Transcript Part 4 - Minimal Winning Coalitions

Chapter 13 – Weighted Voting
Part 4
• Appropriate applications of measures of power
• Minimal winning coalitions
• Classification of weighted voting systems
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 ?
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.)
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
The system has a dictator
Consensus is required
Minimal Winning Coalitions – Only Two Voters
One possibility
The other possibility
The system has a dictator
Consensus is required
Examples: All of these systems are
equivalent
Examples: All of these are equivalent
[ 5 : 6,1 ]
[ 10 : 10, 5 ]
[ 51 : 60, 40 ]
[ 51 : 51, 49 ]
[ 51 : 99,1 ]
[ 5 : 4,1 ]
[ 10 : 9, 5 ]
[ 51 : 50, 50 ]
[ 40 : 20, 20 ]
[ 22 : 20, 20 ]
Minimal Winning Coalitions – 3 Voter Systems
Can you think of how to draw the last two possibilities?
Minimal Winning Coalitions – 3 Voter Systems
Dictator
Chair Veto
Clique
Consensus
Majority
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
– [ 10: 10, 5, 1 ]
– [ 5: 6, 2, 1 ]
– [ 51: 60, 20, 20 ]
Banzhaf power
( 8, 0, 0 )
( 8, 0, 0 )
( 8, 0 , 0 )
Shapley-Shubik power
( 1, 0, 0 )
( 1, 0, 0 )
( 1, 0, 0 )
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
– [ 4: 2, 2, 1 ]
– [ 51: 48, 3, 1]
– [ 12: 8, 5, 1 ]
Banzhaf power
( 4, 4, 0 )
( 4, 4, 0 )
( 4, 4, 0 )
Shapley-Shubik power
( 1/2, 1/2, 0 )
( 1/2, 1/2, 0 )
( 1/2, 1/2, 0 )
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
–
–
–
–
[3: 1, 1, 1 ]
[100: 60, 39, 1]
[10: 8, 1, 1]
[12: 5, 5, 2]
Banzhaf power
( 2, 2, 2 )
( 2, 2, 2 )
( 2, 2, 2 )
( 2, 2, 2 )
Shapley-Shubik power
( 1/3, 1/3, 1/3 )
( 1/3, 1/3, 1/3 )
( 1/3, 1/3, 1/3 )
( 1/3, 1/3, 1/3 )
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
– [ 3: 2, 1, 1 ]
– [ 4: 3, 2, 1 ]
– [ 20: 15, 7, 5]
Banzhaf power
( 6, 2, 2 )
( 6, 2, 2 )
( 6, 2, 2 )
Shapley-Shubik power
( 2/3, 1/6, 1/6 )
( 2/3, 1/6, 1/6 )
( 2/3, 1/6, 1/6 )
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
– [ 38: 30, 30, 15]
– [ 2 : 1, 1, 1 ]
– [ 11 : 7, 5, 6 ]
Banzhaf power
( 4, 4, 4 )
( 4, 4, 4 )
( 4, 4, 4 )
Shapley-Shubik power
(1/3, 1/3, 1/3)
(1/3, 1/3, 1/3)
(1/3, 1/3, 1/3)