Brexit – Europe’s politicians don’t know their game

The tragedy of Brexit is that neither the United Kingdom nor, especially, the European Union saw fit to convincingly spell out to the public what the future would look like after the referendum. A credible precommitment, not excluding credible threats, ought to have promoted a cooperative outcome in the game. Instead, the uncertainty drove each of the British peoples to what they believed to be the safest option.

Brexit is sometimes argued to represent a situation where mutual cooperation is in everyone’s best interests but still the individual players “defect” and go solo. To model such situations, a game called “prisoner’s dilemma” has been conceived. In such a game, two partners-in-crime are interrogated separately. If both cooperate – with each other that is, not with the authorities – they can only be convicted on a lesser charge; if both defect, they face heavy charges; but in case one partner remains silent and the other betrays his partner, the partner ends up in jail for an even longer time whereas the other is released. Cooperation in the prisoner’s dilemma simply is not a strong equilibrium, i.e. a player can do better by unilaterally changing strategy. Defection is the dominant strategy. That is not the game you want to play.

It is implied that the prisoners have no recourse to rewarding or punishing their partner-in-crime after the game has played out. By advertising beforehand an expected penalty for players that defect against cooperators, the prisoner’s dilemma morphs into another game that has been known since Jean-Jacques Rousseau and David Hume as the stag hunt. Two hunters must decide whether to combine their skills to try and hunt down a stag or go it alone and settle for an easier hare. (For those that dislike hunting: Hume gave the example of two rowing a boat; if one does not cooperate, the other rows in vain.)

Crucially, the stag hunt game has two (pure-strategy)equilibria: the more rewarding stag hunt in the event both cooperate, and the alternative of the hare that is less risky as it does not depend on coordination. The crux is not whether to cooperate is an equilibrium but rather which equilibrium to choose: potential solidarity or perceived safety. Precommitting to a penalty mechanism – ranging from the European Union precluding any cherry-picking to full-out retaliatory measures – allows at least a cooperative equilibrium to arise by transforming the prisoner’s dilemma into a stag hunt. But uncertainty surrounding the willingness to cooperate and the payoff from doing so pushes the hunters to individually settle for the hare. Resolving such uncertainty may in addition nudge them towards the higher-potential stag. We leave it to the reader to identify which is which in brexit country.

Or maybe brexit is best captured by a game called “battle of the sexes”? Suppose, for argument’s sake, that Britain preferred to leave the Union whereas the continentals would have liked the islanders to remain, but that in any case all of Europe’s politicians wanted to be ‘in the same place’: either the United Kingdom leaves and the European Union reluctantly accomodates the departure, or the United Kingdom remains and puts on a happy face. In either coordination equilibrium only one player is pleased, but both equilibria are preferred to the situation in which a mismatch occurs. How then to avoid the parties choosing different strategies?

One possible way out is to “refine” the equilibrium concept by means of a randomising device that correlates the parties’ strategies. If both parties decide to flip a coin (say the United Kingdom organises a referendum) and agree beforehand that “heads” means “remain” and “tails” implies “leave,” neither would rationally want to alter his strategy after the referendum for fear of getting stuck in a coordination failure. Which brings us back to the tragedy we started from.

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