Mathematical Modeling: Weighted Voting: Session 5

prepared by:

Joseph Malkevitch
Mathematics Department
York College (CUNY)
Jamaica, NY 11451


web page:

Weighted voting has been one way that political bodies have tried to provide a "fair" way to create a "legislative" environment.

In addition to the material in Chapter 11 of For All Practical Purposes (7th or 8th editions), there is a nice brief treatment below:

(note, however, that one need not require that the value of the quota be an integer. Also, this article has not been updated for all the changes that have occurred with regard to the European Union.)

Every time a new country enters the European Union (EU) a negotiation must occur to determine what weights to assign the countries that are going to constitute the "new" European Union. At various times there has been controversy that these weights have been set in a way that is unfair to "powerful" countries or to "weak" countries. Every time there is a census in the United States, voting bodies tied to representation by population (one of the principles laid down by the US Constitution, Congress, and the Supreme Court) the weights in various weighted voting games (including the Electoral College, and NY State County Boards of Supervisors) must be altered to be in compliance with legal requirements. In a famous incident in American history there was a time when the Congress could not agree on the number of seats each state was to get in the House of Representatives and so, for a period, the Constitution was being "disregarded."

In the discussion below, for convenience, I will assume we have voting games which have no dummy players.

Two voting games (whether or not they are weighted voting games, and not all voting games have weighted voting representations) are isomorphic or equivalent if they have the same minimal winning coalitions.


[51; 49, 47, 4} has minimal winning coalitions [1, 2}, {1,3} and {2, 3}.

[2; 1, 1, 1] has minimal winning coalitions [1, 2}, {1,3} and {2, 3}.

[18; 11, 10, 8] has minimal winning coalitions [1, 2}, {1,3} and {2, 3}.

All three of these games have the same minimal winning coalitions so they are really the same "same" game in different disguises.

As the number of players increases the number of weighted voting games with n players grows quickly, but there is not much "room" for games with very few players, to approximate the varied populations that might occur between players for the purpose of designing a fair legislative situation. One gets more choice by moving to voting games that can not be represented by a weighted voting scheme. There is beautiful work of Alan Taylor and William Zwicker that builds on ideas from earlier workers on switching circuits that provides a characterization of weighted voting games within the category of "simple games" (monotonic with respect to winning coalitions) that involves the idea of "trading" players in coalitions.

Perhaps the biggest issue with weighted voting is that when weights are assigned proportional to some indicator such as population, gross national product, etc., there is no guarantee that the resulting weighted voting game will yield "power" to the player in the same proportions. Thus, in the example above [51; 49, 47, 4} the total weights cast by the players is 100. Thus, player 3 has 4% of the total weight. However, player 3 has 1/3 of total power, the same as each of the other players, by any reasonable notion of power. We have seen examples where a player can have positive weight and no power, in the sense of not being a member of any minimal winning coalition. Such a player is usually referred to as a "dummy."

Political scientists, psychologists, and sociologists are concerned with the reality and perception of power in political voting situations. There are many articles which deal with empirical questions here. Does the Banzhaf or Shapley-Shubik Power Index better reflect perceived and/or actual power relationships in voting games. Note, that these indices can be computed from the minimal winning coalitions and, thus, do not depend on whether or not the voting game has a weighted voting representation.


William Poundstone has written several "mass market" books that you might enjoy related to game theory and voting.

Prisoner's Dilemma (1992)

Gaming the Vote: Why Elections aren't Fair (and What We can Do About It)

The book by Alan Taylor and William Zwicker, Simple Games, Princeton University Press has many interesting results related to the pure and applied mathematics related to voting games. However, the book is quite technical.

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