MODULE 9: Mathematics in the Social Sciences

Question 1

Voting Theory

Answer 1

process of producing a single choice from varied and conflicting choices that reflects the desire of each individual

Question 2

Order Theory

Answer 2

area of mathematics that studies ways in which objects can be ordered

Question 3

Voting System

Answer 3

a way of a group to select one winner from among several candidates

Question 4

Ranked Voting System

Answer 4

when the voting system asks for ranking

Question 5

Transitivity

Answer 5

when x is preferred than y, and y is preferred than z, then x is preferred than z

Question 6

Voting Method

Answer 6

a process wherein mathematics is used to count and consolidate votes to produce one winner

Question 7

The Four Voting Methods

Answer 7

Plurality
Borda Count
Pairwise Comparison
Plurality with Elimination

Question 8

The Four Voting Methods and their Problems

Answer 8

Plurality –> Condorcet Criterion
Borda Count –> Majority Criterion
Pairwise Comparison –> Independence of Irrelevant Alternatives
Plurality with Elimination –> Monotonicity Criterion

Question 9

Fairness Principles

Answer 9

Condorcet Criterion –> 1 to 1 comparison
Majority Criterion –> 50% + 1
Monotonicity Criterion –> If votes are changed to the previous winner, then the previous winner should still be the winner
Independence of Irrelevant Alternatives Criterion –> If a candidate is removed, the winner should still be the winner
Unanimity –> the winner is the candidate everyone prefers

Question 10

Arrow’s Impossibility Theorem

Answer 10

Theorem by Kenneth Arrow states that is impossible to have a voting system that would satisfy all fairness principles

Question 11

Approval Voting

Answer 11

An unranked voting system wherein voters can approve any number of candidates.

Question 12

Weighted Voting Systems

Answer 12

voting systems wherein voting rights are not equally divided such as shareholder voting, bloc voting, and committee voting

Question 13

Motion

Answer 13

any vote only involving two alternatives and no abstentions

Question 14

Three Factors of Motion

Answer 14

Quota –> min. no. of votes needed to win
Players –> voters
Weight –> number of votes for a player

Question 15

Invalid Voting System

Answer 15

total number of votes is less than the quota

Question 16

Illegal Voting System

Answer 16

the quota is less than half the total number of votes

Question 17

One person-one vote

Answer 17

equal number of votes/weight

Question 18

Dictator

Answer 18

A player whose weight is bigger than the quota

Question 19

Dummy

Answer 19

A non-critical player

Question 20

Veto Power

Answer 20

the power of a player who can single-handedly prevent any player to pass a motion (very close to the quota)

Question 21

Unanimous Vote

Answer 21

All players are needed to reach the quota

Question 22

Banzhaf Index

Answer 22

A mathematical measurement of power by John Banzhaf wherein the player who can influence the outcome of the election has the most power

Question 23

Coalition

Answer 23

players who join forces

Question 24

Weight of the Coalition

Answer 24

total number of votes of all the players in the coalition

Question 25

Winning Coalitions

Answer 25

Coalitions with enough votes to win

Question 26

Losing Coalitions

Answer 26

Coalitions that do not have enough votes to win

Question 27

Grand Coalition

Answer 27

All players joining forces

Question 28

Critical Player/Pivotal Player

Answer 28

The player whose desertion can turn a winning coalition into a losing one

Question 29

Banzhaf Power Distribution

Answer 29

the power each player holds

Question 30

Steps in Computing the Banzhaf Power Index

Answer 30

1. list all coalitions (2^N -1)
2. winning coalitions
3. critical players
4. count the number of times each player is critical
5. total number of times every critical player is critical
   step 4/step 5

Question 31

Unanimous Vote Mathematical Results

Answer 31

1/N

Question 32

Proportional Systems Mathematical Results

Answer 32

same Banzhaf Power Distribution

Question 33

Three-Player Voting System with no veto power

Answer 33

1/3, 1/3, 1/3

Question 34

Tolle’s Theorem

Answer 34

Possible Power Distribution of any Four-Player Voting System:
1. (1/4, 1/4, 1/4, 1/4) –> unanimous
2. (5/12, 1/4, 1/4, 1/12)
3. (1/2, 1/6, 1/6, 1/6)
4. (1/3, 1/3, 1/6, 1/6)
5. (1/3, 1/3, 1/3, 0) –> dummy