Apportionment and Voting in statistics ppt

APPORTIONMENT AND VOTING

1. Session 2 Apportionment and Voting

2. The Plurality Method of Voting Each voter votes for one candidate, and the candidate with the most votes wins. The winning candidates does not have to have the majority of the votes.

3. Example: A newly opened restaurant ask their first 50 customer to rank the food they serve according to their preferences.   Ranking 1 st place votes Adobo 2 1 4 3   7 Pancit Guisado 3 3 2 4   Grilled Tuna 1 2 3 2   10 Chicken BBQ 4 4 1 1   18+15=33 No of voters 10 7 18 15     thus, the most preferred food is Chicken BBQ and 66% of the voting people preferred Chicken BBQ.

4. Borda Count Method of Voting If there are n candidates or issues in an election, each voter ranks the candidates or issues by giving n points to the voter’s first choice, n-1 points to the voter’s second choice, and so on, with the voter’s least favorite choice receiving 1 point. The candidate or issue that receives the most total points is the winner.

5. Example: The board of directors of a company would have to decide on what investment plan will they have to invest the company extra money.   Ranking Investment A 3 3 1 Investment B 1 2 3 Investment C 2 1 2 No of voters 12 3 10

6. Using Borda Method   Investment A: 3(10) = 30 2(0) = 0 total is 45 1(15) = 15   Investment B: 3(12) = 36 2(3) = 6 total is 52 1(10) = 10   Investment C: 3(3) = 9 2(22) = 44 total is 53 1(0) = 0 Thus, Investment C wins.

7. Plurality with Elimination This method is a variation of the plurality method in which we will consider the voter’s alternative choices.

8. Example: Consider the data below,   Ranking Investment A 3 3 1 Investment B 1 2 3 Investment C 2 1 2 No of voters 12 3 10 Using plurality with elimination method, we will 1 st eliminate investment C since it has the fewest 1 st place vote.   Ranking Investment A 2 2 1 Investment B 1 1 2         No of voters 12 3 10 The board will repeat the process and eliminate the investment with fewest 1 st -place votes. In this case it is investment A. therefore, the investment B wins.

9. Pairwise Comparison Voting Method Also considered as the head-to-head method where each candidate is compared one-on-one with each of the other candidate. A candidate receives 1 point for a win, 0.5 for a tie, and 0 point for a loss.

10. Condorcet Criterion   A candidate who wins all possible head-to-head match ups should win, an election when all candidates appear on the ballot.

11. Example: Use pairwise comparison voting method to determine which investment should the company invest.   Ranking Investment A 3 3 1 Investment B 1 2 3 Investment C 2 1 2 No of voters 12 3 10

12. Compare Investment A to B A is preferred over B by 10 votes B is preferred over A by 12+3=15 votes Compare Investment A to C A is preferred over C by 10 votes C is preferred over A by 12+3=15 votes Compare Investment B to C B is preferred by 12 votes C is preferred by 3+10=13 votes

13. Versus Investment A Investment B Investment C Investment A   B C Investment B     C Investment C              

14. Fairness Criteria MAJORITY CRITERION: The candidate who receives a majority of the first-place votes is the winner.   MONOTONICITY CRITERION: If candidate A wins an election, then candidate A also win the election if the only change in the voters’ preferences is that supporters of a different candidate change their votes to support candidate A.   CONDORCET CRITERION: A candidate who wins all possible head-to-head matchups should win an election when all candidates appear on the ballot.  

15. INDEPENDENCE OF IRRELEVANT ALTERNATIVES: If a candidate wins an election, the winner should remain the winner in any recount in which losing candidates withdraw from the race.   Arrow Impossibility Theorem There is voting method involving three or more choices that satisfies all four fairness criteria.

16. Quota The weight of the voters vote Coalition Winning coalition Critical voter Weighted Voting System

17. Weighted Voting System A weighted voting system of n voters is written where q is the quota and w1 through represent the weights of each of the n voters. The weight of a voter is the number of votes controlled by the voter. Also, quota is the required number of votes to pass.   .  

18. This notation can describe various voting systems One person, one vote: each person has one vote and a majority is required to pass a measure. Dictatorship: when even the other voters are present, the sum of their votes does not meet the quota. Null system: when even all the members vote but still the sum does not meet the quota. Veto power system: when each vote has a veto power, that is every one should vote for a measure to pass

19.   Number of Possible Coalitions of n Voters :   A coalition is a set of votes each of whom votes the same way. A winning coalition is a set of voters the sum of whose votes is greater than the quota and a losing coalition is a set of voters whose votes is less than the quota. The critical voter is the voter who leaves a winning coalition turns into a losing coalition.  

20. Example: Determine the winning coalitions and the critical voters of each winning coalition of the data that follows: {521: 408, 230, 102, 300}

21. Solution: The winning coalition must have at least 521 votes. Winning Coalition Number of Votes Critical Voters {A,B} 638 A,B {A,D} 708 A,D {A,B,C} 740 A,B {A,B,C,D} 1040 NONE {B,C,D} 632 B,D {B,D} 530 B,D {A,C,D} 810 A,D

22. Banzhaf Power Index   Computed using  

23. Example: Consider {2:1,1,1} of A, B, C in a one person, one vote system Winning Coalition Number of Votes Critical Voters {A, B} 2 A, B {A, C} 2 A, C {B, C} 2 B, C {A, B, C} 3 None In this case each voter has equal power.