Game theory, maths and the EU referendum

by Tim Gowers | 19.06.2016

Does mathematics have anything to say about whether we should stay in the EU? In one sense, of course, it does. Numbers matter – for example, the fact that we do not send £350 million a week to Brussels.

For a mathematician, however, mathematics is not so much about numbers, important though they are, as it is about abstraction: that is, getting to the heart of problems by stripping away their inessential features and searching for widely applicable general principles. So instead of asking whether we should stay in the EU, I am drawn to a more general question: what is the point of cooperative agreements?

This question, which belongs to the branch of mathematical economics known as game theory, has been much studied. In brief, the answer is that when you have a group of interacting agents, the following situation often arises.

  1. Each agent has a choice of two options, A and B.
  1. It is better for everybody if everybody chooses A than if everybody chooses B.
  1. Each individual agent gains by switching its choice from A to B if the other agents do not switch.

This situation is called the Prisoner’s Dilemma, after a famous illustrative example that concerns two prisoners who have to decide whether to betray each other. Whenever it arises, there is a case for an enforceable cooperative agreement: without an agreement, each agent will tend to act in its own selfish interest and choose B, which will be worse for everybody; with an agreement that forces the agents to choose A, they will all be better off.

Here are just a few of the many examples that apply when the agents are European nation states.

A country is better off if its workers are decently paid, do not work excessively long hours, and work in a safe environment. (If this is not a high priority for you, that just means that you will need other examples to illustrate the abstract principle.)

However, treating workers decently costs money, so if a company is competing with companies from other countries, it will be tempted to gain a competitive advantage by paying its workers less, making them work longer hours, and cutting back on health and safety measures, which will enable it to reduce the price of its product.

More generally, national governments will be tempted to gain a competitive advantage for their whole country by allowing companies to treat their workers less well. And it may be that that competitive advantage is of net benefit to the country: yes, some workers suffer, but the boost to the economy in general reduces unemployment, helps the country to build more hospitals, and so on.

In such a situation, it may benefit an individual country to exercise a B-type option by becoming “the sweatshop of Europe”. If that is the case, then in the absence of a supranational organisation that forbids this, there is a pressure on all countries to do it, and if they do, then there is no competitive advantage any more, so workers end up worse off and there is no compensating gain.

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Another example — so obvious that I won’t dwell on it — is the need to combat climate change. Here, option A is to take significant steps to reduce emissions, and option B is not to bother. The world as a whole will be much better off if we all choose option A, but any individual country stands to gain by being selfish and choosing option B. So again we need enforceable supranational agreements. (While the EU cannot combat climate change on its own, it can negotiate with the rest of the world much more efficiently as a bloc.)

A third example is tax harmonisation. Without it, countries are free to compete by setting low rates of VAT and corporation tax. This distorts markets and creates a pressure to race to the bottom. If governments want revenue from these taxes, then they need agreements to set minimum rates, as we have in the EU for VAT but not, so far, for corporation tax.

Next time you hear a Leave campaigner complain about EU control and regulation, ask yourself whether what they really want is to defect from an agreement that is there to deal with an instance of the Prisoner’s Dilemma.

Why do 92% of UK fishermen want to leave the EU? Because they want to overfish while the rest of the EU agrees to quotas. Why does farming minister George Eustice want to leave the EU? Because he wants to be free of the EU’s birds and habitat directive while the rest of the EU’s farmers spend money looking after their birds and wildlife habitats.

Can we trust our government not to succumb to these kinds of agreement-breaking temptations if it is free to do so? No, which is why in areas where international cooperation is needed, international organisations such as NATO, the World Trade Organisation and the EU are needed to enforce it.

Tim Gowers is British mathematician, a professor at the University of Cambridge and winner of a Fields Medal in 1998

Edited by Hugo Dixon

12 Responses to “Game theory, maths and the EU referendum”

  • While the points in the article are generally well-presented, and certainly convincing in theory, the whole argument falls apart when one notices that wage dumping and tax heavens through regulations have been going on within the EU for decades, and there does not appear to be aby action to change the situation.

    • True there are offshore tax loopholes. But it’s not logical to conclude this means the ‘system’ or Prisoners Dilemma model is somehow flawed. it just means the EU Commission has not yet managed to gain the political support from Member States to address the loopholes. It’s perfectly possible that such a consensus can be achieved, and indeed the political momentum seems to be slowly gathering pace. So it’s an argument for _more_ cooperation through EU mechanisms rather than less.

      • Not just offhore tax loopholes. Honest-to-goodness policy in some European countries is to be tax heavens (Luxembourg and the Netherlands spring to mind), has been for a significant period of time without any serious -or successful- attempt from the EU to crack down on this practice. Similarly with wage dumping; despite grumbling and the occasional ineffectual stern note the situation persists.
        I will happily agree that it is theoretically possible that the EU will tackle these problems. The empirical evidence however is that it has persistently failed to do so- and from what we read in our corner of the world, *how* this will change has not featured at all in the Remain campaign.

  • This model assumes agents (countries) are ‘exchangeable’.

    Nothing specifically relates to the UK, so this argument could be applied identically for Turkey to join the EU or the UK to remain.

    This is an reasoned advocacy of Federalism, which has many advantages; but to deduce (for example) that corporation tax harmonisation should be enforced seems to be a dubious extrapolation.

    • “this argument could be applied identically for Turkey to join the EU..”

      The Prisoners Dilemma model doesn’t apply for EU expansion to Turkey as it not in any Member State’s individual national interest for (a) EU to have a shared border with Syria and Iraq (for security), or (b) for Turkey’s large population to have freedom of movement. So there will not be the necessary inherent conflict between individual Member State interests and collective interests.

      • I think the point is that if the argument is meant to establish that it’s better for the UK to participate in the EU because it is better to have enforceable agreements about wages, environmental standards or tax havens, then the same argument also supports having Turkey ( and Mexico and China and Somalia and North Korea and everywhere else) in the EU as well.

  • According to this article it would be better for UK to leave EU then? Or are you suggesting EU would get involved in a race to the bottom with UK?

  • While the text is sound, it’s credibility could be improved if Norway wasn’t coloured in as part of the EU in the graphic 🙂

  • The ‘prison’, in this case, is Fortress Europe, which we must now trust to make more rational choices in the ‘prison’ of Earth. Your model doesn’t explain why we should trust it to do that, and frankly, I don’t.

    Ask EU-ophiles about their motives, and first you’ll hear guff about harmony and brotherhood, and then, when you challenge them, the urgency will enter their voices and they’ll start on the overwhelming need to make a common stand against the United States and China. And underlying it all, never mentioned but always present, is the core EU-xenophobia: Germanophobia. I don’t feel those particular phobias, but I do fear the diminution of democracy, and in a world where less- or un-democratic powers are rising, democracy is the last thing we should be abandoning.

    Also, A and B in your model are Member States of the European Union. The alphabet is longer than you seem to realise, and weakening tariff barriers between rich countries in order to strengthen them against poor countries shows your model of cooperation to be just another conspiracy.

  • Please share, even if your answers are no.
    Before you go to the polls for the EU referendum, there are some things I would like you to do.
    Just for a while, ignore everything that the Remain and Leave campaigns have told you. Ignore the promises and assurances, the threats and the fearmongering. These are, after all, the same economists who didn’t predict the Economic Crisis, or the multiple recessions we’ve been through since the century turned. These are the same historians who couldn’t predict the rise of the so-called Islamic State, or the various terrorist groups that preceded them. And these are the same politicians who have made various promises to their constituents to bring them to and keep them in power, and failed to keep them when they reach their goals. And no matter what any of them say, the nations of the UK will survive, the islands will not crumble into the ocean, and the next few years are going to be tough ones, no matter the outcome.
    Instead, think on your own personal experiences of a UK in the EU, then ask yourself just two questions:
    1. Do you think that the UK is, or could be, strong enough to stand on its’ own?
    2. Are you personally willing to put the effort in to make it happen and keep it that way?

    • Those aren’t the questions on the ballot. The question to ask yourself is exactly as the article says; are we better off cooperating in a bigger group, or competing alone?

  • Hello !
    as an admirer of Great britain and of Her Majesty The Queen, I would really love Great Britain to stay in the EU. However with some pre-conditions: most important is the commitment to promote Peace and stability and abandon all actions designed to destabilize governments in foreign countries like it was unfortunately done In Irak, Ukraine, Syria.