Game Theory Says Don’t Bet Against a No-Deal Brexit

The U.K. parliament’s votes on Tuesday night may not have made it easy for anyone involved in the Brexit process, but they mark a moment of clarity from a game theory point of view.

It is apparent that the U.K., represented by the parliament, wants to play a one-off prisoner’s dilemma game with the European Union. The EU, for its part, is playing an infinitely repeated prisoner’s dilemma. Both parties are acting rationally in their own way — within the constraints of the different games they are playing. A cooperative outcome is only possible if one of the sides accepts the other’s vision of what game is being played.

On Tuesday, the U.K. parliament made it clear that it doesn’t want the country simply to give up and stay in a customs union with the EU; it doesn’t want an extension of the Brexit process without a clear idea of what to do with the extension; and it doesn’t want to exit without a deal. It wants Prime Minister Theresa May to go back to the negotiating table with the EU and get a better deal (specifically, one that doesn’t land the U.K. in a permanent customs union with the EU to avoid a hard border in Ireland).

In more abstract terms, the parliament wants May to play the one-shot prisoner’s dilemma. In this game, both sides are prisoners and have the opportunity to cooperate or to defect (compromise on the Irish backstop or face a no-deal Brexit). The dominant strategy in the prisoner’s dilemma — the best one to adopt without knowing the other side’s intentions — is to defect (show readiness to leave without a deal).

If the other side chooses to cooperate, so much the better. The parliament is, effectively, telling May to assume that the payoff matrix in this game is skewed and the EU benefits more from cooperating (for example, going for a time-limited backstop) than the U.K. would from caving to EU demands and staying indefinitely in the customs union.

In experiments, people assigned higher payoffs cooperate more often than the ones with low payoffs (although it must be said that cooperation rates still don’t rise above one-third).

The parliament’s game is risky, then, but it’s not irrational.

If the EU, too, were playing that game, it would make its move on the basis of an assessment whether the U.K. would really be prepared to leave without a deal or whether it would blink at the last moment. But the EU isn’t playing that game.

Instead, it’s playing an infinitely repeated prisoner’s dilemma. The other player in this game is not the U.K. but any member state. It doesn’t end with Brexit: After the U.K., other countries (we don’t know how many, so the game is played as an infinite one) may be tempted to seek an EU exit on favorable terms, that is, keeping the trade benefits but not following lots of common rules.

In such a game, always defecting is still the fail-safe strategy for both players, but cooperation makes more sense (and occurs more frequently in experiments) than in the one-off prisoner’s dilemma: Repetition makes it possible that the other side, too, will cooperate and a mutually beneficial pattern will emerge.

But once Player Member State defects — that is, refuses to blink and compromise — it makes sense for Player EU to default to a punishment strategy, for example, what’s known as the grim trigger strategy, refusing to cooperate in every interaction after the other player’s defection.

In the game the EU is playing, some concessions make sense if the U.K. cooperates — for example, if the U.K. accepts the Irish backstop or some other form of sovereignty loss in exchange for continued trade on favorable terms. They make little sense, however, if the U.K. defects (i.e., effectively threatens to leave without a deal and refuses the backstop).

If the two parties keep playing their separate games, no deal is the likeliest scenario. A compromise (on one or the other side’s terms) is only possible if one of the parties accepts the other’s version of the game. That’s still possible at this very late stage in the Brexit process. But accepting the EU’s perspective appears to be difficult for the U.K. parliament because it sees Brexit as a one-off deal.

For the same reason, it’s tough for the EU to accept the U.K.’s perspective: The EU knows the game may well continue with other members.

If I were betting on the outcome based purely on game theory considerations, and not on the complex web of personal preferences and ambitions apparent from the blow-by-blow coverage of the Brexit process, I’d place my money on no deal.

Bershidsky is Bloomberg Opinion’s Europe columnist. He was the founding editor of the Russian business daily Vedomosti and founded the opinion website