Wednesday, 15 February 2017

Brexit negotiations as a game of chicken

In a post last month, Tim Harford perceptively characterised the posturing between Britain and the European Union over Brexit as a game of chicken:
First: to be an effective negotiator often means accepting some risk of disaster. The simplest model of this is the game of “Chicken”, in which two leather-clad rebels get into their cars, and drive towards each other at a furious pace. The first one to veer off the road loses his dignity, unless neither of them swerve, in which case both of them will lose a lot more than that.
Chicken is an idiotic game, whose players have little to gain and much to lose. But Chicken teaches us that you can gain an advantage by limiting your own options. Imagine detaching your steering wheel and flamboyantly discarding it as you race headlong towards your opponent. Victory would be guaranteed. Nobody would drive straight at a car that cannot steer out of the way. But here’s a worrisome prospect: what if, as you hurl your own steering wheel out of the window, you notice that your rival has done exactly the same thing?
All this matters because both the UK and the EU are doing their best to give the impression that they’ve thrown their steering wheels away. Control of immigration is non-negotiable, says Theresa May. Fine, says the EU — in that case membership of the single market is out of the question. Fine, says May: we’re out. Don’t let the door hit you as you leave, says the EU.
It’s easy to see why both sides are behaving like this — it’s the logic of Chicken. But the eventual result may be something no sane person wants: a car crash. In May’s recent speech, she set out her willingness to risk such a crash by saying she might walk away without a deal. That does make some sense: it’s how you act if you want to win a game of Chicken. But there are games of Chicken that nobody wins.
The Brexit negotiations chicken game is laid out in the table below. The EU and the UK can choose to 'make concessions', or to 'play hardball'. If both make concessions, the outcome is essentially pretty neutral (a payoff of zero for both of them). However, if either the EU or the UK plays hardball while the other makes concessions, whichever of them plays hardball comes out better off (positive payoff) at the expense of the other (negative payoff).  Finally, if both play hardball, both will be much worse off (very negative payoffs).


Where are the Nash equilibriums in this game? To identify them, we can use the 'best response' method. To do this, we track: for each player, for each strategy, what is the best response of the other player. Where both players are selecting a best response, they are doing the best they can, given the choice of the other player (this is the definition of Nash equilibrium).

For our game outlined above:
  1. If the UK makes concessions, the EU's best response is to play hardball (since + is better than 0) [we track the best responses with ticks, and not-best-responses with crosses; Note: I'm also tracking which payoffs I am comparing with numbers corresponding to the numbers in this list];
  2. If the UK plays hardball, the EU's best response is to make concessions (since - is better than --);
  3. If the EU makes concessions, the UK's best response is to play hardball (since + is better than 0); and
  4. If the EU plays hardball, the UK's best response is to make concessions (since - is better than --).
Note that there are two Nash equilibriums, where one of the EU or the UK plays hardball, and the other makes concessions. However, both of them want to be the one playing hardball. This is a type of coordination game, and it is likely that both the EU and the UK will try to play hardball (but leading both to incur big losses!).

The solution to getting your preferred equilibrium outcome in the chicken game is to make a credible commitment (such as removing the steering wheel that Harford suggests for the classic chicken game). In the case of Brexit though, it isn't clear how either side can make a credible commitment to the hardball strategy, and both are already moving their feet towards the accelerator. But if neither are willing to make concessions, the outcome is clear.

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