# Voting Games: Part I

When Americans awoke the morning after Election Day, they were surprised to learn that the results of the balloting for president had been indecisive.

Joseph Malkevitch
York College (CUNY)
malkevitch@york.cuny.edu

1. Introduction

In effect the Electoral College involves "weighted voting," a situation where votes are cast in blocks. Weighted voting systems are more common than many people realize. There are the voting schemes involving the European Union, the United Nations, amending the Canadian Constitution, county governments in New York State, and in many other environments which affect people's lives in many ways. Sometimes there are layers of complexity in a voting system. For example, if one looks at the United States House of Representatives or the Senate, it may seem relatively easy to understand what is going on. However, the passing of a new law in the United States needs the approval of the House of Representatives, the Senate, and the President; when the President vetoes a proposed law, there is a method to override the veto. Is there an easier way to think of this "complex voting" situation and to get insight into the power relationships involved? Can mathematics help in understanding what is going on with these voting systems and in designing systems that are more fair and equitable?

2. Voting systems and games

In dealing with voting and elections in a democratic setting we are interested in having the systems be as fair as possible. What does it mean to have a fair voting system? Voting is a complex activity and there are many fairness principles and points of view that might come into play. These include issues as to whether or not a felon should be allowed to vote, the pros and cons of different election decision methods, and using different kinds of ballots.

Most democracies use a direct system of translating voter feelings into the selection of a single winner (e.g. elections for U.S. Senator) or group of winners (e.g. members of a faculty search committee) for the election. However, for a variety of complex reasons, when the U.S. Constitution was created it did not call for the direct election of the President, but for an indirect system using an Electoral College . (Up until 1913, when the 17th Amendment went into effect, even Senators were not directly elected by the American people. They were "chosen" by state legislators.) Thus, strictly speaking, when Americans vote for president they are not voting for a candidate for president but rather for a statewide group of electors committed to voting for the candidate the voter chose. The number of these electors is equal to the total number of senators and members of the House of Representatives for that state. There are also electors for the District of Columbia.

Since the number of members each state has in the House of Representatives can change every 10 years, which reflects the number of seats assigned by the Huntington-Hill Method of Apportionment , the relative power of the states in the Electoral College can change with time. Furthermore, historically the electors who are selected for each state are chosen on a "winner take all" basis. Thus, if in Florida George Bush beat Albert Gore by a single vote in 2000, all of Florida's electors would have been committed to vote for George Bush. Technically, the issue of whom electors vote for is complex, too, and some states have experimented or are considering experimenting with dividing the number of electors between the candidates who run for president on the basis of the popular vote rather than using "winner take all."

Thus, in practice the way the Electoral College works is that there are 51 "players." Each player is either a state or the District of Columbia, and each casts a block of votes: the District of Columbia casts 3 votes and each state casts 2 (the number of senators for the state) plus the number of seats the state has in the House of Representatives, which will be thought of as a weight.

In order to determine if a group of players has "won" the game or been able to achieve its goal, there is a number known as the quota associated with the game. Any group of players that casts blocks of votes, the sum of whose weights exceed the quota, "wins." In the Electoral College there are different groups of states such that if a candidate wins those states, the candidate will be elected president because the weights associated with these states exceed the required quota (270). Thus, the Electoral College is an example involving weighted voting. Sometimes the phrase "weighted voting game" is used. For the case of the Electoral College, weighted voting did not arise directly because of concern with equity issues. Some would argue that it arose for opposite reasons! However, let us look at a variety of settings where equity seems to be the driving force behind using voting games or weighted voting.

In the Electoral College it is natural to describe the way voting proceeds in terms of having players who cast blocks of votes. However, in other situations there are "players" who vote and there are rules which specify when a group of players can "win." For example, the United Nations Security Council consists of players of two kinds: the five so-called permanent members: the United States, the United Kingdom, France, Russia, and China, and ten other members who are chosen on a rotating basis. This makes for a total of 15 countries. To take substantive action requires 9 votes including those of all five permanent members. The Security Council voting game can be represented as a weighted voting game, even though this voting game is not described using weights for the players.

Here are some other voting game examples. Canada is a federation of 10 provinces. For the Canadian Constitution to be amended requires that at least 7 of the 10 provinces approve and that these 7 provinces contain at least one-half of the Canadian population. Voting where complicated rules might want to be implemented also occurs in situations involving organizations with less visibility. The World Bank and the International Monetary Fund are examples where different countries of unequal status are involved with the governance of the organizations. The reason for the unequal status arises from the fact that although in one sense one may say a country is a country, this view oversimplifies the dynamics of the organizations. The U.S. and Japan have relatively small populations but tremendous influence on the world economy. India is a country with a very large population but with a lesser influence when compared to the United States and Japan. Thus, when voting occurs in an organization such as the World Bank or International Monetary Fund, one might want to have complicated rules for passing policies so that the different realities of the countries involved are taken into account. A large group of small countries should not be able to set a policy telling a few rich countries what to do with funds that often come from these richer countries.

In some legislative circumstances, such as some county governments in New York State, one may want each physical person (representing, perhaps, a city in the county) to cast one vote, even though the counties the people represent are very different on many scales (e.g. population, gross product, or wealth). In other cases it may be that one wants to take into account that the representative from a very "small city" (say, as measured by population) does not have the same "influence" as the representative from a very large city within the county government. In New York State, typically, a county legislature is known as a County Board of Supervisors. These considerations explain why many counties in New York use weighted voting.

3. Basic mathematical ideas

When mathematics is applied to a real-world situation, one needs to abstract and simplify the most important aspects of the situation to get insight. In the case of weighted voting it is convenient to have a name for a group of players who come together to achieve a common goal. We will use the term coalition for such a group.

For simplicity we will assume that the weights cast by players are nonnegative integers. However, there are interesting mathematical issues involved in whether or not it makes any difference to use more general weights. Furthermore, since we have a voting situation in which decisions will be made on the basis of the groups of players who come together to cast votes, we need to know how we can tell when a group of players can "get their way." One way to do this is to just specify a list of all those groups of players who, when they work in concert, can "win." This, however, does not take advantage of the numerical weights assigned to the players, and the list of groups that can win may be very, very long. A more practical way involves a positive integer quota Q. One way to produce Q is to have it set at one-half the sum of the weights, add 1, and if the sum is not an integer, round down to the nearest integer. However, for some situations, where one wants to make it hard to change from the status quo, the quota might be set higher than this "simple majority" level. For example, amending the U.S. Constitution requires more than a simple majority and in many legislative bodies certain kinds of action (e.g. approval of a treaty) require more than a simple majority. However, as we saw with examples such as the United Nations Security Council, it is not clear when voting games can be represented as weighted voting games.

In light of these ideas it is natural to define the idea of a simple game . We are given a collection of people P, the players in the game, and a collection of subsets W of these players. The sets in W are known as winning coalitions. The intuitive idea is to think of W as those sets of players that can accomplish their goal in a voting situation. It is natural to assume a monotonicity axiom, that is, if X is a member of the set of winning coalitions W, then any set Y that contains X is also in W. The motivation is that having more players in a group that can already accomplish its goal can do no harm in terms of "winning." A subset of players which is not winning is called losing . Of special interest are those winning coalitions, denoted MW, which have the property that removing a single player from such a coalition changes it from a winning coalition to a losing coalition. Such coalitions are known as minimal winning coalitions .

The definitions above are very general. The definition we have given for a winning coalition does not preclude that if some coalition C is winning, the coalition of players not in C is not a losing coalition, even though in most but not all real world situations this would usually be the case. An example of a situation where one can have a winning coalition whose complement (i.e. elements not in this coalition) contains a collection of players who can also win is in the procedure that the United States Supreme Court uses to approve certain cases it wants to have brought before the Court. Only 4 justices are needed to approve such a request. Thus, the complement of a winning coalition can also contain a winning coalition.

In light of the fact that there are so many situations that involve voting and decision making, political scientists and the lay public have developed many terms that capture special aspects of either the players in such games or their coalitions. For example, people talk about dictators, veto players, and blocking coalitions. These terms can be given meaning within the framework of voting games. For example, a dictator is a person who can "win" without the assistance of other players. So if a voting game has a minimal winning coalition with a single player, the player in that coalition is a dictator. However, unless there is a rule preventing this, there may be more than one dictator in a game. Several minimal winning coalitions might consist of a single player. If one is interested in a kind of game where this would not be possible, one would have to make an assumption which rules out having more than one minimal winning coalition with a single player.

What is the intuitive meaning of a veto player? This is someone without whose approval one can not win. This idea can be captured by saying that a player is a veto player in a game if that player is a member of every minimal winning coalition. Can a voting game with two dictators (weird thought, right?) have a veto player? If one is not content with the relationship between definitions one has made, one must either change the definition or restrict the kinds of games one wants to deal with.

Another common term in voting is the idea of a blocking coalition. The intuitive idea here is that one has a collection of players who are not winning yet can prevent any other collection of players from being a winning coalition. Based on this intuition one can formalize the definition of a blocking coalition. After defining these words one can try to find interesting examples or prove theorems involving these concepts. The important thing to notice is that mathematics often provides unexpected insights by examining situations that seem artificial. Another reason for sometimes allowing general structures that may not occur in the real world is that it is often easier to prove general assertions, which may allow for things that one may not see in practice, than to prove assertions restricted only to the cases one sees in practice! Although this is not used in common parlance, if a voting game has a player who is not the member of any minimal winning coalition, the player is known as a dummy . A dummy may have influence on other players in terms of their behavior, but there is no voting pattern where the dummy's presence or absence makes any difference.

Notation is often a handmaiden of mathematical progress. For weighted voting games the following notation has become increasingly standard. One denotes the players by 1, 2, ..., n , and the weights they cast by w 1 , w 2 , ..., w n . If the players, without loss of generality, have been named so that the weights are in decreasing order we can write:

Given a weighted voting game in this form it is a straightforward process to write out the winning coalitions and minimal winning coalitions for the game.

Example:

Consider the game:

which has three players who cast 4, 2, and 1 vote(s), respectively. If we set the quota Q at the "majority" level, namely, 4, then we have the following collection of minimal winning coalitions: {1}! So the game has a dictator in the sense that there is no minimal winning coalition with more than one player, and that player 1 by him/herself constitutes a minimal winning coalition. Players 2 and 3 are dummies in this game because they are not members of any minimal winning coalition. Even though it is not perhaps of much "political interest," it is of mathematical interest to note that as the value of Q varies from 1 to 7 one gets a different set of minimal winning coalitions each time (and one might even want to consider the "degenerate" game where the quota is zero!).

Q = 1; minimal winning coalitions: {1}, {2}, {3}

Q = 2; minimal winning coalitions: {1}, {2}

Q = 3; minimal winning coalitions: {1}, {2, 3}

Q = 4; minimal winning coalition: {1}

Q = 5; minimal winning coalition: {1, 3}

Q = 6; minimal winning coalition: {1,2}

Q = 7; minimal winning coalition: {1, 2, 3}

Note that because the players' vote totals are different powers of 2 and every number between 1 and 7 has a unique representation in the binary number system, we get a different set of minimal winning coalitions for each number from 1 to 7. One can generalize what happened here to games with four players where the weights are the powers of two. This idea allows one to get a lower bound on a count for the number of "different" weighted voting games with n players.

4. Unintuitive behavior

Simple, seemingly realistic examples show that without paying attention to some simple mathematical issues weighted voting can go awry.

Let us begin with the following example. Suppose we have a county with five cities as in the diagram below.

These cities have rather different populations: City 1 has 70,000 people, City 2 has 40,000 people, and Cities 3, 4, and 5 have, respectively, 30,000, 30,000, and 10,000 people.

It does not seem fair to have a single legislator in the county government for each city because the different cities have quite different populations. This suggests that the representative for each city cast a vote which is proportional to the population of that city. Thus, the legislature for the county would have 5 representatives, each of whom casts a weighted vote. If we "equate" 10,000 people to a weight of 1, we would have the representative of City 1 cast 7 votes, the representative of City 2 cast 4 votes, and the representatives of Cities 3, 4, and 5 cast 3, 3, and 1 vote(s), respectively. I will use the phrase cast 7 votes and cast a vote of weight 7 interchangeably. In the notation developed earlier we can write this game as:

Since the sum of the weights is 18, if half of this number plus 1 is used, the quota for this game would be 10. What are the minimal winning coalitions for this game:? They are: {1, 2}, {1, 3}. {1, 4}, and {2, 3, 4}. Unintuitively, we see that player 5 though having positive weight is never a member of any minimal winning coalition. Player 5 is a dummy. Furthermore, although player 2 has a different weight from players 3 and 4, it seems intuitively clear that players 2, 3, and 4 have the same influence. We see that players with positive weight can have no influence and players with unequal weight can have the same influence. This phenomenon can sometimes be very dramatic. Consider the two voting games below:

It should be clear that in both games all three players have equal "influence" because any two-player coalition is a winning coalition. Even though in the second game two of the players have much more weight, they do not really have more power.

What this example shows is that it is often not easy to tell how much influence or power the players exert by merely looking at the weights of the players in a voting game. Weighted voting illustrates unintuitive/paradoxical behavior as has also been the case with other aspects of voting .

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).

 Country name 1999 to 2004 2004 to 2007 2007 to 2009 Belgium 25 24 24 Bulgaria 18 Cyprus 6 6 Czech Republic 24 24 Denmark 16 14 14 Germany 99 99 99 Greece 25 24 24 Spain 64 54 54 Estonia 6 6 France 87 78 78 Hungary 24 24 Ireland 15 13 13 Italy 87 78 78 Latvia 9 9 Lithuania 13 13 Luxembourg 6 6 6 Malta 5 5 Netherlands 31 27 27 Austria 21 18 18 Poland 54 54 Portugal 25 24 24 Romania 36 Slovakia 14 14 Slovenia 7 7 Finland 16 14 14 Sweden 22 19 19 United Kingdom 87 78 78 Total 626 732 786

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:

 Germany, France, Italy and the United Kingdom 10 Spain 8 Belgium, Greece, the Netherlands and Portugal 5 Austria and Sweden 4 Denmark, Ireland and Finland 3 Luxembourg 2 Total 87

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

 Germany, France, Italy and the United Kingdom 29 Spain and Poland 27 Netherlands 13 Belgium, Czech Republic, Greece, Hungary and Portugal 12 Austria and Sweden 10 Denmark, Ireland, Lithuania, Slovakia and Finland 7 Cyprus, Estonia, Latvia, Luxembourg and Slovenia 4 Malta 3 Total 321

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, WG 1 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,  WG 2  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]

WG 3 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 WG 1 , WG 2 , ..., 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 WG 1 , ..., 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.

6. References

Algaba, E. and J. Bilbao, J. Garcia, J. Lopez, Computing power indices in weighted multiple majority games, Math. Social Sciences 46 (2003) 63-80.

Amer, R. and F. Carreras, and A. Magana, Extension of values to games with multiple alternatives, Annals of OR, 84 (1998) 63-78.

Banzhaf, J., Weighted voting does not work: a mathematical analysis, Rutgers Law Review, 19 (1965) 317-343.

Banzhaf, J., Multi-member electoral districts - do they violate the "one Man, one Vote' principle?, Yale Law J., 75 (1966) 1309-1338.

Banzhaf, J., One man, 3.312 votes: A mathematical analysis of the electoral college, Villanova Law Review, 13 (1968) 304-332.

Banzhaf, J., One man, ? votes: Mathematical analysis of political consequences and judicial choices, George Washington Law Rev., 36 (1968) 808-823.

Barua, R. and S. Chakravarty, S. Roy, P. Sarkar, A characterization and some properties of the Banzhaf-Coleman-Dubey-Shapley sensitivity index, Games and Economic Behavior, to appear.

Bilbao, J. and J. Fernandez, N. Jimenez, J. Lopez, Voting power in the European Union enlargement, European J. of Oper. Research 143 (2002) 181-196.

Bolger, E., Power indices for multicandidate voting games, Inter. J. of Game Theory, 14 (1986) 175-186.

Brams, S. and A. Doherty, M. Weidner, Game theory and multilateral negotiations: the Single European Act and the Uruguay round, In, I. Zartman (ed.), International and Multilateral Negotiation: Approaches to the Management of Complexity, Jossey-Bass, San Francisco, p. 95-112.

Chakravarty, N. and A. Goel, T. Sastry, Easy weighted majority games, Math. Social Sciences 40 (2000) 227-235.

Chvatal, V. and P. Hammer, Aggregation of inequalities in integer programming, Ann. Discrete Math., 1 (1977) 145-162.

Coleman, J., Control of collectivities and the power of a collectivity to act, In, B. Lieberman (ed.), Social Choice, Gordon and Breach, New York, 1971, p. 269-300.

Deegan, J. and E. Packel, A new index of power for simple n-person games, Int. J. Game Theory, 7 (1978) 113-123.

Diffo, L. and J. Moulen, Ordinal equivalence of power notions in voting games, Theory and Decision 53 (2002) 313-325.

Dubey, P., On the uniqueness of the Shapley value, International J. of Game Theory, 4 (1975) 131-139.

Dubey, P., and L. Shapley, Mathematical properties of the Banzhaf power index, Mathematics of Operations Research 4 (1979) 99-131.

Einy, E., The desirability relation of simple games, Math. Soc. Sci., 10 (1985) 155-158.

Felsenthal, D. and M. Machover, The weighted voting rule in the EU's council of ministers, 1958-1995: Intentions and Outcomes, Electoral Studies 16 (1997) 33-47.

Felsenthal, D. and M. Machover, W. Zwicker, The bicameral postulates and indices of a priori voting power, Theory and Decision 44 (1998) 83-116.

Felsenthal, D. and M. Machover, The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes, Edward Elgar Publishing, Cheltenham, 1998.

Frixas, J., The Shapley-Shubik power index for games with several levels of approval in the input and output, Decision support systems, to appear.

Freixas, J. and W. Zwicker, Weighted voting, abstention, and multiple levels of approval, Soc. Choice Welf., 21 (2003) 399-331.

Garrett, G. and I. McLean, M. Machover, Power, Power Indices and Blocking Power: A comment, British J. of Political Science, 25 (1995) 563-568.

Garrett, G. and G. Tsebelis, Why resist the temptation to apply power indices to the European Union?, J. of Theoretical Politics, 11 (1999) 291-308.

Garrett, G. and G. Tsebelis, More reasons to resist the temptation of power indices in the European Union, J. of Theoretical Politics 11 (1999) 331-338.

Gianaris, W., Weighted voting in the international monetary fund and the world bank, Fordham International Law Journal, 14 (1991-1992) 910-945.

Herne, K. and H. Nurmi, The distribution of a priori voting power in the EC Council of Ministers and the European Parliament, Scandinavian Political Studies, 16 (1993) 269-284.

Hilliard, M., Weighted voting theory and applications, Tech. Report No. 609, School of Operations Research and Industrial Engineering, Cornell University, 1983.

Hollwer, M. and M. Widgren, Why power indices for assessing European Union decision-making? J. of Theoretical Politics 11 (1999) 321-330.

Hosli, M., Admission of European Free Trade Association states to the European Community: effects on voting power in the European Community and Council of Ministers, International Organization, 47 (1993) 629-643.

Hosli, M., The balance between small and large: effects of a double-majority system on voting power in the European Union, International Studies Quart., 39 (1995) 351-370.

Hosli, M., Voting strength in the European Parliament: The influence of national and/or partisan actors, European J. of Political Research, 31 (1997) 351-356.

Imrie, R., The impact of weighted vote on representation in municipal governing bodies of New York State, In, L. Papayanopoulos, (ed.), Democratic Representation and Apportionment: Quantitative Methods, Measures, and Criteria, Annals of the New York Academy of Sciences, volume 219, 1973, p. 192-199.

Isbell, J., A class of majority games, Quart. J. Math. Oxford Series, 7 (1956) 183-187.

Johnson, R., The conflict over qualified majority voting in the European Union Council of Ministers: An analysis of the UK negotiating stance using power indices, British J. of Political Science, 25 (1995) 245-254.

Johnson, R., Can power be reduced to a quantitative index - and if so, which one? A response to Garrett, McLean and Machover, British J. of Political Science, 25 (1995) 568-572.

Johnson, R. On keeping touch with reality and failing to be befuddled by mathematics, British J. of Political Science, 26 (1996) 598-599.

Kilgour, D., A formal analysis of the amending formula of Canada's Constitution Act,, Canadian J. of Political Science, 16 (1983) 771-777.

Krohn, I. and P. Sudhslter, Directed and weighted majority games, Mathematical Methods of Operations Research 42 (1995) 189-216.

Lane, J.-E., and R. Maeland, Voting power under the constitution, J. of Theoretical Politics, 7 (1995) 223-230.

Lapidot, E., The counting vector of a simple game, Proceedings of the Amer. Math. Soc., 31 (1972) 228-231.

Lane, J. and S. Berg, Relevance of voting power, J. of Theoretical Politics 11 (1999) 309-320.

Lucas, W., Measuring Power in Weighted Voting Games, Case Studies in Applied Mathematics, Mathematical Association of America, Washington, 1976, pp. 42-106.

Lucas, W., Measuring power in weighted voting games, Chapter 9, in Political and Related Models, S. Brams, W. Lucas, and P. Straffin, Jr. (eds.), Springer-Verlag, New York, 1983, p. 183-238.

Lucas, W., Fair Voting, Consortium for Mathematics and its Applications (COMAP), Lexington, 1993.

Maatsui, T. and Y. Matsui, A survey of algorithms for calculating power indices of weighted majority games, J. of the Operations Research Society of Japan 43 (2000) 71-86.

Morriss, P., Power: A Philosophical Analysis, Manchester U. Press, Manchester, 1987.

Nowak, A., On an axiomatization of the Banzhaf value without the additivity axiom, Int. J. Game Theory, 26 (1997) 137-141.

Owen, G., Political games, Naval Res. Logistics Quart., 18 (1971) 345-355.

Peleg, B., A theory of coalition formation in committees, J. Math. Econ., 7 (1980) 115-134.

Penrose, L., The elementary statistics of majority voting, J. of the Royal Statistical Society, 109 (1946) 53-57.

Saari, D., The ultimate of chaos resulting from weighted voting systems, Advances in Applied Math. 5 (1984) 286-308.

Shapley, L. and M. Shubik, A method for evaluating the distribution of power in a committee system, American Political Science Review, 48 (1954) 787-792.

Shapley, L., Simple games: An outline of the descriptive theory, Behav. Sci., 7 (1962) 59-66.

Shapley, L., A comparison of power indices and a nonsymmetric generalization, Rand Paper P-5872, The Rand Corporation, Santa Monica, 1977.

Steunenberg, B. D. Schmidtchen, C. Koboldt, Strategic power in the European Union: evaluating the distribution of power in policy games, J. of Theoretical Politics 11 (1999) 339-366.

Straffin, P., Homogeneity, independence and power indices, Public Choice, 30 (1977) 107-118.

Straffin, P., Topics in Voting, Birkhäuser, Boston, 1980.

Tannenbaum, P., Power in weighted voting systems, The Mathematica Journal 7 (1997) 59-63.

Taylor, A., Mathematics and Politics, Springer-Verlag, New York, l995.

Taylor, A. and W. Zwicker, A characterization of weighted voting, Pro. American Math. Soc. 115 (1992) 1089-1094.

Taylor, A. and W. Zwicker, Weighted voting, multicameral representation, and power, Games and Economic Behavior, 5 (1993) 170-181.

Taylor, A. and W. Zwicker, Simple Games and Magic Squares, J. Combinatorial Theory, ser. A., 71 (1995) 67-88.

Taylor, A. and W. Zwicker, Quasi-weightings, trading, and desirability relations in simple games, Games and Economic Behavior 16 (1996) 331-346.

Taylor, A. and W. Zwicker, Interval measures of power, Math. Social Sci., 33 (1997) 23-74.

Taylor, A. and W. Zwicker, Simple Games, Princeton U. Press, Princeton, 1999.

Tong, Z. and R. Kain, Vote assignments in weighted voting mechanisms, IEEE Transactions on Computers 40 (1991) 664-667.

Walther, E., An analysis of weighted voting systems using the Banzhaf value, Master of Science Thesis, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, 1977.

Widgren, M., Voting power in the EC decision making and the consequences of two different enlargements, European Economic Review, 38 (1994) 1153-1170.

Widgren, M., Probabilistic voting power in the EU Council: the cases of trade policy and social regulation, Scandinavian J. of Economics, 97 (1995) 345-346.

Winder, R., Threshold Logic, Doctoral Thesis, Princeton University, Princeton, 1962.

Young, S. and A. Taylor, W. Zwicker, Counting quota systems: a combinatorial question from social choice theory, Math. Mag., 68 (1995) 331-342.

(Unsigned) How the European Union Works, European Communities, 2003 (ISBN 92-894-5283-8)

Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials. Some of the items above can be accessed via the ACM Portal , which also provides bibliographic services.

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