By Neil McCulloch

In May, I wrote a blog that used Game Theory – the branch of economics invented by John Nash, the Nobel prize winner featured in the film The Beautiful Mind – to explore the likely outcome of the Brexit negotiations. I showed that, no matter what the UK negotiating stance, the outcome of the negotiations would depend overwhelmingly on what is politically and economically acceptable in the EU. So it has proven, with the UK being forced to accept the EU's position on the divorce settlement, the rights of EU citizens, the European Court of Justice and mush else.

But last week's debacle in Brussels over the Irish border – and the fudge agreed in the early hours of Friday morning to enable negotiations to proceed – provides another opportunity for game theory to have its revenge. No matter what form of words has been agreed to enable talks to move on, I show below why, when push comes to shove, we are likely to end up with some kind of hard(ish) border in Ireland.

In the previous game, I characterised the two 'players' as the UK and the EU.  Now let's forget the EU for a second and imagine that the two players are the UK and the Republic of Ireland. Say that each side has one of two possible strategies that it might push for: hard border; and soft border.

Hard border strategy: 

For the UK this means that the government would accept the DUP's position that there must be no regulatory divergence between Northern Ireland and the UK with the result that there must be some kind of hard(ish) border between Northern Ireland and the Republic of Ireland.
For the Republic of Ireland, adopting a 'hard border strategy' amounts to acquiescing in the need for a hard border to enable a successful long-term trade deal between the Republic (and the rest of the EU) and the UK.

Soft border strategy: 

For the UK this means that the government would reject the DUP's position (potentially endangering its own majority in the commons) and agree to some kind of regulatory alignment between Northern Ireland and the Republic in order to achieve a soft border between the two.
For the Republic of Ireland, adopting a 'soft border strategy' means holding fast to its insistence on regulatory alignment even at the cost of a long-term trade deal between the Republic (and the rest of the EU) and the UK.

Now consider the long-term payoffs to both the UK and the Republic of Ireland from adopting these two strategies.  Note I focus on the long-term payoffs – not the payoff this week, but the payoffs at the end of the whole process. They are shown below.

Consider first the approach that the UK government was taking before the DUP's dramatic intervention this week.  It had adopted a soft border strategy.  If the UK adopts a soft border strategy, then the logical thing for the Irish government to do is to agree. Ireland is happy – it gets a soft border and regulatory alignment. The UK is less happy – it has to accept some regulatory alignment which will upset the DUP,  but this may be worth it if it gets a final deal on trade that it is happy with ( I’ve shown this by giving a score of 2 for Ireland and 1 for the UK in this case although, of course, the actual values of the scores is arbitrary – it is only their relative value that matters).  This is an equilibrium.

However, notwithstanding the form of words agreed on Friday, that equilibrium will continue to present difficulties for the DUP.  So let us consider the alternative – that, in the end, the UK government is forced by the DUP (and its Brexiters) to adopt a hard border stance i.e. insisting on no regulatory divergence between Northern Ireland and the rest of the UK.  What is the Irish government's best long-term strategy in this situation? This depends on whether the political cost of accepting some kind of border is higher or lower than the economic cost of failing to do a trade deal with the UK.  However, the UK represents the largest single country trading partner for the Republic of Ireland so trade relations with the UK affect the entire population of the Republic, whereas, although it contravenes an important point of principle, the Irish border actually affects a smaller share of the population of the Republic than are affected by its economic ties with the rest of the UK. 

Given this, it could be argued that it is in Ireland's long-term interests to acquiesce to a hard(ish) border in return for a good long-term trade deal with the UK.  In other words, a situation in which both sides play a hard border strategy is also an equilibrium.  In this case, the UK is much happier than Ireland, but both are better off than no deal at all. Hence Brexit has created an agonising dilemma. There are two equilibria – one in which the UK wins more than the Republic and one in which the Republic wins more than the UK. Both sides will push hard for their preferred position, but no progress will be made unless one side backs down.

So who will win?  Now let's broaden the game to include the rest of the EU and we see that life gets much more difficult for Ireland. While the rest of the EU has backed Ireland out of solidarity with a fellow EU member – most of the EU has little political or economic interest in the issue of the Irish border.  However, they do have a significant interest in an effective trade deal with the UK.  Hence if the Republic of Ireland continues to push for a soft border as talks progress, then expect to see it being put under considerable pressure from its European partners.  Hence if the UK maintains a 'hard border' stance, it's quite possible that the Irish government will be pushed to acquiesce. 

But the Irish government has one final card up its sleeve – they have the legal right to veto the entire deal between the UK and the EU. If they are willing to play hard ball for a soft border – to be willing to damage their own economic interests for the sake of preventing a hard border – then it is still possible that the UK would be forced to acquiesce.

The sad logic of game theory suggests that a far more likely outcome is the re-emergence of some kind of border between Northern Ireland and the Republic.

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.

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