Will Theresa May’s Brexit deal survive? Game theory has an answer

The European Union may have approved Theresa May’s withdrawal agreement, but getting her own MPs to do so seems impossible. And her approach is confusing. On the one hand, she has been telling pro-Remain MPs that they need to vote for her deal or they’ll be left with no deal at all. On the other, she’s been warning eurosceptics that they could face another referendum or election unless they back her deal. As has been pointed out, both of these threats cannot be true simultaneously; either the prime minister has gone crazy or there is a reason.

Assuming the latter, why on Earth would this be a viable strategy? Mathematics offers one way to understand the situation. In game theory, there is something called the the prisoner’s dilemma that can help explain what May is up to. This particular game demonstrates that two rational players – in this case, the prisoners – will choose to snitch on each other even though it would appear to be in their best interests to cooperate.

Here is the set up: two thieves are arrested for a jewellery theft. They are guilty, but the police can’t find the jewels and there is no hard evidence to link them with the crime. The police therefore need a confession. Each prisoner has two strategies – blame their partner or stay loyal. If they both point the finger at the other one, they both get convicted with four years of prison. If both stay silent, circumstantial evidence will put them in prison for one year. If one cooperates and the other doesn’t, the one who stays silent gets five years of prison and the one who points the finger is free.

These rules are known to the thieves, and the police keep them in separate rooms. And now the paradox reveals itself. Cooperating and staying loyal would be the best option for them as a pair, as it would lead to the least amount of prison time (two years in total). But for the individual, it is more complicated. If you trust your partner to cooperate by staying silent, you may as well snitch and get off free. And if you don’t trust your partner, you should definitely snitch. Ultimately the cost of staying silent is the highest, with potentially five years in prison. Rational players will therefore always defect rather than cooperate.

While counter intuitive, this works because the numbers are rigged. The prisoner’s dilemma wouldn’t necessarily lead prisoners to snitch if when both confessed, for example, they’d end up in prison for ten years (rather than four) and get one to five years in prison for staying silent.

The Tory MPs dilemma

The prisoner’s dilemma can be used to model many social and political situations. With the Brexit strategy, the role of the prisoners is played by the two Tory factions, the Remainers and Brexiteers. Each group has the option of voting for or against the deal – cooperating or not. The cost of each decision is a political one.

If both factions vote against the deal, there will be some political cost since the deal probably won’t pass. But it is completely unclear what will happen afterwards. This might trigger a leadership election, a general election, a new referendum or crashing out with no deal. All MPs like at least one of these options – so if everybody thinks there is a good chance they will get what they actually want (which is impossible), the political cost will be perceived as low.

If both factions vote in favour of this deal, it will have a high probability of passing (there are always willing Labour MPs to vote against their party line). Each of the Tory groups can blame the other for the “bad deal” that they were “forced” to sign, but it could arguably be a higher cost strategy as everybody fundamentally dislikes something about it. So it may seem the best bet would be for the two factions to cooperate to vote against it.

But the PM could place her MPs in a prisoner’s dilemma situation to make sure they don’t cooperate with each other to scupper her deal. She could do this by convincing both factions that voting in favour of her deal is good for them but bad for the others. To do that, she would need to rig the value of the perceived cost so that it will be higher if they vote against it than if they vote in favour – independent of what the political opponents in the other group choose to do. In this way, if both factions behaved in the most rational way possible, they would end up voting in favour, not only to get the other group in trouble, but to save grace themselves.

May is already telling MPs that if they vote for her then they get what they want (or rather they don’t get what they don’t want). The other group can be cast as fanatical or enemies of the people (depending which faction she is talking to). The cost modulation involves removing each team’s favourite option from the table – getting Remainers to think a second referendum is impossible and getting Brexiteers to think EU will never improve on the deal – which is exactly what the PM has been doing.

For the prisoners dilemma, the assumptions are simple: both players act rationally to their best interest and there are only two possible strategies involved. If each group believes what May has been telling them – so it is her deal or no other deal – then she has set up the prisoner’s dilemma perfectly. Remainers will support her deal as the lowest cost option and the one that will stop the Brexiteers getting a no deal outcome. The Brexiteers will back her too, partly to stop the Remainers getting a second referendum.

So if May is persuasive, game theory suggests the deal will pass. Of course, this is just one of several mathematical models. And there are certain assumptions involved that may not completely capture the complexity of reality. The players are not individuals but groups. Each MP has different levels of ambition and dedication to the country. And whether she can guarantee that her factions will act rationally is the biggest question of all. After all, Brexit is a highly emotive issue.

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