Voting Games: Part I
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 twoplayer 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.

Introduction

Voting systems and games

Basic mathematical ideas

Unintuitive behavior

More voting games

References