Report

Group Decision Making Y. İlker TOPCU, Ph.D. www.ilkertopcu.net www.ilkertopcu.org www.ilkertopcu.info www.facebook.com/yitopcu twitter.com/yitopcu Decision Making? Decision making may be defined as: • Intentional and reflective choice in response to perceived needs (Kleindorfer et al., 1993) • Decision maker’s (DM’s) choice of one alternative or a subset of alternatives among all possible alternatives with respect to her/his goal or goals (Evren and Ülengin, 1992) • Solving a problem by choosing, ranking, or classifying over the available alternatives that are characterized by multiple criteria (Topcu, 1999) Group Decision Making? • Group decision making is defined as a decision situation in which there are more than one individual involved (Lu et al., 2007). • These group members have their own attitudes and motivations, recognise the existence of a common problem, and attempt to reach a collective decision. • Moving from a single DM to a multiple DM setting introduces a great deal of complexity into the analysis (Hwang and Lin, 1987). • The problem is no longer the selection of the most preferred alternative among the nondominated solutions according to one individual's (single DM's) preference structure. • The analysis must be extended to account for the conflicts among different interest groups who have different objectives, goals, criteria, and so on. Group Decision Making • 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 • Process-oriented approaches • Focuses on the process of making a group decision. The main objective is to generate new ideas. Content-Oriented Methods • These techniques operate under the following assumptions: • All participants of the group problem solving share the same set of alternatives, but not necessarily the same set of evaluation criteria • Prior to the group decision-making process, each decision maker or group member must have performed his own assessment of preferences. • The output of such analysis is a vector of normalized and cardinal ranking, a vector of ordinal ranking, or a vector of outranking relations performed on the alternatives. Content-Oriented Approaches • Implicit Multiattribute Evaluation • (Social Choice Theory) • Explicit Multiattribute Evaluation SOCIAL CHOICE THEORY • Voting • Social Choice Function Voting Methods • Nonranked Voting System • Preferential Voting System Nonranked Voting System • One member elected from two candidates • One member elected from many candidates • Election of two or more members 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 One member elected from many candidates • The first-past-the-post system • Election by simple majority • Majority representation system • Repeated ballots Voting goes on through a series of ballots until some candidate obtains an absolute majority of the votes cast • The second ballot On the first ballot a candidate can’t be elected unless he obtains an absolute majority of the votes cast The second ballot is a simple plurality ballot involving the two candidates who had been highest in the first ballot 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 Election of two or more members cont. • 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 • List systems Voter chooses between lists of candidates • Highest average (d’Hondt’s rule) • Greatest remainder Election of two or more members cont. • 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 EXAMPLE 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 B C D 86,000 56,000 38,000 20,000 Solution with “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 /3 /2 A B C D 86,000 56,000 38,000 20,000 43,000 56,000 38,000 20,000 43,000 28,000 38,000 20,000 28,667 28,000 38,000 20,000 28,667 28,000 19,000 20,000 3 1 1 0 Solution with “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 quotient and each list is given as many seats as its poll contains the quotient. • If any seats remain, these are allocated successively between the competing lists according to the sizes of the remainder 200,000 / 5 = 40,000 List A B C D Votes 86.000 56.000 38.000 20.000 Seats Remainder 2 6.000 1 16.000 0 38.000 0 20.000 Seats 2 1 1 1 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 preference V1 1 2 3 4 A C D B V2 A C D B V3 A C D B V4 B A C D V5 B A C D Voters V6 B A C D V7 B A C D V8 C A D B V9 V10 V11 C C D A A A D D C B B B Case 2 of Dodgson • Contradiction in absolute majority: Candidate A and B Order of Preference V1 1 2 3 4 B A C D V2 B A C D V3 B A C D V4 B A D C V5 B A D C Voters V6 B A D C V7 A C D B V8 A C D B V9 V10 V11 A A A C D D D C C B B B Case 3 of Dodgson • Contradiction in absolute majority, the second ballot : Elimination of candidate A Order of Preference V1 1 2 3 4 B A D C V2 B A C D V3 B A D C V4 C A B D V5 C A B D Voters V6 C A B D V7 C A D B V8 D A C B V9 V10 V11 D A A A B D B D C C C B Preferential Voting System The voter places 1 on the ballot paper against the name of the candidate whom he considers most suitable He/she places a figure 2 against the name of his second choice, and so on... The votes are counted and the individual preferences are aggregated with the principle of simple majority rule • Strict Simple Majority xPy: #(i:xPiy) > #(i:yPix) • Weak Simple Majority xRy: #(i:xPiy) > #(i:yPix) • Tie xIy: #(i:xPiy) = #(i:yPix) Preferential Voting System • More than Two Alternative Case: • According to Condorcet Principle, if a candidate beats every other candidate under simple majority, this will be the Condorcet winner and there will not be any paradox of voting EXAMPLE • Suppose the 100 voters’ preferential judgments are as follows: 38 votes: aPcPb 32 votes: bPcPa 27 votes: cPbPa 3 votes: cPaPb • All candidates are compared two by two: a P b: 41 votes; b P a 59 votes a P c: 38 votes; c P a 62 votes cPbPa b P c: 32 votes; c P b 68 votes C is Condorcet winner Advantages of Preferential Voting • If nonranked voting is utilized for the previous example: 38 votes: a P c P b a: 38 votes 32 votes: b P c P a b: 32 votes Simple 27 votes: c P b P a c: 27+3=30 votes Majority 3 votes: c P a P b Second ballot Absolute majority is 51 votes: c is eliminated The second ballot is a simple plurality ballot (Suppose preferential ranks are not changed) a: 41 votes b: 59 votes Disadvantages of Preferential Voting • Committee would have a circular preference among the alternatives: would not be able to arrive at a transitive ranking 23 votes: a P b P c 17 votes: b P c P a 2 votes: b P a P c 10 votes: c P a P b 8 votes: c P b P a b P c (42>18), c P a (35>25), a P b (33>27) Intransitivity (paradox of voting) Disadvantages of Preferential Voting cont. • Aggregate judgments can be incompatible Voters V1 V2 V3 BP D DP A AP B AP B Order of preference 1 2 3 4 A B C D D A B C B C D A AP B BP D DP A AP C AP C BP C CP D DP A Winner A B C D Social Choice Functions • • • • • • • Condorcet’s function Borda’s function Copeland’s function Nanson’s function Dodgson’s function Eigenvector function Kemeny’s function EXAMPLE • Suppose the 100 voters’ preferential judgments are as follows: 38 votes: ‘a P b P c’ 28 votes: ‘b P c P a’ 17 votes: ‘c P a P b’ 14 votes: ‘c P b P a’ 3 votes: ‘b P a P c’ Condercet’s Function • The candidates are ranked in the order of the values of fC fC(x) = min #(i: x Pi y) yA\{x} ‘a P b’ 55 votes & ‘b P a’ 45 votes ‘a P c’ 41 votes & ‘c P a’ 59 votes ‘b P c’ 69 votes & ‘c P b’ 31 votes a b c a 45 59 b 55 31 c 41 69 - fC 41 45 31 b PaPc Borda’s Function • The candidates are ranked in the order of the values of fB fB(x) = #(i: x Pi y) yA a b c fB a - 55 41 96 b 45 - 69 114 c 59 31 - 90 b PaPc Borda’s Function (alternative approach) A rank order method is used. • With m candidates competing, assign marks of m–1, m–2, ..., 1, 0 to the first ranked, second ranked, ..., last ranked but one, last ranked candidate for each voter. • Determine the Borda score for each candidate as the sum of the voter marks for that candidate a: 2 * 38 + 0 * 28 + 1 * 17 + 0 * 14 + 1 * 3 = 96 b: 2 * ( 28 + 3 ) + 1 * ( 38 + 14 ) + 0 * 17 = 114 c: 2 * ( 17 + 14 ) + 1 * 28 + 0 * ( 38 + 3 ) = 90 Copeland’s Function • The candidates are ranked in the order of the values of fCP • fCP(x) is the number of candidates in A that x has a strict simple majority over, minus the number of candidates in A that have strict simple majorities over x fCP(x) = #(y: yA x P y) - #(y: yA y P x) #(i: a Pi b) = 55 > #(i: b Pi a) = 45 ‘a P b’ #(i: a Pi c) = 41 < #(i: c Pi a) = 59 ‘c P a’ #(i: b Pi c) = 69 > #(i: c Pi b) = 31 ‘b P c’ fCP(a) = 1 - 1 = 0, fCP(b) = 1 - 1 = 0, fCP(c) = 1 - 1 = 0 Nanson’s Function Let A1 = A and for each j > 1 let Aj+1 = Aj \ {xAj: fB(x) < fB(y) for all yAj, and fB(x) < fB(y) for some yAj} where fB(x) is the Borda score Then fN(x) = lim Aj gives the winning candidate j A1 = A = {a, b, c} fB(a) = 96 fB(b) =114 fB(c) = 90 Nanson’s Function Candidate c is eliminated as s/he has the lowest score: A2 = {a, b} 38 votes: ‘a P b’ 28 votes: ‘b P a’ 17 votes: ‘a P b’ 14 votes: ‘b P a’ 3 votes: ‘b P a’ fB(a) = 55 fB(b) = 45 Candidate b is eliminated and candidate a is the winner: aPbPc Dodgson’s Function • Based on the idea that the candidates are scored on the basis of the smallest number of changes needed in voters’ preference orders to create a simple majority winner (or nonloser). a b a - b 45/55 c 59/41 31/69 c 55/45 41/59 - 69/31 - change 9 5 19 bPaPc Eigenvector Function • Based on pairwise comparisons on the number of voters between pair of alternatives • The idea is based on finding the eigenvector corresponding to the largest eigenvalue of a positive matrice(pairwise comparison matrix: D) …. X1 X2 Xm X1 1 n12 / n21 n1m / nm1 X2 n21 / n12 1 n2m / nm2 nm1 / n1m nm2 / n2m 1 … Xm Eigenvector Function • First construct the pairwise comparison matrix D: a b c a b 1 55/45 41/59 45/55 1 69/31 c 59/41 31/69 1 • Then find the eigenvector of D a b c a a b 1 1.2222 0.6949 0.8182 1 2.2258 a b c 1.439 0.4493 c sum 1 3.2572 2.6715 3.9207 b c 0.307 0.4575 0.1772 0.314 0.2512 0.3743 0.5677 0.398 0.4418 0.1682 0.2551 0.288 1 bPaPc 1 1 Which one to choose? • The most appropriate compromise or consensus ranking should be defined according to • Kemeny’s function Kemeny’s function • Based on finding the maximization of the total amount of agreement or similarity between the consensus rankings and voters’ preference orderings on the alternatives • Let L be the consensus ranking matrix • E be a translated election matrix: M-Mt • fK= max <E, L> • where <E, L> is the (ordinary inner product of E and L) Kemeny’s function Social Choice Functions Ranking Condercet’s Function Borda’s Function Dodgson’s Function bPaPc bPaPc bPaPc Nanson’s Function Eigenvector Function aPbPc bPaPc • Evaluate two rankings according to Kemeny’s function: • bPaPc • aPbPc Kemeny’s function • fK= max <E, L> M a b c a 0 45 59 b 55 0 31 E = M-MT c 41 69 1 bPaPc L a b c a 0 1 -1 b -1 0 -1 c 1 1 0 aPbPc L a b c a 0 -1 -1 b 1 0 -1 c 1 1 0 E a b c a 0 -10 18 b 10 0 -38 c -18 38 0 Fk (bPaPc) = -10 -18 -10 +38 -18 +38 = 20 Fk (aPbPc) = 10 -18 +10 +38 -18 +38 = 60 Example – Voting, List System • Suppose the results of the last election for Muğla is as follows. If Muğla is represented by 8 deputies in the parliment, How many deputies should each party get? Parties A B C D E Total Votes 150.000 95.000 76.000 47.000 32.000 400.000 Example – Social Choice Functions The professors of ITU The Industrial Engineering department wants to select the head of the department. The preferences of 60 professors are listed in the Table. Who should be selected as the head? aPbPc bPcPa bPaPc cPaPb cPbPa 23 17 2 10 8 60