Chapter 13 – Weighted Voting Part 4

1. Chapter 13 – Weighted VotingPart 4 • Appropriate applications of measures of power • Minimal winning coalitions • Classification of weighted voting systems

2. Appropriate Applications of the Measures of Power • When should we use the Banzhaf measure of power ? • When should we use the Shapley-Shubik measure of power ? • What assumptions do we make in using either of these measures of power ?

3. Appropriate Applications of the Measures of Power • We can consider the various permutations of voters within a weighted voting system to represent a spectrum of opinions on any given issue. • Voters may rearrange themselves on that spectrum in different ways for different issues. • At one end of the spectrum is the voter who is very much in favor of passing a measure and at the other end is the voter who is very much in opposition. • We can use the Shapley-Shubik analysis to measure power within a weighted voting system in this type of situation.

4. Appropriate Applications of the Measures of Power • We use the Shapley-Shubik measure of power for situations where it makes sense to consider the various permutations or arrangements of the voters. • We can use the Shapley-Shubik analysis to measure power when we model situations in which expect voters to influence each other, building coalitions, one voter at a time.

5. Appropriate Applications of the Measures of Power • An important assumption in the Shapley-Shubik analysis which is as yet unmentioned is the assumption that all permutations are equally likely. • In fact, in many real-life situations, perhaps there are some voters who are very often at the extremes of the spectrum of opinions on many issues. And perhaps there are other voters who are generally somewhere in the middle of the spectrum of opinions. • If this is the case, then perhaps not all permutations of voters are equally likely. This means that, in reality, those voters in the middle of the spectrum may be the critical voters on many issues more often than those at the extremes. • Further, because voters who are regularly in the middle of the spectrum of opinion are more likely to be critical on most issues, in a sense, this increases there effective power within that voting system.

6. Appropriate Applications of the Measures of Power • The Banzhaf measure of power does not consider any spectrum of opinion. • With the Banzhaf measure of power there is no consideration of the dynamics of building a coalition – where a voter might try to persuade another to join a coalition. • In the Banzhaf analysis of power distribution there is also an assumption that all of the various combinations (coalitions) of voters are equally likely. • Again, in some situations, this may not be the case.

7. Appropriate Applications of the Measures of Power • With the Banzhaf analysis of power within a weighted voting system, we are making an assumption that all coalitions are equally likely. This may not be the case. • For example, consider the system [7: 6, 6, 1]. • With Banzhaf’s analysis of the power distribution within this system, it is determined that all voters are equally powerful. • The Banzhaf index is (4, 4, 4). • However, in reality, perhaps the two voters with 6 votes, maybe for political reasons, always form coalitions in opposition to the third voter. In effect, that voter then has no power in this system.

8. Appropriate Applications of the Measures of Power • The previous example illustrates that one important assumption implicit in the Banzhaf analysis of power is this: • It is assumed that the voters within a weighted voting system are equally likely to form any possible coalition. • This assumption is valid when it is understood that the purpose of the Banzhaf power index (just like the Shapley Shubik power index) is to measure in some way the distribution of power inherent in the system itself, regardless of who the participants are within that system, and other political considerations.

9. Appropriate Applications of the Measures of Power • Is Banzhaf or Shapley-Shubik a better analysis of power? We might answer one way or another depending on the situation. • For example, with the example of power within the United Nations Security Council, it could be argued that the Shapely-Shubik analysis is a good measure of the power distribution because of the way in which resolutions are drafted. Supposing that resolutions are drafted with the intention that they will pass, they will be written in such a way that they will appeal to the five permanent members who will then look for support among the nonpermanent members. • Ultimately, there is no right or wrong answer to which is better. The decision about which method of analysis is better is subjective. Perhaps the best way to summarize this situation is as follows: • Imagine you are a lawyer and you represent a client who is participating in a weighted voting system. Why analysis do you consider better? The answer may depend on which is in the interest of your client. And to prepare a strong case, you will need to understand both types of analysis.

10. Minimal Winning Coalitions • The reason we study the minimal winning coalitions for weighted voting systems is that it will provide a way for us to classify weighted voting systems. • Using this approach, for example, we will demonstrate there are essentially only two different weighted voting systems for systems of two voters and that there are essentially only five different possible weighted voting systems for systems of three voters. This approach can be extended for classifying systems of any number of voters. • Definition: A winning coalition in which every voter is critical is called a minimal winning coalition. • Fact: The minimal winning coalitions for a weighted voting system uniquely determine that system. Given two weighted voting systems with the same number of voters, we need only consider the minimal winning coalitions to decide if the systems are equivalent.

11. Minimal Winning Coalitions • Weighted voting systems are equivalent if and only if, regardless of the names of the voters of each system, they have the same minimal winning coalitions. • Any collection of sets of voters can serve as the minimal winning coalitions for a given weighted voting system provided that all of the following are true: • There is at least one minimal winning coalition. • If two minimal winning coalitions are distinct, each must have a voter who does not belong to the other. (Neither is a subset of the other). • Every pair of minimal winning coalitions has to overlap, with at least one voter in common.

12. Minimal Winning Coalitions • Any collection of sets of voters can serve as the minimal winning coalitions for a given weighted voting system provided that all of the following are true: • There is at least one minimal winning coalition • If two minimal winning coalitions are distinct, each must have a voter who does not belong to the other. (Neither is a subset of the other). • Every pair of minimal winning coalitions has to overlap, with at least one voter in common. • Consider each of the three requirements listed above: • Without at least one minimal winning coalition no measure could ever pass. • Neither of two minimal winning coalitions can be a subset of the other, otherwise the larger set would not be minimal. • If any two coalitions did not overlap it would be possible for two opposing coalitions to both win. (The nonempty intersection of two coalitions will guarantee their agreement.)

13. Minimal Winning Coalitions – Only Two Voters • There is at least one minimal winning coalition • If two minimal winning coalitions are distinct, each must have a voter who does not belong to the other. (Neither can be a subset of the other.) • Every pair of minimal winning coalitions has to overlap, with at least one voter in common. One possibility The other possibility A B A B The system has a dictator Consensus is required

14. Minimal Winning Coalitions – Only Two Voters One possibility The other possibility A B A B The system has a dictator Consensus is required Examples: All of these systems are equivalent [ 5 : 6,1 ] [ 10 : 10, 5 ] [ 51 : 60, 40 ] [ 51 : 51, 49 ] [ 51 : 99,1 ] Examples: All of these are equivalent [ 5 : 4,1 ] [ 10 : 9, 5 ] [ 51 : 50, 50 ] [ 40 : 20, 20 ] [ 22 : 20, 20 ]

15. Minimal Winning Coalitions – 3 Voter Systems A A A B B C B C C A A B C B C Can you think of how to draw the last two possibilities?

16. Minimal Winning Coalitions – 3 Voter Systems A A A B C B B C C Consensus Dictator Clique A A B C B C Majority Chair Veto

17. Minimal Winning Coalitions – 3 Voter Systems • There are only 5 different possible weighted voting systems with 3 voters: • Dictator • Clique • Consensus • Chair Veto • Majority Rules • Examples of 3 voter systems with a Dictator: All of these are equivalent. System Banzhaf power Shapley-Shubik power • [ 10: 10, 5, 1 ] ( 8, 0, 0 ) ( 1, 0, 0 ) • [ 5: 6, 2, 1 ] ( 8, 0, 0 ) ( 1, 0, 0 ) • [ 51: 60, 20, 20 ] ( 8, 0 , 0 ) ( 1, 0, 0 )

18. Minimal Winning Coalitions – 3 Voter Systems • There are only 5 different possible weighted voting systems with 3 voters: • Dictator, Clique, Consensus, Chair Veto, and Majority Rules • Examples of 3 voter systems with a Clique : All of these are equivalent. System Banzhaf power Shapley-Shubik power • [ 4: 2, 2, 1 ] ( 4, 4, 0 ) ( 1/2, 1/2, 0 ) • [ 51: 48, 3, 1] ( 4, 4, 0 ) ( 1/2, 1/2, 0 ) • [ 12: 8, 5, 1 ] ( 4, 4, 0 ) ( 1/2, 1/2, 0 )

19. Minimal Winning Coalitions – 3 Voter Systems • There are only 5 different possible weighted voting systems with 3 voters: • Dictator, Clique, Consensus, Chair Veto, and Majority Rules • Examples of 3 voter systems with Consensus rule : All of these are equivalent. System Banzhaf power Shapley-Shubik power • [3: 1, 1, 1 ] ( 2, 2, 2 ) ( 1/3, 1/3, 1/3 ) • [100: 60, 39, 1] ( 2, 2, 2 ) ( 1/3, 1/3, 1/3 ) • [10: 8, 1, 1] ( 2, 2, 2 ) ( 1/3, 1/3, 1/3 ) • [12: 5, 5, 2] ( 2, 2, 2 ) ( 1/3, 1/3, 1/3 )

20. Minimal Winning Coalitions – 3 Voter Systems • There are only 5 different possible weighted voting systems with 3 voters: • Dictator, Clique, Consensus, Chair Veto, and Majority Rules • Examples of 3 voter systems with Chair Veto : All of these are equivalent. System Banzhaf power Shapley-Shubik power • [ 3: 2, 1, 1 ] ( 6, 2, 2 ) ( 2/3, 1/6, 1/6 ) • [ 4: 3, 2, 1 ] ( 6, 2, 2 ) ( 2/3, 1/6, 1/6 ) • [ 20: 15, 7, 5] ( 6, 2, 2 ) ( 2/3, 1/6, 1/6 )

21. Minimal Winning Coalitions – 3 Voter Systems • There are only 5 different possible weighted voting systems with 3 voters: • Dictator, Clique, Consensus, Chair Veto, and Majority Rules • Examples of 3 voter systems with Majority Rules : All of these are equivalent. System Banzhaf power Shapley-Shubik power • [ 38: 30, 30, 15] ( 4, 4, 4 ) (1/3, 1/3, 1/3) • [ 2 : 1, 1, 1 ] ( 4, 4, 4 ) (1/3, 1/3, 1/3) • [ 11 : 7, 5, 6 ] ( 4, 4, 4 ) (1/3, 1/3, 1/3)