SOCIAL CHOICES (Voting Methods) THE PROBLEM In a society, decisions are made by its members in order to come up with a situation that benefits the most. What is the best voting method of arriving at a decision as a group? By: Dr. Ederlina G.Nocon Ms. Michele G.Tan What makes a decision process fair? 1 2 Social Choice and Voting Terminologies Social Choice theory looks at processes by which different and conflicting choices of members of a group consolidated into a single choice of the group. Voting is the main vehicle by which decisions are arrived at in a democratic society. A preference ballot is a ballot on which each voter ranks all eligible candidates, from first to last place, with no tied ranks. 3 4
Activity Terminologies Consider the presidential candidates for May 2016 election: Binay, Duterte, Poe, Roxas and Santiago A preference table is a table showing how many times each possible ballot was submitted. Rank 1st 2nd 3rd 4th 5th Candidate 5 6 Terminologies Fairness Criteria A fairness criterion is a mathematical statement about our expectations for a voting system. The Majority Criterion If a candidate receives a majority of first-place votes, that candidate should win the election. 7 8
Fairness Criteria Fairness Criteria The Condorcet Criterion If a candidate beats any other candidate in a head-to-head contest, that candidate should win the election. Such a candidate, if one exists, is called a Condorcet candidate or Condorcet winner. Not every election necessarily has one. The Independence of Irrelevant Alternatives Criterion If a re-election is held with the same ballots and non-winning candidates are removed, the previous winner should still win. 9 10 Fairness Criteria Fairness Criteria The Pareto Criterion If there is at least one candidate, say Candidate A, that every voter prefers to another candidate, say Candidate B, then it should be impossible for Candidate B to win the election. The Pareto condition is named after Vilfredo Pareto (1848 1923), an Italian economist. The Monotonicity Criterion It should be impossible for a winning candidate to lose in a re-election if the only changes in the votes were changes that were favorable to that candidate. 11 12
Fairness Criteria Fairness Criteria All voters should be treated equally. No voter has special influence, only the ballot counts. If voters exchange ballots, the result of the election should still be the same. All candidates should be treated equally. No candidate has more privilege than any other. In the case of two candidates, this means if every voter reversed their vote, the election result would be reversed as well. 13 Unanimity If every individual prefers a certain option to another, then so must the resulting societal choice. Non-dictatorship The social choice function should not simply follow the preference order of a single individual while ignoring all others. 14 Terminologies Some Methods of Voting A voting method is a mathematical procedure that uses data from the preference table to determine a winner. Majority Rules In this method, the candidate which receives the majority (more than 50%) of first-place votes wins the election. 15 16
Majority Rules Majority Rules Illustration A is the winner, having obtained majority first-place votes (10 of 17). Majority rules is a good method of voting but only guarantees a winner if there are two candidates and an odd number of voters. If there are more than two candidates, it is possible that none of the candidates receives a majority and thus no winner could be determined by majority rules. 17 18 Majority Rules Method versus criterion Illustration: The Majority Criterion: If a candidate receives a majority of firstplace votes, that candidate should win the election. Majority Rules Method: The candidate which receives the majority of first-place votes wins the election. A has the most number of first-place but cannot be the winner under majority rules. 6 out of 15 is not a majority. 19 We note here that Majority Rules Method guarantees that the process satisfies the Majority Criterion. 20
May's Theorem Some Methods of Voting If the number of voters is odd, there are only two candidates, and we want a voting method that never results in a tie, then majority rules is the only voting method that satisfies the following three criteria: 1) All voters are treated equally. Condorcet Method A candidate is a winner when he would, on the basis of the ballots cast, defeat every other candidate in a one-on-one contest using majority rule. In fact, if there are only two candidates, the Condorcet Method is exactly the same as Majority Rules. 2) Both candidates are treated equally. 3) Monotonicity This was popularized in the 18th century by Marquis de Condorcet. 21 22 Condorcet Method Condorcet Method B versus A B is over A on 5+3=8 of the ballots, while A is over B on 6+1=7 of the ballots. Thus, B defeats A by a score of 8 to 7. B versus C B is over C on 6+5+3=14 of the ballots, while C is over B on 1 of the ballots. Thus, B defeats C by a score of 14 to 1. 23 24
Condorcet Method Condorcet's Voting Paradox B versus D B is over D on 6+5+1=12 of the ballots, while D s over B on 1 of the ballots. Thus, B defeats D by a score of 12 to 1. Since B defeated all other candidates, B is the condorcet winner. The Condorcet's method may not produce a winner when there are more than two candidates. 25 26 Some Methods of Voting Plurality Method Plurality Method Illustration: In this method, whoever receives the most first-place votes is declared the winner. This is by far the most simple and widelyused voting method. It may require a tie-breaker though. 20% 20% 10% 10% 40% Winner 27 1 2 3 4 5 A has 6 first-place votes, B has 5 first-place votes, C has 1 first-place votes, and D has 3 first-place votes. A wins using plurality. 28
Plurality versus majority Plurality Method Voting systems which make use of the plurality method include barangay, local, and national elections in the Philippines. However, voters simply choose candidates without ranking them. 29 30 Plurality versus majority Condorcet Criterion Illustration: Since there are 37 votes cast (14+4+11+8), the majority is 19 votes (37/2=18.5). A received a plurality but not a majority. Three locations for the new season of a TV program are being considered: Baler (B), Camiguin (C), and Vigan (V). Studio executives and the show s production team vote using the Plurality Method. The preference table for the election is shown below. 31 32
Condorcet Criterion Plurality fails the Condorcet Criterion Since B wins versus C by a score of 16 to13 (12+4 to 9+4) And B wins versus V by a score of 16 to 13 also Now, observe that B has 12 first-place votes, V has 13 first-place votes, and C has 4 first-place votes. Therefore, V wins using plurality. ( 12+4 to 9+4 ), B is the Condorcet winner. 33 34 Plurality fails the Condorcet Criterion Fairness Criteria The Plurality Method also fails the Condorcet Criterion. In some elections held using the Plurality Method, there is a candidate who would beat every other candidate head-to-head, but does not win the election. The Independence of Irrelevant Alternatives Criterion If a re-election is held with the same ballots and non-winning candidates are removed, the previous winner should still win. 35 36
Plurality fails the IIA Criterion Plurality fails the IIA Criterion Since A has 7 firstplace votes, B has 4 first-place votes, and C has 4 first-place votes, A wins by plurality. Now suppose C dropped out. B would now win by plurality. 37 38 Plurality fails the IIA Criterion Plurality is vulnerable to Strategic Voting The Plurality Method also fails the Independence of Irrelevant Alternatives Criterion. A voter voting strategically reasons as follows: If my preferred candidate has no chance to win, then I will vote instead for someone I like less, but who has a chance to win. Whenever voters vote differently from their true preferences, the voting method may be flawed. 39 40
Plurality is vulnerable to Strategic Voting Borda Count Illustration # Voters 3 3 1 A B C B A A C C B A voter prefers candidate C, but since candidate C has no chance of winning, instead the voter would chose the candidate he like less. 41 The Borda Count Method is named after Jean-Charles de Borda (1733-1799). Borda Cound Method assigns points in a non-increasing manner to the ranked candidates on each voter s preference list. Given n candidates, below is the point system Rank Points 1st n 1 2nd n - 2 nth 0 42 Borda Count Borda Count Borda Score is the total points received for each candidate from all voters are added. The winner is the candidate with the highest Borda score. No. of Votes Rank 6 5 3 1 No. of Votes Rank 6 5 3 1 1 st : 3 points A: 18 B: 15 D: 9 C: 3 2 nd :2 points B: 12 D: 10 B: 6 A: 2 3 rd : 1 point C: 6 A: 5 A: 3 B: 1 4 th : 0 point D: 0 C: 0 C: 0 D: 0 43 1 st : 3 points A: 18 B: 15 D: 9 C: 3 2 nd :2 points B: 12 D: 10 B: 6 A: 2 3 rd : 1 point C: 6 A: 5 A: 3 B: 1 4 th : 0 point D: 0 C: 0 C: 0 D: 0 A: 18 + 5 + 3 + 2 = 28 B: 12 + 15 + 6 + 1 = 34 (winner) C: 6 + 0 + 0 + 3 = 9 D: 0 + 10 + 9 + 0 = 19 44
Some Methods of Voting Hare System The Hare System In this method, a winner is determined by repeatedly deleting candidates, in stages, that are least preferred. A winner may emerge after all other candidates have been deleted or when there is a tie among two or more candidates. The Hare System A 14, B 4, C 11, D - 8 The Hare system was introduced by Thomas Hare in 1861 and is used in, among others, choosing the site of the Olympics, the Academy Awards, and elections in Australia and Ireland. 45 46 Hare System Hare System Eliminate B A 14, C 11, D 12 47 48
Hare System Hare System Eliminate C A 14, D 23 D Wins 49 50 SUMMARY EXPLORING Majority Condorcet Criterion Criterion IIA Criterion Pareto Monotonicity Criterion Criterion Plurality Yes No No Yes Yes Borda Count No No No Yes Yes Sequential Pairwise Yes Yes No No Yes Hare System Yes No No Yes No Make a survey on a topic that will ask at least 15 persons to rank 4 choices and apply 4 voting methods to determine the winner from this survey. For example, let your classmates rank their preference on the following choices of fast food stores: Jollibee, McDonalds, KFC, Burger King Approval Voting No No Yes No Yes 51 52
Sample result from students Sample result from students MCDO VS. JOLLIBEE VS. KFC VS. BON CHON SURVEY RESULTS GRACE VS. MAR VS. BINAY VS. DUTERTE SURVEY RESULTS No. of Votes 1st 2nd 3rd 4th 1 K B M J 1 K B J M 1 K M J B 3 K J M B 1 M K B J 4 M K J B 3 M J K B 1 M B K J 1 J K M B 1 J K B M 2 J M B K 1 J M K B 1 J B K M 3 B J K M 1 B J M K 1 B M K J Declared Winner Majority NO Winner Plurality McDo Condorcet NO Winner No. of Votes 1 st 2nd 3rd 4 th 1 G M B D 6 G M D B 2 G D M B 3 M D G B 2 M D B G 1 M G B D 1 M B G D 1 B M D G 1 B M G D 1 B D G M 5 D G M B 1 D G B M 1 D M G B Declared Winner Majority NO Winner Plurality Grace Condorcet NO Winner 53 54 BLOG YOUR THOUGHTS Voting systems are If I were to design a voting system in the Philippines for the choice of the President then I would If I were to design a voting system in the Philippines for the choice of senators then I would A fair voting system for me is the one that 55