The games people play

Brexit is often described as an economic game-changer, whilst Prime Minister Theresa May has accused the Scottish First Minister of playing a game with her call for a second Scottish independence referendum. But politics is a form of game and Brexit certainly warrants examination in a game theoretic context, which can be described as “the study of mathematical models of conflict and cooperation between intelligent rational decision-makers.” Leaving aside the question of the rationality of the Brexit decision, we can perhaps use it to gain some additional insight on what the optimal (or least sub-optimal) outcomes are likely to be, not only for the UK but also for the rest of Europe.

Mathematical games can be split into two broad categories: cooperative and non-cooperative. Whatever the EU may be, it is a consensus based institution in which decisions are traditionally arrived at in a cooperative manner (although the Greeks may not see it in this way). In principle, the EU is a cooperative environment in which players are constrained to act according to the legal rules. Those who breach the rules are subject to sanctions. To take the example of environmental protection, there were 3464 infringements of EU law over the period 2007 to 2015, of which the UK accounted for 179, or 5%. Italy was the biggest offender accounting for 9% of all infringements followed by Spain (8.6%) and Greece (7%). Without making any judgement on individual nations’ degree of compliance with the law, the rules are known and the EC publishes data to name and shame the transgressors who are expected to comply with sanctions.

Similar rules apply to trade. But not everyone thinks the EU applies the rules consistently. Alan Halsall owns a company (Silver Cross) which makes prams, and in 2015 the French government banned his company from selling its products in France on safety grounds despite being cleared for sale elsewhere across the EU. Not surprisingly, Mr Halsall argued strongly in favour of Brexit. But the problem is not the EU rules: it is that the French government applies higher safety standards than other countries. In fact, Mr Halsall had a case for applying to the European Commission to appeal this decision although he chose not to do so, arguing that he will benefit more from applying his energies to markets where he is able to generate sales.

But if he thinks the current rules are stacked against him, Brexit will change the rules of the game completely. The process of renegotiating trade deals with the rest of the EU implies entering into a non-cooperative bargaining “game” where the EU has no incentive to cooperate with the UK, primarily because it wishes to avoid giving support to the idea that leaving the EU is an easy option. Although the ultimate outcome is likely to result in a cooperative situation in which a new set of rules apply, the process of getting there will be non-cooperative. And even if we do reach a cooperative solution, the end result is likely to produce an outcome which is worse than the position we started from. In game theoretic terms, the outcome will be inefficient – at least for the UK.

Those of you who have seen the film A Beautiful Mind will be aware of the work of John Nash, who offered significant insights into the mathematics of bargaining problems. His key insight was that equilibrium is reached when no player can unilaterally change their strategy and get a better result, given that they know the strategy of the other player(s).   In other words two parties should cooperate when non-cooperation leads to results where at least one side is worse off (so-called Pareto inefficient outcomes). Clearly, the EU is not going to allow the UK to have free access to the single market: such a strategy will weaken the EU because it implies there is no cost to exit, which will endanger the EU’s long-term existence. But the UK cannot accept access to the single market whilst continuing to pay into the EU budget and accept ongoing free movement of labour (the Norwegian solution), because this gives the UK the same system as before Brexit but without any control over the legislative process.

The problem the EU faces is to trade off punishing the UK against the harm that non-cooperation inflicts upon itself. Likewise, the UK must trade off the best deal against the political costs of giving away too many concessions. Abstracting from deals regarding the exit costs, one possible Nash equilibrium is for the EU to offer the UK continued access to the single market for an annual fee which is lower than the UK's current net EU contributions but which offers no say over drafting legislation. The UK should accept this because although it is a worse deal than the current arrangements, it is economically less damaging than relying on WTO tariffs and preserves market access for exporters on both sides of the table.

One of the best known forms of game is the zero sum option. Many Brexit supporters appear to believe they are operating in a positive-sum game: if the UK leaves the EU, any disadvantages from leaving will be more than offset by the gains. There is no evidence to support this sunny optimism. Indeed any action which harms UK trade with the rest of the EU, such as the imposition of trade tariffs, will result in a negative sum outcome. 

Looking at this in a wider perspective, it is evident from recent polling evidence that the degree of dissatisfaction with the EU is rising across the whole continent. We can thus perhaps think of the decision of whether to leave as a sequential game, in which the action of one country influences the decisions of others. It is yet possible that the UK’s decision to be the first mover in this game will trigger other countries to go down the same path. But it is not in the UK’s interest to be the first mover because it will face all the adverse consequences as a result. Far better in this case to allow others to make the first move – this may be a game where there is no first mover advantage.

Contrary to what politicians tell us, Brexit really is a game – admittedly one with high stakes. However, a central assumption in many variants of game theory is that the players are rational. In other words, they always choose an action which gives their most preferred outcome, given what they expect their opponents to do. I have a pretty fair idea how the European Commission is likely to act: Based on recent rhetoric I am less certain of the rationality of the British position.