Chapter 8 :

Apportionment
and Voting
GROUP 6
ANDAL, ABARCA, BELEN, CABRERA, DELCO
 After completing this chapter, the students will be able to:

 Compare and contrast the different apportionment methods.

 Understand apportionment problem.
 Produce a valid and fair apportionment.
 Review an election result using the different fairness criterion.
 Apply the methods in apportionment.
 Define important terms in voting.
 Differentiate and apply plurality, Borda count, plurality-with-
elimination, and pairwise comparison voting methods.
 Determine the most appropriate voting method in a given situation.
 Apply the concepts of apportionment and voting in real life situation.
Use mathematical concepts and tools in apportionment and voting.

LEARNING COMPETENCIES
Chapter Outline:
Unit 8.1: Introduction to Apportionment
Unit 8.2: Introduction to Voting
Unit 8.3: Weighted Voting System
 Review on the Philippine Constitution of 1987
This section discusses the two of the most fundamental principles of
democracy: the right and duties to vote and the value of the vote of each
individual and how to properly apportion the representations of groups and the
like. Apportionment can also be seen in the Philippine Constitution of 1987,
Article VI, Section 5, which states that —

"(1) The House of Representatives shall be composed of not more than two hundred
and fifty members, unless otherwise fixed by law, who shall be elected from
legislative districts apportioned among the provinces, cities, and the Metropolitan
Manila area in accordance with the manner of their respective inhabitants, and on
the basis of a uniform and progressive ratio and those who, as provided by law,
shall be elected through a party-list system of registered national, regional, and
sectoral parties or organizations.'

Unit 8.1: Introduction to Apportionment

 Review on the Philippine Constitution of 1987
"(3) Each legislative district shall comprise, as far as practicable, contiguous,
compact; and adjacent territory. Each city with a population of at least two
hundred fifty thousand, or each province, shall have at least one representative."

"(4) Within three years following the return of every census, the Congress shall make
a reapportionment of legislative districts based on the standards provided in this
section.
This article requires that representatives (or congressman) be apportioned among
the several districts according to their respective numbers. Meaning, the number of
representatives each area sends to Congress (lower house) should be based on its
population. In this case, apportionment determines the number of the House of
Representatives according to the proportion of the population of provinces, cities,
and the Metropolitan Manila area to the population of the Philippines. The current
apportionment used is based from the 1980 census. Currently, there is a total of 297
representatives, composed of 234 district representatives and 63 sectoral
representatives.

Unit 8.1: Introduction to Apportionment

 Defining the Terminologies
Apportionment is the act of dividing items between different groups according to
some plan especially to make proportionate distribution in a fair manner.
Mathematically, an apportionment is a function which takes as input the values q,
n, p3, p2,….pn,
where q and n are positive integers, pn's are positive numbers, and whose output
is a sequence of non-negative integers ql, q2,…, qn such that q = ql + q2 + .... +
qn

Some of the basic elements of every apportionment problem are states, seats,
populations, standard divisor, and standard quota. State is a term used to
describe the parties having a stake in the apportionment, we denote it as ql + q2
+ ... + qn. Seat is a set of k identical, indivisible objects that are being
divided among n states. It is assume that there are more seats than there are
states but does not guarantee that every state can potentially get a seat.
Population is a set of n positive numbers that are used as the basis for the
apportionment Of the seats to the states, we denote p1, p2,...,pn as state's
respective populations and p denotes the total population p = p1 + p2 +…+ pn.

Unit 8.1: Introduction to Apportionment

 Defining the Terminologies

Apportionment problem is to find a method for rounding standard quota into whole
numbers so that the sum of the numbers is the total number of allocated items.
The Quota Rule is an apportionment method that apportion to state/group has a
fractional part, either the integer immediately above, or the integer part of, that
state's/group's standard quota.
The lower quota is the standard quota rounded down to the nearest whole number, while
the upper quota is the standard quota rounded up to the nearest whole number.
A standard divisor is the ratio of population to seats and it is found by dividing
the total population under consideration by the number of seats.

Standard Divisor (d) = Total Population (p) + Number of Seats

Unit 8.1: Introduction to Apportionment

 Defining the Terminologies
Standard quota of a state is the exact fractional number of seats that the state would
get if fractional seats were allowed; we use the notation q1, q2, …., qn to denote the
standard quotas of the respective states. (In general, the standard quotas are
expressed in fraction or decimal-two or more decimal places).

A standard quota for particular group is found by dividing that group's population by
the standard divisor.

Standard Quota (qn) = Population of a Particular Group (pn) + Standard Divisor (d)

We let,

q = total number to be allocated (q=q1+q2+…+qn),

n = the number of groups or state,

p1, p2, …, pn = population of each group or states (objects to the nth state),

qn = standard quota or allocation of each group or states (objects to the nth state),

d = standard divisor, and

m = modified divisor.

Unit 8.1: Introduction to Apportionment

 Defining the Terminologies

A function of a variable is monotone if this function will not decrease in value

whenever the variable increases in value. The types of monotonicity are house
monotone, population monotone, and quota monotone.
A house monotone is an apportionment method if no state/group can lose a seat when
the size of the total number of allocation increases.
Secondly, a population monotone is n apportionment method when no state/group can
lose a seat when only the population increases.
Lastly, a quota monotone, is an appropriation method when no state/group can
lose a seat whenever its quota increases.

Unit 8.1: Introduction to Apportionment

 Apportionment Methods

A. Hamilton's Method
Hamilton's method was the first apportionment method to be approved in the United
States of America Congress, but was vetoed by President George Washington in
1792. The method was proposed by Alexander Hamilton. The Hamilton's method was
later used off and on between 1852 and 1901. Hamilton's method tends to favor
larger states.

Unit 8.1: Introduction to Apportionment

 Apportionment Methods
Steps in solving apportionment using Hamilton's method:
1. Solve for the standard divisor.
2. Determine the standard quota for, each group by dividing its population by
the standard divisor.
3. Round each group's standard quota down to the nearest whole number. This is
called lower quota.
4. Sum the lower quota to find how many leftover seats exist,
5. Allocate the leftover seats (one at a time) to the group with the largest
decimal remainders (or largest fractional parts) in their standard quotas until
no leftovers remain.

Unit 8.1: Introduction to Apportionment

 Apportionment Methods

B. Jefferson's Method

Jefferson's method differs from Hamilton's method on how to resolve the situation
when the lower quota or initial quota is less than the actual seats available which
is referred to as modified lower quota. Jefferson's method tends to favor larger
states. If a state/group gets more than the integer immediately above its quota,
Jefferson's violates the upper quota rule. The method proposed by Thomas Jefferson
in 1792 and was first used in US Congress in 1792 until 1840. He is one of the
founding father of US and became its third president. Steps in solving apportionment
using Jefferson's method:

Unit 8.1: Introduction to Apportionment

 Apportionment Methods

Steps in solving apportionment using Jefferson's method:

1. Solve for the standard divisor

2. Determine the standard quota for each group by dividing its population
by the standard divisor.
3. Round each group’s standard quota down to the nearest whole number.
This is called modified lower quota.
4. Choose a number m, which represents a desired approximate size for
congressional districts.
5. Compute the modified quotas for each group, and round these numbers
down to obtain qn. If q1 + q2 + … + qn = q , then we have the
apportionment. Otherwise, change m and try it again. (The divisor we
end up using is called the modified divisor or adjusted divisor).

Unit 8.1: Introduction to Apportionment

 Apportionment Methods

B. Adams' Method

Adams' method differs from Jefferson's method on how to resolve the situation
when the upper quota or initial quota is greater than the actual seats available
which is referred to as modified upper quota. Adams' method always apportions at
least 1 seat to each state/group, unlike Jefferson's method. Adams' method tends to
favor smaller states/groups. If a state/group gets less than the integer part of it
standard quota, Adams' violates the lower quota rule. The method was proposed by
John Quincy Adams but was never used in the US Congress, he became the sixth US
president.

Unit 8.1: Introduction to Apportionment

 Apportionment Methods

Steps in solving apportionment using Adam's method:

1. Solve for the standard divisor.

2. Determine the standard quota for each group by dividing its population by
the standard divisor.
3. Round each group's standard quota up to the nearest whole number. This is
called modified upper quota.
4. Choose a number m, which represents a desired approximate size for
congressional districts.
5. Compute the modified quotas for each group, and round these numbers up to obtain
qn. If ql + q2 + ... + qn = q, then we have the apportionment. Otherwise, change
m and try it again.

Unit 8.1: Introduction to Apportionment

 Apportionment Methods

D. Webster’s Method
Webster’s method is based on ordinary rounding, a state's quota has
just as much chance of having a remainder at or above 0.5 as it does
having one below 0.5. The method was proposed by Daniel Webster in
1830 and was first adopted in 1842 and also used in the late 19th
century and early 20th century in the US Congress. Webster's method
tends to favor smaller states.

Unit 8.1: Introduction to Apportionment

 Apportionment Methods
steps in solving apportionment using Webster's method:

1. Solve for the standard divisor.

2. Determine the standard quota for each group by dividing its population by the
standard divisor.
3. Round each group's standard quota to the nearest whole number (conventional
rounding rules). This is called modified rounded quota.
4. Choose a number m, which represents a desired approximate size for congressional
districts.
5. compute the modified quotas for each group, and round these numbers conventionally
to obtain qn. If ql + q2 + …. + qn = q, then we have the apportionment. Otherwise,
change m and try it again.

Unit 8.1: Introduction to Apportionment

 Note:
a. Hamilton's method satisfies the quota rule, but
violates population (and house) monotone.

b. Jefferson's (violates upper quota),

c. Adam’s (Violates lower quota), and

d. Webster’s (violates upper and lower quota) can lead

to violation of the quota rule, but satisfy population
(and house monotonicity).

Unit 8.1: Introduction to Apportionment

 Example 2: Suppose the newly bought 30 police patrol cars of the Philippine
National Police—National Capital Region are selected according to the 2015
census of most populated cities in Metro Manila, as shown in the table below.
Use the four methods of apportionment (Hamilton's, Jefferson's, Adams' and
Websters) to determine the number of board members each city should have.

Compare the results of the using the four appropriation methods.

Unit 8.1: Introduction to Apportionment

 Example 2: Suppose the newly bought 30
police patrol cars of the Philippine
National Police—National Capital Region
are selected according to the 2015 census
of most populated cities in Metro Manila,
as shown in the table below. Use the four
methods of apportionment (Hamilton's,
Jefferson's, Adams' and Websters) to
determine the number of board members each
city should have.

Unit 8.1: Introduction to Apportionment

 Example 2: Suppose the newly bought 30
police patrol cars of the Philippine
National Police—National Capital Region
are selected according to the 2015 census
of most populated cities in Metro Manila,
as shown in the table below. Use the four
methods of apportionment (Hamilton's,
Jefferson's, Adams' and Websters) to
determine the number of board members each
city should have.

Unit 8.1: Introduction to Apportionment

 Example 2: Suppose the newly bought 30
police patrol cars of the Philippine
National Police—National Capital Region
are selected according to the 2015 census
of most populated cities in Metro Manila,
as shown in the table below. Use the four
methods of apportionment (Hamilton's,
Jefferson's, Adams' and Websters) to
determine the number of board members each
city should have.

Unit 8.1: Introduction to Apportionment

 Example 2: Suppose the newly bought 30
police patrol cars of the Philippine
National Police—National Capital Region
are selected according to the 2015 census
of most populated cities in Metro Manila,
as shown in the table below. Use the four
methods of apportionment (Hamilton's,
Jefferson's, Adams' and Websters) to
determine the number of board members each
city should have.

Unit 8.1: Introduction to Apportionment

 Example 2: Suppose the newly bought 30
police patrol cars of the Philippine
National Police—National Capital Region
are selected according to the 2015 census
of most populated cities in Metro Manila,
as shown in the table below. Use the four
methods of apportionment (Hamilton's,
Jefferson's, Adams' and Websters) to
determine the number of board members each
city should have.

Unit 8.1: Introduction to Apportionment

 Example 2: Suppose the newly bought 30
police patrol cars of the Philippine
National Police—National Capital Region
are selected according to the 2015 census
of most populated cities in Metro Manila,
as shown in the table below. Use the four
methods of apportionment (Hamilton's,
Jefferson's, Adams' and Websters) to
determine the number of board members each
city should have.

Unit 8.1: Introduction to Apportionment

 Example 2: Suppose the newly bought 30
police patrol cars of the Philippine
National Police—National Capital Region
are selected according to the 2015 census
of most populated cities in Metro Manila,
as shown in the table below. Use the four
methods of apportionment (Hamilton's,
Jefferson's, Adams' and Websters) to
determine the number of board members each
city should have.

Unit 8.1: Introduction to Apportionment

 Example 2: Suppose the newly bought 30
police patrol cars of the Philippine
National Police—National Capital Region
are selected according to the 2015 census
of most populated cities in Metro Manila,
as shown in the table below. Use the four
methods of apportionment (Hamilton's,
Jefferson's, Adams' and Websters) to
determine the number of board members each
city should have.

Unit 8.1: Introduction to Apportionment

 Example 2: Suppose the newly bought 30
police patrol cars of the Philippine
National Police—National Capital Region
are selected according to the 2015 census
of most populated cities in Metro Manila,
as shown in the table below. Use the four
methods of apportionment (Hamilton's,
Jefferson's, Adams' and Websters) to
determine the number of board members each
city should have.

Unit 8.1: Introduction to Apportionment

 Example 2: Suppose the newly bought 30
police patrol cars of the Philippine
National Police—National Capital Region
are selected according to the 2015 census
of most populated cities in Metro Manila,
as shown in the table below. Use the four
methods of apportionment (Hamilton's,
Jefferson's, Adams' and Websters) to
determine the number of board members each
city should have.

Unit 8.1: Introduction to Apportionment

 Example 2: Suppose the newly bought 30 police patrol cars of the Philippine National
Police—National Capital Region are selected according to the 2015 census of most
populated cities in Metro Manila, as shown in the table below. Use the four methods of
apportionment (Hamilton's, Jefferson's, Adams' and Websters) to determine the number of
board members each city should have.
Compare the results of the using the four appropriation methods.
ANSWER

Unit 8.1: Introduction to Apportionment

 Apportionment Methods
E. Paradoxes in Apportionment

An apportionment exists when the rules for apportionment produce results which
violate quota rules. The different paradoxes are Alabama paradox, population paradox,
and new state, paradox.

The Alabama paradox occurs when an increase in the available number of seats/items
decrease in the number of seats/items for a given group. An apportionment method that
avoids the Alabama paradox is the house monotone.

The population paradox occurs when state's/group's population increases but its
allocated number of seats/items decreases. The methods of apportionment that avoids
the population paradox is said to be population monotone.

The new states paradox (or Oklahoma paradox) occurs when the additional of a new
state/group, with corresponding. increase in the number of available seats/items can
cause change in the apportionment of items among the other states/groups.

Unit 8.1: Introduction to Apportionment

 Apportionment Methods
F. Huntington-Hill Method
Huntington-Hill method was proposed by Edward Huntington (mathematician)
and Joseph Hill (Chief Statistician at the Census Bureau). This
appropriation method guarantees that additional transfer of a seat from one
state/group to another will reduce the ratio between degrees of
representation in any two states/groups. The method was derived from
apportionment principle and was used in the US Congress since 1941.
The Huntington-Hill method is quite similar to Websters method, but
attempts to minimize the percent differences on how many people each
representative will represent.

Unit 8.1: Introduction to Apportionment

 Steps in solving apportionment using Huntington-Hill's method:
1. Solve for the standard divisor.
2. Determine the standard quota for each group by dividing its population
by the standard divisor.
3. Round down each group's standard quota to the nearest whole number.
4. Compute for the geometric mean = of each group and compare it with the
quota in each group. (i) Round up the quota, if the lower quota is greater
than the geometric mean; and (ii) round down the quota, if the lower quota is
less than the geometric mean.
5. If the total allocation from Step 4 is not equal to the required number
of representatives recalculate the quota and allocation by choosing another
modified divisor m and try it again.

Unit 8.1: Introduction to Apportionment

 Table 8.1 shows the first 15 geometric
mean equivalent of n quota.

Unit 8.1: Introduction to Apportionment

 Example 3: Suppose the newly bought 30 police patrol cars of the Philippine National
Police — National capital Region are selected according to the 2015 census of most
populated cities in Metro Manila, as shown in the table below. Use the Huntington-
Hill's method of apportionment determine the number of police patrol cars each city
should have.

Unit 8.1: Introduction to Apportionment

 Example 3: Suppose the newly
bought 30 police patrol cars
of the Philippine National
Police — National capital
Region are selected
according to the 2015 census
of most populated cities in
Metro Manila, as shown in
the table below. Use the
Huntington-Hill's method of
apportionment determine the
number of police patrol cars
each city should have.

Unit 8.1: Introduction to Apportionment

 Example 3: Suppose the newly
bought 30 police patrol cars
of the Philippine National
Police — National capital
Region are selected
according to the 2015 census
of most populated cities in
Metro Manila, as shown in
the table below. Use the
Huntington-Hill's method of
apportionment determine the
number of police patrol cars
each city should have.

Unit 8.1: Introduction to Apportionment

 Example 3: Suppose the newly
bought 30 police patrol cars
of the Philippine National
Police — National capital
Region are selected
according to the 2015 census
of most populated cities in
Metro Manila, as shown in
the table below. Use the
Huntington-Hill's method of
apportionment determine the
number of police patrol cars
each city should have.

Unit 8.1: Introduction to Apportionment

 Example 3: Suppose the newly
bought 30 police patrol cars
of the Philippine National
Police — National capital
Region are selected
according to the 2015 census
of most populated cities in
Metro Manila, as shown in
the table below. Use the
Huntington-Hill's method of
apportionment determine the
number of police patrol cars
each city should have.

Unit 8.1: Introduction to Apportionment

 Allocating Additional Seat
When there is a choice of adding one seat/representative to one of several
states/groups, the seat/representative should be allocated to the state/group with the
highest Huntington-Hill number. The value of Huntington-Hill Number (HHN), where p1 is
the population of state/group i and a is the number of representatives from a
state/group.

Unit 8.1: Introduction to Apportionment

 Example 4: The table below shows the number of public ambulances that are assigned
to the four districts and the average number of emergencies a month in Laguna
Province. Use the Huntington-Hill apportionment principle to determine to which
district a new ambulance should be assigned.

Unit 8.1: Introduction to Apportionment

 Example 4: The table below shows the
number of public ambulances that are
assigned to the four districts and the
average number of emergencies a month in
Laguna Province. Use the Huntington-Hill
apportionment principle to determine to
which district a new ambulance should be
assigned.

Unit 8.1: Introduction to Apportionment

 Example 5: Suppose the newly bought 30 police patrol cars of the
Philippine National Police – NCR are apportioned as showed in the table.
Use Huntington-Hill Apportionment Principle to determine to which city the
additional police patrol car should be assigned.

Unit 8.1: Introduction to Apportionment

 Example 5: Suppose the newly bought 30
police patrol cars of the Philippine
National Police – NCR are apportioned as
showed in the table. Use Huntington-Hill
Apportionment Principle to determine to
which city the additional police patrol
car should be assigned.

Unit 8.1: Introduction to Apportionment

Unit 8.1: Introduction to Apportionment
 END of
Unit 8.1:
Introduction to
Apportionment

Unit 8.1: Introduction to Apportionment

INTRODUCTION
TO VOTING
THEORYAbarca, Jan Kathryn
Andal, Angelika
Marie
Belen, Reynaldo Jr.
Cabrera, April Anne
Delco, Kuh-kai
Introduction to Voting Theory
Voting is being applied in selecting our
leaders in the different aspects of society. The
paradox is that the more opportunities a voter has
to vote, the less he/she seems to appreciate and
understand the meaning of votes. Leaders are
usually chosen for whoever gets the most number
of votes. The Philippine constitution for example
applies such principle in selecting the president,
vice president, senators, representatives and other
elected government officials.
Introduction to Voting Theory
The notion of majority rule in election still
holds in some cases but it is mostly effective in two
competing candidates. We will discuss different
voting methods that we can apply to determine a
winner in an election and be limited on the basic
voting methods.
What is Preference Ballot?
A preference ballot is a ballot in which the
voters list their candidates in order of preference
from 1st to last. A preference ballot always gives a
complete ranking of the candidates from 1st to last
place, unlike in a typical Philippine election ballot.
A preference schedule is a table which
summarizes the result of all the individual
preference ballots for an election result.
 1
Plurality Voting
Method
Plurality Voting Method
The candidate with the most first place votes
wins. The winning candidate does not have to have
a majority of the votes. The weakness of plurality
method is it fails to take into account a voter’s other
preferences beyond first choice and can lead to
some bad election results
Example
 Advantages

Easy system of Doesn't require People Requires fewer

voting complicated understand and operational and
repeat voting navigate the monetary
procedures to system with resources to
declare a ease. hold and
winner. execute than
other systems
of voting.
 Disadvantages

winning reflecting poorly The Absolute Majority and

candidate might on the choice of Proportional Representation
secure his the people. alternatives are employed as
victory by a devices to overcome the
very small disadvantages of Plurality
margin of votes Voting Systems
Borda Count
Voting
Method 2
Borda Count Voting Method
If there are n candidates in an election, each
voter ranks the candidates, the last choice
candidates gets 1 point, the second-to-the last
candidate gets 2 points,(n-1) points for the second
place and n points for first place. The Borda Count
method violates two basic fairness criteria: the
majority criterion and the Condorcet criterion.
Borda Count Voting Method
Despites its Borda Count is
flaws,
considered one of the best voting method for
deciding elections with many candidates . The
candidates with the highest total point is the
winner.
This methos was introduced by Jean-Charles
de Borda (1733-1799).
Example
 3
Plurality with Elimination
Voting or Instant Runoff
Voting
Plurality with Elimination Voting or
Instant Runoff Voting
The plurality-with-elimination method is
earned out in round and it will take n-1 rounds
when there are n candidates. The method counts
the first-place votes of each candidate . The
plurality –with-eliminaton method is used to
address insincere voting (or tactical voting or
strategic voting) when a voter supports a candidate
other than his/her sincere preference for a specific
purpose or to prevent an undesirable outcome.
Plurality with Elimination Voting or
Instant Runoff Voting
Insincere voting happens when more than
one candidate shares somewhat similar point of
view. In a closely contested election, a few
insincere elections can completely change the
outcome of an election. The following are the steps
to generate the winner in a plurality-with-
elimination method:
Plurality with Elimination Voting or
Instant Runoff Voting
Insincere voting happens when more than
one candidate shares somewhat similar point of
view. In a closely contested election, a few
insincere elections can completely change the
outcome of an election.
 The following are the steps to generate the
winner in a plurality-with-elimination method:

1. 1.If a candidate has a majority of first-place

votes, the candidate is the winner;
2. 2. If no candidate has a majority, eliminate the
candidate with the fewest first-place votes
3. 3. Set up new round of voting with the
remaining candidates. Repeat step 1 and step
2
4. 4. When only two candidates remain in a round,
the candidate with the most votes wins the
Example
 4
Pairwise
Comparison
Voting Method
Pairwise Comparison Voting Method

Each pair of candidates is matched head-to-

head (one-on-one); that is, each candidate gets 1
point for a head to head win and half a point for a
tie. We need to go through all possible pairs of
candidates with the most total points wins the
election.
Example
Fairness Criteria
Fairness is often subjective
through the following criteria must
be met for a fair voting system.
Majority Criterion
If there is a candidate that
has majority of the first place
votes, then that candidate is
the winner of the election.
 Monotonicity
If candidate ACriterion
is a winner
of an election and, in a re-
election, all changes in the
ballots are changes
favorable only to A. then
candidate A is the winner
of the election
Condorcet Criterion
If there is a candidate that
is preferred by the voters
over each the other
candidates (all possible
head-to-head matchups)
then the candidate is the
winner of the election
(Condorcet candidate).
Independence of
Irrelevant Alternatives
Criterion
If candidate A is a winner of an
election, and one (or more) of
the other choices is
disqualified/withdraw and the
ballots recounted. Then
candidate A is still the winner of
the election.
THANK
YOU
Weighted Voting
System
UNIT 8.3
 When it comes to voting rights, the democratic ideal of equality translates into
the principle one person – one vote but not in the case of voting in a corporate
world in other institutions that voting rights are equated to the value of one
individual.

Example: In a corporate voting, where shareholders vote counts are proportional

to the amount of shares they invest in the company.
Each individual with one share gets one vote , while individual with 10 shares
gets the equivalent of 10 votes.

Weighted Voting System – Each vote has some weight attached to it.
Mathematically, a weighted voting system of n votes are written {q: w1, w2,...,wn}
, where q is the quota and w1,w2,....,wn is the weight of each of n voters.
 Key Terms:

Quota – The minimum weight needed for the votes or weight needed for the
resolution to be approved.

Coalition – A group of players/voters voting the same way, either for or against
motion.
The number of possible coalition of n votes is 2n – 1 .
Winning Coalition – A coalition if a coalition has enough weight to meet quota.

Losing Coalition – A coalition with vote less than the quota.

Grand Coalition – The coalition consisting of all the players.

Critical Voter (Critical Player)– A member of a winning coalition and thereby if
that
member’s vote makes the difference between winning and losing.
Dummy – A player whose vote is not critical in any winning coalition.

Dictatorship – A system when a player weight is equal or greater than the quota,
such a player
is called a dictator.
Null Voting System – Happens if all members’ vote is less than a quota.

Veto Power– A system when a players support is necessary for the quota to met.
 Banzhaf Power Index (BPI)

Was originally created by Lionel Penrose in 1946 and was reintroduced by John
Banzhaf in 1965.
The Banzhaf power index of a voter v is given by

𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒕𝒊𝒎𝒆𝒔 𝒗𝒐𝒕𝒆𝒓 𝒗 𝒊𝒔 𝒂 𝒄𝒓𝒊𝒕𝒊𝒄𝒂𝒍 𝒗𝒐𝒕𝒆𝒓

𝑩𝑷𝑰(𝒗) =
𝑵𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒕𝒊𝒎𝒆𝒔 𝒂𝒏𝒚 𝒗𝒐𝒕𝒆𝒓 𝒗 𝒊𝒔 𝒂 𝒄𝒓𝒊𝒕𝒊𝒄𝒂𝒍 𝒗𝒐𝒕𝒆𝒓

Steps in Computing the Banzhaf Power Index

1. List all winning coalitions.
2. In each coalition, Identify the critical voters.
3. Count the number of times each voter is critical.
4. Compute for the Banzhaf Power Index.
Example 1:

Suppose the senators are voting Collectively based from their political affiliation.
Currently, there are 24 senators who are from four different political affiliations
namely: Nationalista Party (NP), Nationalists People's Coalition (NPC), Liberal
Party' (LP), and United Nationalist Alliance (UNA) with the following number of
senators 11, 7, 5, and 1, respectively. To be able to pass a senate resolution it
needs at least 13 votes.
Determine the following
(a) voting system in the senate,
(b) total possible coalition,
(c) list of winning coalition,
(d) the critical voters for each winning coalition, and (e) the Banzhaf power
index for each political affiliation.
Solution:
Let NP = 11 votes; NPC = 7 votes; LP = 5 votes; UNA = 1 vote; n = 4 political
affiliation

(a) voting system in the senate

[13: 11, 7, 5, 1]

(b) total possible coalition

2n – 1 = 24 – 1 = 16 -1 = 15
(c) list of winning
coalition

Winning Coaltion Number of Votes

{NP, NPC} 18
{NP, LP} 16
{NP, NPC, UNA} 23
{NP, LP, UNA} 19
{NP, LP, UNA} 17
{NPC, LP, UNA} 13
{NP, NPC, LP, UNA} 24
(d) the critical voters for each winning
coalition
Winning Coaltion Number of Votes Critical Voters No. of Critical Votes
{NP, NPC} 18 NP, NPC 2
{NP, LP} 16 NP, LP 2
{NP, NPC, UNA} 23 NP 1
{NP, LP, UNA} 19 NP, NPC 2
{NP, LP, UNA} 17 NP, LP 2
{NPC, LP, UNA} 13 NPC, LP, UNA 3
{NP, NPC, LP, UNA} 24 None 0
(e) The Banzhaf power index for each political affiliation.

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑣𝑜𝑡𝑒𝑟 𝑣 𝑖𝑠 𝑎 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑜𝑡𝑒𝑟

𝐵𝑃𝐼(𝑣) =
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑎𝑛𝑦 𝑣𝑜𝑡𝑒𝑟 𝑣 𝑖𝑠 𝑎 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑜𝑡𝑒𝑟

Number of times any voter is critical = 2 + 2 + 2 + 1 + 1+ 3+ 0 + 1 = 12

5
𝐵𝑃𝐼(𝑁𝑃) = = 0.4167
12

4
𝐵𝑃𝐼(𝑁𝑃𝐶) = = 0.3333
12

2
𝐵𝑃𝐼(𝐿𝑃) = = 0.1667
12

1
𝐵𝑃𝐼(𝑈𝑁𝐴) = = 0.0833
12
Example 2:

Suppose the four owners of SJAURR Inc., Winston (W), Redentor (R), Marisol and
Soledad (S), own, 700 shares, 550 shares, 315 shares, and 435 shares,
respectively. There is a total of 2,000 votes; half of this is 1,000 votes, so the quota
is 1,001 votes. The weighted voting for the company is [1,001: 700, 550, 315, 425].

(a) list the winning coalition,

(b) determine the critical voters for each winning coalition, and
(c) find the Banzhaf power index for each voter.
Solution:
Let W = 700 votes R = 550 votes M = 315 votes S = 435 votes
The voting system is (1,001: 700, 550, 315,
425)
(a) List of winning coalition. The total possible coalition is 2n - 1 = 24 - 1
Winning Coaltion Number of Votes
= 16 - 1 = 15.
{W, R} 1,250
{W, M} 1,015
{W, S} 1,135
{W, R, M} 1,565
{W, M, S} 1,450
{W, R, S} 1,685
{R, M, S} 1,300
{W, R, M, S} 2,000
(b) determine the critical voters for each winning
coalition
Winning Coaltion Number of Votes Critical Values Number of critical
values
{W, R} 1,250 W, R 2
{W, M} 1,015 W, M 2
{W, S} 1,135 W, S 2
{W, R, M} 1,565 W 1
{W, M, S} 1,450 W 1
{W, R, S} 1,685 R, M, S 3
{R, M, S} 1,300 None 0
{W, R, M, S} 2,000 W 1
(c) Find the Banzhaf power index for each political affiliation.

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑣𝑜𝑡𝑒𝑟 𝑣 𝑖𝑠 𝑎 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑜𝑡𝑒𝑟

𝐵𝑃𝐼(𝑣) =
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑎𝑛𝑦 𝑣𝑜𝑡𝑒𝑟 𝑣 𝑖𝑠 𝑎 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑜𝑡𝑒𝑟

Number of times any voter is critical = 2 + 2 + 2 + 1 + 1+ 3+ 0 + 1 =

12
6
𝐵𝑃𝐼(𝑊) = 12
= 0.5000

2
𝐵𝑃𝐼(𝑅) = 12
= 0.1667

2
𝐵𝑃𝐼(𝑀) = 12
= 0.1667

2
𝐵𝑃𝐼(𝑆) = 12
= 0.1667
Example 3:

Four voters on a council, A, B, C, and D make decision using the voting system {5:
3, 2, 2,1}, except when there is a tie. In the event of a tie, the chairman of a
council serves as the fifth voter, E, and casts a vote to break the tie. For this
voting scheme, find the Banzhaf power Index for each voter, including voter E.

Solution:

Let A = 3 votes B = 2 votes C =2 votes D =1 vote

The voting system is. (5: 3, 2, 2, 1).

The total possible coalition for the four voters on a council is 2n - 1 = 24 - 1 = 16 -
1 = 15.
Step 1: List all possible winning coalitions and determine the
critical voters.

Winning Coaltion Number of Votes Critical Voters

{A, B} 5 A, B < Without
chairman
{A,C} 5 A, C < Without
chairman
{A, B, C} 7 A < Without
chairman
{A, C, D} 6 A, C < Without
chairman
{B, C, D} 5 None < Without
chairman
{A, B, C, D} 8 None < Without
chairman
{A, D, E} 5 A, D, E < With chairman

{B, C, E} 5 B, C, E < With chairman

Step 2: Determine the number of times any voter is a critical voter. The number
of times any voter is critical is 2 + 2 + 1 + 2 + + 0 + 3 + 3 = 13.

Winning Coaltion Number of Votes Critical Voters No. of Critical Voters

{A, B} 5 A, B 2
{A,C} 5 A, C 2
{A, B, C} 7 A 1
{A, C, D} 6 A, C 2
{B, C, D} 5 None 0
{A, B, C, D} 8 None 0
{A, D, E} 5 A, D, E 3
{B, C, E} 5 B, C, E 3
Step 3: Compute for the Banzhaf power index of each voter.

5
𝐵𝑃𝐼(𝐴) = 13
= 0.3846

2
𝐵𝑃𝐼(𝐵) = 13
= 0.1538

3
𝐵𝑃𝐼(𝐶) = 13
= 0.2308

1
𝐵𝑃𝐼(𝐷) = 13
= 0.0769

2
𝐵𝑃𝐼(𝐸) = = 0.1538
13
THANK YOU!