The voting rules used in the European Council (EU) under the Treaty of Lisbon (which became effective on 1 November 2014) is quite unique within the current, global range of electoral systems. It not only requires a “qualified majority” of both the number of member states supporting a proposal and the population of the supporting member states, but also specifies a “blocking minority” which can block a proposal if certain condition is satisfied. It is the existence of such a “blocking minority” that makes the system complicated and interesting to study from the mathematical point of view. In fact, Kurz & Napel (2015) showed that this voting system has dimension between and and therefore this dimension sets a world record of the dimension of the real-world voting bodies. Indeed, the previous record holders are EU under Treaty of Nice and the Legislative Council of Hong Kong and both of them have dimension (see Freixas (2004) and Cheung & Ng (2014)). In this article, we will reduce Kurz & Napel’s upper bound to or even (depending on the populations of the countries in EU). This is achieved by finding a new way to represent the union of two weighted games as an intersection of certain weighted games (Theorem 1). It is expected that Theorem 1 will be useful for computing the dimension of other real world voting systems. Finally, assuming that the United Kingdom is no longer a member of the European Union Council, we will show that our upper bound will jump to while Kurz & Napel’s lower bound will only increase to .
2 Notations and definitions
A simple game is a pair where is the set of players described as and is the characteristic
function which satisfies:
is the characteristic function which satisfies:
or , for all ;
A coalition is winning if and losing if and we let be the set of winning coalitions of .
A weighted majority game is a simple game which can be realized by a vector
A weighted majority game is a simple game which can be realized by a vectortogether with a threshold which makes the representation so that is a winning coalition if and only if .
Let be two simple games with identical player set . The intersection of and , denoted by is the simple game with and the union of and , denoted by is the simple game with .
The dimension of is the smallest such that coincides with the intersection of weighted games.
The codimension of is the smallest such that coincides with the union of weighted games.
The boolean dimension of is the smallest such that can be represented as unions and intersections of weighted games.
3 Data sets and the voting rule
Let us first state the rule of the voting game of the EU systems: We numbered the 28 EU members by (see Table or ). A motion will be passed if
at least of EU members support the motion (and we let the weighted majority game )
the EU members that support the motion represent at least of the total population (and we let where is the population of the -th country), or
no more than four EU members reject the motion (and we let ).
Therefore and hence the boolean dimension of is (see ). Note that we also have .
To compute the dimension of , we need to know the populations of the 28 EU members. Here we will use four data sets. The 2014 data (Table 1) is from Kurz & Napel (2015)  , it will provide a clear comparison between their estimation and ours. The 2016, 2017, 2018 data (Table 2,3,4) are taken from
, it will provide a clear comparison between their estimation and ours. The 2016, 2017, 2018 data (Table 2,3,4) are taken fromhttp://ec.europa.eu/eurostat on 7 March, 2019. Indeed, according to Kurl & Napel (2015), their data set was also taken from http://ec.europa.eu/eurostat, but it seems the website had adjusted the data afterward. Note that the orders of the countries are not the same in the four tables, as we arranged the countries in descending populations, which will make it easier to study the voting game mathematically.
4 Realization of games as intersection of weighted games
In this section, we explain two constructions which are useful to realize a game as intersections of weighted games.
Suppose that we have two simple games and with identical player set such that . Let be the set of maximal coalitions which is winning in but losing in . For each , define to be a weighted game with quota and the weight if and if . Note that a coalition is losing in if and only if . Therefore, we have
Note that if we set then the construction is essentially the proof that every simple game is the intersection of weighted games.
Let and be two weighted games. We want to realize as an intersection of weighted voting systems.
Let for or and .
If then and we are done. Now suppose and let and .
Suppose that . Then for each , we define
Note that if is winning in , then it is winning in for each . Now suppose is winning in but not winning in . In this case, by the definition of and hence for each , we have and . Hence is a winning coalition of .
Therefore . If we let and and apply the construction I, we have
Note that this method may not yield good results when is not big, as we will see in the next section.
We conclude this section by stating the result we just obtained.
Theorem 1. Let and be two weighted games with the same set of players. Then
5 Upper bound of the dimension of the EU system.
Recall that . Hence the dimension of is plus the dimension of . In this section, we will obtain a two digit upper bound on the dimension of based on the populations of the EU members in 2014,2016-18 given in Table 1-4 respectively.
5.1 2014 data
Let and . For any , define . Using the populations given in Table 1, one can check that . Hence , and .
Apply Theorem 1 to so that can be realized as the intersection of the following weighted games:
Therefore, the dimension of in is at most .
5.2 A remark
If we take and using 2014 data, then contains , , , and hence and our method produces no good upper bound.
5.3 2016 data
We take and for the data set. Just like the 2014 data set, we have and . However, and . Hence can be realized as the intersection of the following weighted games:
Therefore, the dimension of in 2016 is at most .
5.4 2017 data
For this case, and but we still have and .
As a result, can be again realized as the intersection of weighted games as in the case for the data set.
5.5 2018 data
In this case, everything is identical to that of the data set, except . Therefore, the dimension of in 2018 is again at most .
5.6 2018 data without UK
The United Kingdom has been seriously considering the possibility of leaving the EU during the preparation of this paper. Therefore, it would be interesting to see what happens to our upper bound if UK is no longer a member of the EU. Using the 2018 data set without UK, we found that and . Hence, the upper bound of the dimension of then jumps to .
Also, we would like to point out that Kurz & Napel lower bound of will also change with the absence of UK. Using the similar method introduced in section 5 of the paper by Kurz & Napel (2015), we find the following set of losing coalitions with the ’pairwise incompatibility property’ (see Kurz & Napel (2015), section 4):
where the numbers correspond to each country’s population ranking in 2018 data. Note that the number (which represents the UK) is absent. This can be extended by adding the maximal losing coalition of the largest countries. Therefore, the lower bound of will increase to if UK leaves EU.
Kurz & Napel (2015) used integer linear programing to estimate the upper bound of the voting system of EU and gets a large bound. Our estimation made use of the structure of the voting system to get a much smaller upper bound . Moreover, our estimation indicates that the dimension is sensitive to the populations of the countries.
In general, our method works best if is close to . From the fact that , one may notice that our method can actually be extended to all simple games.
-  Cheung, W. S. and Ng, T. W. (2014). A three-dimensional voting system in Hong Kong. European Journal of Operational Research, 236(1), 292-297.
-  Freixas, J. (2004). The dimension for the European Union Council under the Nice rules. European Journal of Operational Research, 156(2), 415-419.
-  Kurz, S., and Napel, S. (2015). Dimension of the Lisbon voting rules in the EU Council: a challenge and new world record. Optimization Letters, 1-12.