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Voting Games: Part II

In attempting to make sense of what goes on in complex voting situations it is tempting to think of the participants as players in a game and investigate those players who can come together to "win."

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

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1. Introduction

Democratic societies and international organizations use a wide variety of complex rules to reach decisions. Examples where it is not always easy to understand the consequences of the way voting is done include the Electoral College used to elect the President of the United States, the United Nations Security Council, the governance structure of the World Bank and the International Monetary Fund, and various councils of the European Union.

In attempting to make sense of what goes on in complex voting situations it is tempting to think of the participants as players in a game and investigate those players who can come together to "win." Such groups of winning players are known as winning coalitions. Furthermore, if each of the players in such a game could be assigned a weight, and the winning coalitions could be determined by finding those groups of players whose weight exceeds a fixed quota, then it would appear that it would be easier to see the relative power of the players involved. Unfortunately, this approach turns out to be unsuccessful because many games can not be represented as weighted voting games. Even in weighted voting games a player's influence or power is not always easily related to the weighted vote that the player casts. Thus, a player i can have positive weight but never be a member of a minimal winning coalition (a coalition that wins and the removal of a single player does not allow the coalition to win any longer). Such players are known as dummies. It can also happen that players with unequal weights do not have the same "power." Here I will examine the insights that mathematics has provided into complex voting game systems, thereby providing political scientists, the lay public, and the participants in these games insights which they would not otherwise have had.

2. Voting games

In a situation where there is a voting body where the legislators represent geographical regions, say, of different sizes, it is tempting to believe that equity can be achieved in the following way: Each representative will be assigned a weight proportional to the population of the geographic region that the legislator represents. A quota is established by summing these weights, dividing by 2, and moving up to the next largest integer. When this is done we have an example of a weighted voting game. The winning coalitions are precisely those coalitions whose weights sum to at least the quota. More generally one might set the quota in a more arbitrary manner (typically larger than the quota just defined) and take the winning coalitions to be those whose members have weights that sum to at least the quota. This approach captures the idea of having more than a simple majority for an action to be taken - for example, to approve a treaty or amend a constitution. Games that are weighted voting games seem appealingly simple. Let us look at an actual case history to see what can happen if one proceeds in a naive fashion.

This case history involves Nassau County, New York, which is the geographic region on Long Island just to the east of New York City. Nassau County's government took the form of a Board of Supervisors, one representative for each of various municipalities, who cast a block of votes. Not only did Nassau County use weighted voting, but weighted voting was also used for the counties of New York State other than the 5 counties in New York City.

Here are the weighted voting systems used at various times by Nassau County. The passing quota shown reflects the number of votes needed to pass "ordinary legislation." Some special voting situations required larger quotas than the numbers shown.

  1958 1964 1970 1976 1990
Hempstead (H1) 9 31 31 35 29
Hempstead(H2) 9 31 31 35 29
North Hempstead 7 21 21 23 15
Oyster Bay 3 28 28 32 22
Long Beach 1 2 2 2 7
Glen Cove 1 2 2 2 6
Total votes 30 115 115 129 108
Quota to pass 16 58 63 71 65

The numerical weights were chosen to try to take into account the populations of the different municipalities, which were quite disparate. It is easy to see that in 1958, Oyster Bay, Long Beach and Glen Cove were dummies (!) and that North Hempstead had the same power as Hempstead 1 and Hempstead 2 because any two of these communities formed a minimal winning coalition. You should check for yourself that in 1964 there were also dummies. After 1964 the quota was raised to guarantee that no community was a dummy. However, eventually Nassau County converted to having a county legislature because of the difficulties of using a weighted voting system in a situation where there were so few players and such large differences in the populations of the communities.

Still, this approach to governance has its own problems. District lines cut across what had often been natural historical political entities. Furthermore, though the districts served by each member of the county legislature could be made approximately equal in population, this could not be done in a way that avoided some members of the legislature being tempted to vote in an identical way because, though they are different people, they have identical interests, given the geographical communities that they serve. To the extent that it is desirable that every member of the legislature act in an independent manner, this has been hard to achieve.

3. When are voting games weighted voting games?

When a voting game is described in a complex verbal way it is often difficult for the participants in the game and the people whom the game serves to know what the "influence" or "power" relations of the players are. If a game is given in weighted voting form at least one can relatively easily check if a particular set of players is a winning coalition or not. Thus, it is a natural question to ask whether or not a game can be represented as a weighted voting game. Again, this means that the players cast weights and there is a value Q such that a coalition is winning if and only if the sum of the weights cast by the players in the coalition is Q or greater.

For simple games with few players all simple voting games are weighted. However, for larger simple games this is no longer true.

Suppose we have a weighted voting game G and two winning coalitions of players C1 and C2 where C1 contains a player i who is not in C2, and C2 contains a player j not in C1. Now form two new coalitions by having i leave C1 and join C2, and j leave C2 to join C1. There are two possibilities: either i and j have equal weight or they do not. In the first case the new coalitions after the switch must still both be winning coalitions since the weights of the players in the two coalitions have not changed after the switch. However, in the case of unequal weights for i and j, after the switch one of the new coalitions has increased in weight. Hence, after the switch it can not be that both of the new coalitions are losing. Therefore, if we can find a voting game having coalitions where switching a pair of players, as described above, changes two winning coalitions to two losing coalitions, the game we started with can not be a weighted voting game! The argument above can be extended in a natural way to exchanging groups of players instead of individual players. Using this idea of trading, it is possible to show that the "game" of passing a bill by the U.S. Congress and the President can not be represented as a weighted voting game. (This game involves 435 members of the House of Representatives, 100 Senators, the Vice-President (who votes when there is a tie in the Senate but not otherwise) and the President!)

Although not every voting game is weighted, Alan Taylor and William Zwicker showed that every simple voting game can be given a "vector" representation. The idea in a vector-representation of a voting game is that one has an n-tuple of numbers and a quota associated with each of the positions of the n-tuple. The winning coalitions are found by vector addition (adding the components) for the players having total weight exceeding the quotas for each component. For example, with the following vector weights:

A member of the House of Representatives:

(0, 1)

A member of the Senate:

(1, 0)

The Vice-President:

(.5, 0)

The President:

(16.5, 72)

and quotas of (67, 290), one successfully captures the winning coalitions of the "passing a law" game. Thus, while the voting game for passing a federal law is not weighted, it can be understood as a relatively simple vector game. These ideas have been used to try to measure the "complexity" of the rules involved in various real-world voting games. Taylor and Zwicker also obtained many fascinating results about the relationship between "trading" and the structure of voting games and weighted voting games.

4. Power indices

While the issue of being able to represent a voting game as a weighted voting game is a convenience for the players in thinking about the game, the advantage is deceptive. The reason is that the weight relations that are involved in a weighted voting game do not reflect the "influence" or "power" relations. We have already seen that a player may have positive weight and yet be a dummy. Thus, it is natural to ask whether there is a way to find the power or influence of the players in a voting game. If this were possible in a way that people could agree on, then one could try to assign weights to players that gave rise to "power," i.e. that was proportional to population as a way of ensuring equity in the weighted voting game setting.

Relatively early in the development of game theory, proposals were put forward to measure the strength, influence, or power of the players in voting games. Among the earliest power indices that were developed for voting games used a game theory idea now known as the Shapley Value, named for Lloyd Shapley. The power index which grows out of this concept is known as the Shapley-Shubik Index which is named for the mathematician Lloyd Shapley (below, left) and the economist Martin Shubik (below, right). Both Shapley and Shubik have had influential careers in creating and applying game theory ideas.

Photo of Lloyd Shapley Photo of Martin Shubik

Although the Shapley-Shubik index can be approached axiomatically, let me explain the idea in a more intuitive way. Suppose that in a voting game one imagines all the players being "polled" in a particular order. When a winning coalition forms the first time, the person who changed the situation to a winning coalition is given a "pivot point." Thus, i gets a pivot point if a yes vote of the players who are polled before i do not form a winning coalition, but with i, they do. This means that whether a player gets a pivot or not depends on the player's role in the winning coalition structure and the ordering picked.

Consider all the orderings of the n players in a voting game. Repeating the idea above in slightly different terms, the players vote in a particular order, and it is noted at what point some player i, together with the previous players (if there were any), form a winning coalition. We will give player i in this ordering a pivot point if the players up to that point do not form a winning coalition but the players together with i do form a winning coalition. If there are n players, there are n! orderings and one pivot point per ordering. The Shapley-Shubik power index for player i is the total number of pivots for i divided by n!.

Here is a simple example with four players:

The game with quota 4, and 4 players casting weights of 4, 3, 2, and 1

Note that the quota required here (7) is more than a simple majority, which would be 6. To avoid confusion between the weight a player casts and the names of the voters we will use the names of w, x, y, and z for the players who cast votes of weight 4, 3, 2, and 1, respectively.

Since there are 4 players, we have 24 different ways to arrange them. Reading from left to right I have placed a star to the right of that player's name whose vote together with those of the previous players first creates a winning coalition (i.e. weight sum of 7 or more).


The Shapley-Shubik power index (written as a fraction in lowest terms) of each of the players is below:

w power 7/12,

x power 1/4,

y power 1/12, and

z power 1/12.

Photo of John Banzhaf IIIFirst, note that the fractions sum to 1, and thus one can interpret these numbers as probabilities. Note also that the power reflects the "symmetries" present in the way that the players enter into the minimal winning coalitions. The minimal winning coalitions are {w,x} and {w,y,z}. Thus, it should not be surprising that y and z have the same power though they have different weights.

More important for political games was a power index proposed by the mathematically trained (but lawyer by profession) John Banzhaf III. Banzhaf realized - and took the issue to the public and the courts - that it was naive to try to get equity for voting games by merely assigning weights proportional to population.

Banzhaf also developed a way of computing the power of the players in voting games that is both intuitively appealing and seems more natural for political situations than that of Shapley and Shubik. Suppose that we have a voting game and consider a "bill" that has been placed before the players in the game. Each of the players can be thought of as voting yes or no on the bill. Now, consider a particular yes-no voting pattern P. We can see if the bill passes or not for pattern P. When the bill does pass, this means that we must have a winning coalition for the players who voted in this pattern. Now, for each player s in P who voted yes we can determine what would happen if s had voted no. If the change from yes to no changes the bill from passing to failing to pass, we assign s a pivot point. Note there may be several pivots for a given pattern P or possibly no pivots assigned for that pattern. (This contrasts with the Shapley-Shubik situation where for each of the n! orderings of the players one gets one pivot per ordering. In the Banzhaf case there are 2n yes/no patterns for n players, but not all of these patterns result in a pass.) We will define the Banzhaf power index of player i to be the number of pivots that i receives for all patterns P where the bill passes and i favored the bill, and where, if i were to switch his/her vote (the other votes staying fixed), the bill would no longer pass, divided by the total number of pivots.

Shown below is the game for which earlier we computed the Shapley-Shubik power index of the players.


Here is how the calculation goes for the Banzhaf power index. We list below all the winning coalitions (e.g. yes/no patterns which result in a pass), and next to each player (on the right) we put a "^" symbol if that player, by changing a vote from yes to no, would change the winning coalition to a losing one:


There are a total of ten "^" symbols in the list above.

Hence the Banzhaf power indices are:

w has power 1/2,

x has power 3/10,

y has power 1/10, and

z has power 1/10.

Again the equality of power for y and z reflects the symmetry with which these players enter into the minimal winning coalitions. Although it might appear that the Shapley-Shubik and Banzhaf indices give different numerical values but qualitatively the same picture, this is not always the case. (The analysis of a proposed way to amend the Canadian Constitution, which was never adopted, illustrates this point, as does the analysis of the game to pass a bill in the United States (Congress and President interacting)). Note that the powers that we are talking about here do not accrue from the person casting the vote but from the structural properties of the game. It is conceivable that some who structurally would be dummies might have considerable influence in the way that actions actually get taken because of their charisma.

Which is better, the Shapley-Shubik or Banzhaf approach to power indices? Before you decide, you should realize that these are not the only candidates for power indices. There are additional indices, a notable one due to J. Deegan and E. Packel. This index is intriguing for treating coalitions of different sizes in a different way. This might well make sense, since one might think that smaller coalitions are more likely to form than larger coalitions. Yet, one can also imagine scenarios where the dynamics of how coalitions form are very complex, so that it is hard to sort out how power and winning coalition size interact. A variety of scholars have provided interesting axiomatizations of a variety of power indices (e.g. J. Deegan, P. Dubey, E. Packel, L. Shapley). The value of axiomatizations is that if one can characterize different power indices axiomatically, one can analyze a particular real-world situation, thereby helping one decide which power indices make sense.

What is tantalizing is that clever mathematical analysis and the attempts of social scientists (such as Shubik) further both the insights we have into fairness and equity questions as well as intriguing aspects of mathematics. When Taylor and Zwicker collected information for their recent book Simple Games, they came across many independent rediscoveries of ideas related to games in many areas of knowledge. As so often has been the case, mathematics and its applications progress together.

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

5. 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.

Aumann, R. and S. Hart, (eds.), Handbook of Game Theory with Economic Applications, Volumes 1 and 2, North-Holland, Amsterdam, 1992, 1994.

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 P. Affuso, New paradoxes of voting power on the EC Council of Ministers, Electoral Studies, 4 (1985) 134-139; 290.

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.

Brams, S. and W. Riker, Models of coalition formation in voting bodies, in Herndon and Bernd, (eds.), Mathematical Applications in Political Science, VI, U. Of Virginia Press, Charlottesville, 1972.

Carreras, F. and J. Freixas, Complete simple games, J. Math. Social
Sciences, 32 (1996) 139-155.

Carreras, F. and J. Freixas, A power analysis of linear games with
consensus, J. of Math. Social Sciences, 48 (2004) 207-221.

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.

Freixas, J., The dimension for the European Union Council under the Nice rules, European J. of Operations Research, 156 (2004) 415-419.

Freixas, J., The Shapley-Shubik power index for games with several levels of approval in the input and output, Decision Support Systems, 39 (2005)

Freixas, J., The Banzhaf index for games with several levels of approval in the input and output, Annals of Operations Research, 137 (2005) 45-66.

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.

Kahan, J. and A. Rapoport, Theories of Coalition Formation, Lawrence Erlbaum Associates, Hillsdale, 1984.

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

Kilgour, D. and T. Levesque, The Canadian constitutional amending formula: bargaining in the past and future, Public Choice, 44 (1984) 457-480.

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.

Miller, D., A Shapley value analysis of the proposed Canadian constitutional amendment scheme, Canadian J. of Political Science, 4 (1973) 140-143.

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

Morrow, J., Game Theory for Political Scientists, Princeton U. Press, Princeton, 1994.

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.

Owen, G. and L. Shapley, Optimal location of candidates in ideological space, International J. of Game Theory, 18 (1989) 339-356.

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.

Roth, A. (ed.), The Shapley Value: Essays in Honor of Lloyd S. Shapley, Cambridge U. Press, London, 1988.

Saari, D., The ultimate 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., The power of voting blocs: an example, Mathematics Magazine, 50 (1977) 22-24.

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

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

Straffin, P., Power indices in politics, in S. Brams, W. Lucas, and P. Straffin, (eds.), Political and Related Models, Springer-Verlag, New York, 1983.

Straffin, P., The Shapley-Shubik and Banzhaf indices as probabilities, in A. Roth, (ed.), The Shapley Value: Essays in Honor of Lloyd S. Shapley, Cambridge U. Press, London, 1988.

Straffin, P., Game Theory and Strategy, Mathematical Association of America, Washington, 1993.

Straffin, P., Power and Stability in Politics, in Handbook of Game Theory with Economic Applications, Volume 2, R. Aumann, and S. Hart, (eds.), North-Holland, Amsterdam, 1994, p. 1128-1151.

Straffin, P. and B. Grofman, Parliamentary coalitions: a tour of models, Mathematics Magazine, 57 (1984) 259-274.

Straffin, P. and M. Davis, and S. Brams, Power and satisfaction in an ideologically divided voting body, in M. Holler (ed.), Power, Voting and Voting Power, Physica Verlag, Wurzburg, 1981.

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

Taylor, A., Mathematics and Politics, Springer-Verlag, New York, 1995.

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)


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