The Game Theory of Brexit

"Applying a game theory approach is actually quite insightful since it provides a logical way of picking through the various claims and counter-claims of those in favour of and against leaving"
"Applying a game theory approach is actually quite insightful since it provides a logical way of picking through the various claims and counter-claims of those in favour of and against leaving"

By Neil McCulloch

The most important question in British politics right now is: Will we get a good deal at the end of Article 50? Leavers argue that Europe needs us more than we need them and that a good deal is therefore likely. Remainers point to strong incentives for the EU to 'punish' the UK in order to send a signal to other would be exiters. Who is right?

As an economist, the question cries out for the use of one of the standard tools of the trade – game theory. Game theory is the branch of economics invented by John Nash, the Nobel prize winner featured in the film The Beautiful Mind. Nash was the first to formalise mathematically a way of determining the likely outcomes when two or more parties are playing a game and each has various strategies that it can deploy and different valuations for the various outcomes that might arise.

Whether or not it works, applying a game theory approach is actually quite insightful since it provides a logical way of picking through the various claims and counter-claims of those in favour of and against leaving.


Consider a game with two players: the UK and the EU. For simplicity, let's say that the EU has just two possible strategies it can pursue. It can give the UK a good deal, or it can give the UK a bad deal. Given this, the UK also has two strategies. It can either accept the deal it finally negotiates, or it can reject it and exit the EU with no deal.

The reality is obviously more complex than this simple framework. The EU can offer lots of options between good and bad and the UK is involved in the negotiations so can help to shape what is on offer. But this simple framework can still reveal some important insights.

Now let's put some valuations on the possible outcomes from the EU and the UK playing all the possible combinations of these strategies. Figure 1 shows one possible set of values.

Figure 1: Outcomes in the Brexit game

The numbers in brackets in each cell indicate the benefit for each player given the strategies adopted. The first number in each cell is the outcome for the UK; the second is the outcome for the EU (the reason for the question marks are explained below). The numbers chosen are illustrative. They don't correspond to an actual estimate of gains or losses – all that matters is their relative size.

Take the top right cell where the EU offer a good deal and the UK accepts it. This gives an outcome of, say, minus 2 for the UK. I've chosen -2 simply to reflect the fact that pretty much any realistic deal with the EU is unlikely to give the same full access to the single market as the UK currently has, but under a good deal I assume that loss isn't too bad. If you are a Leaver and believe that the EU will give a deal that is better than our current access, you can make the numbers positive if you like – none of the results I will show depend on the absolute numbers I've chosen.

Now consider the top left cell. Obviously, if the UK accepts a bad deal it will be worse off than under a good deal. I've arbitrarily said that the payoff for the UK under this situation is -8. Now compare this with the payoff in which the UK drops out of the EU with no deal. This is clearly very bad for the UK, so I've given that -10. So far the strategy for the UK is clear: accept whatever deal is on offer, unless the EU offers something worse than -10, in which case the UK would walk away.

Now consider the game from the EU's perspective. The EU has two competing incentives. From an economic perspective, it would like to give the UK as good a deal as possible, since a good deal for the UK will be better economically for the EU too. On the other hand, it may wish to punish the UK for leaving in order to dissuade future leavers. The tension between these two factors is illustrated in Figure 2.

Figure 2: The political and economic costs of a Brexit deal for the EU

Figure 2a: Economic and political costs exactly counterbalance

Figure 2b: Economic and political costs result in an optimum position for the EU

Figure 2a shows that a Brexit deal that is bad for the UK (-10 quality on the horizontal axis) would be good politically for the EU (the blue line) – but also terrible economically for the EU (the orange line). Conversely a Brexit deal that is good for the UK (0 quality) would also be economically good for the EU, but terrible politically.

If the two effects are exactly equal and opposite to one another, then the overall economic and political outcome for the EU would be the same (-5) regardless of the quality of the final deal for the UK. The economics and the politics cancel each other out.

If this is the case, then the Leavers are right. The EU will basically be indifferent between a good and a bad deal for the UK because of the competing economic and political costs for themselves, so all the UK needs to do is to offer some additional small inducement to the EU and the rational response for the EU will be to go along with a good deal. After all, they are going to get -5 regardless of what happens, so they might as well give the UK -2 instead of -8, if the UK can do something to help improve the -5 somewhat.

Before Leavers get excited, this outcome is critically dependent on the assumption that the economic and political costs exactly counterbalance one another. If the economic costs to the EU of giving the UK a bad deal outweighed the political costs of giving the UK a good deal, then the EU would offer the UK a good deal. This is what the Leave campaign have always argued would happen.

Conversely, if the political costs of giving the UK a good deal outweighed the economic ones, then the EU would give the UK a bad deal. And Britain, as long as it was better than crashing out with no deal, would accept it. This is what many Remainers have often argued.

In reality, there is likely to be an optimum deal from the EU perspective. Consider Figure 2b. The situation is the same, but the curves are no longer straight. This means that the EU would be very keen to avoid a terrible deal for the UK which would also harm them, but the additional economic benefits of a good deal reduce as the deal gets better. Conversely, the additional political cost of giving the UK a good deal also rises as the deal gets better. Intuitively, this is a more likely scenario.

However, if we now look at the average political and economic cost to the EU, we see that there is a clear optimal quality of deal – not too bad, so that the EU economy is harmed, but not so good that it would encourage other member states to leave.

Now looking back at Figure 1, we can see that, as long as the optimal deal for the EU is better than -10, then the UK will accept that deal. In other words, the equilibrium is that the EU will offer the UK a mediocre deal – and the UK will accept it. This strikes me as a much more plausible outcome. The deal will not be as good as the Leavers claim but neither will it be the Armageddon that Remainers have warned about.

Note however that the UK has no control at all in this game. The outcome is determined entirely by the balance of political and economic costs in the EU. As long as the deal is not worse than crashing out, it is rational for the UK to accept it.

Now let's add a wildcard into the mix. Barrister Jolyon Maugham and a number of others are attempting to get the Irish courts to refer the question of whether triggering Article 50 is irrevocable to the European Court of Justice (ECJ). If they are successful, the ECJ will decide whether the UK could, if it wanted, decide that the best alternative to a bad deal would be to stay in the EU rather than to crash out without a deal. If it turns out that the UK has this right, how might this change the negotiating dynamic?

Figure 3 shows the same information as Figure 1, but with an additional strategy for the UK of staying in the EU.

Figure 3: Outcomes of Brexit game with the option to stay

What scorings should be put for the UK and the EU for the option to Stay? Remainers would argue that this is the status quo and hence the payoffs should be (0,0). But there is a significant political cost for the UK government associated with changing its mind and staying. I've therefore chosen a value of -8. This choice is significant because it is less bad than the cost of dropping out of the EU without a deal. In effect, I am assuming that, while the majority of the UK would be happy with a good or even a mediocre deal, they might prefer staying in the EU to crashing out with no deal.

Now consider the payoffs to the EU as a result of the UK staying. If the EU offered a good deal, but the UK decided instead to stay in the EU, then the situation would revert to the status quo and the payoff for the EU would be zero. However, now consider the situation in which the EU offers a very bad deal (I've chosen -15, but anything worse than -10 will do). Given this, the UK's best option is to stay in the EU. It only gets -8 doing this, whereas all of the other options are worse. This would be humiliating for the UK, but it would also be a political win for the EU so I have scored the payoff for the EU above zero at 5 (again the number doesn't matter as long as it is above zero).

Now consider the equilibrium of this new situation and, in particular, the impact of the UK having the option to stay. The answer is not good for the UK. If the UK has the legal right to stay, and the EU is better off if it does so, then the optimal strategy for the EU is to offer a truly terrible deal. It no longer makes sense for it to offer its own preferred equilibrium of -5 as before. Now it will offer something worse than -10. Why? Because if it offers a deal that is so bad that the UK could not possibly accept it, then the UK will be forced to choose between crashing out without a deal and the status quo. If (and it is an important if) the shame of changing its mind is less bad than a hard Brexit, the equilibrium of this game is the bottom left hand corner in which the UK stays in the EU.

This is why Maugham and his co-claimants strategy is both very clever and also very risky. It moves the equilibrium from one in which the EU provides a mediocre deal which the UK then accepts, to one where the UK is forced to choose between two extremely uncomfortable options – one which is economically disastrous and another which would be politically disastrous, at least for the current government. In short: Game on.

Dr. Neil McCulloch is an Oxford-based economist focusing on developing countries. He was previously the director of the Economic Policy Program at Oxford Policy Management and the lead economist of the Australian Aid program in Indonesia. He also led the Globalisation Research Team in the Institute of Development Studies in the UK and was a senior economist for the World Bank in Indonesia.

The opinions in politics.co.uk's Comment and Analysis section are those of the author and are no reflection of the views of the website or its owners.

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