Back to Mathematics and Politics
In
many situations, voting is not "one personone vote." For example,
the stockholders of a company usually have one vote for each share of stock
owned in the company. Thus, some people may have many votes (have much power)
while other voters may have only a few votes (have little power).
Here we will make matters simpler by looking only at cases where the voting
is between two alternatives. For the case of only two alternatives, all
reasonable voting methods can be shown to be equivalent to a simple
"majority rules." This allows us to remove our focus from
particular voting methods and, instead, examine the concept of power
in situations where voters' votes carry different weights (socalled weighted
voting systems). A weighted voting system is a decisionmaking procedure in which the participants have varying numbers of votes. Examples of such systems include shareholder elections and the election of the president of the United States by the electoral college. Some legislative bodies have such strong discipline that each legislator always votes as dictated by his or her party. These legislatures are weighted voting systems, each of which is entitled to a number of votes equal to the size of its delegation in the legislature. The power of a participant in a weighted voting system can be roughly defined as the ability of the participant to influence a decision. There are several ways to measure mathematically the power of a participant in a weighted voting system. We will study two ways, the Banzhaf power index and ShapleyShubik power index. Either of these indices provides a much more accurate measure of a participant's power than the number of votes that the participant is entitled to cast. Example 1: The system to amend
the Canadian Constitution[1]
Since 1982, and amendment
to the Canadian constitution becomes law only if it is approved by at least
seven of the ten Canadian provinces subject to the proviso that the approving
provinces have among them, at least half of Canada’s population.

In 1971 the Premiers of the ten provinces of Canada met in Victoria to negotiate an amendment procedure for the Canadian constitution. The history is interesting. The constitution of Canada was contained in the British North America Act of 1867, by which Britain granted independence to Canada. However, no procedure had been given by which Canadians could amend their constitution other than by petitioning the British Parliament to enact the amendment. This was a strange situation, and "patriation of the constitution", in other words, bringing the constitution under Canadian control, had been a patriotic issue in Canada since the 1920's. In order to patriate the constitution, the Canadian provinces had to agree on an amending procedure, and this problem was to be addressed by the Victoria Conference.
The problem is complicated by the diversity of the Canadian provinces, in size as well as in politics and culture. In 1970 the two largest provinces, Ontario and Quebec, contained 64% of the Canadian population. Any scheme which treated all provinces equally (as in the United States, where a constitutional amendment must be approved by 3/4 of the states, with large and small states treated equally) would surely be unfair to residents of these provinces.
The amending procedure proposed by the Victoria Conference recognized provincial disparity. A constitutional amendment would have to be approved by
Notice that both Ontario and Quebec have veto power over constitutional amendments. British Columbia also seems to have considerable power. The "prairie provinces" are Alberta, Saskatchewan and Manitoba; the "Atlantic provinces" are New Brunswick, Nova Scotia, Prince Edward Island and Newfoundland. How fair is this scheme? Does the unequal power of the provinces mirror, at least roughly their relative populations? To answer questions like these, we need to formalize and quantify the notion of the power of a voter in a voting body. A mathematical model of a voting body strips away all personalities and ideologies, and considers only which groups of voters can pass bills (or constitutional amendments in the above case). Those subsets of voters which can pass bills are called winning coalitions.
On March 25, 1957, the Treaties establishing the
European Economic community (EEC) were signed by the six[2]
founders of EEC (Belgium, France, Germany, Italy, Luxembourg, Netherlands) in
Rome ("Treaties of Rome"). However, to represent different levels of
influence and power within Europe, France, Germany, and Italy were given four
votes each, while Belgium and the Netherlands were given two votes and
Luxembourg one.
In
order to pass a law into effect, twelve of the seventeen votes were required.
In 1965, the Nassau County Board of Supervisors in New York had allocated 115 votes to its six districts, based on population. This gave the six districts the following number of votes:
District 
Votes 
Hempstead #1 
31 
Hempstead #2 
31 
Oyster Bay 
28 
North Hempstead 
21 
Glen Cove 
2 
Long Beach 
2 
In order for a county law to pass, 58 votes are needed.
The
Council has 15 members five permanent members and 10 elected by the General
Assembly for twoyear terms. Each Council member has one vote. Decisions on
substantive matters require nine votes, including the concurring votes of all
five permanent members. This is the rule of "great Power unanimity",
often referred to as the "veto" power. For example:
Membership of
the Security Council in 2000 



Term Ends


Term Ends


United States 
Permanent 
Mali 
12/31/2001 

Argentina 
12/31/2000 
Namibia 
12/31/2000 

Bangladesh 
12/31/2001 
Netherlands 
12/31/2000 

Canada 
12/31/2000 
Russian Federation 
Permanent 

China 
Permanent 

France 
Permanent 
Tunisia 
12/31/2001 

Jamaica 
12/31/2001 
Ukraine 
12/31/2001 

Malaysia 
12/31/2000 
United Kingdom 
Permanent 
For
future discussion we will need several definitions:
Definition 1: A collection of voters is called a coalition. If a coalition has enough votes to pass the bill, it is called a winning coalition. If a coalition is not winning it is called a losing coalition.
Definition 2: The quota is the minimum number of votes needed to win an election.
In Example 1, the quota is 12 votes, and some winning coalitions are:
A: France, Germany and Italy
B: France, Italy, Belgium and Netherlands
C: France, Italy, Belgium, Netherlands and Luxembourg.
There
are other winning coalitions in this example. Note that last winning coalition,
coalition C, remains winning even if Luxembourg is taken away. On the other
hand, if any of the members is deleted from the coalitions A and B, they become
losing coalitions.
Definition 3: A winning coalition with the property that the deletion of any of its member renders it losing is called a minimal winning coalition.
Coalitions A and B in the Example 1 are minimal winning coalitions.
Definition 4: A yesno voting system is said to be weighted voting system if it can be described by specifying the number of votes each voter has (weight), and quota. The shorthand notation
_{} is used. The quota, q, is given first, followed by a colon, and then a list of weights of each of the voters in the system, usually in decreasing order.
In this notation, the voting system given in example 1 by the European Economic Community is a weighted voting system _{}, and the voting system in described by example 4 in Nassau County is a weighted voting system _{}.
Under the United Nations Security Council voting rules, each Council member has one vote, while the five permanent members of the council have the power of veto. Decisions on substantive matters require nine votes. Surprisingly enough, this is a weighted voting system. We can show this by producing the quota, q, for this voting system, and the weights of each of the Security Council members. Obviously, all of the nonpermanent members have the same weight, and all of the permanent members of the council have a larger weight than nonpermanent members. Lets assume than nonpermanent members have weight 1, and that permanent members have weight x. Then, all five permanent members need only four nonpermanent members to go along in order to get nine votes necessary for passage of a proposal. In other words, _{}. At the same time, if a single permanent member of the Security Council is against a certain proposal, the proposal will not reach the necessary quota for passage, i.e., _{}. Putting these two inequalities together, we get _{}, or _{}. The smallest value x can be is therefore 7, and the quota is then larger than _{}, and at most _{}, so _{}.
In order to prove that quota is really 39, we now must show that each losing coalition has weight at most 38, and that each winning coalition has weight at least 39.
Any losing coalition will either have at least one permanent member missing, so even if everybody else is a member of this coalition, its weight is _{}, or it will have less than nine votes in total, i.e., it will have five permanent members, and at most three nonpermanent members with the combined weight _{}.
Any winning coalition must have all five permanent members on board, and must have at least nine total members. That means that in addition to five permanent members at least four nonpermanent members must be in a winning coalition, with the combined weight _{}.
Therefore, United Nations Security Council is weighted voting system: _{}.
The Treaty on European Union increased the European Parliament's say through a codecision procedure, which means that a wide range of legislation (such as internal market, consumer affairs, transEuropean networks, education and health) is adopted both by the Parliament and the Council.
In the vast majority of cases (including agriculture, fisheries, internal market, environment and transport), the Council decides by a qualified majority vote with Member States carrying the following weightings:
Germany, France, Italy and the United Kingdom 
10 votes 
Spain 
8 votes 
Belgium, Greece, the Netherlands and Portugal 
5 votes 
Austria and Sweden 
4 votes 
Ireland, Denmark and Finland 
3 votes 
Luxembourg 
2 votes 
Total 
87 votes 
When a Commission proposal is involved, at least 62 votes must be cast in favor. In other cases, the qualified majority is also 62 votes, but these must be cast by at least 10 Member States. In practice, the Council tries to reach the widest possible consensus before taking a decision so that, for example, only about 14% of the legislation adopted by the Council in 1994 was the subject of negative votes and abstentions.
Therefore, European Community is a weighted voting system: _{}.
a)
[17:5, 4, 4, 2]
Quota is too high. Nothing will pass.
b)
[11:4,4,4,4,4]
In essence, this is oneperson, onevote.
c)
[15:5,4,3,2,1]
Must have unanimous consent of the voters to pass anything.
Equivalent to [5:1,1,1,1,1].
d)
[11:12, 5, 4]
Example of a dictator, a voter whose weight is bigger than or equal to
the quota. A voter without power is called a dummy.
e)
[12:11, 5, 4, 2]
The voter with a weight of 11 has veto power, even though the voter is
not a dictator.
f)
[101:99, 98, 3]
Any two of the voters can reach the quota; no voter can do it alone.
ANOTHER KEY POINT
The weight of a player is not a good measure of a player’s POWER.
For instance, it is possible for a weighted
voting system to actually reduce to a oneperson, onevote situation in which
case all voters have the same POWER even though they don’t all have the same
weight. For example, for the weighted voting system [101:99, 98, 3] none of the voters can pass a motion alone and any two voters can join together to pass a motion; so, although the
voters have different weights,
each voter has the same amount
of power.
SO, how do we define power?
Let's
now look at an example to motivate a measure of power. Consider the example
voting system [6:5, 3, 1]. What are all of the possible coalitions?
{P_{1},P_{2},P_{3}}
{P_{1}, P_{2}}
{P_{1}, P_{3}}
{P_{2},P_{3}}
{P_{1}}
{P_{2}}
{P_{3}}
Now
which of these are winning coalitions?
{P_{1},P_{2},P_{3}}
{P_{1}, P_{2}}
{P_{1}, P_{3}}
In each of these winning coalitions, which voter (s) are critical?
P_{1} is critical 3 times.
P_{2} is critical 1 time.
P_{3} is critical 1 time.
Now
P_{1} is critical in 3 out of 5 times, or 60%. P_{2} and P_{3}
are each critical 1 times out of five, or 20% each. This measure of power is
called Banzhaf index of power.
Definition: Suppose that p is a voter in a yesno voting system. Then the total Banzhaf power of p, denotes here by TBP(p), is the number of coalitions C satisfying the following three conditions:
Definition: Suppose that p_{1} is a voter in a yesno voting system and that the other voters are denoted by p_{2}, p_{3}, ...,p_{n} . Then the Banzhaf index of p_{1}, denoted here by BI(p_{1}), is the number given by:
_{}
Remark: The Banzhaf index of a voter is between 0 an 1 if we add up the Banzhaf indices of all n voters, we get the number 1.
There are two (equivalent) methods to calculate total Banzhaf power:
Procedure 1: Assign each voter a plus one for each winning coalition of which it is a member, and assign it a minus one for each winning coalition of which it is not a member. The sum of these plus and minus ones turns out to be the total Banzhaf power of the voter.
Procedure 2: Assign each voter a plus two for each winning coalition of which it is a member. Subtract the total number of winning coalitions form this sum. The answer turns out to be the total Banzhaf power of the voter.
Important Note: To determine the Banzhaf power index, we will have to count all the possible coalitions and then only keep the winning ones. (Given N voters, there are a total of 2^{N} possible coalitions, including the “empty” coalition, which has no one in it.)
A committee consists of four voters, P_{1}, P_{2}, P_{3}, and P_{4}. Each committee member has one vote, and a motion is carried by majority vote except in the case of a 22 tie. In this case, if P_{1} voted for the motion, then it carries. (P_{1} plays the tiebreaker here.) Determine the Banzhaf power index of each of these four players.
How many winning coalitions are there?
{P_{1},P_{2},P_{3}, P_{4}}
{P_{1}, P_{2}, P_{3}}
{P_{1}, P_{2}, P_{4}}
{P_{1},P_{3},P_{4}}
{P_{2},P_{3},P_{4}}
{P_{1}, P_{2}}
{P_{1}, P_{3}}
{P_{1}, P_{4}}
How many times are each of the voters critical?
P_{1} is critical 6 times.
P_{2} is critical 2 times.
P_{3} is critical 2 times.
P_{4} is critical 2 times.
Then the Banzhaf power index of each is given by
P_{1}: [6/12] or 50 percent
P_{2}: [2/12] or 16.67 percent (approximately)
P_{3}: [2/12] or 16.67 percent (approximately)
P_{4}: [2/12] or 16.67 percent (approximately)
One final idea is needed before we can present the formal definition of the ShapleyShubik index. Suppose, for example, that we have a yesno voting system with seven voters: p_{1}, p_{2}, p_{3}, p_{4}, p_{5}, p_{6}, p_{7}. We want to identify one of the players as being pivotal for this ordering. To explain this idea, we picture a larger and larger coalition being formed as we move from left to right. That is, we first have p_{3} alone, then p_{5} joins to give us the twomember coalition p_{3}, p_{5}. Then p_{1} joins, yielding the threemember coalition : p_{3}, p_{5}, p_{1}. And so on. The growing coalition changes from a nonwinning one to a winning one. Since the empty coalition is losing and the grand coalition is winning, it is easy to see that one voter must be pivotal.
Definition: Suppose p is a voter in a yesno voting system for a set of voters. Then the ShapleyShubik index of p, denoted here by SSI(p), is the number given by:
_{}
Remarks:
Consider the example voting system: [6:5, 3, 1]
What are the sequential coalitions? There should be 3!=6 coalitions with three voters:
áP_{1},P_{2},P_{3}ń
áP_{1},P_{3},P_{2}ń
áP_{2},P_{1},P_{3}ń
áP_{2},P_{3},P_{1}ń
áP_{3},P_{1},P_{2}ń
áP_{3},P_{2},P_{1}ń
In each of these sequential coalitions, which voter is the pivotal one?
P_{2} is pivotal in the first sequential coalition.
P_{3} is pivotal in the second sequential coalition.
P_{1} is pivotal in the third sequential coalition.
P_{1} is pivotal in the fourth sequential coalition.
P_{1} is pivotal in the fifth sequential coalition.
P_{1} is pivotal in the sixth sequential coalition.
Now P_{1} is given a ShapleyShubik power index of 4/6 or 66.66% (approximately). P_{2} and P_{3} each have a ShapleyShubik power index of 1/6 or 16.66% each (approximately).
a) {4: 3, 2, 2} Which player has the most power?
§
from a Banzaf point of view:
Look at all the winning coalitions.
{P_{1}}{P_{1}, P_{2}} (winning)
{P_{2}}
{P_{3}}
{P_{1}, P_{3}} (winning)
{P_{2}, P_{3}} (winning)
{P_{1}, P_{2}, P_{3}} (winning)
Find the critical players in each.
winning coalition {P_{1}, P_{2}} 
critical players P_{1 }&_{ }P_{2 }P_{1 }&_{ }P_{3 }P_{2 }&_{ }P_{3 }none 
BPI(P) = (# of
times player P is critical) / (# times all are critical)
BPI(P_{1 }) = 2/6 = 1/3 ; BPI(P_{2 }) = 2/6 = 1/3
; BPI(P_{3 }) = 2/6 = 1/3
·
from a ShapleyShubik point of view:
Find all sequential coalitions containing all players (N! of them).
{P_{1}, P_{2}, P_{3}}
{P_{1}, P_{3}, P_{2}}
{P_{2}, P_{1}, P_{3}}
{P_{2}, P_{3}, P_{1}}
{P_{3}, P_{2}, P_{2}}
{P_{3}, P_{2}, P_{1}}
Find the pivotal player in each.
sequential coalition {P_{1}, P_{2}, P_{3}} 
pivotal player P_{2 }P_{3 }P_{1 }P_{3 }P_{2 }P_{2} 
SSPI(P) = (# of times player P is pivotal) / (# times all are pivotal)
SSPI(P_{1 }) = 1/6 ; SSPI(P_{2 }) = 3/6 = 1/2 ; SSPI(P_{3}) = 2/6 = 1/3
There
are 159 winning coalitions under the Canadian Constitution (fortunately, this
scan be done with a computer). The breakdown of the Banzhaf Index of Power is
given in the table below. Contrast this with the table in the next example
discussing Victoria Conference Proposal. Which breakup of the power is more
fair? Why?

Population 
Percentage of population 
Seats in parliament 
Percents of Seats 
TBP 
Banzhaf Index 
Ontario 
10,084,885 
37.06% 
103 
34.22% 
101 
13% 
Quebec 
6,895,963 
25.34% 
75 
24.92% 
85 
11% 
British Columbia 
3,282,061 
12.06% 
34 
11.30% 
81 
11% 
Alberta 
2,545,553 
9.35% 
26 
8.64% 
75 
10% 
Saskatchewan 
988,928 
3.63% 
14 
4.65% 
73 
9% 
Manitoba 
1,091,942 
4.01% 
14 
4.65% 
73 
9% 
New Brunswick 
723,900 
2.66% 
10 
3.32% 
71 
9% 
Nova Scotia 
899,942 
3.31% 
11 
3.65% 
71 
9% 
Prince Edward Island 
129,765 
0.48% 
4 
1.33% 
69 
9% 
Newfoundland 
568,474 
2.09% 
7 
2.33% 
71 
9% 
Yukon Territory 
27,797 
0.10% 
1 
0.33% 
0 
0% 
Northwest Territories 
57,649 
0.21% 
2 
0.66% 
0 
0% 
TOTAL 
27,211,413 
100% 
301 
100.00% 
770 
100% 
From the Example 1, we know that, under Victoria Conference proposal, a constitutional amendment would have to be approved by
The
following is the list of all of the winning coalitions:
So a winning coalition will have three parts:
Ontario and Quebeck have to be a part of the winning coalition 
BC and A; BC and S BC and M A and S and M BC and A and S BC and A and M BC and S and M BC and A and S and M TOTAL of eight winning groups 
NB and NS NB and PEI NB and N NS and PEI NS and N PEI and N NB and NS and PEI NB and NS and N NB and PEI and NS NS and PEI and N NB and NS and PEI and N TOTAL of eleven winning groups 
In total this produces _{} winning coalitions.
To calculate individual indices, notice that Ontario and Quebec are critical in all 88 of the winning coalitions.
British Columbia is in 7 of the eight winning groups in its column. Therefore, British Columbia is a member of 7H11 = 77 winning coalitions. It is not a member of the remaining 11 winning coalitions. Therefore, the Total Banzhaf Power of British Columbia is 7711 = 66.
In a similar way, we can evaluate the total Bazhaf power for the other 7 Canadian provinces.
We obtain the following table:

Population 
Percentage of population 
Seats in parliament 
Percents of Seats 
TBP 
Banzhaf Index 
Ontario 
10,084,885 
37.06% 
103 
34.22% 
88 
22% 
Quebec 
6,895,963 
25.34% 
75 
24.92% 
88 
22% 
British Columbia 
3,282,061 
12.06% 
34 
11.30% 
66 
16% 
Alberta 
2,545,553 
9.35% 
26 
8.64% 
22 
5% 
Saskatchewan 
988,928 
3.63% 
14 
4.65% 
22 
5% 
Manitoba 
1,091,942 
4.01% 
14 
4.65% 
22 
5% 
New Brunswick 
723,900 
2.66% 
10 
3.32% 
24 
6% 
Nova Scotia 
899,942 
3.31% 
11 
3.65% 
24 
6% 
Prince Edward Island 
129,765 
0.48% 
4 
1.33% 
24 
6% 
Newfoundland 
568,474 
2.09% 
7 
2.33% 
24 
6% 
Yukon Territory 
27,797 
0.10% 
1 
0.33% 
0 
0% 
Northwest Territories 
57,649 
0.21% 
2 
0.66% 
0 
0% 
TOTAL 
27,211,413 
100% 
301 
100.00% 
404 
100% 