote.

This system is a bit like a system that siblings sometimes use when their parents have died and they want to share out a number of objects of sentimental value. If they have also been left money, then they can make bids for the various items and then take those bids into account when they share out the money. It can, I am told, be a good way of avoiding acrimony.

When I thought of the above idea (I make no claim to be the first to do so, by the way), I worried that it would have the potential to lead to game playing of an undesirable kind. And since writing the last few paragraphs I have realized that indeed it does have that potential. One party could suggest an outrageously unfair piece of legislation that would be disastrous for the people represented by the other party, and then bid it up in order to force the other party to waste valuable credit ensuring that it does not pass. For instance, a party that principally represents voters in a certain region could propose a hugely expensive program of improving public transport in that region, paid for by taxpayers all round the country. Or in a country with more than one language, a party that represents speakers of one language could propose a motion to ban all use of other languages in schools.

At the time of writing, I have not come up with a good system for dealing with this problem. The difficulty I have is that the obvious ideas seem to involve having to have some measure of how “reasonable” a piece of proposed legislation is, in order to attach a cost to proposing it, whereas I was looking for a system where the cost would be determined automatically by the “market forces” that arise from the need to spend political credit.

So let me conclude, slightly limply, with the assertion that it seems wrong for a majority to be able to call all the shots, and that if one does not care about simplicity then it ought to be possible to devise a system that does not have this defect.

It is worth mentioning that in many countries with sharp ethnic or religious divisions, minorities are guaranteed ministerial posts. That is a crude way of sharing out power more fairly: I am wondering whether there are other ways.

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