Here is a very short, fairly non-technical description of the Brexit crisis seen through the game theory lens.

I start with some assumptions about the strategic payoffs (not in standard form for those acquainted with game theory) of the EU-UK Brexit negotiations. The payoffs tells us the net benefit from the possible strategic outcomes of the Brexit negotiations. Here we assume four strategic outcomes based on two decisions from two parties: offer a compromise
Brexit deal or do not offer a compromise Brexit deal. I use ‘UK’ to stand in for the United Kingdom, and ‘EU’ to stand in for the European Union. The smaller the number (in brackets), the lower (and worse) the payoff (e.g., -10 is worse than -7). Here is a payoff table:

1. No compromises (no-deal Brexit): -10 (UK), -10 (EU)
2. Deal with UK compromising: -8 (UK), -5 (EU)
3. Deal with EU compromising: -5 (UK), -8 (EU)
4. Deal with both parties compromising: -7 (UK), -7 (EU)

The exact numbers don’t matter for these purposes, simply their relative sizes. What this payoff table says is that a no-deal Brexit is the worst situation for both parties (both get -10). It’s hard to know if this is true in the real world, but it does not seem unreasonable. It also tells us that we have two ‘pure strategy’ Nash equilibria: #2 (UK compromise, No EU compromise) and #3 (No UK compromise, EU compromise). These are equilibria because if forced into one of these two situations, neither party would be better off opting out. For example, say the US forced the #3 deal somehow (the UK does not compromise, but the EU does). Once that agreement is in place, the UK is not better off by switching its position from no-deal to deal, since switching would result in a worse payoff of -7 (#4). Similarly, the EU does not benefit from switching from deal to no-deal, since that would result in a worse payoff of -10 (#1).

Unfortunately, there is no easy way to ‘force’ a deal, so we get a standoff; both parties seem to be waiting for the other to compromise. We also have a deadline; if a deal is not made before October 31, 2019 , then #1 happens by default. Both parties know that a no-deal Brexit is the worst-case scenario, but do not seem compelled to give an inch. This is not a surprise for game theorists because unlike #2 and 3, #1 and #4 are not the same kind of equilibria. If both parties were somehow nudged into a #4 deal, for example, they both would have an incentive to defect to another option that has a higher payoff–where they do not compromise but the other side does. In this standoff, #1 may not be a desirable strategy, or optimal in any sense, but could still happen if a deal isn’t struck by the deadline.

Does Boris Johnson have a way out?

Boris Johnson, the current British PM has been a belligerent on the question of Brexit, and has made a number of decisions that signal preparation for a no-deal Brexit–promising money to help mitigate the financial consequences, and shutting down debate in Parliament. To many, this may seem like insanity (particularly if the no-deal Brexit really is the worse case scenario), but it could also be a clever strategy.

Once elected as PM, Johnson may have reasoned that if he can create a ‘credible threat’ of a no-deal Brexit, he will force the EU to compromise and solve the deadlock. He can create this credible threat by seeming to proactively remove #2 and #4 from the list of negotiating options (leaving only #1 and #3). He did this by reducing opposition debate in the House of Commons through parliamentary prorogation. This was a rare parliamentary maneuver (though not unfamiliar to Canadians), that could have been intended to signal that a compromise deal (that would require Parliamentary discussion) is now impossible. Once a British compromise deal is removed as a possibility, then the EU would have no choice but to compromise themselves, leading to agreement #3. This is because if the UK doesn’t compromise, the EU would be worse off with a no-deal Brexit (-10) (#1).

So Johnson’s quick work to prorogue Parliament may have been a tactic to convince the EU that the UK was creating a situation where no-deal was inevitable, thereby forcing their hand…

Another layer in the game…

However, now with the recent no-confidence vote in British Parliament, the threat of no UK compromise suddenly seems less credible. The British Parliament has changed its signal, effectively saying that they do not want a no-deal Brexit (or, for many, that they don’t want Brexit at all, and are trying to force conditions that will roll back the plan altogether).

In response to this, Johnson is pushing for an election. Why might he be doing this? Well, it is possible that the EU might also see an election as a credible threat of no compromise, particularly if it looks like Johnson might win with more support than his government currently has. Once an election is underway, there will be no government to negotiate with the EU, and less time to come up with a compromise deal acceptable to the UK parliament. It’s risky for Johnson–his government could easily lose an election. Nevertheless, if the election creates the credible threat of no-deal Brexit, he might again provoke some EU concessions.

It’s hard to know definitively what is going to happen in the short or long term. It’s also possible that there is no actual strategic intent here–maybe Johnson is just bluffing his way through this without much thought. What is clear, however, is that game theory gives us some language for understanding possible ways to think about the problem, and perhaps anticipate how the future will unfold…