Voting Games: Part I

5. More voting games

In trying to get insight into a phenomenon it is often interesting to study one case in detail, especially in a situation which has had to deal with ongoing or evolving problems. A good example of this for voting games are the problems faced by the countries in Europe that were trying to cooperate for mutual gain by agreeing on common policies. The outcome of these evolving efforts is currently known as the European Union (EU). The European Union recently voted a series of expansions. Based on what has already been voted it will grow to 27 member countries which account for approximately 450 million people. The EU has faced some very singular challenges recently. When, after the breakup of the Soviet Union, countries that had been under Soviet domination wanted to join the EU, the existing EU had to decide on what basis and timetable to admit them. Also, countries of very different physical size, population size, and economic power were involved in trying to find a fair way to deal with countries that were already part of the EU and those wanting membership. Some of these countries had been involved in armed conflict within the lifetime of their leaders. Some major countries waxed hot and cold about how much involvement to have.

The three major governance structures of the EU are the European Parliament, the Council of the European Union, and the European Commission. This governance is carried out through two "councils": the European Council and the Council of the European Union. The European Council essentially treats member nations as equals. Its members are the heads of state of the different countries together with the President of the European Commission. By contrast the Council of the European Union consists of government ministers from the member countries and uses a complex voting game to carry out its decisions. Weighted voting is used in conjunction with the Council of the European Union.

The European Parliament assigns differing numbers of seats for the member countries, a situation that calls for mathematical analysis. The table below shows the different number of seats assigned to different member countries, for the specified time periods, as provided by the Nice Treaty which was signed in 2001 and came into force in 2003. The blanks indicate that the country involved did not have representation in the European Parliament at that time. Considerable controversy within the EU was created when the implications of the power relations that were implicit in what was agreed to in Nice started to be debated. Mathematical analysis can be used to determine the "power" of the different countries which were parties to the Nice agreement. Furthermore, additional mathematical analysis (some statistically based) makes it possible to determine the power relations among the countries under the assumption that voting patterns followed historical patterns of cooperation. Thus, it would be possible to compute the "power" of countries C and D assuming that they decided what to do independently on a certain issue. However, if historically C and D always agreed on policy, and, thus, were likely to always vote in the same way, it might be that by voting together this would significantly enhance their power. It turns out that in one of the early voting games developed in conjunction with EU governance, Luxembourg was a dummy (i.e. never a member of any winning coalition).
 

In the recent period prior to May 1, 2004 the weighted voting game for the Council of the European Union (which in the early days of the Union had a different name) involved the numbers below:
 

From May 1, 2004 to October 31 there will be an interim scheme and the scheme below will go into effect November 1, 2004.

 

There have been a variety of different analyses of various scenarios associated with EU changes. In order to describe the weighted voting situation for the European Union one can adopt a convention which allows the players to be numbered from 1 to 27 according to the following list: Germany, United Kingdom, France, Italy, Spain, Poland, Romania, The Netherlands, Greece, Czech Republic, Belgium, Hungary, Portugal, Sweden, Bulgaria, Austria, Slovak Republic, Denmark, Finland, Ireland, Lithuania, Latvia, Slovenia, Estonia, Cyprus, Luxembourg, and Malta. The weights assigned to the 27 players are 29, 29, 29, 29, 27, 27, 14, 13, 12, 12, 12, 12, 12, 10, 10, 10, 7, 7, 7, 7, 7, 4, 4, 4, 4, 4, 3.

A weighted voting game,WG₁ which attempts to loosely and historically represent influence among the countries is given by:

[255; 29, 29, 29, 29, 27, 27, 14, 13, 12, 12, 12, 12, 12, 10, 10, 10, 7, 7, 7, 7, 7, 4, 4, 4, 4, 4, 3 ]


Another game, WG₂  requires the approval of a majority of the countries involved is given below (for some votes, however, approval of 2/3 of the countries would be required).

[14; 1, 1, ..., 1]

WG₃ reflects an approval of 62% of the population of the countries that are in the EU, and is given by:

[620; 170, 123, 122, 120, 82, 80, 47, 33, 22, 21, 21, 21, 21, 18, 17, 17, 11, 11, 11, 8, 8, 5, 4, 3, 2, 1, 1 ]

Using this framework one can examine the proposed idea that in order for the EU to act, a voting game requiring three independent things to happen would be required. A coalition that "won" in each of the three games would have to hold! On the surface this would seem to be a rather complex game. Thinking about the complexity of voting games led mathematicians to the concept of the dimension of a voting game. I have already mentioned that some voting games can be represented as weighted voting games and some can not (though this requires, using reasonable assumptions, that there be at least 4 players). The dimension of a voting game G is the minimum number of weighted voting games WG₁, WG₂, ..., WG_k on the same set of players as G, such that the winning coalitions of G are precisely those winning coalitions that are common to all of the games WG₁, ..., WG_k. It turns out that every simple game has a dimension. The dimension of the game for passing laws in the United States (Senate, House of Representatives and President) is 2. However, the dimension of the European Union voting games above (with either a simple majority or 2/3 majority requirement for the number of countries) is 3. Some wonder if this complexity spells problems for the EU in making future progress, and some are nervous that it will hamper the evolution of further cooperation among the European countries.

Next month we'll discuss how to tell when a voting game can be represented as a weighted voting game, and how to measure the power of players in a voting game.

1. Introduction
2. Voting systems and games
3. Basic mathematical ideas
4. Unintuitive behavior
5. More voting games
6. References