GROUP DECISION MAKING UNDER MULTIPLE CRITERIA SOCIAL CHOICE THEORY Group Decision Making Group Decision Making Process Oriented Content Oriented Approaches Approaches Ö Ö Implicit Explicit Game-Theoretic Multiattribute Multiattribute Approach Evaluation Evaluation Ö Ö X Content-oriented approaches Focuses on the content of the problem, attempting to find an optimal or satisfactory solution given certain social or group constraints, or objectives Implicit Multiattribute Evaluation ­ (Social Choice Theory) Explicit Multiattribute Evaluation Game-Theoretic Approach Social Choice Theory Voting is a group decision making method in a democratic society, an expression of the will of the majority. It is a multiple criteria decision making process whenever a voter casts a vote to select a candidate or alternative policy. The candidate's qualifications may be judged by multiple criteria such as trustworthiness and/or honesty, capabilities, general political stance -- conservative, moderate, or liberal -- and positions on specific issues. These criteria are summarized, in a voter's mind, to be a value function (utility function), but in the counting of votes cast, the multiple criteria do not appear explicitly. A counting method is also called a social choice function. But if we apply the Arrow's impossibility theorem to the choice function then it is called a social welfare function. Voting Nonranked Voting System Preferential Voting System ­ One member elected from two ­ Simple majority decision rule candidates ­ Two-alternative case ­ One member elected from many ­ More than two alternatives case candidates ­ Paradox of voting ­ The first-past-the-post system ­ Majority representation system ­ Election of two or more members ­ The single non-transferable vote ­ Multiple vote ­ Limited vote ­ Cumulative vote ­ List Systems ­ Approval vote Nonranked Voting System One member elected from two candidates Election by simple majority ­ Each voter can vote for one candidate ­ The candidate with the greater vote total wins the election Nonranked Voting System One member elected from many candidates The first-past-the-post system ­ Simple plurality system ­ Election by simple majority Majority representation system ­ Repeated ballots (exhaustive ballot) ­ Voting goes on through a series of ballots until some candidate obtains an absolute majority of the votes cast ­ The second ballot (Plurality with Runoff) ­ On the first ballot a candidate can’t be elected unless he obtains an absolute majority (or a certain percentage, such as 40%) of the votes cast ­ The second ballot is a simple plurality ballot involving the two candidates who had been highest in the first ballot Nonranked Voting System Election of two or more members The single non-transferable vote ­ Each voter has one vote Multiple vote ­ Each voter has as many votes as the number of seats to be filled ­ Voters can’t cast more than one vote for each candidate Limited vote ­ Each voter has a number of votes smaller than the number of seats to be filled ­ Voters can’t cast more than one vote for each candidate Cumulative vote ­ Each voter has as many votes as the number of seats to be filled ­ Voters can cast more than one vote for candidates Nonranked Voting System Election of two or more members Approval voting ­ Each voter can vote for as many candidates as he/she wishes ­ Voters can’t cast more than one vote for each candidate ­ The winner is the candidate with the greater vote total. ­ If there is a strict Condorcet winner - a candidate who satisfies Condorcet's Principle, by defeating all others in pairwise contests - approval voting is shown to be the only non ranked voting system that is always able to elect the strict Condorcet candidate when voters use sincere admissible strategies. Approval voting (AV) Some commonsensical arguments for AV (Brams, 2008) ­ It gives voters more flexible options ­ It helps elect the strongest candidate ­ It will reduce negative campaigning ­ It will increase voter turnout (participation) ­ It will give minority candidates their proper due ­ It is eminently practicable Nonranked Voting System Election of two or more members List systems ­ (system of voting the party ticket) ­ Voter chooses between lists of candidates Methods: ­ Highest average ­ Greatest remainder ­ The idea is to have a common ratio between seats & votes. EXAMPLE – List Systems Suppose an constituency in which 200,000 votes are cast for four party lists contesting five seats and suppose the distribution of votes is: A 86,000 B 56,000 C 38,000 D 20,000 Highest average method d’Hondt’s rule The seats are allocated one by one and each goes to the list which would have the highest average number of votes At each allocation, each list’s original total of votes is divided by one more than the number of seats that list has already won in order to find what its average would be /2 /3 A 86.000 43.000 28.667 21.500 17.200 3 B 56.000 28.000 18.667 14.000 11.200 1 C 38.000 19.000 12.667 9.500 7.600 1 D 20.000 10.000 6.667 5.000 4.000 0 Highest average method Sainte-Laguë method The Sainte-Laguë method divides the number of votes for each party by the odd numbers (1, 3, 5, 7 etc.). This system does not inherently favour larger parties over smaller (or vice versa), and may thus be considered "more proportional" than d'Hondt. /3 /5 A 86.000 28.667 17.200 12.286 9.556 2 B 56.000 18.667 11.200 8.000 6.222 1 C 38.000 12.667 7.600 5.429 4.222 1 D 20.000 6.667 4.000 2.857 2.222 1 Greatest remainder method An electoral quotient is calculated by dividing total votes by the number of seats Each list’s total of votes is divided by the quota and each list is given as many seats as its poll contains the quota. ­ Hare quota: ����� ����� / ����� ���� ­ Droop quota : 1+ [����� ����� / (1+����� ����)] If any seats remain, these are allocated successively between the competing lists according to the sizes of the remainder Solution with “Greatest remainder” method Hare quota: List Votes Seats Remainder Seats 200,000 / 5 A 86.000 2 6.000 2 = 40,000 B 56.000 1 16.000 1 C 38.000 0 38.000 1 D 20.000 0 20.000 1 List Votes Seats Remainder Seats Droop quota: A 86.000 2 19.332 2 1 + B 56.000 1 22.666 2 [200,000 / (1+5)] C 38.000 1 4.666 1 = 33,334 D 20.000 0 20.000 0 Comparison of Highest Average and Greatest Remainder The greatest remainder favors small parties and the highest average favors the larger ones. Whether a list gets an extra seat or not may well depend on how the remaining votes are distributed among other parties: Greatest remainder: ­ it is quite possible for a party to make a slight percentage gain yet lose a seat if the votes for other parties also change. ­ A related feature is that increasing the number of seats may cause a party to lose a seat (the so-called Alabama paradox). Higest average: ­ The highest averages methods avoid this latter paradox but since no apportionment method is entirely free from paradox, they introduce others like quota violation Alabama paradox - Example Seats 25 Hare Quota 204 Party A B C D E F Total Votes 1500 1500 900 500 500 200 5100 Automatic seats 7 7 4 2 2 0 22 Remainder 72 72 84 92 92 200 Surplus seats 0 0 0 1 1 1 3 Total Seats 7 7 4 3 3 1 25 Seats 26 Hare Quota 196 Party A B C D E F Total Votes 1500 1500 900 500 500 200 5100 Automatic seats 7 7 4 2 2 1 23 Remainder 128 128 116 108 108 4 Surplus seats 1 1 1 0 0 0 2 Total Seats 8 8 5 2 2 1 26 Disadvantages of Nonranked Voting Nonranked voting systems arise serious questions as to whether these are fair and proper representations of the voters’ will Extraordinary injustices may result unless preferential voting systems are used Contradictions (3 cases of Dodgson) Case 1 of Dodgson Contradiction in simple majority: Candidate A and B Order of Voters preference V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 1 A A A B B B B C C C D 2 C C C A A A A A A A A 3 D D D C C C C D D D C 4 B B B D D D D B B B B Case 2 of Dodgson Contradiction in absolute majority: Candidate A and B Order of Voters Preference V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 1 B B B B B B A A A A A 2 A A A A A A C C C D D 3 C C C D D D D D D C C 4 D D D C C C B B B B B Case 3 of Dodgson Contradiction in absolute majority, The second ballot : Elimination of candidate A Order of Voters Preference V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 1 B B B C C C C D D A A 2 A A A A A A A A A B D 3 D C D B B B D C B D C 4 C D C D D D B B C C B Preferential Voting System What is needed, in place of the spot vote, is a method of voting that allows the voter to indicate not only which of the three or more candidates he would most desire to see elected, but also in what order of preference he would place the candidates.