1. The Mathematics of Power
Chapter 2:
The Mathematics of Power
2.1 An Introduction to
Weighted Voting

2. Weighted Voting 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 fair?
Should one person-one vote apply when the voters
are institutions or governments, rather than
individuals?

3. Weighted Voting 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
formally called weighted voting. Weighted voting
is not uncommon; we see examples of weighted
voting in shareholder votes in corporations, in
business partnerships, in legislatures, in the United
Nations, and, most infamously, in the way we
elect the President of the United States.

4. Electoral College The Electoral College consists of 51 “voters” (each
of the 50 states plus the District of Columbia), each
with a weight determined by the size of its
Congressional delegation (number of
Representatives and Senators). At one end of the
spectrum is heavyweight California (with 55
electoral votes); at the other end of the spectrum
are lightweights like Wyoming, Montana, North
Dakota, and the District of Columbia (with a paltry
3 electoral votes). The other states fall somewhere
in between.

5. Electoral College 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 a presidential
election.
In this chapter we will look at the mathematics
behind weighted voting, with a particular focus
on the question of power.

6. Weighted Voting System
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. We will only consider
voting on yes–no votes, known as motions. Note
that any vote between two choices (say A or B)
can be rephrased as a yes–no vote (a Yes is a vote
for A, a No is a vote for B). Every weighted voting
system is characterized by three elements:

7. Weighted Voting System
The players.
We will refer to the voters in a weighted voting
system as players. Note that in a weighted voting
system the players may be individuals, but they
may also be corporations, governmental
agencies, states, municipalities, or countries. We
will use N to denote the number of players in a
weighted voting system, and typically (unless
there is a good reason not to) we will call the
players P1, P2,…, PN .

8. Weighted Voting System
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 use w1, w2,…, wN to denote the weights of P1,
P2,…, PN, respectively. We will use V = w1 + w2 +…+
wN to denote the total number of votes in the
system.

9. Weighted Voting System
The quota.
In addition to the player’s weights, every
weighted voting system has a quota. The quota is
the minimum number of votes required to pass a
motion, and is denoted by the letter q. While the
most common standard for the quota is a simple
majority of the votes, the quota may very well be
something else.

10. Weighted Voting System
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 weighted voting systems the rules may
stipulate a quota of three-fourths of the votes, or
four-fifths, or even unanimity (100% of the votes).

11. Notation The standard notation used to describe a
weighted voting system is to use square brackets
and inside the square brackets to write the
quota q first (followed by a colon) and then the
respective weights of the individual players
separated by commas. It is convenient and
customary to list the weights in numerical order,
starting with the highest, and we will adhere to
this convention throughout the chapter.

12. Notation GENERIC WEIGHTED VOTING SYSTEM WITH N PLAYERS
Thus, a generic weighted voting system with N players can be written as:
GENERIC WEIGHTED VOTING SYSTEM WITH N PLAYERS
[q: w1, w2,…, wN]
(with w1 ≥ w2 ≥ … ≥ wN)

13. Example 2.1 Venture Capitalism
Four partners (P1, P2, P3, and P4) decide to start a
new business venture. In order to raise the
$200,000 venture capital needed for startup
money, they issue 20 shares worth $10,000 each.
Suppose that P1 buys 8 shares, P2 buys 7 shares, P3
buys 3 shares, and P4 buys 2 shares, with the usual
agreement that one share equals one vote in the
partnership. Suppose that the quota is set to be
two-thirds of the total number of votes.

14. Example 2.1 Venture Capitalism
Since the total number of votes in the
partnership is V = 20, and two-thirds of 20 is
13 1/3, we set the quota to q = 14. Using the
weighted voting system notation we just
introduced, the partnership can be described
mathematically as [14: 8, 7, 3, 2].

15. Example 2.2 Anarchy 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 of this partnership as the
weighted voting system [10:8,7,3,2], but there is a
problem here: the quota is too small, making it
possible for both Yes’s and No’s to have enough
votes to carry a particular motion.

16. Example 2.2 Anarchy (Imagine, for example, 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
meet the quota.)
In general when the quota requirement is less than
simple majority (q ≤ V/2) we have the potential for
both sides of an issue to win–a mathematical version
of anarchy.

17. Example 2.3 Gridlock
Once again, let’s look at the partnership introduced
in Example 2.1, but suppose now that the quota is
set to q = 21, more than the total number of votes in
the system. This would not make much sense. Under
these conditions no motion would ever pass and
nothing could ever get done–a mathematical
version of gridlock.

18. The Range of Values of the Quota
Given that we expect our weighted voting systems
to operate without anarchy or gridlock, from here
on we will assume that the quota will always fall
somewhere between simple majority and
unanimity of votes. Symbolically, we can express
these requirements by two inequalities: q > V/2 and
q ≤ V. These two restrictions define the range of
possible values of the quota:

19. The Range of Values of the Quota
V/2 < q ≤ V
(where V = w1 + w2 + … + wN)

20. 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 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 have 18 votes–they still need
P4’s votes to pass a motion.)

21. Example 2.4 One Partner–One Vote?
In a practical sense this weighted voting system is
no different from a weighted voting system in
which 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]).

22. Equal Say versus Equal Number of Votes
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.

23. Example 2.5 Dictators
Consider the weighted voting system [11: 12, 5, 4].
Here one of the players owns enough votes to
carry a motion single handedly. In this situation P1 is
in complete control–if P1 is for the motion, then the
motion will pass; if P1 is against it, then 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.

24. Dictator and Dummies In general a player is a dictator if the player’s
weight is bigger than or equal to the quota. Since
the player with the highest weight is P1, we can
conclude that if there is a dictator, then 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 their votes really count. A player who never
has a say in the outcome of the voting is a player
who has no power, and we will call such players
dummies.

25. Example 2.6 Unsuspecting Dummies
Four college friends (P1, P2, P3, and P4) decide to go
into business together. Three of the four (P1, P2, and
P3) invest $10,000 each, and each gets 10 shares in
the partnership. The fourth partner (P4) is a little short
on cash, so he invests only $9000 and gets 9 shares.
As usual, one share equals one vote.
The quota is set at 75%, which here means q = 30 out
of a total of V = 39 votes. Mathematically (i.e.,
stripped of all the irrelevant details of the story), this
partnership is just the weighted voting system
[30: 10, 10, 10, 9].

26. Example 2.6 Unsuspecting Dummies
Everything seems fine with the partnership until one
day P4 wakes up to the realization that with the
quota set at q = 30 he is completely out of the
decision-making loop: For a motion to pass P1, P2,
and P3 all must vote Yes, and at that point it makes
no difference how P4 votes. Thus, there is never
going to be a time when P4’s votes are going to
make a difference in the final outcome of a vote.
Surprisingly, P4–with almost as many votes as the
other partners–is just a dummy!

27. Example 2.7 Veto Power
Consider the weighted voting system [12: 9, 5, 4, 2].
Here P1 plays the role of a “spoiler”–while not having
enough votes to be a dictator, the player has
enough votes to prevent a motion from passing. This
happens because if we remove P1’s 9 votes the sum
of the remaining votes ( = 11) is less than
the quota q = 12. Thus, even if all the other players
voted Yes, without P1 the motion would not pass. In
a situation like this we say that has veto power.

28. Veto Power A player who is not a dictator has veto power if a
motion cannot pass unless the player votes in
favor of the motion. In other words, a player with
veto power cannot force a motion to pass (the
player is not a dictator), but can force a motion to
fail. If we let w denote the weight of a player with
veto power, then the two conditions can be
expressed mathematically by the inequalities
w < q (the player is not a dictator), and V – w < q
(the remaining votes in the system are not enough
to pass a motion).

29. Veto Power
VETO POWER
A player with weight w has veto power if and only if w < q and V – w < q (where V = w1 + w2 + … + wN).

30. The Mathematics of Power
Chapter 2:
The Mathematics of Power
2.2 Banzhaf Power

31. 2.2 Banzhaf Power We can already draw an important lesson:
In weighted voting the players’ weights can be
deceiving. Sometimes a player can have a few
votes and yet have as much power as a player
with many more votes; sometimes two players
have almost an equal number of votes, and yet one player has a lot of power and the other one has none.

32. 2.2 Banzhaf Power To pursue these ideas further we will need a
formal definition of what “power” means and how it can be measured. In this section we will introduce a mathematical method for measuring the power of the players in a weighted voting system. This method was first proposed in 1965 by a law professor named John Banzhaf III. 32

33. 2.2 Banzhaf Power
We start our discussion of Banzhaf’s method with an application to the U.S. Senate.
The circumstances are fictitious, but given party politics these days, not very far-fetched. 33

34. Example: The US Senate The U.S. Senate has 100 members, and a simple
majority of 51 votes is required to pass a bill. Suppose the Senate is composed of 49 Republicans, 48 Democrats, and 3 Independents, and that Senators vote strictly along party lines—Republicans all vote the same way, so do Democrats, and even the Independents are sticking together. 34

35. Example: The US Senate
Under this scenario the Senate behaves as the weighted voting system
351: 49, 48, 34. The three players are the Republican party (49 votes), the Democratic
party (48 votes), and the Independents (3 votes).
There are only four ways that a bill can pass:
All players vote Yes. The bill passes unanimously, 100 to 0.
Republicans and Democrats vote Yes; Independents vote No. The bill passes 97 to 3. 35

36. Example: The US Senate
Republicans and Independents vote Yes; Democrats vote No. The bill passes 52 to 48.
Democrats and Independents vote Yes; Republicans vote No. The bill passes 51 to 49.
That’s it! There is no other way that a bill can pass this Senate. The key observation now is that the Independents have as much influence on the outcome of the vote as the Republicans or the Democrats (as long as they stick together): To pass, a bill needs the support of any two out of the three parties. 36

37. Example: The US Senate
Republicans and Independents vote Yes; Democrats vote No. The bill passes 52 to 48.
Democrats and Independents vote Yes; Republicans vote No. The bill passes 51 to 49.
That’s it! There is no other way that a bill can pass this Senate. The key observation now is that the Independents have as much influence on the outcome of the vote as the Republicans or the Democrats (as long as they stick together): To pass, a bill needs the support of any two out of the three parties. 37

38. The European Union The legislative body for the EU (called the EU
Council of Ministers) operates as a weighted voting
system where the different member nations have
weights that are roughly proportional to their
respective populations (but with some tweaks that
favor the small countries). The second column
shows each nation’s weight in the Council of
Ministers. The total number of votes is V = 345, with
the quota set at = 255 votes.

39. The European Union The third column gives the relative weights
(weight/345) expressed as a percent. The last
column shows the Banzhaf power index of each
member nation, also expressed as a percent. We
can see from the last two columns that there is a
very close match between Banzhaf power and
weights (when we express the weights as a
percentage of the total number of votes).

40. The European Union
This is an indication that, unlike in the Nassau County
Board of Supervisors, in the EU votes and power go
hand in hand and that this weighted voting system
works pretty much the way it was intended to work.

41. The Mathematics of Power
Chapter 2:
The Mathematics of Power
2.3 Shapley to
Shubik Power

42. Shapley-Shubik Measure of Power
A different approach to measuring power, first
proposed by American mathematician Lloyd
Shapley and economist Martin Shubik in The
key difference between the Shapley-Shubik
measure of power and the Banzhaf measure of
power centers on the concept of a sequential
coalition. In the Shapley-Shubik method the
assumption is that coalitions are formed
sequentially:

43. Shapley-Shubik Measure of Power
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, etc.).
Thus, to an already complicated situation we are
adding one more wrinkle–the question of the
order in which the players join the coalition.

44. 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 {P1, P2, P3}, but
we can also write in other ways such as {P3, P2, P1}, {P2, P3, P1}, and so on. In a coalition the only thing
that matters is who are the members–the order in
which we list them is irrelevant.

45. Example 2.13 Three-Player Sequential Coalitions
In a sequential coalition, the order of the players does matter–there is a “first” player, a “second” player, and so on. For three players P1, P2, and P3, we can form six different sequential coalitions:
P is the first player, the P joined in, and last came P3;
as shown on the next slide.

46. Example 2.13 Three-Player Sequential Coalitions

47. Shapley-Shubik Approach
Pivotal player. In every sequential coalition there is a
player who 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.

48. Shapley-Shubik Approach
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. The Shapley-Shubik 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%.

49. Shapley-Shubik Approach
The Shapley-Shubik power index.
We will use 1 (“sigma-one”) to denote the Shapley-
Shubik power index of P1, 2 to denote the Shapley-
Shubik power index of P2, and so on.

50. Shapley-Shubik Approach
The Shapley-Shubik power distribution. The complete
listing of the Shapley- Shubik 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.

51. Summary of Steps 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. (For bookkeeping purposes underline the pivotal players.)

52. Summary of Steps
Step 3. Count the total number of times that P1 is pivotal. This gives SS1, the pivotal count for P1. Repeat for each of the other players to find SS2, SS3, . . .,SSN .
Step 4. Find the ratio 1 = SS1/T. This gives the Shapley-Shubik power index of P1. Repeat for each of the other players to find 2, 3, , N . The complete list of ’s gives the Shapley-Shubik power distribution of the weighted voting system.

53. Multiplication Rule THE MULTIPLICATION RULE
The Multiplication Rule is one of the most useful rules of basic mathematics.
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.

54. 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?

55. Example 2.15 Cones and Flavors Plus Toppings
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!

56. Example 2.16 Counting Sequential Coalitions
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.

57. Factorial
If we have five 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  (N – 1)  …  3  2  1. The number
N  (N – 1)  …  3  2  1 is called the factorial of N and is written in the shorthand form N!.

58. Sequential Coalitions
THE NUMBER OF SEQUENTIAL COALITIONS
N! = N  (N – 1)  …  3  2  1 gives the number of sequential coalitions with N players.

59. 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.
Step 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. (Each
column corresponds to the sequential coalitions
with a given first player.)

60. Example 2.18 Shapley-Shubik Power and the NBA Draft
Step 3. The pivotal counts are SS1 = 10, SS2 = 6, SS3 = 6 and SS4 = 2.

61. Example 2.18 Shapley-Shubik Power and the NBA Draft
Step 4. The Shapley-Shubik power distribution is given by

62. Example 2.18 Shapley-Shubik Power and the NBA Draft
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 Shapley- Shubik 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.

63. The Mathematics of Power
Chapter 2:
The Mathematics of Power
2.4 Subsets
and Permutations

64. Subsets and Permutations
Subsets and Coalitions
Coalitions are essentially sets of players that join
forces to vote on a motion. (This is the reason
we used set notation in Section 2.2 to work with
coalitions.) By definition a subset of a set is any combination of elements from the set. This
includes the set with nothing in it (called the
empty set and denoted by 5 6) as well as the
set with all the elements (the original set itself).

65. Subsets of {P1, P2, P3} and {P1, P2, P3, P4}
The following table shows the eight subsets of the set {P1, P2, P3} and the 16 subsets of
the set {P1, P2, P3, P4}.
The subsets of {P1, P2, P3, P4} are organized into two groups—the “no P4” group and the “add P4” group. The subsets in the “no P4”
group are exactly the 8 subsets of {P1, P2, P3}; the subsets of the “add P4” group
are obtained by adding P4 to each of the 8 subsets of {P1, P2, P3}. 65

66. Subsets of {P1, P2, P3} and {P1, P2, P3, P4}66

67. Subsets of {P1, P2, P3} and {P1, P2, P3, P4}
The key observation in Example 2.20 is that by
adding one more element to a set we double the number of subsets, and this principle applies to sets of any size. Since a set with 4 elements has 16 subsets, a set with 5 elements must have 32 subsets and a set with 6 elements must have 64 subsets. This leads to the following key fact: 67

68. Subsets of {P1, P2, P3} and {P1, P2, P3, P4}
Number of subsets.
A set with N elements has 2N subsets.
Now that we can count subsets, we can also count coalitions. The only difference between coalitions and subsets is that we don’t consider the empty set a coalition (the purpose of a coalition is to cast votes, so you need at least one player to have a coalition). 68

69. Subsets of {P1, P2, P3} and {P1, P2, P3, P4}
Number of coalitions.
A weighted voting system with N players has 2N – 1 coalitions.
Finally, when computing critical counts and Banzhaf power, we are interested in listing just the winning coalitions. If we assume that there are no dictators, then winning coalitions have to have at least two players and we can, therefore, rule out all the single-player coalitions from our list. 69

70. Subsets of {P1, P2, P3} and {P1, P2, P3, P4}
Number of coalitions.
A weighted voting system with N players has 2N – 1 coalitions.
Finally, when computing critical counts and Banzhaf power, we are interested
in listing just the winning coalitions. If we assume that there are no dictators, then
winning coalitions have to have at least two players and we can, therefore, rule out
all the single-player coalitions from our list. 70

71. Subsets of {P1, P2, P3} and {P1, P2, P3, P4}
Number of coalitions of two or more players.
A weighted voting system with N
players has 2N - N - 1 coalitions of two or more players. 71