Brexit: The Belated Threat

Debates on an EU-leaving referendum arose in several member states after Brexit. We want to highlight how the exit of an additional country affects the power distribution in the Council of the European Union. We inspect the power indices of the member states both with and without the country which might leave the union. Our results show a pattern connected to a change in the threshold of the number of member states required for a decision. An exit that modifies this threshold benefits the countries with high population, while an exit that does not cause such a change benefits the small member states. According to our calculations, the threat of Brexit would have worked differently before the entry of Croatia.


Introduction
Since the European Union membership referendum of the United Kingdom (UK) in 2016, Brexit (the withdrawal of the United Kingdom from the European Union (EU)), its possible effects have become a subject of political debates in several countries, like the Czech Republic, France, or Greece (Lyons and Darroch, 2016). Although it might be worth inspecting several political and economic effects of an exit from the European Union, in this paper, we look at one aspect: how the power distribution changes in the Council of the European Union. Kóczy (2016) has shown that Brexit will mainly benefit large countries. In this paper, we firstly try to explore whether the same will remain true if another country will left. Secondly, we want to answer the question: what would be the effect of Brexit if Croatia had not joined the EU?
The Council of the European Union, often referred to as the Council of Ministers, is an institute that represents the governments of the member states. It accepts EU law and synchronizes the policy of the EU. Along with the European Parliament, the Council of the European Union is

Methodology
It is popular to study voting situations as simple cooperative games, where the players are the voters. The value of any coalition (a subset of the player set) is 1 if its players can decide a question, or 0 if not. According to Felsenthal and Machover (2004), there are two interpretations of voting power. One conception, the influence power (I-power) focuses on voting power conceived of as a voter's potential impact on the result of divisions of the decision-making institute: whether the policies proposed are adopted or rejected. The second conception, prize power (P-power) focuses on a voter's expected share of a fixed prize given to the winning coalition. It is more appropriate to use P-power if one wants to calculate the part of the EU budget a member can expect to control, however, it is not suitable to compare different voting situations.
The most common measures of power in voting games are the Shapley-Shubik and the Banzhaf indices. They are used extensively for determining power in the Council of the European Union (Felsenthal and Machover, 2001;Herne and Nurmi, 1993;Kóczy, 2012;Widgrén, 1994). Since we investigate a phenomenon that belongs to the P-Power, it is best to focus more on analyzing the power distribution of the Council of the European Union with the Shapley-Shubik index (Felsenthal and Machover, 1998).
The Shapley-Shubik index is an application of the Shapley value (Shapley, 1953) for simple voting games. Its principle can be described as follows: voters arrive in a random order, and when a coalition becomes winning, the full credit is given to the pivotal player arriving last. A player's power is specified by the proportion of orders in which it plays this role. The index shows that if a decision is made, what probability a particular player has in being instrumental in making that decision. Let N denote the set of players and let S ⊆ N be an arbitrary subset of N . We use the corresponding lower-case letters to denote the cardinality of sets, so that s = |S| and n = |N |.
Definition 1. (Shapley-Shubik index ) For any simple voting game v, player i's Shapley-Shubik index is as follows: The Banzhaf index, which is the normalized Banzhaf-value (Banzhaf, 1965;Coleman, 1971;Penrose, 1946), uses a different approach. A player is called critical if it can turn a winning coalition into a losing one. The index shows what is the probability that a player influences a decision.
Definition 2. Player i's Banzhaf value is: Usually its normalized value is reported as the measure of voting power.
Definition 3. The Banzhaf index is the normalized Banzhaf-score: .
The index shows the voter's expected relative share of the total payoff. When a country leaves, we assume that its payment to the EU budget ceases, therefore the others do not share the same prize as before. 1 Taking this into account, we correct the power index by the following fraction: Original budget -the payment of the leaving country Original budget . (1) We compute for every country and each exit the adjusted power index as a percentage of the pre-exit power index. Since the adjusted power indices may not sum to 1, we cannot define them as probabilities, however, they still express a reasonable financial share. The following example presents an application of the new indices.
Example 1. In the predecessor of the EU (the European Economic Community (ECC)), the six founding states already used a weighted voting system. The weight of the large countries (France, Germany, Italy) was 4, the weight of the medium-sized states (Belgium, the Netherlands) was 2, and the weight of the smallest state (Luxembourg) was 1. The decision threshold was 12.
According to Table 1, Luxembourg's power was 0. France, Germany, and Italy contributed 28% to the EEC budget, Belgium and the Netherlands paid 7.9%, while Luxembourg paid only 0.2%. If Luxembourg would have exited and the decision-threshold had not changed, the remaining countries' Shapley-Shubik and Banzhaf indices would have remained the same, but the adjusted indices would have decreased (see Table 2).
If a large country, for example, France, departs, and the threshold decreases to 9, then the change is more spectacular. The correction ratio, according to formula (1), is 0.72. Table 3    In the following, we will call a country large or small regarding its population. We observe a general pattern, which connects the change in the member state quota to a change in the power distribution: when this threshold is modified by the departure, the power indices of the large countries increase. When such a change is not evoked by the departure, the power indices of the small countries increase.

The impact of additional departures to Brexit
In the computations which investigate the results of an additional departure to Brexit, we base our calculations on the 27-member Union without the UK, because Brexit appears to be a fact if any further exit happens. As mentioned in the previous section, it is also considered that an exit of a country causes a change in the budget. The example of the Czech Republic is presented first because the EU-skeptical sentiment has become stronger recently in this country. 3 The budget correction ratio is 0.989 according to formula (1). Figure 1 shows the budget-adjusted change in power indices due to Czexit as the function of the population.
We find that in the case of Czexit, the power indices of the small countries increase, and the power indices of the large countries such as France, Germany, Italy, Poland and Spain slightly decrease. The main winners from Czexit are Cyprus, Estonia, Luxembourg and Malta. The same can be said if one investigates Czexit in a farsighted sense, meaning to repeat the analysis with population predictions for 2020 and 2030. The only country which power index change differs is Romania: from a slight decrease (see Figure 1), its power modestly increases (see Figure 2).
We get similar results for other departures from a 27-member EU: the power indices of the small countries increase significantly. The detailed results for all member states can be seen in Appendix C. What has created more variation in these cases is the contribution of the particular country to the EU budget. To illustrate this point, let us look at the exit of Germany.
In the case of Germany's exit (Figure 3a), the Shapley-Shubik indices of the smallest countries and Poland increase, while the other countries all lose power. This is because countries with large populations are also the ones that contribute the most, so the budget loss exceeds the power gains caused by the departure of Germany. The correction ratio (1) is 0.711.
The results concerning Poland are especially interesting. If one of the four large countries (Germany, France, Italy, or Spain) leaves, Poland is much better off than Romania or Spain which are the closest countries in size of the population. In all four cases, its Shapley-Shubik index increases despite the other remaining large countries' power decrease. Calculations of another country leaving the 26-member EU, for instance, if the Czech Republic leaves after Germany show a similar pattern to Brexit (Figure 4). This can be elucidated by the fact that as the number of member states decreases from 26 to 25, the Council of the European Union's threshold for the number of supporting member states (determined by the member quota) decreases from 15 to 14. In this case, small countries would lose while the power of the large countries would increase.

The effect of Brexit before the accession of Croatia
Since our findings on an additional departure show an impact that is the inverse of Brexit's (Kóczy, 2016), Brexit might have a different impact before the accession of Croatia compared to the exit from the 28-member EU.
This has significance because if Brexit had decreased the power of large countries such as France and Germany, the impact of the potential Brexit would have been calculated differently by these states that usually dominate the policy of the EU: Brexit would have been a greater risk for them. In other words, if Brexit would have had the reverse impact before Croatia's membership, it could be seen as a belated threat.
We find that Brexit before the accession of Croatia would have favored smaller and would have harmed larger countries ( Figure 5). The results are similar not only for Brexit but for the case of an exit of any other member state from the EU without Croatia (see Appendix D).

A generalization of the results
Note that an additional departure to Brexit has an inverted impact compared to Brexit's impact from the 28-member EU, but it is similar to the potential effect of Brexit if it had happened before Croatia's membership. Results for a departure from the hypothetical 26-member European Union have a strong resemblance to the consequences of Brexit. The inverted impact of an additional departure to Brexit is due to the fact that 15 countries are necessary to make a voting successful in the case of both 26 and 27 members. However, after an additional exit the population threshold decreases.
The voting rule states two main requirements: the support of a given number of countries and a certain percentage of the population. A country will turn a losing coalition into a winning one if (a) the coalition just misses a member state to pass the threshold, and/or (b) if the coalition has the required participation, but the supporting countries are too small to reach the population quota.
With Czexit after Brexit, the population threshold decreases while the member state threshold remains the same, so coalitions with smaller countries become winning, which shifts power from the large to the small member states. This pattern is quite prevalent, we find similar results using population projections for 2020 and 2030 ( Figure 2).
It seems to be a general pattern that an exit triggering a decrease in the quota benefits large, while an exit not triggering such a change benefits small member states.
In the case of 27 member states, voting is successful if at least 15 countries, having together at least a population of 288 million vote in favor. We have examined the number of countries whose power increases if a particular country leaves, which can be considered as a yes vote for the exit of the departing country. Figure 6 presents the number of countries and their total population with an increasing power. Most of the countries would get a positive vote for leaving from 20 or 21 countries, but without the required population. However, in the case of Poland, both thresholds are met, because the power of small and large countries increase, and merely some medium countries lose power.

Conclusions
Inspired by Brexit, the goal of our investigation has been to examine what would happen in the Council of the European Union after a country's exit from the EU. For this purpose, the potential changes in the influence of each country have been measured with the use of power indices.
We find that, not just Brexit, but any other exit from the 28-member EU would favor countries with high population. However, an additional exit would increase the power of small countries. Furthermore, we observe a general pattern which is linked to the change in the member-state threshold. An exit, which changes the number of member states required for a decision, benefits the large, while an exit that does not cause such a change benefits the small countries. Thus a hypothetical Brexit before the accession of Croatia would harm large countries' power in the Council.

Appendix C -The impact of additional departures to Brexit
The following table presents the impact of any member state leaving the 27-member EU, after the United Kingdom departs. The country labels in the columns refer to the country that is leaving the EU, the rows show the remaining member states. The values represent the change (new adjusted S-S power index)/(old adjusted S-S power index) in basis points (1/100th of 1%). Bold indicates increasing, while italic signs decreasing power.  Appendix D -The impact of any member state leaving before the accession of Croatia The following table presents the impact of any member state leaving the 27-member EU before the accession of Croatia. The country labels in the columns refer to the country that is leaving the EU, the rows show the remaining member states. The values represent the change (new adjusted S-S power index)/(old adjusted S-S power index) in basis points (1/100th of 1%). Bold indicates increasing, while italic signs decreasing power.