## Friday, June 09, 2017

### Brexit and Rubinstein’s bargaining model

A Google search for Brexit and game theory finds a number of web pages that discuss models of the negotiation process, constructing payoff matrices in which the parties to the negotiation choose to cooperate (or not) over issues such as free movement of people; these models aim to predict the outcome of the negotiations. The models are rather simplistic, but are probably more realistic than the efforts by certain members of government to treat Brexit as a game of poker.

Rubinstein’s classic paper Perfect Equilibrium in a Bargaining Model (wikipedia page) models a negotiation between 2 players who have to share a pie; there is no limit on the number of rounds, but there’s a cost of delaying, which in one version consists of a player’s discounting factor, which is the rate at which the value of (a share of) the pie goes to zero, each time a round of bargaining takes place without agreement. It could possibly be taken to represent a situation following the March 2019 deadline, supposing that no agreement has been reached, and both sides suffer ongoing economic costs while an agreement is being constructed. The “pie” would consist of the collection of (many) issues that have to be resolved in favour of one side or the other.

Suppose player 1 (the UK) has a discounting factor δ1 and player 2 (the EU) has discounting factor δ2. (δi denotes the fraction of value of remaining after a round, not the fraction lost.) According to the model, the UK’s share of the pie should be (1-δ2)/(1-δ1δ2). It just remains for us to decide the values of these discounting factors, and we’re in good shape to figure out what share of the pie the UK should accept before the deadline expires.

According to this page, Britain depends on the EU for half of its exports, while Britain accounts for only one-sixth of Europe’s. If we give the UK a discounting factor of 1/2 and the EU a discounting factor of 5/6, we get that the UK should receive 2/7 of the pie — so, not the lion’s share. Still, those discounting factors may look harsh; there’s more to life than exports to our geographical neighbours. Let’s average them with 1, and use δ1=3/4 and δ2=11/12. In that case, the UK’s share of the pie unfortunately goes down to 4/15, and I don’t think it gets any better when you adjust them further! To conclude, the UK should cave in on most of the issues that need to be resolved in a Brexit settlement. As the above-mentioned web page notes (based on a different model): The EU has an incentive to offer a bad deal, and the UK has an incentive to take it.