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2
Weighted Voting
Systems
The Power Game
In a democracy we take many things for granted, not the least of
which is the idea that we are all equal. When it comes to voting
rights, the democratic ideal of equality translates into the principle
of one person–one vote. But is the principle of one person–one
vote always justified? Must it apply when the voters are something
other than individuals, such as organizations, states, and even
countries?
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I
n a diverse society, it is in the very nature of things that voters—be
they individuals or institutions—are not equal, and sometimes it is
actually desirable to recognize their differences by giving them different amounts of say over the outcome of an election. What we are talking
about here is the exact opposite of the principle of one voter–one vote, a
principle best described as one voter–x votes, and more formally known as
weighted voting.
One of the best known and most controversial examples of weighted
voting is the Electoral College, that uniquely American institution bequeathed to the nation by the Founding Fathers through Article II, Section 1
of the U.S. Constitution. In the Electoral College, each of the 50 states controls a number of votes equal to its number of Representatives plus Senators
in Congress (in addition, the District of Columbia controls 3 votes). At one
end of the spectrum is California, an electoral heavyweight with 55 electoral
votes; at the other end of the spectrum are the little guys (Alaska, Montana,
North Dakota, etc.) with a meager 3 electoral votes each. The other states
fall somewhere in between. (See the appendix at the end of this chapter for
full details.) The 2000 and 2004 presidential elections brought to the surface,
in a very dramatic way, the vagaries and complexities of the Electoral College system, and, in particular, the pivotal role that a single state (Florida in
2000, Ohio in 2004) can have in the final outcome of the election.
Even in democratic societies, weighted voting is quite common—we
can find examples of weighted voting in shareholder elections, in business partnerships, in international legislative bodies, such as the United
Nations Security Council and the European Union Council of Ministers,
and, of course, in American presidential elections. In this chapter we will
look at the mathematics behind weighted voting. We will start by introducing and illustrating the concept of a weighted voting system, a simple
mathematical formalism that will allow us to describe most
weighted voting situations (Section 2.1). In the remaining
sections we will focus on two different mathematical measures of a voter’s power in a weighted voting system.
Sections 2.2 and 2.3 discuss what is known as the Banzhaf
power index of a voter; Sections 2.4 and 2.5 discuss what is
known as the Shapley-Shubik power index of a voter. In so
doing we will be introduced to several basic but important
mathematical ideas—coalitions, sequential coalitions, and
factorials. Enjoy the ride!
Each State shall appoint a Number of
Electors equal to the whole Number
of Senators and Representatives to
which the state may be entitled in
the Congress.
Article II, Section 1
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50 2 Weighted Voting Systems
2.1 Weighted Voting Systems
We start this section by introducing some terminology. We will use the term
weighted voting system to describe any formal voting arrangement in which voters are not necessarily equal in terms of the number of votes they control. To
keep things simple, we will only consider yes-no votes, generally known as
motions (a vote between two candidates or alternatives can be rephrased as a
yes-no vote and thus is a motion).
Every weighted voting system is characterized by three elements:
■
The players. The voters in a weighted voting system are called players. (From
now on we will stick to the usual convention of using “voters” when we are
dealing with a one person–one vote situation as in Chapter 1, and “players” in
the case of a weighted voting system.) Note that in a weighted voting system
the players may be individuals, but they may also be institutions, agencies,
corporations, and even nations. We will let N denote the number of players
and P1 , P2 , Á , PN the names of the players. (Think of P1 as short for “player
1,” P2 as short for “player 2,” etc.—it is a little less personal but a lot more
convenient than using Archie, Betty, Charlie, etc.)
■
The weights. The hallmark of a weighted voting system is that each player
controls a certain number of votes, called the weight of the player. We will assume that the weights are all positive integers, and we will let w1 , w2 , Á , wN
represent the weights of P1 , P2 , Á , PN , respectively. (A weighted voting system is only interesting when the weights are not all the same, but in principle
we do not preclude the possibility that the weights are all equal. Painted with
this broad brush, weighted voting includes ordinary voting as a special case—
just set all the weights equal to 1.)
■
The quota. The quota is the minimum number of votes needed to pass a motion. We will use q to denote the quota. It is important to realize that the
quota q can be something other than a simple majority of the votes. There are
many voting situations in which the requirement to pass a motion is set higher than just a majority. In the U.S. Senate, for example, it takes a simple majority to pass an ordinary law, but it takes a minimum of 60 votes to stop a
filibuster, and it takes a minimum of two-thirds of the votes to override a
presidential veto. In other organizations the rules may stipulate that threefourths (75%) of the votes are needed, or four-fifths (80%), or even unanimity (100%). In fact, any integer q is a possible choice for the quota as long as
(i) it is more than 50% of the total number of votes, and (ii) it is not more
than 100% of the votes.
Notation and Examples
Every weighted voting system can be described by the generic form
3q: w1 , w2 , Á , wN4. In this notation, we use square brackets to indicate we are
dealing with a weighted voting system. Inside the square brackets we always list
the quota first, followed by a colon and then the respective weights of the individual players. It is customary to list the weights in numerical order, starting with
the highest, and we will adhere to this convention throughout the chapter.
We will now look at a few examples to illustrate some basic concepts in
weighted voting.
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2.1 Weighted Voting Systems 51
EXAMPLE 2.1 Weighted Voting in a Partnership
Four partners—which we will call P1 , P2 , P3 , and P4—decide to start a new business venture. In order to raise the $200,000 needed as startup money, they issue
20 shares worth $10,000 each. Suppose that P1 owns 8 shares, P2 owns 7 shares, P3
owns 3 shares, and P4 owns 2 shares, and that each share equals one vote in the
partnership. Imagine now that they set up the rules of the partnership so that
two-thirds of the partner’s votes are needed to pass any motion. Using our new,
simplified notation, this partnership can be described as the weighted voting system [14: 8, 7, 3, 2]. (Note that the quota is 14 because 14 is the first integer larger
than two-thirds of 20.)
EXAMPLE 2.2 The Quota Can’t Be Too Small Á
Imagine the same partnership discussed in Example 2.1, with the only difference
being that the quota is changed to 10 votes. We might be tempted to think that
this arrangement can be described by [10: 8, 7, 3, 2], but this is not a viable weighted voting system. The problem here is that the quota is not big enough to allow
for any type of decision making. Imagine that an important decision needs to be
made and P1 and P4 vote yes and P2 and P3 vote no. Now we have a stalemate,
since both the Yes’s and the No’s have enough votes to win. This is a mathematical version of anarchy.
Example 2.2 illustrates why in a weighted voting system the quota must be
more than half of the total number of votes. This can be expressed mathematically
by the inequality
q 7 1w1 + w2 + Á + wN2>2.
EXAMPLE 2.3
Á
or Too Large
Once again, let’s look at the partnership introduced in Example 2.1, but this time
with a quota q = 21. Given that there are only 20 votes to go around, even if
every partner were to vote Yes, a motion is doomed to fail. This is the mathematical version of gridlock.
Example 2.3 illustrates why in a weighted voting system the quota cannot be
more than the total number of votes. This can be expressed mathematically by the
inequality
q … 1w1 + w2 + Á + wN2.
EXAMPLE 2.4 One Partner–One Vote?
Let’s consider the partnership introduced in Example 2.1 one final time. This
time the quota is set to be q = 19. Here we can describe the partnership as the
weighted voting system [19: 8, 7, 3, 2]. What’s interesting about this weighted
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52 2 Weighted Voting Systems
voting system is that the only way a motion can pass is by the unanimous support
of all the players. (Note that P1 , P2 , and P3 together can only muster 18 votes—
they still need P4’s votes to pass a motion.) In a practical sense, this weighted voting system is no different from a weighted voting system where each partner has
1 vote and it takes the unanimous agreement of the four partners to pass a motion (i.e., [4: 1, 1, 1, 1]).
The surprising conclusion of Example 2.4 is that the weighted voting system
[19: 8, 7, 3, 2] describes a one person–one vote situation in disguise. This seems
like a contradiction only if we think of one person–one vote as implying that all
players have an equal number of votes rather than an equal say in the outcome of
the election. Apparently, these two things are not the same! As Example 2.4
makes abundantly clear, just looking at the number of votes a player owns can be
very deceptive.
Power; More Terminology; More Examples
Let’s look at a few more examples of weighted voting systems and start to informally focus on the notion of power.
EXAMPLE 2.5 The Making of a Dictator
Consider the weighted voting system [11: 12, 5, 4]. Here one of the players 1P12
owns enough votes to carry a motion singlehandedly. In this situation, P1 is in
complete control—if P1 is for the motion, the motion will pass; if P1 is against it,
the motion will fail. Clearly, in terms of the power to influence decisions, P1 has all
of it. Not surprisingly, we will say that P1 is a dictator.
In general, a player is a dictator if the player’s weight is bigger than or equal to
the quota. Notice that there can only be one dictator (having two players with
weights bigger than or equal to the quota would contradict the requirement that
the quota be bigger than half of the total number of votes). This observation, coupled with the assumption that we list the players by descending weights, implies
that if there is a dictator, it must be P1 .
When P1 is a dictator, all the other players, regardless of their weights, have
absolutely no say on the outcome of the voting—there is never a time when any
of their votes are needed. A player that never has a say in the outcome of the voting is a player that has no power and is called a dummy. When there is a dictator,
all the other players are dummies, but there can be dummies even when there is
no dictator. This is illustrated in our next example.
EXAMPLE 2.6 The Curse of the Dummy
Four college students (P1 , P2 , P3 , and P4) decide to go into business together.
P1 , P2 , and P3 each invest $10,000 in the business, and each gets 10 votes in the
partnership. P4 is a little short on cash, so he invests only $9000 and thus gets 9
votes. Suppose the quota is set at q = 30 (don’t ask why). Under these assumptions the partnership can be described as the weighted voting system [30: 10, 10,
10, 9]. Everything seems right, right?
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2.2 The Banzhaf Power Index 53
Wrong! In this weighted voting system, P4 turns out to be a dummy! Why?
Notice that a motion can only pass if P1 , P2 , and P3 are for it, and then it makes no
difference whether P4 is for or against it. Thus, there is never going to be a time
when P4’s 9 votes are going to make a difference in the outcome of the voting.
Poor P4—in spite of having almost as many votes as the other partners, he has no
power to affect any partnership decisions!
EXAMPLE 2.7 The Power of the Veto
In the weighted voting system [12: 9, 5, 4, 2], P1 plays an interesting role—while
not having enough votes to be a dictator, he has the power to obstruct by
preventing any motion from passing. This happens because without P1’s votes, a
motion cannot pass—even if all the remaining players were to vote in favor of the
motion. In a situation like this we will say that P1 has veto power.
In general, a player P that is not a dictator is said to have veto power if a
motion cannot pass unless P votes in favor of the motion. (This, of course, does not
mean that if P votes in favor of the motion the motion must pass!)
2.2 The Banzhaf Power Index
From the preceding set of examples we can already draw an important
lesson—in weighted voting, a player’s weight does not always tell the full
story of how much power the player holds. Sometimes a player with lots of
votes can have little or no power (see Example 2.6), and conversely, a player with just a couple of votes can have a lot of power (see Example 2.4).
And yet, we have not even formally defined what having “power” means.
We will do so in this section, introducing a mathematical interpretation of
power first suggested by the American lawyer John Banzhaf III in 1965.
To get us started, let’s look at a simple example.
John F. Banzhaf III is Professor of
Public Interest Law at George Washington University, as well as founder
and executive director of the national
antismoking group Action on Smoking and Health.
EXAMPLE 2.8 The Weirdness of Parliamentary Politics
The Parliament of Icelandia has 200 members, divided among three political
parties—the Red Party 1P12, the Blue Party 1P22, and the Green Party 1P32. The
Red Party has 99 seats in Parliament, the Blue Party has 98, and the Green Party
has only 3. Decisions are made by majority vote, which in this case requires 101
out of the total 200 votes. Since in Icelandia members of Parliament always vote
along party lines (voting against the party line is extremely rare in parliamentary
governments), we can think of Icelandia’s Parliament as the weighted voting
system [101: 99, 98, 3].
By just looking at the numbers of votes, we would guess that most of the
power is shared between the Red and Blue parties, with the Green Party having
very little power, if any. On closer inspection, however, a strikingly different story
emerges when we look at the different ways that a motion might pass.
Let’s consider all the possible winning combinations (remember that we are
assuming that voting is strictly along party lines). There are only four:
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54 2 Weighted Voting Systems
(i)
Reds and Blues vote Yes; Greens vote No. The motion passes 197 to 3.
(ii) Reds and Greens vote Yes; Blues vote No. The motion passes 102 to 98.
(iii) Blues and Greens vote Yes; Reds vote No. The motion passes (just barely) 101 to 99.
(iv) Reds, Blues, and Greens all vote Yes. The motion passes 200 to 0.
The surprising fact that emerges from the preceding list is that for a motion to
pass it must have the support of two of the parties, and it makes no difference
which two. Based on this observation we conclude that in the Icelandia Parliament
the little Green Party holds as much power as either of the other two parties!
Building on Example 2.8, we now introduce some additional notation and
terminology.
■
■
■
Coalitions. We will use the term coalition to describe any set of players that
might join forces and vote the same way. In principle, we can have a coalition with as few as one player and as many as all players. The coalition consisting of all the players is called the grand coalition. Since coalitions are just
sets of players, the most convenient way to describe coalitions mathematically is to use set notation. For example, the coalition consisting of players
P1 , P2 , and P3 can be written as the set 5P1 , P2 , P36 (or 5P3 , P1 , P26, or
5P2 , P1 , P36, etc.—the order in which the members of a coalition are listed is
irrelevant).
Winning coalitions. Some coalitions have enough votes to win and some don’t.
Quite naturally, we call the former winning coalitions and the latter losing
coalitions. A single-player coalition can be a winning coalition only when
that player is a dictator, so under the assumption that there are no dictators
in our weighted voting systems (dictators are boring) a winning coalition
must have at least two players. At the other end of the spectrum, the grand
coalition is always a winning coalition, since it controls all the votes. In some
weighted voting systems (see Example 2.4) the grand coalition is the only
winning coalition.
Critical players. In a winning coalition, a player is said to be a critical player
for the coalition if the coalition must have that player’s votes to win. In
other words, when we subtract a critical player’s weight from the total
weight of the coalition, the total of the remaining votes drops below the
quota. (In more formal terminology, P is a critical player for a winning coalition if and only if W - w 6 q, where W is the total weight of the coalition
and w is the weight of P.) Sometimes a winning coalition has no critical
players (the coalition has enough votes that no single player’s desertion can
keep it from winning), sometimes a winning coalition has several critical
players, and when the coalition has just enough votes to make the quota,
then every player is critical.
EXAMPLE 2.8(B) Example 2.8 Revisited
We can think of Example 2.8 in a purely mathematical context and leave the details of Icelandian politics out of the picture. We have a weighted voting system
that can be described by [101: 99, 98, 3]. In this weighted voting system we have
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2.2 The Banzhaf Power Index 55
four winning coalitions, listed in the first column of Table 2-1. In the two-player
coalitions, both players are critical players (without both players the coalition
wouldn’t win); in the grand coalition no player is critical, since any two out of
three would have enough votes to win. Each of the three players is critical for two
coalitions. End of story.
TABLE 2-1 Winning Coalitions in [101: 99, 98, 3]
Coalition
Votes
5P1 , P26
197
P1 and P2
102
P1 and P3
101
P2 and P3
200
None
5P1 , P36
5P2 , P36
5P1 , P2 , P36
Critical players
When John Banzhaf introduced his mathematical interpretation of power in
1965, he had a key insight—a player’s power should be measured by how often the
player is a critical player. Thus, the key to getting a measure of a player’s power is
to count the number of winning coalitions in which that player is critical. From
Table 2-1 we can clearly see that in [101: 99, 98, 3] each player is critical twice.
Since there are three players, each critical twice, we can say that each player holds
two out of six, or one-third of the power.
The preceding ideas lead us to the final definitions of this section.
■
The Banzhaf power index. For the sake of simplicity, let’s start with P1—
exactly the same idea applies to each of the other players. To measure the
power (according to Banzhaf) of P1 , we first count the total number of
winning coalitions in which P1 is a critical player. Let’s call this number B1 .
While useful, B1 does not tell us how much power P1 has in relation to the
other players. A better way to tell the story is to think of power as a pie,
and to describe P1’s power as a fraction or percentage of that pie. To do
this we find how many times each of the other players is critical
1B2 , B3 , Á , BN2, and the total number of times all players are critical
1T = B1 + B2 + Á + BN2. The ratio B1>T (number of times P1 is a critical player over the total number of times all players are critical) is called
the Banzhaf power index of P1 . This number measures the size of P1’s
“slice” of the “Banzhaf power pie” and can be expressed either as a fraction or decimal between 0 and 1 or, equivalently, as a percent between 0
and 100%. For convenience, we will use the symbol b 1 (read “beta-one”)
to denote the Banzhaf power index of P1 .
■
The Banzhaf power distribution. Just like P1 , each of the other players in the
weighted voting system has a Banzhaf power index, which we can find in a
similar way. The complete list of power indexes b 1 , b 2 , Á , b N is called the
Banzhaf power distribution of the weighted voting system. The sum of all the
b’s is 1 (or 100% if they are written as percentages).
Following is a summary of the steps needed to compute the Banzhaf power
distribution of a weighted voting system with N players.
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Computing a Banzhaf Power Distribution
■
■
■
■
■
Step 1. Make a list of all possible winning coalitions.
Step 2. Within each winning coalition determine which are the critical
players. (To determine if a given player is critical or not in a given winning coalition, we subtract the player’s weight from the total number of
votes in the coalition—if the difference drops below the quota q, then
that player is critical. Otherwise, that player is not critical.)
Step 3. Count the number of times that P1 is critical. Call this number B1 .
Repeat for each of the other players to find B2 , B3 , Á , BN .
Step 4. Find the total number of times all players are critical. This total
is given by T = B1 + B2 + Á + BN .
Step 5. Find the ratio b 1 = B1>T. This gives the Banzhaf power index of
P1 . Repeat for each of the other players to find b 2 , b 3 , Á , b N . The complete list of b’s gives the Banzhaf power distribution of the weighted voting system.
In the next set of examples we will illustrate how to implement the preceding
sequence of steps.
EXAMPLE 2.9 Banzhaf Power in [4: 3, 2, 1]
Let’s find the Banzhaf power distribution of the weighted voting system [4: 3, 2, 1]
using Steps 1 through 5.
Step 1. There are three winning coalitions in this weighted voting system.
They are 5P1 , P26 with 5 votes, 5P1 , P36 with 4 votes, and the grand coalition
5P1 , P2 , P36 with 6 votes.
Step 2. The critical players in each winning coalition are shown in the following table.
Winning coalition
t
Critical players
5P1 , P26
5
P1 and P2
4
P1 and P3
6
P1 only
5P1 , P36
5P1 , P2 , P36
Step 3. B1 = 3 (P1 is critical in three coalitions); B2 = 1 and B3 = 1 (P2 and
P3 are critical once each).
Step 4. T = 3 + 1 + 1 = 5.
Step 5. b 1 = B1>T = 3>5; b 2 = B2>T = 1>5; b 3 = B3>T = 1>5. (If we want
to express the b’s in terms of percents, then b 1 = 60%, b 2 = 20%,
b 3 = 20%.)
Of all the steps we must carry out in the process of computing Banzhaf
power, by far the most demanding is Step 1. When we have only three players, as
in Example 2.9, we can list the winning coalitions on the fly—there simply aren’t
that many—but as the number of players increases, the number of possible win-
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2.2 The Banzhaf Power Index 57
ning coalitions grows rapidly, and it becomes necessary to adopt some form of
strategy to come up with a list of all the winning coalitions. This is important, because if we miss a single one, we are in all likelihood going to get the wrong
Banzhaf power distribution. One conservative strategy is to make a list of all possible coalitions and then cross out the losing ones. The next example illustrates
how we would use this approach.
EXAMPLE 2.10 Banzhaf Power and the NBA Draft
When NBA teams prepare for the annual draft of college players, the decision on
which college basketball players to draft may involve many people, including the
management, the coaches, and the scouting staff. Typically, not all of these people
have an equal voice in the process—the head coach’s opinion is worth more than
that of an assistant coach, and the general manager’s opinion more than that of a
scout. In some cases, this arrangement is formalized in the form of a weighted
voting system. Let’s use a fictitious team—the Akron Flyers—for the purposes of
illustration.
In the Akron Flyers draft system, the head coach 1P12 has 4 votes, the general manager 1P22 has 3 votes, the director of scouting operations 1P32 has 2 votes,
and the team psychiatrist 1P42 has 1 vote. Of the 10 votes cast, a simple majority
of 6 votes is required for a yes vote on a player to be drafted. In essence, the
Akron Flyers operate as the weighted voting system [6: 4, 3, 2, 1].
We will now find the Banzhaf power distribution of this weighted voting system using Steps 1 through 5.
Step 1. Table 2-2 starts with the complete list of all possible coalitions (first
column) and the number of votes in each (second column). From this information we can immediately determine which are the winning coalitions. (Notice that the list of coalitions is organized systematically—one-player
coalitions first, two-player coalitions next, and so on. Also notice that within
each coalition the players are listed in numerical order from left to right.
Both of these are good bookkeeping strategies, and you are encouraged to
use them when you do your own work.)
TABLE 2-2 Coalitions in [6: 4, 3, 2, 1]
Coalition
5P16
5P26
5P36
5P46
5P1 , P26*
5P1 , P36*
5P1 , P46
Number of votes
Coalition
Number of votes
4
5P2 , P36
5
3
2
1
7
6
5
Note: * indicates winning coalitions.
5P2 , P46
5P3 , P46
5P1 , P2 , P36*
5P1 , P2 , P46*
5P1 , P3 , P46*
5P2 , P3 , P46*
5P1 , P2 , P3 , P46*
4
3
9
8
7
6
10
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58 2 Weighted Voting Systems
Step 2. Now we disregard the losing coalitions and go to work on the winning
coalitions only. For each winning coalition, we determine which players are
critical. This is indicated in Table 2-2 by underlining the critical players.
(Don’t take someone else’s word for it—please check that these are all the
critical players!)
Step 3. We now count carefully how many times each player is underlined in
Table 2-2. The counts are B1 = 5, B2 = 3, B3 = 3, and B4 = 1.
Step 4. T = 5 + 3 + 3 + 1 = 12.
Step 5. b 1 = 5>12 = 41 23%; b 2 = 3>12 = 25%; b 3 = 3>12 = 25%, and
b 4 = 1>12 = 8 13%.
An interesting and unexpected result of these calculations is that the team’s
general manager 1P22 and the director of scouting operations 1P32 have the same
Banzhaf power index—not exactly the arrangement originally intended.
How Many Coalitions?
Attention! Brief mathematical detour
begins here!
Before we go on to the next example, let’s consider the following mathematical
question: For a given number of players, how many different coalitions are possible? Here, our identification of coalitions with sets will come in particularly
handy. Except for the empty subset 5 6, we know that every other subset of the
set of players can be identified with a different coalition. This means that we can
count the total number of coalitions by counting the number of subsets and subtracting one. So how many subsets does a set have?
A careful look at Table 2-3 shows us that each time we add a new element we
are doubling the number of subsets—the same subsets we had before we added
the element plus an equal number consisting of each of these subsets but with the
new element thrown in.
TABLE 2-3 The Subsets of a Set
Set
Number
of subsets
Subsets
5P1 , P26
5P1, P2 , P36
5P1 , P2, P3 , P46
5P1 , P2 , P3, P4, P56
4
8
16
32
5 6
5
6
5P36
5 6
5P46
5P26
5P2 , P46
5P36
5P3 , P46
5P2 , P36
5P2 , P3 , P46
5P16
5P16
5P1 , P36
5P16
5P1 , P26
5P1 , P26
5P1 , P2 , P36
5P1 , P26
5P26
5P26
5P2 , P36
5P1 , P36
5P1 , P2 , P36
5P1 , P46
5P1 , P2 , P46
The 16 subsets
from the previous column
along with each
of these with P5
thrown in.
5P1 , P3 , P46
5P1 , P2 , P3 , P46
Since each time we add a new player we are doubling the number of subsets,
we will find it convenient to think in terms of powers of 2. Table 2-4 summarizes
what we have learned.
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2.2 The Banzhaf Power Index 59
TABLE 2-4 The Number of Coalitions
Players
Number of subsets
P1 , P2
4 = 22
22 - 1 = 3
P1 , P2 , P3
8 = 23
23 - 1 = 7
P1 , P2 , P3 , P4
16 = 2 4
2 4 - 1 = 15
P1 , P2 , P3 , P4 , P5
32 = 2 5
2 5 - 1 = 31
o
o
o
2N
P1 , P2 , Á , PN
End of detour.
Number of coalitions
2N - 1
We will now return to the problem of computing Banzhaf power distributions. Given what we now know about the rapid growth of the number of coalitions, the strategy used in Example 2.10 (list all possible coalitions and then
eliminate the losing ones) can become a little tedious (to say the least) when we
have more than a handful of players. Sometimes we can save ourselves a lot of
work by figuring out directly which are the winning coalitions.
EXAMPLE 2.11 Cutting Loose the Losing Coalitions
The disciplinary committee at George Washington High School has five members: the principal 1P12, the vice principal 1P22, and three teachers (P3 , P4 , and P5).
When voting on a specific disciplinary action, the principal has three votes, the
vice principal has two votes, and each of the teachers has one vote. A total of five
votes are needed for any disciplinary action. Formally speaking, the disciplinary
committee is the weighted voting system [5: 3, 2, 1, 1, 1].
There are 2 5 - 1 = 31 possible coalitions. We can save a fair amount of effort
by skipping the losing coalitions and listing only the winning coalitions. A little organization will help: We will go through the winning coalitions systematically according to the number of players in the coalition. Obviously, there are no
one-player winning coalitions—we would have to have a dictator for that to happen. There is only one two-player winning coalition, namely 5P1 , P26. The only
three-player winning coalitions are those that include the principal P1 . All fourplayer coalitions are winning coalitions, and so is the grand coalition.
Steps 1 and 2. Table 2-5 shows all the winning coalitions, with the critical
players underlined. (You should double-check and make sure that these are
the right critical players in each coalition—it’s good practice!)
TABLE 2-5 Winning Coalitions in [5: 3, 2, 1, 1, 1] and Critical Players
Two players
Three players
Four players
5P1 , P26
5P1 , P2 , P36
5P1 , P2 , P3 , P46
5P1 , P2 , P56
5P1 , P2 , P4 , P56
5P1 , P2 , P46
5P1 , P3 , P46
5P1 , P3 , P56
5P1 , P4 , P56
5P1 , P2 , P3 , P56
5P1 , P3 , P4 , P56
5P2 , P3 , P4 , P56
Five players
5P1 , P2 , P3 , P4 , P56
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60 2 Weighted Voting Systems
Step 3. The critical player counts are B1 = 11, B2 = 5, B3 = 3, B4 = 3, and
B5 = 3.
Step 4. T = 25.
Step 5. b 1 = 11>25 = 44%; b 2 = 5>25 = 20%; b 3 = b 4 = b 5 = 3>25 = 12%.
EXAMPLE 2.12 Weighted Voting Without Weights
The Tasmania State University Promotion and Tenure committee consists of five
members: the dean (D) and four other faculty members of equal standing
(F1 , F2 , F3 , and F4). (For convenience we are using a slightly different notation
for the players.) In this committee faculty members vote first, and motions are
carried by simple majority. The dean only votes to break a 2-2 tie. Is this a weighted voting system? If so, what is the Banzhaf power distribution?
The answer to the first question is Yes, and the answer to the second question
is that we can apply the same steps we used before even though we don’t have
specific weights for the players.
Step 1. The possible winning coalitions fall into two groups: (i) three or
more faculty members vote Yes. In this case the dean does not vote. These
winning coalitions are listed in the first column of Table 2-6. (ii) Only two
faculty members vote Yes, but the dean breaks the tie with a Yes vote.
These winning coalitions are listed in the second column of Table 2-6.
TABLE 2-6 Winning Coalitions and Critical Players
5F1 , F2 , F36
5D, F1 , F26
5F1 , F3 , F46
5D, F1 , F46
5F1 , F2 , F46
5F2 , F3 , F46
5F1 , F2 , F3 , F46
5D, F1 , F36
5D, F2 , F36
5D, F2 , F46
5D, F3 , F46
Step 2. In the winning coalitions consisting of just three faculty members, all
faculty members are critical. In the coalition consisting of all four faculty
members, no faculty member is critical. In the coalitions consisting of two
faculty members plus the dean, all players, including the dean, are critical.
Table 2-6 shows the details, with the critical players underlined.
Step 3. The dean is critical six times (in each of the six coalitions in the second column). Each of the faculty members is critical three times in the first
column and three times in the second column. Thus, all players are critical an
equal number of times.
Steps 4 and 5. No counts are really necessary here. Since all five players
are critical an equal number of times, they all have the same Banzhaf
power index: b 1 = b 2 = b 3 = b 4 = b 5 = 20%.
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2.3 Applications of Banzhaf Power 61
The Vice President of the United
States shall be the President of the
Senate, but shall have no Vote, unless they be equally divided.
Article I, Section 3, U.S. Constitution
The surprising conclusion of Example 2.12 is that although the rules appear
to set up a special role for the dean (the role of tie-breaker), in practice the dean
is no different than any of the faculty members. A larger scale version of Example 2.12 occurs in the U.S. Senate, where the Vice President of the United States
can only vote to break a tie. An analysis similar to the one used in Example 2.12
shows that as a member of the Senate the vice president has the same Banzhaf
power index as an ordinary senator.
2.3 Applications of Banzhaf Power
The Nassau County Board of Supervisors John Banzhaf first introduced the
concept of Banzhaf power in 1965. In an article entitled Weighted Voting Doesn’t
Work (see reference 2) Banzhaf used mathematics to argue that the weighted
voting system used by the Board of Supervisors of Nassau County, New York,
violated the “equal protection” guarantees of the Fourteenth Amendment.
In the 1960s, Nassau County was divided into six uneven districts,
four with high populations and two rural districts with low
TABLE 2-7 Nassau County Board (1964)
populations. Table 2-7 shows the names of the districts and their
District
Weight
weights based on their respective populations. The quota was a simple majority of 58 (out of 115) votes.
Hempstead #1
31
When we look at these weights carefully, as Banzhaf did, we disHempstead #2
31
cover something remarkable—all the power in the County Board was
concentrated in the hands of the largest three districts (Hempstead #1,
Oyster Bay
28
Hempstead #2, and Oyster Bay), with Glen Cove, Long Beach and
North Hempstead unwittingly playing the role of dummies! North
North Hempstead
21
Hempstead a dummy? How is it possible? Here is how the numbers
Long Beach
2
happen to work out: (i) Any two of the top three districts can form a
winning coalition, and (ii) there can be no winning coalition without
Glen Cove
2
two out of the top three districts.The implications of this are that none
of the last three districts can ever be critical.
Banzhaf’s analysis of power in the Nassau County Board set the stage for a
long series of legal challenges to the use of weighted voting in county boards, culminating with a federal court decision in 1993 abolishing weighted voting in New
York State (see Project D). In 1996, after a protracted fight, the Nassau County
Board of Supervisors became a 19-member legislature, each member having just
one vote and representing districts of roughly equal population.
The United Nations Security Council The main body responsible for maintaining
This arrangement is likely to change
in the near future, as Germany,
Japan, Brazil, India, and possibly one
African nation are being considered
for permanent membership in the
Security Council.
the international peace and security of nations is the United Nations Security
Council. The Security Council is a classic example of a weighted voting system.
It currently consists of fifteen voting nations—five of them are the permanent
members (Britain, China, France, Russia, and the United States); the other ten
nations are nonpermanent members appointed for a two-year period on a
rotating basis. To pass a motion in the Security Council requires a Yes vote from
each of the permanent members (in effect giving each permanent member veto
power) plus additional Yes votes from at least four of the ten nonpermanent
members. Thus, a winning coalition must include all five of the permanent
members plus four or more nonpermanent members.
All in all, there are 848 possible winning coalitions in the Security
Council—too many to list one by one. However, it’s not too difficult to figure
out the critical player story: In the winning coalitions with nine players (five
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62 2 Weighted Voting Systems
For more details see Exercise 67.
The calculations were done using a
computer program available at this
book’s Web site
http://prenhall.com/tannenbaum.
(From the “Jump to” menu, jump to
Chapter 2 and then click on Applets.)
permanent members plus four nonpermanent members) every member of the
coalition is a critical player; in all the other winning coalitions (ten or more
players) only the permanent members are critical players. Using a few simple
(but not elementary) calculations, we can find that of the 848 winning coalitions, there are 210 with nine members and the rest have ten or more members.
Carefully piecing together this information leads to the following surprising
conclusion: The Banzhaf power index of each permanent member is 848/5080
(roughly 16.7%), while the Banzhaf power index of each nonpermanent member is 84/5080 (roughly 1.65%). Notice the discrepancy in power between the
permanent and nonpermanent members: A permanent member has more than
ten times as much power as a nonpermanent member. One has to wonder if
this was the original intent of the United Nations charter or perhaps a miscalculation based on a less than clear understanding of the mathematics of
weighted voting.
The European Union (EU) The EU is a political and economic confederation of
25 member nations, a sort of United States of Europe. The main legislative body
of the EU is the Council of Ministers, and this council operates as a weighted
voting system where different member nations are assigned weights based on a
formula that takes into account their populations as well as other historical and
economic factors. The first column of Table 2-8 lists the 25 member nations of the
EU, and the second column shows each nation’s weight in the Council of
Ministers. The total number of votes in the Council of Ministers is 321, and the
quota is q = 232 (roughly 72.3% of the votes). The “Relative weight” column
shows the weights expressed as a percent of the 321 total votes. For example,
France has 29 out of 321 votes, for a relative weight of 9.03%. The last column in
Table 2-8 shows the Banzhaf power index of each member nation.
TABLE 2-8 The European Union Council of Ministers (2005)
Country
Weight
Relative weight
Banzhaf power index
France, Germany,
Italy, United Kingdom
29
9.03%
8.57%
Spain, Poland
27
8.41%
8.13%
Netherlands
13
4.05%
4.23%
12
3.74%
3.91%
10
3.12%
3.27%
7
2.18%
2.31%
4
1.25%
1.32%
3
0.93%
0.99%
Belgium, Greece,
Czech Republic,
Hungary, Portugal
Austria, Sweden
Denmark, Ireland,
Lithuania, Slovakia,
Finland
Cyprus, Estonia,
Latvia, Luxembourg,
Slovenia
Malta
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2.4 The Shapley-Shubik Power Index 63
One way to determine how well a weighted voting system accomplishes its intended purpose is by matching relative weights with power indexes. Comparing the
“Relative weight” and “Banzhaf power index” columns of Table 2-8, we see that
there is a very close match between the respective entries in the two columns, an
indication that as a weighted voting system, the European Union Council of
Ministers—unlike the Nassau County Board of Supervisors—works pretty much
the way it was meant to work.
2.4 The Shapley-Shubik Power Index
Lloyd Shapley is Professor Emeritus
of Mathematics and Economics at
UCLA (see the biographical profile of
Shapley at the end of this chapter).
Martin Shubik is Professor of Economics at Yale University.
In this section we will discuss a different approach to measuring power, first proposed by American economists Lloyd Shapley and Martin Shubik in 1954. The key
difference between the Shapley-Shubik measure of power and the Banzhaf measure of power has to do with how coalitions are formed. In the Shapley-Shubik
method, the assumption is that coalitions are formed sequentially: Players join the
coalition and cast their votes in an orderly sequence (there is a first player, then
comes a second player, then a third, and so on). If the order in which the players join
the coalition is taken into consideration, we call the coalition a sequential coalition.
Let’s illustrate the difference with a simple example.
EXAMPLE 2.13 Three-Player Sequential Coalitions
When we think of the coalition involving three players P1 , P2 , and P3 , we think in
terms of a set. For convenience we write the set as 5P1 , P2 , P36, but we can also write
in other ways such as 5P3 , P2 , P16, 5P2 , P3 , P16, etc. In a coalition the only thing that
matters is who are the members—the order in which we list them is irrelevant.
In contrast, with three players P1 , P2 , and P3 we can form six different sequential
coalitions: 8P1 , P2 , P39 (this means that P1 is the first player, then P2 joined in, and last
came P3); 8P1 , P3 , P29; 8P2 , P1 , P39; 8P2 , P3 , P19; 8P3 , P1 , P29; 8P3 , P2 , P19. The six sequential coalitions of three players are graphically illustrated in Fig. 2-1. [Note the
change in notation. From now on the notation 8 9 will indicate that we are dealing
with a sequential coalition, and the order in which the players are listed does matter!]
Sequential Coalitions
Ordinary Coalition
P1, P2, P3
P1, P3, P2
P2, P1, P3
P3, P2, P1
P2, P3, P1
P3, P1, P2
P1, P2, P3
FIGURE 2-1 Three players form six
different sequential coalitions.
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64 2 Weighted Voting Systems
The Shapley-Shubik approach to measuring power is parallel to the
Banzhaf approach, but it is based on the concept of the pivotal player in a sequential coalition.
■
Win
Lose
…
Pivotal
Player
FIGURE 2-2 The pivotal player tips
the scales.
Pivotal player. In every sequential coalition
there is a player that contributes the votes
that turn what was a losing coalition into a
winning coalition—we call such a player
the pivotal player of the sequential coalition. Every sequential coalition has one
and only one pivotal player. To find the
pivotal player we just add the players’
weights from left to right, one at a time,
until the tally is bigger or equal to the
quota q. The player whose votes tip the
scales is the pivotal player. This idea is illustrated in Fig. 2-2.
■
The Shapley-Shubik power index. For a given player P, the Shapley-Shubik
power index of P is obtained by counting the number of times P is a pivotal
player and dividing this number by the total number of times all players are
pivotal (just like what we did with the Banzhaf power index but now using
pivotal players instead of critical players and sequential coalitions instead of
ordinary coalitions). As in the case of the Banzhaf power index, the ShapleyShubik power index of a player can be thought of as a measure of the size of
that player’s “slice” of the “power pie” and can be expressed either as a fraction between 0 and 1 or as a percent between 0 and 100%. We will use s1
(“sigma-one”) to denote the Shapley-Shubik power index of P1 , s2 to denote
the Shapley-Shubik power index of P2 , and so on.
■
The Shapley-Shubik power distribution. The complete listing of the ShapleyShubik power indexes of all the players is called the Shapley-Shubik power
distribution of the weighted voting system. It tells the complete story of how
the (Shapley-Shubik) power pie is divided among the players.
Following is a summary of the steps needed to compute the Shapley-Shubik
power distribution of a weighted voting system with N players.
Computing a Shapley-Shubik Power Distribution
■
Step 1. Make a list of all possible sequential coalitions of the N players.
Let T be the number of such coalitions. (We will have a lot more to say
about T soon!)
■
Step 2. In each sequential coalition determine the pivotal player.
■
Step 3. Count the total number of times that P1 is pivotal. Call this number SS1 . Repeat for each of the other players to find SS2 , SS3 , Á , SSN .
Step 4. Find the ratio s1 = SS1>T. This gives the Shapley-Shubik power
index of P1 . Repeat for each of the other players to find s2 , s3 , Á , sN .
The complete list of s’s gives the Shapley-Shubik power distribution of
the weighted voting system.
■
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2.4 The Shapley-Shubik Power Index 65
This is the beginning of a brief but
extremely important mathematical
detour. At the end of this detour we
will be able to answer the question,
How many sequential coalitions are
possible?
The Multiplication Rule and Factorials
We will start by introducing one of the most useful rules of basic mathematics,
stunning in its simplicity:
The Multiplication Rule
If there are m different ways to do X, and n different ways to do Y, then X
and Y together can be done in m * n different ways.
EXAMPLE 2.14 Cones and Flavors
An ice cream shop offers 2 different choices of cones and 3 different flavors of ice
cream (this is a boutique ice cream shop!). Using the multiplication rule, we can
conclude that there are 2 * 3 = 6 different choices for a single (cone and one
scoop of ice cream) order. The choices are illustrated in Fig. 2-3.
Figure 2-3 also clarifies the logic behind the multiplication rule: If we had m
different types of cones and n different flavors of ice cream, the corresponding
arrangement would show the cone/flavor combinations laid out in m rows and n
columns, illustrating the m * n possibilities.
FIGURE 2-3 The scoop on the
multiplication rule.
EXAMPLE 2.15 Cones and Flavors Plus Toppings
A bigger ice cream shop offers 5 different choices of cones, 31 different flavors of
ice cream (this is not a boutique ice cream shop!), and 8 different choices of topping. The question is, If you are going to choose a cone and a single scoop of ice
cream but then add a topping for good measure, how many orders are possible?
The number of choices is too large to list individually, but we can find it by using
the multiplication rule twice: First, there are 5 * 31 = 155 different cone/flavor
combinations, and each of these can be combined with one of the 8 toppings into
a grand total of 155 * 8 = 1240 different cone/flavor/topping combinations. The
implications—as well as the calories—are staggering!
We will now use the multiplication rule to count sequential coalitions.
Specifically, we want to know, how many sequential coalitions with N players
are possible?
EXAMPLE 2.16 Counting Sequential Coalitions
Let’s start with a warm-up—how do we count the number of sequential coalitions with four players? Using the multiplication rule, we can argue as follows:
We can choose any one of the four players to go first, then choose any one of
the remaining three players to go second, then choose any one of the remaining
two players to go third, and finally the one player left goes last. Using the multiplication rule, we get a total of 4 * 3 * 2 * 1 = 24 sequential coalitions with
four players.
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66 2 Weighted Voting Systems
If we have 5 players, following up on our previous argument, we can count on
a total of 5 * 4 * 3 * 2 * 1 = 120 sequential coalitions, and with N players the
number of sequential coalitions is N * 1N - 12 * Á * 3 * 2 * 1. The number N * 1N - 12 * Á * 3 * 2 * 1 is called the factorial of N and is written in
the shorthand form N!.
This is the end of our mathematical
detour. We will return to factorials in
Chapter 6, but for the time being, if
you have a business or scientific calculator, you might want to find the
factorial key and familiarize yourself
with its use. This is also a good time
to give Exercises 35–40 a try!
The Number of Sequential Coalitions
The number of sequential coalitions
N! = N * 1N - 12 * Á * 3 * 2 * 1.
with
N
players
is
Back to the Shapley-Shubik Power Index
We will illustrate the procedure for computing Shapley-Shubik power by revisiting a couple of examples done earlier in this chapter.
EXAMPLE 2.17 Shapley-Shubik Power in [4: 3, 2, 1]
In this example we will compute the Shapley-Shubik power distribution of the
weighted voting system [4: 3, 2, 1]. (We computed the Banzhaf power distribution
of this weighted voting system in Example 2.9, so at the end we will be able to
compare the results and see if there is a difference.)
Steps 1 and 2. Table 2-9 shows the six sequential coalitions of three players,
with the pivotal player in each sequential coalition underlined. The second
column shows the weights as they are being added from left to right until the
tally reaches or exceeds the quota q = 4.
TABLE 2-9 Sequential Coalitions for [4: 3. 2. 1] (Pivotal
Players Underlined)
Sequential coalition
8P1 , P2 , P39
8P1 , P3 , P29
8P2 , P1 , P39
8P2 , P3 , P19
8P3 , P1 , P29
8P3 , P2 , P19
Weight tallies (from left to right)
3 + 2 = 5
3 + 1 = 4
2 + 3 = 5
2 + 1 + 3 = 6
1 + 3 = 4
1 + 2 + 3 = 6
Step 3. A count of how many times each player is pivotal yields
SS1 = 4, SS2 = 1, and SS3 = 1.
Step
s1 =
4
6
4. The Shapley-Shubik power
= 66 23%, s2 = 16 = 16 23%, and s3 =
1
6
distribution
= 16 23%.
is
given
by
When we compare this power distribution with the Banzhaf power distribution found in Example 2.9 ( b 1 = 35 = 60%, b 2 = 15 = 20%, and b 3 = 15 = 20%),
we see that the two interpretations of power are indeed different.
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2.4 The Shapley-Shubik Power Index 67
EXAMPLE 2.18 Shapley-Shubik Power and the NBA Draft
We will now revisit Example 2.10, the NBA draft example. The weighted voting
system in this example is [6: 4, 3, 2, 1], and we will now find its Shapley-Shubik
power distribution.
Steps 1 and 2. Table 2-10 shows the 24 sequential coalitions of P1 , P2 , P3 , and
P4 . In each sequential coalition the pivotal player is underlined. (Notice
there is a method to the madness of Table 2-9—each column corresponds to
the sequential coalitions with a given first player. You may want to use the
same or a similar pattern when you do the exercises on Shapley-Shubik
power distributions.)
TABLE 2-10 Sequential Coalitions for [6: 4. 3. 2. 1] (Pivotal Players Underlined)
8P1 , P2 , P3 , P49
8P2 , P1 , P3 , P49
8P3 , P1 , P2 , P49
8P4 , P1 , P2 , P39
8P1 , P3 , P2 , P49
8P2 , P3 , P1 , P49
8P3 , P2 , P1 , P49
8P4 , P2 , P1 , P39
8P1 , P2 , P4 , P39
8P1 , P3 , P4 , P29
8P1 , P4 , P2 , P39
8P1 , P4 , P3 , P29
8P2 , P1 , P4 , P39
8P2 , P3 , P4 , P19
8P2 , P4 , P1 , P39
8P2 , P4 , P3 , P19
8P3 , P1 , P4 , P29
8P3 , P2 , P4 , P19
8P3 , P4 , P1 , P29
8P3 , P4 , P2 , P19
8P4 , P1 , P3 , P29
8P4 , P2 , P3 , P19
8P4 , P3 , P1 , P29
8P4 , P3 , P2 , P19
Step 3. The pivotal player counts are SS1 = 10, SS2 = 6, SS3 = 6, and
SS4 = 2.
Step 4. The Shapley-Shubik power distribution of the weighted voting sys2
6
6
tem is given by s1 = 10
24 = 41 3 %, s2 = 24 = 25%, s3 = 24 = 25%, and
2
1
s4 = 24 = 8 3%.
If you compare this result with the Banzhaf power distribution obtained in
Example 2.10, you will notice that here the two power distributions are the same.
If nothing else, this shows that it is not impossible for the Banzhaf and ShapleyShubik power distributions to agree. In general, however, for randomly chosen
real-life situations, it is very unlikely that the Banzhaf and Shapley-Shubik methods will give the same answer.
EXAMPLE 2.19 Shapley-Shubik in the City
In some cities, the city council operates under what is known as the “strongmayor” system. Under this system, the city council can pass a motion under simple majority, but the mayor has the power to veto the decision. The mayor’s veto
can then be overruled by a “super-majority” of the council members. As an illustration of the strong-mayor system we will consider the city of Cleansburg. In
Cleansburg the city council has four members plus a strong mayor that has a vote
as well as the power to veto motions supported by a simple majority of the council members. On the other hand, the mayor cannot veto motions supported by all
four council members. Thus, a motion can pass if the mayor plus two or more
council members support it, or, alternatively, if the mayor is against it but the four
council members support it.
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68 2 Weighted Voting Systems
See Exercise 68 for an important
generalization of these observations.
Common sense tells us that under these rules, the four council members have
the same amount of power but the mayor has more. We will compute the ShapleyShubik power distribution of this weighted voting system to figure out exactly
how much more.
With 5 players, this weighted voting system has 5! = 120 sequential coalitions to consider. Obviously, we would prefer not to have to write them all down,
and, in fact, we can skip that step and just use our imagination. Specifically, we
will imagine a table in which the 120 sequential coalitions are listed in 5 columns.
The first column has all the sequential coalitions where the mayor (let’s call him
P1) appears as the first player in the coalition, the second column has all the sequential coalitions where P1 is in second place, the third column has all sequential
coalitions with P1 in third place, and so on. Now close your eyes and try to visualize this table in your mind’s eye for a moment or two. Most likely, you visualized
a table that had the same number of sequential coalitions in each of the columns.
This is a key observation, as it allows us to conclude that there are 120 , 5 = 24
sequential coalitions in each column.
We are now in a position to find the pivotal player count of the mayor 1P12.
We first observe that the mayor is the pivotal player only when he is the third or
fourth player in a sequential coalition, as illustrated in Figs. 2-4 (a) and (b) respectively. This means that in our imaginary table, P1 is pivotal in every sequential
coalition in columns 3 and 4, and no others. So there we have it—the pivotal
count for the mayor is 24 + 24 = 48, and his Shapley-Shubik power index is
48>120 = 2>5 = 40%.
Lose
Lose
Win
P1
(a)
P1
(b)
P1
(c)
FIGURE 2-4
For the purposes of comparison, the
reader is encouraged to calculate
the Banzhaf power distribution of
the Cleansburg city council–see
Exercise 71.
Since the Shapley-Shubik power index of the mayor equals 40% and the
four council members must share the remaining 60% of the power equally, it
follows that each council member has a Shapley-Shubik power index of 15%.
In conclusion, the Shapley-Shubik power distribution of the Cleansburg city
council is s1 = 40%, s2 = s3 = s4 = s5 = 15%.
In this example it’s not so much the conclusion but how we got it that is
worth remembering.
2.5 Applications of Shapley-Shubik
Power
The Electoral College Calculating the Shapley-Shubik power index of the states
in the Electoral College is no easy task. There are 51! sequential coalitions, a
number so large (67 digits long) we don’t even have a name for it. Individually
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2.5 Applications of Shapley-Shubik Power 69
checking all possible sequential coalitions is out of the question, even for the
world’s fastest computer. There are, however, some sophisticated mathematical
shortcuts, which, when coupled with the right kind of software, allow the calculations to be done by an ordinary computer in a matter of seconds (see reference 16
for details).
The appendix at the end of this chapter shows both the Banzhaf and the
Shapley-Shubik power indexes for each of the 50 states and the District of
Columbia. Comparing the Banzhaf and the Shapley-Shubik power indexes
shows that there is a very small difference between the two. This example
shows that in some situations the Banzhaf and Shapley-Shubik power indexes
give essentially the same answer. The next example illustrates a very different
situation.
The United Nations Security Council As mentioned in Section 2.3, the United
Nations Security Council consists of fifteen member nations—five are
permanent members and ten are nonpermanent members appointed on a
rotating basis. For a motion to pass it must have a Yes vote from each of the five
permanent members plus at least four of the ten nonpermanent members. It can
be shown that this arrangement is equivalent to giving the permanent members
7 votes each, the nonpermanent members 1 vote each, and making the quota
equal to 39 votes.
We will sketch a rough outline of how the Shapley-Shubik power distribution
of the Security Council can be calculated. The details, while not terribly difficult,
go beyond the scope of this book.
1. There are 15! sequential coalitions of 15 players (roughly about 1.3 trillion).
2. A nonpermanent member can be pivotal only if it is the ninth player in the
coalition, preceded by all five of the permanent members and three nonpermanent members. (There are approximately 2.44 billion sequential coalitions
of this type.)
3. From (1) and (2) we can conclude that the Shapley-Shubik power index of a
nonpermanent member is approximately 0.19% 12.44 billion>1.3 trillion L
0.0019 = 0.19%2. For the purposes of comparison it is worth noting that there
is a big difference between this Shapley-Shubik power index and the corresponding Banzhaf power index of 1.65% obtained in Section 2.3.
4. The ten nonpermanent members (each with a Shapley-Shubik power index
of 0.19%) have together 1.9% of the power pie, leaving the remaining 98.2%
to be divided equally among the five permanent members. Thus, the
Shapley-Shubik power index of each permanent member is approximately
98.2>5 = 19.64%.
This analysis shows the enormous difference between the Shapley-Shubik
power of the permanent and nonpermanent members of the Security Council—
permanent members have roughly 100 times the Shapley-Shubik power of nonpermanent members!
The European Union We introduced the European Union Council of Ministers
in Section 2.3 and observed that the Banzhaf power index of each member
nation matches very closely the percentage of votes that it holds. What about
the Shapley-Shubik power index? Once again, there isn’t much future in trying
to calculate the power indexes directly—there is a total of 25! sequential
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70 2 Weighted Voting Systems
If you are curious as to exactly how
many years, try Exercise 35(d).
coalitions, and even for a supercomputer it would take years to list them all. On
the other hand, through some mathematical shortcuts, the same calculations
can be done in a few seconds by even an ordinary computer. (An applet that
can be used to carry out the calculations is available at this book’s Web site
http://prenhall.com/tannenbaum. From the “Jump to” menu, jump to Chapter 2
and then click on Applets.) The results are shown in the last column of Table 211. Notice, once again, how nicely the power indexes match the relative
weights, confirming our earlier observation that the EU Council is an excellent
working weighted voting system.
TABLE 2-11 The European Union Council of Ministers (2005)
Country
Weight
Relative weight
SS power index
Italy, United Kingdom
29
9.03%
9.29%
Spain, Poland
27
8.41%
8.61%
Netherlands
13
4.05%
3.98%
Hungary, Portugal
12
3.74%
3.65%
Austria, Sweden
10
3.12%
3.02%
7
2.18%
2.10%
Luxembourg, Slovenia
4
1.25%
1.19%
Malta
3
0.93%
0.89%
France, Germany,
Belgium, Greece, Czech Republic,
Denmark, Ireland,
Lithuania, Slovakia, Finland
Cyprus, Estonia, Latvia,
Conclusion
In any society, no matter how democratic, some individuals and groups have
more power than others. This is simply a consequence of the fact that individuals
and groups are not all equal. Diversity is the inherent reason why the concept of
power exists.
Power itself comes in many different forms. We often hear clichés such as
“In strength lies power” or “Money is power” (and the newer cyber version,
“Information is power”). In this chapter we discussed the notion of power as it
applies to formal voting situations called weighted voting systems and saw how
mathematical methods allow us to measure the power of an individual or
group by means of a power index. In particular, we looked at two different
kinds of power indexes: the Banzhaf power index and the Shapley-Shubik
power index. These indexes provide two different ways to measure power, and
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Profile
71
while they occasionally agree, they often differ significantly. Of the two, which
one gives the more accurate measure of a player’s power?
Unfortunately, there is no simple answer. Both of them are useful, and in
some sense the choice is subjective. Perhaps the best way to evaluate them is to
think of them as being based on a slightly different set of assumptions. The idea
behind the Banzhaf interpretation of power is that players are free to come and
go, negotiating their allegiance for power (somewhat like professional athletes
since the advent of free agency). Underlying the Shapley-Shubik interpretation
of power is the assumption that when a player joins a coalition, he or she is making a commitment to stay. In the latter case a player’s power is generated by his or
her ability to be in the right place at the right time.
In practice, the choice of which method to use for measuring power is based
on which of the assumptions better fits the specifics of the situation. Contrary to
what we’ve often come to expect, mathematics does not give us the answer, just
the tools that might help us make an informed decision.
Profile Lloyd S. Shapley (1923–
)
Lloyd Shapley is one of the giants of modern game theory—
which, in a nutshell, is the mathematical study of games. (The
word game is used here as a metaphor for any situation involving competition and cooperation among individuals or
groups each trying to further their own separate goals. This
broad interpretation encompass- es not only real games like
poker and Monopoly, but also economic, political, and military “games,” thus making game theory one of the most important branches of modern applied mathematics.)
Lloyd Shapley was born in Cambridge, Massachusetts to a
prominent scientific family. His father, Harlow Shapley, was a
renowned astronomer at Harvard University and director of
the Harvard Observatory. Shapley entered Harvard in 1942,
but he interrupted his studies to join the Army during World
War II. From 1943 to 1945 he served in the Army Air Corps in
China, where he was awarded a Bronze Star for breaking the
Japanese weather code. In 1945 he returned to Harvard, receiving a Bachelor of Arts in Mathematics in 1948.
Upon graduation from Harvard, Shapley joined the Rand
Corporation, a famous think tank in Santa Monica, California,
where much of the pioneering research in game theory was
being conducted. After a year at Rand, and with a reputation
as a budding young star in game theory in tow, Shapley went to
Princeton University as a graduate student in mathematics. At
Princeton he joined a circle that included some of the most
brilliant mathematical minds of his time, including John von
Neumann, the father of game theory, and fellow graduate student John Nash, the mathematical genius whose life is portrayed in the Academy Award–winning movie A Beautiful
Mind (with Russell Crowe playing John Nash).
As a graduate student, Shapley had a reputation as a
fierce competitor in everything he took on, including card
games and board games, some of his own invention. One of
the games Shapley invented with a couple of other graduate
students (one of which was Nash and the other was Martin
Shubik, who at the time was an economics graduate student
as well as Shapley’s roommate) was a fiendish board game
called So Long Sucker, in which players formed coalitions to
gang up and double-cross each other. Some of the experiences and observations that Shapley and Shubik (who became life-long friends) derived from their graduate student
game-playing days eventually led to their development of
the Shapley-Shubik index for measuring power in a weighted voting system (a weighted voting system, after all, is just a
simple game in which players form coalitions in order to pass
or block a specific action by the group).
After receiving his Ph.D. in Mathematics from Princeton in 1953, Shapley returned to the Rand Corporation,
where he worked as a research mathematician until 1981.
During this time he made many major contributions to game
theory, including the invention of convex and stochastic
games, the Shapley value of a game, and the Shapley-Shubik
power index. The citation when he was awarded the Von
Neumann Prize in Game Theory in 1981 partly read, “His individual work and his joint research with Martin Shubik has
helped build bridges between game theory, economics, political science, and practice.”
In 1981 Shapley joined the Mathematics department at
UCLA, where he is currently Professor Emeritus of Mathematics and Economics.
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72 2 Weighted Voting Systems
Key Concepts
Banzhaf power
distribution, 55
Banzhaf power index, 53
coalition, 54
critical player, 54
dictator, 52
dummy, 52
factorial, 66
grand coalition, 54
losing coalition, 54
motion, 50
multiplication rule, 65
pivotal player, 64
player, 50
quota, 50
sequential coalition, 63
Shapley-Shubik power
distribution, 64
Shapley-Shubik power index, 63
veto power, 53
weighted voting system, 50
weight, 50
winning coalition, 54
Exercises
WALKING
A. Weighted Voting Systems
1. Consider the weighted voting system [13: 7, 4, 3, 3, 2, 1].
Find
5. A committee has four members (P1 , P2 , P3 , and P4). In
this committee P1 has twice as many votes as P2 ; P2 has
twice as many votes as P3 ; P3 has twice as many votes as
P4 . Describe the committee as a weighted voting system
when the requirements to pass a motion are
(a) at least two-thirds of the votes
(a) the total number of players
(b) more than two-thirds of the votes
(b) the total number of votes
(c) at least 80% of the votes
(c) the weight of P2
(d) the minimum percentage of the votes needed to
pass a motion (rounded to the next whole percent)
2. Consider the weighted voting system [62: 10, 10, 10, 10,
8, 5, 5, 5, 5, 4, 4, 3, 3, 3, 2]. (This weighted voting system
describes the voting in the European Union Council of
Ministers prior to 2004.) Find
(a) the total number of players
(b) the total number of votes
(c) the weight of P6
(d) the minimum percentage of the votes needed to
pass a motion (rounded to the next whole percent)
3. Consider the weighted voting system [q: 10, 6, 5, 4, 2].
(a) What is the smallest value that the quota q can take?
(b) What is the largest value that the quota q can take?
(c) What is the value of the quota if at least two-thirds
of the votes are required to pass a motion?
(d) What is the value of the quota if more than twothirds of the votes are required to pass a motion?
4. Consider the weighted voting system [q: 6, 4, 3, 3, 2, 2].
(d) more than 80% of the votes
6. A committee has six members (P1 , P2 , P3 , P4 , P5 , and P6).
In this committee P1 has twice as many votes as P2 ; P2 and
P3 each has twice as many as P4 ; P4 has twice as many
votes as P5 ; P5 and P6 have the same number of votes. Describe the committee as a weighted voting system when
the requirements to pass a motion are
(a) a simple majority of the votes
(b) at least three-fourths of the votes
(c) more than three-fourths of the votes
(d) at least two-thirds of the votes
(e) more than two-thirds of the votes
7. In each of the following weighted voting systems, determine which players, if any, (i) are dictators; (ii) have
veto power; (iii) are dummies.
(a) [6: 4, 2, 1]
(b) [6: 7, 3, 1]
(c) [6: 5, 5, 1]
8. In each of the following weighted voting systems, determine which players, if any, (i) are dictators; (ii) have
veto power; (iii) are dummies.
(a) What is the smallest value that the quota q can take?
(a) [95: 95, 80, 10, 2]
(b) What is the largest value that the quota q can take?
(b) [95: 65, 35, 30, 25]
(c) What is the value of the quota if at least threefourths of the votes are required to pass a motion?
(d) What is the value of the quota if more than threefourths of the votes are required to pass a motion?
(c) [48: 32, 16, 8, 4, 2, 1]
9. In each of the following weighted voting systems, determine which players, if any, (i) are dictators; (ii) have
veto power; (iii) are dummies.
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Exercises
(a)
(b)
(c)
(d)
[19: 9, 7, 5, 3, 1]
[15: 16, 8, 4, 1]
[17: 13, 5, 2, 1]
[25: 12, 8, 4, 2]
10. In each of the following weighted voting systems, determine which players, if any, (i) are dictators; (ii) have
veto power; (iii) are dummies.
(a) [27: 12, 10, 4, 2]
(b) [22: 10, 8, 7, 2, 1]
(c) [21: 23, 10, 5, 2]
(d) [15: 11, 5, 2, 1]
B. Banzhaf Power
11. Consider the weighted voting system [10: 6, 5, 4, 2].
(a) What is the weight of the coalition formed by P1
and P3 ?
(b) Write down all winning coalitions.
(c) Which players are critical in the coalition
5P1 , P2 , P36?
(d) Find the Banzhaf power distribution of this weighted voting system.
12. Consider the weighted voting system [5: 3, 2, 1, 1].
(a) What is the weight of the coalition formed by P1
and P3 ?
(b) Which players are critical in the coalition
5P1 , P2 , P36?
(c) Which players are critical in the coalition
5P1 , P3 , P46?
(d) Write down all winning coalitions.
(e) Find the Banzhaf power distribution of this weighted voting system.
13. (a) Find the Banzhaf power distribution of the weighted voting system [6: 5, 2, 1].
(b) Find the Banzhaf power distribution of the weighted voting system [3: 2, 1, 1]. Compare your answers
in (a) and (b).
14. (a) Find the Banzhaf power distribution of the weighted voting system [7: 5, 2, 1].
(b) Find the Banzhaf power distribution of the weighted voting system [5: 3, 2, 1]. Compare your answers
in (a) and (b).
15. (a) Find the Banzhaf power distribution of the weighted voting system [10: 5, 4, 3, 2, 1].
(b) Find the Banzhaf power distribution of the weighted voting system [11: 5, 4, 3, 2, 1].
(Hint: Note that the only change from (a) is in the
quota, and use this fact to your advantage.)
16. (a) Find the Banzhaf power distribution of the weighted voting system [9: 5, 5, 4, 2, 1].
(b) Find the Banzhaf power distribution of the weighted voting system [9: 5, 5, 3, 2, 1].
73
17. Consider the weighted voting system [q: 8, 4, 2, 1]. Find
the Banzhaf power distribution of this weighted voting
system when
(a) q = 8
(b) q = 9
(c) q = 10
(d) q = 12
(e) q = 14
18. Consider the weighted voting system [q: 5, 3, 1]. Find
the Banzhaf power distribution of this weighted voting
system when
(a) q = 5
(b) q = 6
(c) q = 7
(d) q = 8
(e) q = 9
19. A business firm is owned by 4 partners, A, B, C, and D.
When making decisions, each partner has one vote and
the majority rules, except in the case of a 2-2 tie. Then,
the coalition that contains D (the partner with the least
seniority) loses. What is the Banzhaf power distribution
in this partnership?
20. Consider the weighted voting system [8: 5, 3, 1, 1, 1].
(a) Make a list of all winning coalitions.
(Hint: There aren’t too many!)
(b) Using (a), find the Banzhaf power distribution of
this weighted voting system.
(c) Suppose that P1 , with 5 votes, sells one of her votes
to P2 , resulting in the weighted voting system [8: 4,
4, 1, 1, 1]. Find the Banzhaf power distribution of
this system.
(d) Compare the power index of P1 in (b) and (c). Describe the paradox that occurred.
Exercises 21 and 22 refer to the Nassau County (N.Y.) Board
of Supervisors, as discussed in this chapter.
21. The Nassau County Board of Supervisors (1960s version). In
the 1960s, the Nassau County Board of Supervisors operated as the weighted voting system [58: 31, 31, 28, 21,
2, 2]. Assume the players are P1 through P6 .
(a) List the winning coalitions involving only P1 , P2 , or
P3 .
(b) List the winning coalitions in which P4 is a member.
(c) Use the results in (b) to explain why P4 is a dummy.
(d) Use the results in (a), (b), and (c) to find the
Banzhaf power distribution of the weighted voting system.
22. The Nassau County Board of Supervisors (1990s version). In
the early 1990s, after a series of court decisions, the Nassau County Board of Supervisors was changed to operate as the weighted voting system [65: 30, 28, 22, 15, 7, 6].
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74 2 Weighted Voting Systems
(a) List all the three-player winning coalitions and the
critical players in each.
(b) List all the four-player winning coalitions and the
critical players in each.
30. Consider the weighted voting system [q: 4, 3, 2]. Find the
Shapley-Shubik power distribution of this weighted voting system when
(a) q = 5
(Hint: There are 11 four-player winning coalitions.)
(b) q = 6
(c) List all the five-player winning coalitions and the
critical players in each.
(c) q = 7
(d) Use the results in (a), (b), and (c) to find the
Banzhaf power distribution of the weighted voting system.
(e) q = 9
C. Shapley-Shubik Power
23. Consider the weighted voting system [16: 9, 8, 7].
(a) Write down all the sequential coalitions, and in each
sequential coalition underline the pivotal player.
(b) Find the Shapley-Shubik power distribution of this
weighted voting system.
24. Consider the weighted voting system [8: 7, 6, 2].
(d) q = 8
31. Find the Shapley-Shubik power distribution of each of
the following weighted voting systems.
(a) [51: 40, 30, 20, 10]
(b) [59: 40, 30, 20, 10]
[Hint: Compare this situation with the one in (a).]
(c) [60: 40, 30, 20, 10]
32. Find the Shapley-Shubik power distribution of each of
the following weighted voting systems.
(a) [41: 40, 10, 10, 10]
(a) Write down all the sequential coalitions, and in each
sequential coalition underline the pivotal player.
(b) [49: 40, 10, 10, 10]
(b) Find the Shapley-Shubik power distribution of this
weighted voting system.
(c) [60: 40, 10, 10, 10]
25. Find the Shapley-Shubik power distribution of the
weighted voting system [5: 3, 2, 1, 1].
26. Find the Shapley-Shubik power distribution of the
weighted voting system [60: 32, 31, 28, 21].
27. Find the Shapley-Shubik power distribution of each of
the following weighted voting systems.
(a) [8: 8, 5, 1]
(b) [8: 7, 5, 2]
(c) [8: 7, 6, 1]
[Hint: Compare this situation with the one in (a).]
33. A business firm is owned by four partners, A, B, C, and
D. When making decisions, each partner has one vote
and the majority rules. In case of a 2-2 tie, the tie is broken by going against D (i.e., if D votes yes, the decision
is no, and vice versa). Find the Shapley-Shubik power
distribution in this partnership.
34. A business firm is owned by four partners, A, B, C, and
D. When making decisions, each partner has one vote
and the majority rules. In case of a 2-2 tie, the tie is broken in favor of A (the senior partner). Find the ShapleyShubik power distribution in this partnership.
(d) [8: 6, 5, 1]
(e) [8: 6, 5, 3]
28. Find the Shapley-Shubik power distribution of each of
the following weighted voting systems.
D. Miscellaneous
(a) [6: 4, 3, 2, 1]
For Exercises 35 and 36 you should use a calculator with a factorial key (typically it’s a key labeled either x! or n!). All scientific calculators and most business calculators have such a key.
(b) [7: 4, 3, 2, 1]
35. Use a calculator to compute each of the following.
(c) [8: 4, 3, 2, 1]
(a) 13!
(d) [9: 4, 3, 2, 1]
(b) 18!
(e) [10: 4, 3, 2, 1]
(c) 25!
29. Consider the weighted voting system [q: 5, 3, 1]. Find the
Shapley-Shubik power distribution of this weighted voting system when
(a) q = 5
(b) q = 6
(d) Suppose that you have a supercomputer that can
list 1 million sequential coalitions per second. Estimate (in years) how long it would take the computer to list all sequential coalitions of 25 players.
36. Use a calculator to compute each of the following.
(c) q = 7
(a) 12!
(d) q = 8
(b) 15!
(e) q = 9
(c) 20!
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Exercises
(d) Suppose that you have a supercomputer that can
list 10,000 sequential coalitions per second. Estimate (in years) how long it would take the computer to list all sequential coalitions of 20 players.
The purpose of Exercises 37 through 40 is for you to learn
how to manipulate factorials numerically. If you use a calculator to answer these questions, you are defeating the purpose
of the exercise. Please answer Exercises 37 through 40 without
using a calculator.
37. (a) Given that 10! = 3,628,800, find 9!.
(b) Find 11!>10!.
(c) Find 11!>9!.
(d) Find 9!>6!.
(e) Find 101!>99!.
38. (a) Given that 20! = 2,432,902,008,176,640,000, find
19!.
(b) Find 20!>19!.
(c) Find 201!>199!.
(d) Find 11!>8!.
39. Find the value of each of the following. Give the answer
in decimal form.
(a) 19! + 11!2>10!
(b) 1101! + 99!2>100!
40. Find the value of each of the following. Give the answer
in decimal form.
(a) 119! + 21!2>20!
(Hint: 1>20 = 0.05)
(b) 1201! + 199!2>200!
41. Consider a weighted voting system with six players (P1
through P6).
75
(c) How many coalitions in this weighted voting system do not include P7 or P6 ?
(d) How many coalitions in this weighted voting system include both P7 and P6 ?
[Hint: Use your answers of (a) and (c).]
43. Consider a weighted voting system with six players (P1
through P6).
(a) Find the number of sequential coalitions in this
weighted voting system.
(b) How many sequential coalitions in this weighted
voting system have P1 as the last player?
(Hint: Think of all the possible sequential coalitions
of the remaining players.)
(c) How many sequential coalitions in this weighted
voting system do not have P1 as the last player?
[Hint: Use your answers of (a) and (b).]
44. Consider a weighted voting system with seven players
(P1 through P7).
(a) Find the number of sequential coalitions in this
weighted voting system.
(b) How many sequential coalitions in this weighted
voting system have P7 as the first player?
(Hint: Think of all the possible sequential coalitions
of the remaining players.)
(c) How many sequential coalitions in this weighted
voting system do not have P7 as the first player?
[Hint: Use your answers of (a) and (b).]
45. Consider the weighted voting system [q: 6, 5, 4, 4, 3, 2].
(a) Find all the possible values of q for which no player
has veto power. Explain.
(a) Find the total number of coalitions in this weighted
voting system.
(b) Find all the possible values of q for which P1 is the
only player with veto power.
(b) How many coalitions in this weighted voting system do not include P6?
(c) Find all the possible values of q for which every
player has veto power.
(Hint: Think of all the possible coalitions of the remaining players.)
(d) Find all values of q such that the only winning
coalition is the grand coalition.
(c) How many coalitions in this weighted voting system do not include P5 or P6?
(e) What is the connection between the answers to (c)
and (d)? Explain.
(d) How many coalitions in this weighted voting system include both P5 and P6?
46. Consider the weighted voting system [q: 10, 8, 7, 5, 4, 4, 3].
[Hint: Use your answers of (a) and (c).]
(a) Find all the possible values of q for which no player
has veto power. Explain.
42. Consider a weighted voting system with seven players
(P1 through P7).
(b) Find all the possible values of q for which P1 is the
only player with veto power.
(a) Find the total number of coalitions in this weighted
voting system.
(c) Find all the possible values of q for which every
player has veto power.
(b) How many coalitions in this weighted voting system do not include P7?
(d) Find all values of q such that the only winning
coalition is the grand coalition.
(Hint: Think of all the possible coalitions of the remaining players.)
(e) What is the connection between the answers to (c)
and (d)? Explain.
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76 2 Weighted Voting Systems
JOGGING
47. In a weighted voting system with five players (P1 through
P5) the winning coalitions are as follows: 5P1 , P2 , P36,
5P1 , P2 , P46, 5P1 , P2 , P3 , P46, 5P1 , P2 , P3 , P56, 5P1 , P2 ,
P4 , P56, and 5P1 , P2 , P3 , P4 , P56. Find the Banzhaf
power distribution of this weighted voting system.
48. Consider the weighted voting system [24: 6, 5, 4, 4, 3, 2].
(Note that here the quota equals 100% of the votes.)
(a) If P has veto power, then P is a member of every
winning coalition.
(b) If P is a critical member of every winning coalition,
then P has veto power.
53. Dummies. We defined a dummy as a player that is never a
critical player. Explain why each of the following is true.
(a) If P is a dummy, then any winning coalition that contains P would also be a winning coalition without P.
(a) Write down the winning coalition(s).
(b) If P is a dummy, then P cannot be a member of
every winning coalition.
(b) Find the Banzhaf power index of each player in this
weighted voting system.
(c) If P is a dummy, then P cannot be a pivotal member
in any sequential coalition.
(c) Explain why the following is true: If the grand
coalition is the only winning coalition, the Banzhaf
power index of each player is 1>N, where N denotes the number of players.
(d) If P is never a pivotal member in a sequential coalition, then P must be a dummy.
49. Consider the weighted voting system [24: 6, 5, 4, 4, 3, 2].
(a) How many different sequential coalitions are there?
(b) In how many sequential coalitions is P6 pivotal?
[Hint: See Exercise 43(b).]
(c) What is the Shapley-Shubik power index of P6 ?
(d) What is the Shapley-Shubik power index of each of
the other players? Explain.
(e) A generalization of the results in (d) is that in any
weighted voting system where the quota equals
100% of the votes, the Shapley-Shubik power
index of each player is 1>N, where N denotes the
number of players. Explain why this is true.
[Hint: Generalize your observations in (a)
through (d).]
50. The disciplinary board at Tasmania State University is
composed of five members, two of which must be faculty and three of which must be students. To pass a motion
requires at least three votes, and at least one of the
votes must be from a faculty member.
(a) Find the Banzhaf power distribution of the disciplinary board.
(b) Describe the disciplinary board as a weighted voting system [q: f, f, s, s, s].
51. A professional basketball team has four coaches, a head
coach (H), and three assistant coaches 1A 1 , A 2 , A 32.
Player personnel decisions require at least three yes
votes, one of which must be H’s.
(a) If we use [q: h, a, a, a] to describe this weighted voting system, find q, h, and a.
(b) Find the Shapley-Shubik power distribution of the
weighted voting system.
52. Veto power. We defined a player P to have veto power if
the coalition formed by all the other players is a losing
coalition. Explain why each of the following is true.
54. (a) Consider the weighted voting system [22: 10, 10, 10,
10, 1]. Are there any dummies? Explain your
answer.
(b) Without doing any work [but using your answer for
(a)], find the Banzhaf and Shapley-Shubik power
distributions of this weighted voting system.
(c) Consider the weighted voting system [q: 10, 10, 10,
10, 1]. Find all the possible values of q for which P5
is not a dummy.
(d) Consider the weighted voting system [34: 10, 10,
10, 10, w]. Find all positive integers w which
make P5 a dummy.
55. Consider the weighted voting system [q: 8, 4, 1].
(a) What are the possible values of q?
(b) Which values of q result in a dictator? (Who? Why?)
(c) Which values of q result in exactly one player with
veto power? (Who? Why?)
(d) Which values of q result in more than one player
with veto power? (Who? Why?)
(e) Which values of q result in one or more dummies?
(Who? Why?)
56. Consider the weighted voting system [9: w, 5, 2, 1].
(a) What are the possible values of w?
(b) Which values of w result in a dictator? (Who? Why?)
(c) Which values of w result in a player with veto
power? (Who? Why?)
(d) Which values of w result in one or more dummies?
(Who? Why?)
57. (a) Verify that the weighted voting systems [12: 7, 4, 3,
2] and [24: 14, 8, 6, 4] result in exactly the same
Banzhaf power distribution. (If you need to make
calculations, do them for both systems side by side
and look for patterns.)
(b) Based on your work in (a), explain why the two
proportional
weighted
voting
systems
3q: w1 , w2 , Á , wN4 and 3cq: cw1 , cw2 , Á , cwN4 always have the same Banzhaf power distribution.
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Exercises
58. (a) Verify that the weighted voting systems [12: 7, 4, 3,
2] and [24: 14, 8, 6, 4] result in exactly the same
Shapley-Shubik power distribution. (If you need to
make calculations, do them for both systems side
by side and look for patterns.)
(b) Based on your work in (a), explain why the two
proportional weighted voting systems 3q: w1 ,
w2 , Á , wN4 and 3cq: cw1 , cw2 , Á , cwN4 always
have the same Shapley-Shubik power distribution.
59. Consider the weighted voting system [14: 10, 1, 1, 1, 1, 1].
(a) When P1 is the last player in a sequential coalition,
then P1 is pivotal. In how many ways can P1 be the
last player in a sequential coalition?
(Hint: See Exercise 43.)
(b) When P1 is the fifth player in a sequential coalition,
then P1 is pivotal. In how many ways can P1 be the
fifth player in a sequential coalition?
(c) Without listing any sequential coalitions, determine
the Shapley-Shubik power index of P1 .
(d) Without listing any sequential coalitions, determine
the Shapley-Shubik power distribution of this
weighted voting system.
60. Consider the generic weighted voting system
3q: w1 , w2 , Á , wN4. (Assume w1 Ú w2 Ú Á Ú wN .)
(a) Find all the possible values of q for which no player
has veto power.
(Hint: See Exercises 45 and 46.)
(b) Find all the possible values of q for which every
player has veto power.
(Hint: See Exercises 45 and 46.)
61. The weighted voting system [6: 4, 2, 2, 2, 1] represents a
partnership among five people (P1 , P2 , P3 , P4 , and
you!). You are the last player (the one with 1 vote),
which in this case makes you a dummy! Not wanting to
remain a dummy, you offer to buy one vote. Each of
the other four partners is willing to sell you one of
their votes, and they are all asking the same price.
Which partner should you buy from in order to get as
much power for your buck as possible? Use the
Banzhaf power index for your calculations. Explain
your answer.
62. The weighted voting system [27: 10, 8, 6, 4, 2] represents
a partnership among five people (P1 , P2 , P3 , P4 , and P5).
You are P5 , the one with two votes. You want to increase your power in the partnership and are prepared
to buy one share 11 share = 1 vote2 from any of the
other partners. P1 , P2 , and P3 are each willing to sell
cheap ($1000 for one share), but P4 is not being quite as
cooperative—she wants $5000 for one share. Given that
you still want to buy one share, whom should you buy it
from? Use the Banzhaf power index for your calculations. Explain your answer.
77
63. The weighted voting system [18: 10, 8, 6, 4, 2] represents a partnership among five people (P1 , P2 , P3 , P4 ,
and P5). You are P5 , the one with two votes. You want
to increase your power in the partnership and are prepared to buy shares 11 share = 1 vote2 from any of the
other partners.
(a) Suppose that each partner is willing to sell one
share and they are all asking the same price. Assuming that you decide to buy only one share,
which partner should you buy from? Use the
Banzhaf power index for your calculations.
(b) Suppose that each partner is willing to sell two
shares and they are all asking the same price. Assuming that you decide to buy two shares from a
single partner, which partner should you buy from?
Use the Banzhaf power index for your calculations.
(c) If you have the money and the cost per share is
fixed, should you buy one share or two shares
(from a single person)? Explain.
64. Sometimes in a weighted voting system, two or more
players decide to merge—that is to say, to combine their
votes and always vote the same way. (Notice that a
merger is different from a coalition—coalitions are temporary, whereas mergers are permanent.) For example,
if in the weighted voting system [7: 5, 3, 1] P2 and P3
were to merge, the weighted voting system would then
become [7: 5, 4]. In this exercise, we explore the effects
of mergers on a player’s power.
(a) Consider the weighted voting system [4: 3, 2, 1]. In
Example 9 we saw that P2 and P3 each have a
Banzhaf power index of 1>5. Suppose that P2 and
P3 merge and become a single player P*. What is
the Banzhaf power index of P*?
(b) Consider the weighted voting system [5: 3, 2, 1].
Find first the Banzhaf power indexes of players P2
and P3 and then the Banzhaf power index of P*
(the merger of P2 and P3). Compare.
(c) Rework the problem in (b) for the weighted voting
system [6: 3, 2, 1].
(d) What are your conclusions from (a), (b), and (c)?
65. Decisive voting systems. A weighted voting system is
called decisive if for every losing coalition, the coalition
consisting of the remaining players (called the
complement) must be a winning coalition.
(a) Show that the weighted voting system [5: 4, 3, 2]
is decisive.
(b) Show that the weighted voting system [3: 2, 1, 1, 1]
is decisive.
(c) Explain why any weighted voting system with a
dictator is decisive.
(d) Find the number of winning coalitions in a decisive
voting system with N players.
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78 2 Weighted Voting Systems
66. Equivalent voting systems. Two weighted voting systems
are equivalent if they have the same number of players
and exactly the same winning coalitions.
(a) Show that the weighted voting systems [8: 5, 3, 2]
and [2: 1, 1, 0] are equivalent.
(b) Show that the weighted voting systems [7: 4, 3, 2, 1]
and [5: 3, 2, 1, 1] are equivalent.
(c) Show that the weighted voting system discussed in
Example 2.12 is equivalent to [3: 1, 1, 1, 1, 1].
(d) Explain why equivalent weighted voting systems
must have the same Banzhaf power distribution.
(e) Explain why equivalent weighted voting systems
must have the same Shapley-Shubik power distribution.
67. The United Nations Security Council. The U.N. Security
Council consists of 15 member countries—5 permanent members and 10 nonpermanent members. A motion can pass only if it has the vote of all five of the
permanent members plus at least four of the nonpermanent members.
(a) Describe the critical players in a winning coalition.
(Hint: Consider separately nine-member winning
coalitions and winning coalitions with ten or more
members.)
(b) Use your answer in (a), together with the fact that
there are 210 nine-member coalitions and 638
coalitions with ten or more members, to explain
why the total number of times all players are critical is 5080.
(c) Using the results of (a) and (b), show that the
Banzhaf power index of a permanent member is
given by the ratio 848/5080.
(d) Using the results of (a), (b), and (c), show that the
Banzhaf power index of a nonpermanent member
is given by the ratio 84/5080.
(e) Explain why the U.N. Security Council is equivalent to a weighted voting system in which each nonpermanent member has 1 vote, each permanent
member has 7 votes, and the quota is 39 votes.
RUNNING
68. Consider a weighted voting system with N players, P1
through PN .
(a) Choose a random player P. In how many sequential coalitions is P in the first place? Second place?
Last place?
(b) Choose randomly two players. Call them P and Q.
In how many sequential coalitions is P in the first
place and Q in the second place? How about P in
the last place and Q in the first place? How about P
in place j and Q in place k, where j and k are any
two distinct numbers between 1 and N?
(c) Choose randomly three players. Call them P, Q,
and R. Choose randomly three different numbers
between 1 and N. Call them j, k, and m. In how
many sequential coalitions is P in the jth place, Q in
the kth place, and R in the mth place?
(d) Generalize the results of (a), (b), and (c).
69. Minimal voting systems. A weighted voting system is
called minimal if there is no equivalent weighted voting
system with a smaller quota or with a smaller total number of votes. (For the definition of equivalent weighted
voting systems, see Exercise 66.)
(a) Show that the weighted voting system [3: 2, 1, 1]
is minimal.
(b) Show that the weighted voting system [4: 2, 2, 1] is
not minimal and find an equivalent weighted voting system that is minimal.
(c) Show that the weighted voting system [8: 5, 3, 1] is
not minimal and find an equivalent weighted voting system that is minimal.
(d) Given a weighted voting system with N players and
a dictator, describe the minimal voting system
equivalent to it.
70. The Nassau County Board of Supervisors revisited. Between
1994 and 1995, the Nassau County Board of Supervisors
operated as the weighted voting system [65: 30, 28, 22,
15, 7, 6] (see Exercise 22). Show that the weighted voting system [15: 7, 6, 5, 4, 2, 1] is
(a) equivalent to [65: 30, 28, 22, 15, 7, 6]
(b) minimal (see Exercise 69).
(Hint: First show that all six weights must be positive and all must be different. Then examine possible
quotas that would give the correct results for the
three coalitions 5P1 , P2 , P56, 5P1 , P2 , P66, and
5P2 , P3 , P46. Conclude that the players’ weights
cannot be 6, 5, 4, 3, 2, 1. Use the same coalitions to
conclude that w3 + w4 7 w1 + w6 , and finally that
if w1 = 7, then w4 = 4.)
71. The Cleansburg City Council. Find the Banzhaf power distribution in the Cleansburg City Council. (See Example
2.19 for details.)
72. The Fresno City Council. In Fresno, California, the city
council consists of seven members (the mayor and six
other council members). A motion can be passed by the
mayor and at least three other council members, or by
at least five of the six ordinary council members.
(a) Describe the Fresno City Council as a weighted
voting system.
(b) Find the Shapley-Shubik power distribution for the
Fresno City Council.
(Hint: See Example 2.19 for some useful ideas.)
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Projects and Papers 79
73. Suppose that in a weighted voting system there is a
player A who hates another player P so much that he
will always vote the opposite way of P, regardless of the
issue. We will call A the antagonist of P.
(a) Suppose that in the weighted voting system [8; 5, 4,
3, 2], P is the player with two votes and his antagonist A is the player with five votes. What are the
possible coalitions under these circumstances?
What is the Banzhaf power distribution under
these circumstances?
(b) Suppose that in a generic weighted voting system
with N players there is a player P who has an antagonist A. How many coalitions are there under
these circumstances?
(c) Give examples of weighted voting systems where a
player A can
(i) increase his Banzhaf power index by becoming an antagonist of another player
(ii) decrease his Banzhaf power index by becoming an antagonist of another player
(d) Suppose that the antagonist A has more votes than
his enemy P. What is a strategy that P can use to
gain power at the expense of A?
74. (a) Give an example of a weighted voting system with
four players and such that the Shapley-Shubik
power index of P1 is 3/4.
(b) Show that in any weighted voting system with
four players, a player cannot have a Shapley-Shubik power index of more than 3/4 unless he or she
is a dictator.
(c) Show that in any weighted voting system with N
players, a player cannot have a Shapley-Shubik
power index of more than 1N - 12>N unless he or
she is a dictator.
(d) Give an example of a weighted voting system with
N players and such that P1 has a Shapley-Shubik
power index of 1N - 12>N.
75. (a) Give an example of a weighted voting system with
three players and such that the Shapley-Shubik
power index of P3 is 1>6.
(b) Explain why in any weighted voting system with
three players, a player cannot have a ShapleyShubik power index of less than 1>6 unless he or
she is a dummy.
(c) Give an example of a weighted voting system with
four players and such that the Shapley-Shubik
power index of P4 is 1>12.
(d) Explain why in any weighted voting system with
four players, a player cannot have a ShapleyShubik power index of less than 1>12 unless he or
she is a dummy.
76. (a) Give an example of a weighted voting system with
N players having a player with veto power who has
a Shapley-Shubik power index of 1>N.
(b) Explain why in any weighted voting system with N
players, a player with veto power must have a
Shapley-Shubik power index of at least 1>N.
77. (a) Give an example of a weighted voting system with
N players having a player with veto power who has
a Banzhaf power index of 1>N.
(b) Explain why in any weighted voting system with N
players, a player with veto power must have a
Banzhaf power index of at least 1>N.
78. An alternative way to compute Banzhaf power. As we know,
the first step in computing the Banzhaf power index of
player P is to compute B, the total number of times P is
a critical player. The following formula gives an alternative way to compute B: B = 2 # WP - W where W is the
number of winning coalitions and WP is the number of
winning coalitions containing P. Explain why the preceding formula is true.
Projects and Papers
A. The Johnston Power Index
The Banzhaf and Shapley-Shubik power indexes are not the
only two mathematical methods for measuring power. The
Johnston power index is a subtle but rarely used variation of
the Banzhaf power index in which the power of a player is
based not only on how often he or she is critical in a coalition, but also on the number of other players in the coalition.
Specifically, being a critical player in a coalition of 2 players
contributes 1>2 toward your power score; being critical in a
coalition of 3 players contributes 1>3 toward your power
score; and being critical in a coalition of 10 contributes only
1>10 toward your power score.
A player’s Johnston power score is obtained by adding
all such fractions over all coalitions in which the player is
critical. The player’s Johnston power index is his or her
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80 2 Weighted Voting Systems
Johnston power score divided by the sum of all players’
power scores. (A more detailed description of how to compute the Johnston power index is given in reference 17.)
Prepare a presentation on the Johnston power index.
Include a mathematical description of the procedure for
computing Johnston power, give examples, and compare
the results with the ones obtained using the Banzhaf
method. Include your own personal analysis on the merits
of the Johnston method compared with the Banzhaf
method.
different arguments have been made over the years to support the case that the Electoral College is not nearly as bad
as it seems. Massachusetts Institute of Technology physicist
Alan Natapoff has recently used mathematical ideas (many
of which are connected to the material in this chapter) to
make the claim that the Electoral College is a better system
than a direct presidential election. Summarize and analyze
Natapoff’s mathematical arguments in support of the Electoral College. (Natapoff’s arguments are nicely described in
reference 6.)
B. The Past, Present, and Future of the Electoral
College
D. Banzhaf Power and the Law
Starting with the Constitutional Convention of 1776 and
ending with the Bush-Gore presidential election of 2000,
give a historical and political analysis of the Electoral College. You should address some or all of the following issues:
How did the Electoral College get started? Why did some
of the Founding Fathers want it? How did it evolve? What
has been its impact over the years in affecting presidential
elections? (Pay particular attention to the 2000 presidential
election.) What does the future hold for the Electoral College? What are the prospects that it will be reformed or
eliminated?
C. Mathematical Arguments in Favor of the Electoral College
As a method for electing the president, the Electoral College
is widely criticized as being undemocratic. At the same time,
John Banzhaf was a lawyer, and he made his original arguments on behalf of his mathematical method to measure
power in court cases, most of which involved the Nassau
County Board of Supervisors in New York State. (See the
discussion in Section 2.3.) Among the more significant court
cases were Graham v. Board of Supervisors (1966); Bechtle v.
Board of Supervisors (1981); League of Women Voters v.
Board of Supervisors (1983); and Jackson v. Board of Supervisors (1991). Other important legal cases based on the
Banzhaf method for measuring power but not involving
Nassau County were Ianucci v. Board of Supervisors of
Washington County and Morris v. Board of Estimate (U.S.
Supreme Court, 1989). Choose one or two of these cases,
read their background, arguments, and the court’s decision,
and write a brief for each. This is a good project for pre law
and political science majors, but it might require access to a
good law library.
Appendix 1 Power in the Electoral College (2001–2010)
Electoral votes
State
Power index (%)
Number
Percent
Shapley-Shubik
Banzhaf
Alabama
9
1.673
1.639
1.640
Alaska
3
0.558
0.540
0.546
Arizona
10
1.859
1.824
1.823
Arkansas
6
1.115
1.086
1.092
California
55
10.223
11.036
11.402
Colorado
9
1.673
1.639
1.640
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Appendix 1 81
Connecticut
7
1.301
1.270
1.274
Delaware
3
0.558
0.540
0.546
District of Columbia
3
0.558
0.540
0.546
Florida
27
5.019
5.087
5.012
Georgia
15
2.788
2.761
2.744
Hawaii
4
0.743
0.722
0.728
Idaho
4
0.743
0.722
0.728
Illinois
21
3.903
3.910
3.865
Indiana
11
2.045
2.010
2.007
Iowa
7
1.301
1.270
1.274
Kansas
6
1.115
1.086
1.092
Kentucky
8
1.487
1.454
1.457
Louisiana
9
1.673
1.639
1.640
Maine
4
0.743
0.722
0.728
Maryland
10
1.859
1.824
1.823
Massachusetts
12
2.230
2.197
2.190
Michigan
17
3.160
3.141
3.116
Minnesota
10
1.859
1.824
1.823
Mississippi
6
1.115
1.086
1.092
Missouri
11
2.045
2.010
2.007
Montana
3
0.558
0.540
0.546
Nebraska
5
0.929
0.904
0.910
Nevada
5
0.929
0.904
0.910
New Hampshire
4
0.743
0.722
0.728
New Jersey
15
2.788
2.761
2.744
New Mexico
5
0.929
0.904
0.910
New York
31
5.762
5.888
5.795
North Carolina
15
2.788
2.761
2.744
North Dakota
3
0.558
0.540
0.546
20
3.717
3.717
3.677
7
1.301
1.270
1.274
Ohio
Oklahoma
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82 2 Weighted Voting Systems
Electoral votes
State
Power index (%)
Number
Percent
Shapley-Shubik
Banzhaf
7
1.301
1.270
1.274
Pennsylvania
21
3.903
3.910
3.865
Rhode Island
4
0.743
0.722
0.728
South Carolina
8
1.487
1.454
1.457
South Dakota
3
0.558
0.540
0.546
Tennessee
11
2.045
2.010
2.007
Texas
34
6.320
6.499
6.393
Utah
5
0.929
0.904
0.910
Vermont
3
0.558
0.540
0.546
Virginia
13
2.416
2.384
2.375
Washington
11
2.045
2.010
2.007
5
0.929
0.904
0.910
Wisconsin
10
1.859
1.824
1.823
Wyoming
3
0.558
0.540
0.546
Oregon
West Virginia
References and Further Readings
1. Banzhaf, John F., III, “One Man, 3.312 Votes: A Mathematical Analysis of the Electoral College,” Villanova Law Review, 13 (1968), 304–332.
2. Banzhaf, John F., III, “Weighted Voting Doesn’t Work,” Rutgers Law Review, 19
(1965), 317–343.
3. Brams, Steven J., Game Theory and Politics. New York: Free Press, 1975, chap. 5.
4. Felsenthal, Dan, and Moshe Machover, The Measurement of Voting Power: Theory
and Practice, Problems and Paradoxes. Cheltenham, England: Edward Elgar, 1998.
5. Grofman, B., “Fair Apportionment and the Banzhaf Index,” American Mathematical
Monthly, 88 (1981), 1–5.
6. Hively, Will, “Math Against Tyranny,” Discover, November 1996, 74–85.
7. Imrie, Robert W., “The Impact of the Weighted Vote on Representation in Municipal
Governing Bodies of New York State,” Annals of the New York Academy of Sciences,
219 (November 1973), 192–199.
8. Lambert, John P., “Voting Games, Power Indices and Presidential Elections,” UMAP
Journal, 3 (1988), 213–267.
9. Merrill, Samuel, “Approximations to the Banzhaf Index of Voting Power,” American
Mathematical Monthly, 89 (1982), 108–110.
10. Meyerson, Michael I., Political Numeracy: Mathematical Perspectives on Our Chaotic
Constitution. New York: W. W. Norton, 2002, chap. 2.
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References and Further Readings 83
11. Riker, William H., and Peter G. Ordeshook, An Introduction to Positive Political Theory. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1973, chap. 6.
12. Shapley, Lloyd, and Martin Shubik, “A Method for Evaluating the Distribution of
Power in a Committee System,” American Political Science Review, 48 (1954), 787–792.
13. Sickels, Robert J., “The Power Index and the Electoral College: A Challenge to
Banzhaf’s Analysis,” Villanova Law Review, 14 (1968), 92–96.
14. Straffin, Philip D., Jr., “The Power of Voting Blocs: An Example,” Mathematics Magazine, 50 (1977), 22–24.
15. Straffin, Philip D., Jr., Topics in the Theory of Voting, UMAP Expository Monograph.
Boston: Birkhäuser, 1980, chap. 1.
16. Tannenbaum, Peter, “Power in Weighted Voting Systems,” The Mathematica Journal,
7 (1997), 58–63.
17. Taylor, Alan, Mathematics and Politics: Strategy, Voting, Power and Proof. New York:
Springer-Verlag, 1995, chaps. 4 and 9.