Neither in- nor outside: the UK in the European Union

What is it that is neither inside nor outside, but still it is in the house? The EU membership of the United Kingdom reminds me a lot of this Hungarian riddle (solution later): when the UK was out, it tried to get in, but since the UK is in, it tries to get out. It is especially strange to look at this from the East where countries are begging to be let in the EU - while the UK wants to walk out on the other end. Analysis through the glasses of cooperative game theory.

Cooperative game theory does not deal with strategies or actions, but basically with allocational problems where one wants to distribute a joint value or payoff among the players. The European Union, as an institution, among others, simplifies a lot of things by creating a common market, eliminating the barriers to the free flow of goods and labor. This is the added value why the members entered, and remain members. It does matter, however, how this surplus is divided among the members. In cooperative game theory one would only deal with distributions that satisfy two conditions: first, the entire payoff will be distributed, and secondly, individuals are rational, that is, each member is better off than working alone. In a given situation,  we look at how much each country will receive from the common budget and compare it with how much they can produce on their own. The Britons, when they talk about withdrawal - the Brexit-, basically claim to get less than their stand-alone value.

The UK would like to get a bigger slice of the cake.

In such a case, we say that the United Kingdom will block the current distribution: it may propose a new one, but if that is not accepted, will simply leave the Union. In order for such a block to be credible we must check whether the exit is possible and whether it really makes sense. The first essentially says that if the Britons are really unhappy in the EU, the population will vote to exit - if not, there is nothing to talk about, you can even push them back a bit (more). The outcome of the Brexit referendum is yet unclear, but then a strong campaign may decide the result. Whether it is worth to leave, is a complex issue. Currently, the UK is a net contributor to the EU budget, paying about €4.5 billion for the annual membership. This is not an outstanding value per capita, but the question is what do they get in return. The United Kingdom has since the beginning been the EU's chief whiner who ended up with its policies in minority positions more often than anyone else - well, this is their problem. But what happens after the block?

Suppose that the grand coalition (i.e., all the players together) have a value of 100, of which the player in question gets 9, even though he himself, alone, were also able to achieve 10. The remaining 91 is pocketed by the rest. Suppose also that after the withdrawal of this player the remaining players coalition gets only 80 (we also assume that the game is superadditive, so this is all we need to know). Then the part above 70 + 10 is the gains of cooperation, but out of this extra 20 actually 21 goes to the rest! Then the protest is fully justified. If, however the player's own value is only 8 the failed attempt to obtain a higher payoff may actually lower his payoff. 

The EU's story has an additional twist. The "others" is not a common pot, but it is the sum of all other individual member states' private payments (the game is non-transferable utility) and you may find that the distribution not monotone in the budget, that is, a player may receive less out of a larger budget. This is one of the irregularities that may happen to power indices: larger countries would financially gain despite the smaller budget (see my paper or my earlier writing ), while another analysis by VoteWatch looked at the policy aspect: many countries see the UK as a hindrance to a more uniform, more regulated United States of Europe. These countries do not mind the exit, on the contrary, and will not make (further) concessions. This significantly weakens the UK's bargaining position and seems to have surprised the UK. Analysts explain the surprisingly soft UK position with this.

In sum, the Britons would like to get a larger slice of the cake and threaten with withdrawal otherwise. The EU is worse off without them, but not all countries, so the unsuccessful negotiations may force them to leave. Were they bluffing? If yes, too bad, the exit will probably hurt - not only them but all those in favour of a smaller budget, a smaller Brussels: the Dutch, the Swedes, the Hungarians ....

ps. The solution to the riddle is the window.