If politicians were mathematicians

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.

62 Responses to “If politicians were mathematicians”

  1. Dennis Jackson Says:

    I have not taken the time to think through this completely, but varying the auction method could be the solution. Requiring the second highest bidder to also pay their bid would prevent parties attempting to bleed each other of credit, as the difference would negligible. I am confident there are flaws in this approach though!

    • gowers Says:

      I thought about that, but a larger party could still play silly games in order to increase the ratio of its credit to the credit of the smaller party. But I don’t rule out some such variant working.

    • Steve Linton Says:

      This auction (with everyone paying their bid, more or less) is, in a sense, what happens. If a group cares about something a lot, they spend more of their actual money (on PR to persuade the public to persuade their MPs of the rightness of their cause) or that rather vague thing “political capital” to try anjd make it happen. An example is to hand — electoral reform. Essentially, a rather small group of MPs (the Lib Dems) cares about this a lot. So they are willing to offer their support for lots of other policies which they maybe don’t want, but can live with, in exchange for the Conservatives support on this one.

      The fact that this doesn’t operate to rigid open rules is a problem, of course, allowing people to play all kinds of tricks behind the scenes, but also means that anyone caught “gaming”: the system in front of the scenes automatically loses, which is maybe good.

  2. Sarang Says:

    The US Senate doesn’t have these particular problems: (1) there’s relatively little party discipline (though that’s been changing among the Republicans) and (2) minority rights, at least the rights of _some_ kinds of minorities, are protected through the filibuster. Unfortunately, the main consequence of (1) is egregious and parochial behavior by the median Senator, while (2) leads to the minority obstructing absolutely _anything_ the majority wants to do. To some extent your solutions avoid both these issues by keeping party discipline (albeit with probabilistic “excuses”) and pricing the filibuster. (I remember trying to persuade people that the latter was a good idea during last year’s healthcare debate.) I think the limit of 100% rigid partisans is a bad one here: situations where a majority supports arbitrarily outrageous acts (e.g. Rwanda, 1994) are situations that are too far gone for politics to help. In practice one is constrained by the squishier elements of one’s group; that fact should be built into the model somehow. For instance, at some point legislation gets sufficiently outrageous and unrealistic that the minority needn’t spend capital on opposing it.

    I might prefer a fixed number of vetoes to the bidding scheme. Another thought: a more neutral measure than the “reasonableness” of a bill is whether it’s substantively the same as another bill. It seems to me that there are usually only a smallish number of oppressive things the majority wants to do; the simplest way of exhausting the minority’s political capital is to keep putting up mild variants of the same few bills. (Assuming some legal constraints on the “constitutionality” of bills, there are relatively few ways of hurting the minority without severe collateral damage. Take your roads example: in any country there will be a lot of people from one region that live/work in another.)

  3. Terence Tao Says:

    These are interesting thought-experiments, not necessarily because the proposed alternatives are superior, but because they help give a clearer picture of the parameter space of possibilities, and of the relative strengths and weaknesses of the status quo.

    Some amount of probabilistic noise is probably helpful. Many voting paradoxes (e.g. Arrow’s theorem) ultimately stem from the fact that there is no deterministic way to map [0,1] to {0,1} in any “homomorphic” manner, but this becomes possible once some randomness is permitted. For instance a “random dictator” system in which a randomly selected voter ends up determining the outcome of an election overcomes some of the obstructions posed by Arrow’s theorem, though presumably one would prefer a system that did not give the extremist factions with, say, 1% of the vote anywhere close to a 1% chance of gaining power.

    Your first proposal is based on giving enough plausible deniability to offer cover to any representative. It reminds me of a system I have heard is used for some death penalty executions, in which several people push a button at the same time to make it ambiguous which person actually killed the prisoner. Two issues come to mind. Firstly, this system would reduce the accountability of each representative, not just to their party whip, but also to external watchdogs and to the public at large. There would be a window of opportunity for corruption: one could sell a small fraction of one’s votes and it could be plausibly passed off as statistical noise as long as the fraction remained within the standard deviation of the process.

    Secondly, one would have to guard against the speaker (or whoever gets to call the votes) repeatedly calling a vote on an issue on which one has, say, a 48-52 minority, until the random fluctuations break in one’s favour and the legislation is passed. Admittedly this would be a fairly blatant misuse of one’s power to set the vote, but there is definitely scope for gaming the system by structuring the votes in a way to somehow exploit the randomness.

    With the auction system (which seems to be a formalisation of the nonrigorous concept of “political capital”), I see a tragedy of the commons problem: if the government is formed from a coalition of two self-interested parties, then each one would prefer the other to spend more credits to pass legislation that they both want, in order to have more power down the road. So there is an incentive to drag one’s feet on every issue to make the other party do more of the heavy lifting. This is again a reflection of the non-homomorphic nature of the majority map from [0,1] to {0,1}; the marginal benefit of moving from a 51% majority to a 52% majority is negligible compared to the expense in political capital.

    • gowers Says:

      On that last point, I think negotiation between parties behind the scenes would have to take place in order to avoid precisely that situation. But if two parties happened to agree on one issue and disagree about most other issues, then the negotiations could be tricky. Perhaps there could be a convention that the “norm” was for each party to contribute in proportion to its (starting) political capital.

    • Ori Says:

      Regarding the problem of exploiting random fluctuations to win a vote:
      One could introduces noise that would keep the total tally the same, like randomly swapping some pairs of votes, instead of independent noise.

  4. Tom Ellis Says:

    I think the ethnic division of power has worked particularly poorly in Lebanon, although the way they do it is rather specific.

  5. Paul Says:

    Or Belgium for that matter:
    “The second problem is one that is particularly serious in a country that has a large minority with very different interests from the majority. […] Suppose that 30% of the country belongs to group A and 70% to group B. ”
    “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.”
    Check! (in one part of the country)

  6. Jason Says:

    I was referred to your site by a mathematician friend of mine. Let me first say that as a victim of the first-past-the-post electoral system, I’m a strong advocate for electoral reform. But I’d like to address some assumptions you’re making at the start of your post, hopefully helpfully from the perspective of a person who has both studied and participated in politics.

    The four things you list are frustrating aspects of politics. There seems to be precious little logic involved in the way that politicians operate. But consider: is there a great deal of logic in the way people sell beer? Are we to be convinced that we will be surrounded with beautiful people in exotic locations because of our choice of beer? That more beautiful people will want to have sex with us more frequently? Clearly not.

    Now, is that because the marketers of beer are too stupid to be able to understand an argument of more than one sentence? No. On the contrary, it is because they understand how you and I actually make decisions a good deal better than you and I do. No one would pay what it costs to run a 30 second beer commercial with girls in bikinis during the superbowl without mathematical certainty that it was going to have the desired effect – that it would change how people choose to buy beer. And that’s exactly what they have.

    It’s absolute nonsense, and it works. But the irrationality is not due to a lack of intelligence on the part of the people communicating the message. It’s is carefully designed to match a lack of rationality on the part of the people receiving the message.

    In politics, the things you describe are communications strategies. Beer ads are short, and they tell you something to the effect of “your life will be better with Brand X”. Simple, punchy, and demonstrably false. They don’t go into detail of trying to justify their assertion that they are the better beer. They don’t consider their own downsides, or the upsides of their competitors. They support their own product even when their own product is worse. And they say the same thing, in the same way, over and over and over because it doesn’t matter whether or not you believe it. It matters whether or not you remember it. Because they know that if you do – if they can get their message into your head – you will make decisions differently.

    So that’s the first part of my argument. That the problem with the discourse is not a problem with politicians. The second part of my argument is a little sharper.

    It’s all good fun to call politicians stupid. But what that does is it denigrates public service. It creates the idea in the heads of irrational people – defined as all of us – that any attempt to do good is to be derided. It makes good people avoid the practice of public service, and leaves it to those whose social radar is so poorly tuned that they fail to see that they are aspiring to be something as respected and trusted as a used car salesman. The net result of arrogant assumptions that politicians are stupid is therefore stupid politicians.

    Yes, we need people who think hard and rationally about the issues. People need to think hard about the electoral system and how it is getting us bad results, and what kind of system would get us better results. But in a democracy, there’s going to have to be someone else who comes up with a punchy, short, flawed, repeated, consistent message about why the average voter should give a crap, so that when they get to the voting booth, they will actually make their decisions differently.

    • Mac Says:

      Your argument misses the point I feel, I think the 4 problems are in how political debate is conducted, not why. I think the problem is that your intelligent politicians are pandering to the ‘ignorant masses,’ which creates the exact opposite situation decry here. Your implication is if the politicians can sell it to the people, it’s the right thing to do. I believe @gowers just wants some of the rhetoric and dogma removed from politicking in general.

      I think your response comes from a defensive-politician standpoint and _completely_ misses why most people dislike politicians. In fact you seem to be at fault of the same denigration as you accuse ‘we the people’ of, just directed the other way.

      Lastly, people don’t deride politician’s attempts to do good, none of the author’s four points pertained to that at all. We deride the seeming idiocy of political rhetoric. Even if it is as you say and the politicians are giving simpleminded constituents what affects them most to get elected, is this really how we want our decision makers to act? There should be a difference between a rote stump speech during a campaign and the way politicians debate in session. The politicians we have in America seem to conflate those two and it brings down the intelligence level of the discourse.

    • gowers Says:

      I think there is an important difference between selling beer and selling the merits of a political party. Suppose, for example (and this is by no means purely hypothetical) that party A takes some difficult economic decisions and loses popularity so that party B wins the next election and comes into power. And suppose that as a result of party A’s necessary but unpopular measures, the economy is then in very good shape for a few years. The politicians in party B may be perfectly aware that the credit for the state of the economy is principally due to party A, but that is definitely not what they are going to say.

      Now the average member of the public will have a tendency to think, “Party B is in power and the economy is in good shape. Therefore, party B are good stewards of the economy.” After all, the truth is a bit subtle. So the falsehoods put out by politicians in party B will tend to be believed, and anyone who tries to explain the subtlety will be shouted down. That is quite different from putting out a message that by buying this beer you will suddenly become part of a group of young and beautiful people. People know that that is nonsense (even if their subsequent behaviour is then affected by that nonsense).

      I do realize that everybody is aware that a lot of what politicians say is self-serving nonsense, so the point I am making applies only to a certain fraction of what they say, where the nonsense has a certain superficial plausibility to it.

    • Gil Kalai Says:

      I do not share the politicians-bashing sentiment in the beginning of the post (e.g. items 1-4) and in the comments. Overall, the political debate in places I witnessed (and I even had a taste of it in the UK) seems rather reasonable, and the anti-politics/anti-politicians sentiments have, in my opinion, other reasons. (Probably the situation regarding commercial advertisement/propoganda is worse compared to politics.)

      Before the Israeli elections I wrote an article http://www.haaretz.com/print-edition/opinion/the-political-game-theory-1.271723 . My conclusion was “The prevailing popular feeling is that our politicians are clearly acting irrationally – but my impression is different. The problem is not irrational behavior but a real difficulty in making decisions under conditions of uncertainty, especially in a reality made up of multiple players with genuine – and sometimes immeasurably great – differences in terms of goals, interests and values.”

      I remember that several people stopped me and told me that they were very happy with my article and that they also agree that politicians are completely irrational.

      As for the “mechanism design” ideas of the post. They look interesting but I am not sure there really is a way to devise a system (even a complicated one) that overcome the difficulties described in the post. (But as mathematicians we are in a win/win situation. We can also prove interesting impossibility results. This is a privilege politicians do not have.)

    • gowers Says:

      The reason I find 1-4 annoying is that it makes it almost impossible to push through certain policies that would in fact be in everyone’s interest. For example, there are quite strong arguments in favour of legalizing and regulating drugs: their use could be regulated, the prices would drop dramatically, crime (both by suppliers and by people who need to steal to pay for their habit) would be greatly reduced, many people who are capable of using drugs in a responsible way (as many people use alcohol now) would no longer be criminals, and so on. Of course I recognise that there are counterarguments, especially with more harmful and addictive drugs. But the point I am making is that even if very careful scientific studies demonstrated the country would have less crime AND fewer drug addicts if drugs were legalized, items 1-4 in my post mean that it would be political suicide for any party to advocate legalization. There are many other examples like this, of good policies that are slightly counterintuitive and therefore doomed never to be implemented.

      I should have made it clearer in the post that a large part of the blame for the situation rests with the general public (and, I am tempted to add, certain journalists). Politicians are under pressure to win votes, so presumably their behaviour has evolved as a response to what the general public likes.

      It is interesting to speculate what politicians would be like if all voters were mathematicians. (I’m not saying that the world would be a better place …)

    • Gil Kalai Says:

      I don’t know. I don’t find academic politics that much different than general politics (except that, as the famous saying says, the stakes for academic politics are too low) and I do not see any evidence that mathematicians (or scientists) involved in any kind of politics behave differently compared to other people.

    • Robin Hanson Says:

      Jason, you are close, but don’t quite say it directly: The key source of those mentioned problems is: voter stupidity. It has little to do with the details of the voting rules; those things occur in a wide range of systems with a wide range of rules. If average voters are in charge, and they are stupid, then appealing to voter misconceptions is often a winning strategy for politicians. It is not that politicians can’t know better, it is that they’d better not say better if they want to keep their jobs. You can try to inform voters better, but stupid voters often prefer not to be so informed.

    • FD3SA Says:

      Robin has nailed it; voter education (defined as the probability of making a rational decision) is the foundation upon which our current system was built. Most western civilizations were created to govern masses of uneducated individuals, who needed representative democracy for the simple reason that they were unable to govern themselves directly.

      Representative democracy leads to perversions which inevitably lead to the structural inefficiency we see today. One need look no further than the incentive structure currently in place for public servants. The power of the free market has been demonstrated by the Western world, yet our market incentives for politicians ensure an expansion of government and subsequently increased spending. The term “public service” means just that, ones who volunteer their time at the expense of financial gain.

      With the advent of the internet and the success of science, it is baffling that our current system persists as it does. It causes great cognitive dissonance to those rational citizens who yearn for a say in their country’s future. One would guess that a superior governance system exists somewhere in the entire “governance space” that fits the information age’s needs without the arduous processes in place today.

      I believe Switzerland is on the right track, and I believe the majority of the western world would do well to follow their example. It is the only country with a forward looking system which can evolve into a mode of governance that is satisfactory for an educated electorate.

  7. Noah Snyder Says:

    Coming from the US where our government has much less party discipline and more ability for electoral minorities to block majority priorities, I actually think both are *bad* as goals (regardless of whether your proposals manage to implement them).

    Both lack of party discipline and minority veto rights result in lack of accountability to voters. Most citizens don’t pay that much attention to politics (for perfectly rational reasons). In particular, people can’t really be bothered to pay that much attention to who is actually to blame for what. The upshot is that the majority party is blamed for lack of progress even when that’s the fault of the minority party or of rogue lawmakers within the majority party.

    Furthermore you get a lot of stupid arguments about whether some party “secretly wants to do X” when in a reasonable system the party that won elections would just do what they want to do and you wouldn’t have to care about what they secretly wanted to do.

    Also the main upshot of “minority rights” in US government was that white southerners could block voting rights for black people, preserve segregation, and stop the federal government from militarily intervening against southern white terrorist groups. So it’s worth keeping in mind that often “minority rights” don’t always end up doing what you want them to do.

  8. Anonymous Says:

    A multiparty democracy seems very roughly analogous to an n-body system in the sense it can have very complicated dynamics that depend sensitively on initial conditions (such as history, culture, economic conditions, demographic structure, etc.), even when the dynamical rule itself is relatively simple.This is most easily seen by considering specific examples of multiparty democracies, such as Norway, India, Liberia, the Weimar Republic, Switzerland, Iraq, and Venezuela.

    In the case of politics, that rule is likely much more complicated than an inverse square law. Given the very considerable difficulties encountered in solving the problem of 3 or more point masses moving under the influence of Newtonian gravity, it seems rather unlikely that its political analog is solvable in general at present. It might therefore be useful to focus on the very special case of electoral reform in Britain, using as much detailed knowledge of British history, economics, demographics, and other relevant information as possible.This is inherently a multidisciplinary enterprise, and I would be surprised if insights from combinatorics, dynamics, and control theory, for example, were completely useless.

    It does seem based on historical experience that regimes of elite rule are stable under “small” changes in the voting system, so that “small” changes in this system make little difference, while “large” changes have very unpredictable consequences. Such considerations suggest that for now a very Burkean approach to electoral reform is wise, whatever one thinks of “first-past-the-post”: there are worse things than British politics, even in its present form.

    That said, this is so important a problem that any progress toward its solution is likely to be very helpful. I think your idea of trying a massively collaborative online approach is incredibly clever, and offers the best hope of making progress, especially if experts in the relevant disciplines contribute, and the information management problems that arise can be solved.


  9. Manoj Gopalkrishnan Says:

    India has a very strong “anti-defection law” which makes our political parties very powerful, and our legislators toothless. I quote: “A member of a House belonging to any political party shall be disqualified for being a member of House if he votes or abstains from voting in such House contrary to any direction issued by the Political party to which he belongs or by any person or authority authorised by it in this behalf, without obtaining, in either case, the prior permission of such political party, person or authority and such voting or abstention has not been condoned by such political party, person or authority within fifteen days from the date of such voting or abstention.”

    This version of the law was introduced in 1985 because of a widespread suspicion that members of parliament were being bribed for their votes (horse-trading). Personally, I do not like it very much. I suspect it is to blame for many of the ills of Indian democracy, like dynastic rule, a large number of small political parties and absence of meaningful debate in parliament.

    You propose randomizing the vote and secret ballot. Both these measures would make legislators less accountable for their vote. This brings with it a risk of increased horse-trading.

    • Manoj Gopalkrishnan Says:

      On second thoughts, a secret ballot has some advantages that make it horse trading-proof. One may pay money for a legislator’s vote, but it is difficult to detect a double-cross.

      My concern then would be, echoing a problem Prof Gowers pointed out in his post already, what is to prevent politicians from being “hypocrites.” They could say one thing during elections, and then do another in the secret ballot. Perhaps there is a trade-off between making a system hypocrite-proof and making it horse trading-proof.

    • Anonymous Says:

      Manoj, to what extent, and in what way, would politically possible changes in voting rules in India influence public policy in India?

    • Manoj Gopalkrishnan Says:

      @Anonymous, I am not an expert in this. But here are my impressions. Before 1985, India was effectively a one-and-half party system. In 1985, the anti-defection law was strengthened. The next couple of decades saw the fragmentation of the larger political parties, and emergence of many smaller parties. It appears to me that the heads of political parties became more powerful, compounding the problems of dynastic rule, nepotism and sycophancy. I also have a lurking suspicion that, since the party decides a legislator’s vote, the quality of parliamentary debate has deteriorated.

      What would happen to public policy if voting rules are changed? I have no idea.

  10. Anonymous Says:

    It seems that requiring representatives to vote with their parties reduces the process of corruption to that of corrupting party leaders, who may then be merely the most skillful practitioners of deception. A large and well financed collection media organiizations and “intellectuals” can be recruited to assist such deception. This phenomenon appears particularly striking in the US, where handing money to politicians in slightly indirect ways has conveniently been deemed “constitutionally protected free political speech” by the courts, making it extremely difficult to regulate.

    “Revolving doors” between political elites and corporate and financial elites also appear particularly difficult to close. Since politicians write the laws that regulate such elites, there seems to be a feedback loop, with plutocracy as something like a locally asymptotically stable fixed point. IMO it would take radical changes with highly unpredictable consequences to change this very much.

    • Anonymous Says:

      Fortunately, the relaxation time of the political system following reform may be quite long, as long as a generation or two. However, it may take random shocks of the magnitude of WW2 to produce genuinely significant reform.

  11. Anonymous Says:

    Update:Not all plutocracies are bad. Some can be reasonably enlightened and public spirited, but this depends more on history and culture than on voting schemes. I hope Cameron and Clegg have a healthy sense of noblesse oblige.

  12. Ross Anderson Says:

    There’s a huge research literature on this subject; read Amartya Sen’s book on social choice for an axiomatic treatment.

    Anyway, the idea that you can lie about how you voted with a set probability (say 10%) is an intriguing one. Curiously, the UK census next year plans to protect privacy for citizens by swapping personal data with low probability between similar households. What do you think of that? Probably not much; if your personal data get leaked you have some deniability in a criminal case where the test is “beyond reasonable doubt” – but not in a civil case “on the balance of the evidence”.

    I also doubt it would work in parliament; politicians tend to be empathisers rather than systematisers. If Nick Clegg votes against the Government, David Cameron would ask “did you really do this”? and Nick would say, with sincerity, “no”. (Of course, the 1 in 20 Lib Dems who vote against the government more frequently than the 95% confidence level would have to be shot.)

    • Anonymous Says:

      The question I was raising is: Do we under stand real world political behavior well enough to model it with reasonable accuracy by a handful of simple axioms?

  13. polyanon Says:

    I would say that the “mistake” in your proposals are to consider the problem within a too abstract setting and in this abstract setting consider that political issues are “linear”, can be voted independently one of each other one by one. But political issues are not linear and because political issues are not independent we do have ideologies. Ideologies are (try to be) coherent answers to all the political issues of a given time.

    Long time ago i had the idea of comparing votes and money. Votes in legislatures are not be consumed. 55% of seats of a parliament are 55% of seats during the whole legislature. But money can be consumed. 55 pounds can be transformed in 0 very soon as we all know. So why not consider votes as something which can be consumed during the legislature ? If there were 100 voters and two political parties, your party gets 55%, that is 55 votes, and your oponent gets 45. Voting is anonymous for both parties. If in the first issue, the first party consumes all his votes, let´s say 55 and the other 0, then for the rest of the legislature you have 0 and your oponent 45. It was just an idea. I even started to make some abstract modeling and calculations to inquire how such a system could work, comparing with other systems. But then i realised that ideologies comes in full packages (issues are not linear). A second thing was that you can not change this without changing the whole system. Political institutions, all institutions have some “organic character” (if votes are consumed, why do you need several representants for each party in the parliament in stead of only one or a small group ?) . These two second thouhgts made me to leave this weird idea. And i think the “non-linearity” of political issues is is what makes political discussions to look so irrational.

    So you present a full-package or nothing ideology and if you win with a sufficient majority you can apply it: that´s coherence. Then if the society improves, you win again with more majority. This social algorithm iterates until your ideology made some wrong assumption and something went wrong. Then your opponent, which surely made some changes in his ideology, wins and the algorithm re-starts.

    We must not forget that at the end political institutions are substitutes of war not of mathematical proofs.

  14. Schmoo Says:

    The first system introduces randomness, albiet only a little, in order to fix problems of representation? Go and stand in the corner!

  15. Brian Powell Says:

    What a genuine pleasure to see you consider such issues. I only despair that the type of considerations that you have raised are typical of the broader frustrations of any “rational” approach to sociopolitical issues. How do people of thought challenge people of power ?

  16. The Great Geek Manual » Geek Media Round-Up: May 12, 2010 Says:

    […] If politicians were mathematicians […]

  17. JD Says:

    I’m not sure how well the 10% rule would shield party disloyalty in a strategic setting. The problem is this:

    “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.”

    So what is the threshold for “too much”? For each legislator’s voting record, you can establish the probability p that their number of cross-party votes could have arisen by chance given the 10% error rate. Say a party leader then decides, okay, anyone with a p < .05 has clearly gone against their party rather than just getting bad luck in their errors (eg, more than about 115 cross-party votes out of ~1000 total votes in a year).

    So now everyone who has an inclination to vote against their party more than 1.5% of the time votes against it exactly 1.5% of the time on purpose. But that's no good, since the party leader can easily discern a bump in the histogram of cross-party voting rates at 11.5%, and punish all of them knowing that he will incorrectly punish only 1 in 20. So now they have to back away from 1.5% — but not choose the same rate, since that will produce just as discernible a bump. So they must all choose random values < 1.5% to disguise the group.

    "But why should *I* be the one to vote against my party at .05% just to help disguise those who are voting against it at 1.5% or more?", a party member reasons. So now the faction of would-be disloyalists needs its own internal rule (enforced how?) to make sure they all don't choose the highest possible value and give the game away — a high-risk strategy since, if the party leader discovers them, they're all punished, and all for a payoff of getting to vote against your party on purpose about 1 time in 100. But without such a rule, every disloyalist will just choose the highest cross-party voting rate they can individually get away with, which, with everyone choosing the same rate, will always give the game away whenever the party leader sees that telltale bump in the histogram.

    So all in all, a 10% error rate buys members almost nothing in terms of elbow room to occasionally vote against their party when, as is the case, there are many hundreds of votes every year.

  18. Geek Media Round-Up: May 13, 2010 – Grasping for the Wind Says:

    […] If politicians were mathematicians […]

  19. Jeff Burdges Says:

    I’d agree that reducing representative accountability to party whips and campaign donors sounds like a noble goal. We’d surely lose exactly the same degree of accountability to voters, which dose not sound wise. I suspect the powerful would learn how to manipulate your system better than our current system.

    Deliberative democracy however provides another solution :

    All representatives vote not just yes/no/abstain for a bill, but also name an advocate for their position. A bill no longer becomes law once passed, but must instead then pass a jury trial, and all advocates supported by enough representatives, say 5%, may argue for acceptance, rejection, or modification. If the country has a president, his veto would be eliminated, and instead he’d appoint and advocate too.

    Jurors are selected purely randomly from the voting population, and advocates may not challenge jurors, but instead juries are quite large, say 100–200 people. Jurors’ votes are naturally anonymous, making them accountable only to their own conscience.

    I believe this system would provide exactly the benefits you seek from anonymity while keeping the representatives themselves fully accountable. In particular, we’d see multiple advocates from all the major parties, proposing slightly different modifications. Yes, the party whips could limit diversification, but sending too few advocates would often prove suicidal.

    In addition, we expect this deliberative trial phase would be extremely good at :

    (1) cutting pork barrel spending : Even if your collation partner sends an advocate that supports your health care bill overall, they’ll likely backstab you by advising the jury to cut that new power plant you snuck in.

    (2) giving minorities a voice : If you read the experiments comparing opinion polls with `deliberative opinion polls’, you’ll find an overall 10–20% movement towards the position independent and knowledgeable people more often hold, like being pro-choice or less racist.

  20. F Pait Says:

    It is because of issues like these that the founding fathers in their wisdom have decided to write checks and balances into the constitution, as they say in America. A two-chamber congress and three independent government branches ensures that the US ends up doing what is right, often after trying all the other alternatives. Considering the crooks and idiots going into politics, that is little short of a miracle.

    The best arguments against proportional election or secret votes by representatives can be obtained by observing the workings of Brazil’s congress. If Britain decided to alter somehow its voting system, I as an engineer would suggest direct elections for prime minister and the House of Lords.

  21. Woett Says:

    It seems like that the basics of your article don´t depend that much on assuming what if politicians were mathematicians, because the four rules that annoy you (and me as well btw!) are not caused by politicians not being mathematicians, but more with politicians not being reasonable and logical. Well, not even that; it’s entirely possible that a politician in fact is reasonable and logical, but knows that more people will believe and vote for him, if he makes a certain argument that, although not making every sense, is just very convincing. And in any case, it’s more the community’s fault that politics works the way it works, than the fault of politicians.
    But this may be just nitpicking, because I don’t think it changes the philosophy of your post, which to me is very interesting.

    On a totally unrelated note, but I don’t know where else to put this; since I have been reading a lot of math blogs lately, I decided to start my own blog about (my experience with) math. And because that decision was highly influenced by the succes of yours, I wanted to put a link to your blog on mine. I hope you’re ok with that.

  22. Science Friday, Only Better! (Genetic testing, Irish research events & Meat-cleaver fun) « Ian There Says:

    […] Why election reform seems simple to a mathematician but implausible to a politician. […]

  23. Anonymous Says:

    I suspect that introducing a voting system or parliamentary process comprehensible only to mathematicians would only undermine public confidence in democracy. Moreover, doing so would likely render the Ship of State incomprehensibe even to its captain, causing certain unfortunate accidents. The captain could of course rely on mathematical navigators, but that would produce an even more elite rule by a cult of High Priest-Navigators guided by the charts of an abstract realm with God knows what relation to the high seas.

    • Anonymous Says:

      “High seas” is a special case. I had the Titanic in mind. I should have said “sea”.

  24. vipulnaik Says:

    As has been pointed out, the problem isn’t so much that politicians are irrational, it’s that voters are irrational, and they actually enjoy having false beliefs and enjoy politicians who employ 1-4. This allows two types of politicians to flourish: (a) those who share the voters’ irrationality (b) those who are Machiavellian enough to pander to voters’ irrationality in their rhetoric while secretly pushing more “enlightened” policies.

    Voters reward type (a) more if they can see through politicians’ pretensions (thus, allowing only the honestly deluded to the cynically enlightened) and they reward type (b) more when they care about their irrational beliefs as well as results that couldn’t be achieved through their favorite policies. The politician of type (b) pays lip service to the voters’ delusions while pushing policies that voters hate but that generate results that voters like.

    Your idea might help type (b) gain more over type (a), at least as long as voters can at least make some sense of which politician was broadly responsible for which results (even though they make the wrong policy-result link). But as you’ve pointed out, voters usually don’t even know what policies have short-run versus long-run effects, so if political parties keep taking turns in power, the gains in rewards to type (b) are correspondingly reduced.

    Some related ideas are discussed in detail in Bryan Caplan’s book “The Myth of the Rational Voter” (mostly a US-specific analysis, but the issues are likely to be similar everywhere).

  25. Anonymous Says:

    I think we have some idea of what politics would be like if politicians (and voters) were like mathematicians (cf. Star Trek, Vulcan). But Vulcans are also especially honest and altruistic, so I suspect that rules 1-4 have to be supplemented with additional normative rules to create a civilized representative democracy with reasonably strong efficiency, equity, and stability properties.

    Of course, an approach analogous to one contemplated by Prof. Gowers has been tried in Britain with some success: devolution. But I’m not sure how much further this general sort of approach can carried without doing more harm than good.

    • Anonymous Says:

      I think the same sorts of additional conditions are required even to obtain reasonably civilized political discourse.

    • Anonymous Says:

      Sorry. I meant “abandoning rules 1-4 has to be supplemented by”. I think its right that following the standard rules of reasoned discourse would greatly improve the tone and substance substance of political debate, and that might make it just a tad harder to hijack the ship of state.

      So I’m not opposed to a reasoned approach to politics, only to an overly rationalistic axiomatic-deductive one. After all, the history of the twentieth century doesn’t cast a sunny glow on ideological thinking.

  26. Tim van Beek Says:

    The system in Germany has some of the aspects that are discussed in this thread:
    1. There are parties, one way to get into the parliament is to join one, get on the list of that party at a position n such that the party gets a numer of seats greater than n.
    2. Once you are in the parliament, you are completly free to vote, you are obligated to the greater good of all Germans only, by constitutionaly law.
    3. You may request a voting in secret, which works just like the election of the parliament itself, everyone gets a slip of paper, notes his/her choice on it etc. But by default votings are not secret.
    4. The head of your parliamentary group may request everone to vote in a certain way: If he does not, that means no one will be punished for voting contrary to what the party says. If he does, and you vote for the opposite nevertheless, you will probably have a hard time to make it on the list of that party for the next elections.
    5. But that need not be a problem, if you get the majority of votes in your district, you get into the parliament, no matter if or if not you belong to any party.

    All in all this works pretty well in getting the members of the parliament to vote most of the time in the way the party promised during the elections, and let everyone vote free on matters where there is no “mutual consent” on what the party would say 🙂

  27. Gareth Says:

    “It is this last part that almost no politician would understand, since non-mathematicians have a strong aversion to the probabilistic method.” I would certainly have a strong aversion to a method which could (though it probably wouldn’t) attribute to me a very disloyal voting record when in reality I was being completely loyal. Is it mere mathematical ignorance not to want to leave one’s reputation to chance (even very good chance)?

  28. davidellis2 Says:

    I think the biggest potential problem with the random vote-flipping system is that it would make open debate harder. If an MP agrees with most of his party’s policies (as one would hope!) but with a few important exceptions, surely it’s all the more important that they come out in public with arguments against those policies, with the possibility of influencing the debate and even the outcome.

    [One could argue that this could be got around by politicians making anonymous press releases, but this would throw up many other problems. In the UK system, it’s important that people have an accurate idea of the voting record of their MP on issues that matter to them. If the MP’s vote isn’t flipped, and he is questioned about the way he (is presumed to have) voted, surely he has no other option in the long run than to give his full opinion. Concealing it would both confuse his constituents and make it difficult for him to present a consistent case in future arguments.]

    Generally, I think, the more openness there is in a democracy, the better. The political debate often benefits from `maverick’ MP’s breaking the party lines, despite possible damage to their career. On the other hand, I think one of the biggest problems in democratic politics is when politicians conceal their convictions and tow the party line JUST in order to keep climbing the career-ladder. In the long run, this must be detrimental to their constituents and to the country as a whole, and one would hope that the risk of appearing self-interested would be a strong discouragement. In reality, however, I think this attitude is all too common. Isn’t there a risk that the random vote-flipping system could make it an easier trap to fall into?

    [Interestingly, despite relatively strong party discipline, the Labour party in the UK has consistently had strong dissenting voices; even Robin Cook wasn’t too wedded to his career prospects to resign over the Iraq war…]

  29. davidellis2 Says:

    PS. I’ve explained my position a bit more clearly at http://davidellis2.wordpress.com/2010/05/31/free-votes-and-referendums/

  30. davidellis2 Says:

    Like some other respondents, I suspect that there’s no ‘reasonable’ voting system that can entirely overcome the ‘tyranny of the majority’ problem. In practice, though, even totally self-interested people often realize that it is valuable to promote minority rights as an insurance against finding themselves in a minority over some issue. I’m inclined to think that people’s motives to maintain the ‘social capital’ of a country – trust, and reciprocal respect of rights – is the best safeguard against the tyranny of the majority. When this breaks down entirely, secession seems like the only alternative. A majority will often resist secession, but history shows that this is a great strain on a country in the long run, and not usually sustainable…

  31. ranggaw0636 Says:

    Politician huh?

  32. Peter Says:

    I think most of the contentious political issues are precisely the ones where mathematical reasoning cannot be applied, but instead tend to evoke emotional responses, and thus are thought entirely on the basis of either party’s deeply ingrained feelings on the matter (something that might be difficult to understand for some mathematicians). Even if one side were to know they are wrong, which in practice does not usually happen thanks to something called denial, they would suffer were they to loose the argument (perhaps due to cognitive disonance, loosing ones ideal view of what the world should be like, or maybe something as simple as finding oneself on the loosing team). Thus arguments that seem very flawed for the rest of the us keep being perpertuated by otherwise reasonable people. I think everyone is guilty of this to some extent. Take for example an argument between a husband and a wife over an ugly oversized antique sitting in the corner of a crowded living room. The wife wants to make room for a much needed sofa, but the husband refuses, for the antique is a reminder of his travels during his younger days. It’s not quite clear cut who is right or wrong. One might try to measure the total amount of suffering incured if the antique was to be thrown away as opposed to if it was to stay, and make a decision based on this. But there being two parties, the husband and wife, just as there is a strong dividing line in the American politics these days, the answer might still not be obvious.

  33. Stephen Montgomery-Smith Says:

    Your notion that Mathematicians would act more responsibly than politicians is demonstrably incorrect. My example is the Mathematics Department at the University of Missouri at Columbia. Of course, when we are arguing mathematics, we use all our powers of logic. But when we are arguing issues like who we are going to hire, or who the next department chair should be, we stoop down to a level of logic that would even shame the US Congress or the United Nations.

    Mathematicians are people, just like anyone else. They have personal agendas, and are just as happy to throw out regular logic as anyone else. In fact, they are usually much worse, because they live in this delusion that they do apply logic to real life, and are incapable of seeing themselves as they really are.

    • Stephen Montgomery-Smith Says:

      You know, I need to follow up because you did say “I fully admit that professional mathematicians are as capable as anyone else of making stupid collective decisions.” So my opening statement is too harsh.

    • gowers Says:

      Yes, I tried to get that qualification in quickly in order to forestall that possible criticism.

  34. Elijah Laws Says:

    I’m a student of mathematics from the US, and I have been lurking on your blog for quite some time. Although my perspective is from an American standpoint, I think you are touching upon an issue that exists within governments around the world. Personally, I think new weaknesses in governments are coming to the surface because of modern day inventions. Television, sound recorders, and other devices may be creating a situation where doing business between elections is becoming impossible.

    While watching congress discuss issues on the floor, one finds that debate does not exist; instead, one only sees a repeat in election style talking points. At one time in our government’s history, members of congress could hold more honest and open discussions on the floor, but the presence of video and sound recorders have stifled the possibility for debate. Because these people are surrounded by recording devices, they are using very carefully scripted talking points because they know one misspoken word could cost them their next election.

    Under such conditions, your points 1 through 4 are being used.

  35. Fabien Besnard Says:

    Thank you for this very stimulating post.

    I think your right in pointing out the fact that party discipline has become a burden for representative democracy (In my opinion, political parties as a whole have become such a burden).

    In the first days of modern democracy, Condorcet, who was both a mathematician and a politician, thought a lot about this issues. To sum up, his idea was that democracy was superior to dictatorship because an assembly of (enlighted) people who gather to sincerly seek the truth (that is, the best decision for the general interest) through a voting procedure, is less likely to make mistakes because on the average those who are voting because of bad reasoning will compensate, and those who are correct will shift the equilibrium towards the good decision.

    Of course, the modern assemblies are nowhere close to a “group of people who sincerly seek the truth”. I do not think that changing the voting system will remedy that, although parties voting as blocks certainly kill Condorcet’s insights.

    Theoretically, a deputy who votes “wrongly” takes the risk to be punished by the citizens at the next elections. However, there are too many way to avoid this sort of punishments : people may forget, the deputy may be on a list, or he or she may have an easy-to-win district.

    Thus, the punishment of the party is far more fearful than that of the citizens. I think that this is where the solution to the problem is to be found : the system should be such that a representative would fear the citizens more than his/her party. For instance, every representative and government member could be judged by a random jury of citizens at the end of the mandate, in order to decide if he has been sincerely seeking the general interest. The punishment could go as far as forbiding him to participate in any election whatsoever. I think that this kind of system worked in Athens, long ago.

  36. Quora Says:

    Who are, and who have been, the most intellectually transparent politicians?…

    And has their intellectual transparency proved to be successful for them in their careers? In this context, “intellectual transparency” means being impartial about debating policies; being as prepared to find holes in your arguments as you are in oth…

  37. isomorphismes Says:

    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.

    Dr Gowers, I’m surprised to hear anyone compare mathematical reasoning to politics.

    Rather than needing to parse elaborate arguments where one misplaced sign breaks the whole thing, politicians deal with doubtful arguments from many competing interests. They need to figure out who’s lying or who has what perspective, and to make value judgements which do not reduce to logic.

    Most importantly, they need to make decisions with no good answer at all. When deciding which group to do under there’s really no satisfactory answer, and can be none, yet a choice must be made.

    Lastly, most politicians have law degrees–no casual feat. I’m pretty sure they can in fact understand arguments.

    • isomorphismes Says:

      Here is a real-world example of politics, regarding public transit in San Francisco http://kfarr.files.wordpress.com/2013/04/wiener_ceqa_legislation_package.pdf. I could arbitrarily dig around for something else in another locality such as budget discussions somewhere in the Midlothians or whatever. I’d love to hear how the problems discussed result from lack of reasoning prowess or how the IQ of the politicians is what’s causing the problem, rather than unerasable real-world difficulties.

    • isomorphismes Says:

      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.

      This seems to me like the United States’ reason for separating power into three branches, with the judicial branch being charged with protecting minority interests.

      But this system can in principle lead to decisions being made that are not supported by anything like a majority of members of parliament.

      Persson and Tabellini wrote a (in my opinion, wonderful and engaging) book with different mathematical models of political regimes. They start with the median voter theorem and move on to model political parties, politicians who are unable to credibly commit to post-election behaviour, and voters with unpredictable preferences (teaser: in the end the more predictable groups are satisfied, even if they are fewer in number than harder-to-predict voters).


      Probably if you read this book and get interested in “spatial voting theory” and constitutional design, that field will improve and potentially the design of constitutions around the world could become better.

  38. If politicians were mathematicians – Mathpresso Says:

    […] If politicians were mathematicians […]

  39. If politicians were mathematicians - Esmaeil ASLANI DIRANLOU Says:

    […] If politicians were mathematicians […]

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