The game theory inherent in Brexit
If I were to sit down and play poker with you this morning, I’m not going to show you my cards before we even start playing the game…
So said Priti Patel, the UK’s pro-Brexit international development secretary this past Sunday alluding to the game theory inherent in Brexit negotiations.
The power derived from getting the game theory right, of course, is linked to paying attention to the structure of a problem and abstracting away particular details: if you know what kind of game you are playing you can work out what strategies people will follow.
Despite Patel’s comments, however, Brexit negotiations are not much like poker. Everyone knows everyone else’s hand. What they do not know is what everyone will ask for and what they will give up for it, making it a slightly different kind of problem.
The negotiations fall under cooperative game theory — where the aim is not to beat people but to get them on your side and work to the same purpose.
The UK is essentially involved in a 28-player cooperative game and needs to assemble a coalition to back its aims. The dynamics are closer to those underpinning reality TV game shows or an American high school, where popularity achieves success.
Think of a factory owner and some workers. Combining their labour and capital they can earn some cash. The way in which that cash gets distributed –i.e. whether as profits or wages — depends on the respective strength of the worker’s union, the costs of mothballing the factory for the length of a strike and what they both can earn elsewhere and so on.
You can use the same kind of cooperative strategy to understand Britain’s negotiations with the EU. Both the UK and the remaining 27 countries in the EU profit from cooperation because trade is mutually beneficial to all. Whether a stable outcome exists and allows for the distribution of any surplus, however, depends on who has the bargaining power.
No one yet knows the legal structure of negotiations to leave the EU. For the purposes of this post and to keep it simple I am going to assume the UK government only needs to assemble a qualified majority of member states for a “withdrawal agreement” and ignore the prospect that they could need unanimous consent for a trade deal too.
One of the problems with sustaining cooperation is that it can often be better to have a smaller group and share the gains more narrowly. High school cliques succeed by excluding as much as recruiting — but then the excluded can always get together and make their own group.
Imagine six gangsters break into a bank but then find only four are needed to carry the safe home.
Having all six gangsters work together is unstable as any four could get rid of the other two and each earn more. But a group of four does not work either as the two who are left out could always promise two of the others more and tempt them into their gang.
This new group would not be sustainable as the process can just repeat itself again with two of the others as outsiders.
Call this the Reservoir Dogs scenario: no one trusts each other, so no one gets the money and (nearly) everyone ends up dead.
In cooperative game theory this is referred to as the game having no “core” — meaning there is no feasible and stable coalition where no subset of the coalition have an incentive to leave.
Doing it the Shapley way
Another kind of solution to cooperative games is to pay everyone their expected marginal contribution – in other words weighting their votes by how much they contribute and how likely they are to be the one to make a difference between a deal going ahead and a deal failing.
This is known as the Shapley Value and is often seen as a ‘fair’ way of distributing the gains of cooperation. In the gangster example this would mean just splitting the loot six ways.
The “qualified majority” referred to in Article 50, the law governing how a country can leave the EU, means governments representing about 65 per cent of the EU’s population need to all say yes to a transitional arrangement.
According to Chris Hanretty, a political scientist who is very good at maths, out of the 134m ways in which 27 countries can arrange themselves into coalitions there are 226,670 occasions where just one country could make the difference between a deal failing and a deal going ahead.
These are called the blocking minorities, where countries representing more than 35 per cent of the EU’s population get together to block the deal.
As an example, if Belgium, Austria, France and Germany all worked together they could stop any deal and so could a coalition of Hungary, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Portugal, Slovakia, Slovenia, Spain and Sweden.
The range of possible coalitions indicates the importance of the larger countries to any deal: of the 508 four-country blocking coalitions 432 feature Germany and 313 include France. These are the smallest possible groups who can block a deal with the UK.
That suggests these bigger countries – as you might expect – have a lot of bargaining power.
For the moment this is all just the product of the maths. It could be that, while Spain is often the deciding vote mathematically it is completely irrelevant to the negotiations as they have already made their mind up and no amount of politicking or money can persuade them to change it.
It could also be that countries’ differing goals mean that no agreement is possible and we end up in the Reservoir Dogs scenario.
The British government wants three things out of the negotiations: minimal barriers to trade with the rest of the EU, maximal restrictions on freedom of movement and to pay as little for it as possible.
The rest of the EU wants lots of freedom of movement and free trade – but would like to hive off some of the UK’s high taxpaying financial services – and for the UK to pay them as much as possible.
But not all countries give each goal the same weight. For example, the Visegrad Four – Czech Republic, Hungary, Poland and Slovakia – are really keen on freedom of movement while France seems a bit more interested in shafting the UK on financial services.
For Germany the big priority might be keeping the EU together so the UK cannot be allowed to have any restrictions on free movement. Greece, on the other hand, might not care too much about freedom of movement or free trade; it just wants debt relief.
So the UK could recruit people to its side by making side-deals and buying them off individually – maybe Poland ends up with a shiny new railway, perhaps Ireland gets some of its debt forgiven.
And the rest of the EU can make their own deals so as to present a united front to the UK. Germany can just as easily forgive debt and provide pork barrel infrastructure spending.
To rephrase Ms Patel’s poker metaphor we can work out everyone’s hands but not the bids they are willing to make. Nevertheless it gives us a surface-level framework to understand the scale of the challenge facing the UK government.
Time as a pressure point
And there are other features of the problem that are important too – the UK will automatically leave the EU after two years whether it can get together a stable coalition or not. That puts the pressure on the UK as they will lose 27 export markets while the rest of the EU will only lose one.
In game theory this is known as the disagreement point – what happens if bargaining fails. Even it is never reached it determines the outcome of the game, as the one who suffers most in the event that negotiations break down should have the most incentive to get everyone to agree.
Putting this all together: the UK needs to recruit 20 other countries to agree with it in a period of two years. Its priorities are likely to be restricting immigration and getting the most access to the single market it can. To achieve its goals it will need to pay people off, with the marginal countries able to extract the highest prices.
Or to cut a long story short: either the negotiations end up a lot like Reservoir Dogs, with no agreement and everyone losing the loot, or there is not going to be £350m a week left over in the UK’s budget to give to the NHS.