MODULE 9: Mathematics in the Social Sciences Flashcards
Voting Theory
process of producing a single choice from varied and conflicting choices that reflects the desire of each individual
Order Theory
area of mathematics that studies ways in which objects can be ordered
Voting System
a way of a group to select one winner from among several candidates
Ranked Voting System
when the voting system asks for ranking
Transitivity
when x is preferred than y, and y is preferred than z, then x is preferred than z
Voting Method
a process wherein mathematics is used to count and consolidate votes to produce one winner
The Four Voting Methods
Plurality
Borda Count
Pairwise Comparison
Plurality with Elimination
The Four Voting Methods and their Problems
Plurality –> Condorcet Criterion
Borda Count –> Majority Criterion
Pairwise Comparison –> Independence of Irrelevant Alternatives
Plurality with Elimination –> Monotonicity Criterion
Fairness Principles
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
Arrow’s Impossibility Theorem
Theorem by Kenneth Arrow states that is impossible to have a voting system that would satisfy all fairness principles
Approval Voting
An unranked voting system wherein voters can approve any number of candidates.
Weighted Voting Systems
voting systems wherein voting rights are not equally divided such as shareholder voting, bloc voting, and committee voting
Motion
any vote only involving two alternatives and no abstentions
Three Factors of Motion
Quota –> min. no. of votes needed to win
Players –> voters
Weight –> number of votes for a player
Invalid Voting System
total number of votes is less than the quota
Illegal Voting System
the quota is less than half the total number of votes
One person-one vote
equal number of votes/weight
Dictator
A player whose weight is bigger than the quota
Dummy
A non-critical player
Veto Power
the power of a player who can single-handedly prevent any player to pass a motion (very close to the quota)
Unanimous Vote
All players are needed to reach the quota
Banzhaf Index
A mathematical measurement of power by John Banzhaf wherein the player who can influence the outcome of the election has the most power
Coalition
players who join forces
Weight of the Coalition
total number of votes of all the players in the coalition
Winning Coalitions
Coalitions with enough votes to win
Losing Coalitions
Coalitions that do not have enough votes to win
Grand Coalition
All players joining forces
Critical Player/Pivotal Player
The player whose desertion can turn a winning coalition into a losing one
Banzhaf Power Distribution
the power each player holds
Steps in Computing the Banzhaf Power Index
- list all coalitions (2^N -1)
- winning coalitions
- critical players
- count the number of times each player is critical
- total number of times every critical player is critical
step 4/step 5
Unanimous Vote Mathematical Results
1/N
Proportional Systems Mathematical Results
same Banzhaf Power Distribution
Three-Player Voting System with no veto power
1/3, 1/3, 1/3
Tolle’s Theorem
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
