Dr. Y. İlker TOPCU www.ilkertopcu.net www.ilkertopcu.org www.ilkertopcu.info facebook.com/yitopcu twitter.com/yitopcu instagram.com/yitopcu Dr. Özgür KABAK web.itu.edu.tr/kabak/
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) Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 2
Group Decision Making? Decision situation in which there are more than one decision maker 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 (Hwang and Lin, 1987). The problem is no longer the selection of the most preferred alternative according to one 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. Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 3
Group Decision Making Process Oriented Approaches Content Oriented Approaches Implicit Multiattribute Evaluation Explicit Multiattribute Evaluation Game-Theoretic Approach Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 4
GDM Methods 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. Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 5
Content-Oriented Methods Assumptions: All participants of the GDM share the same set of alternatives, but not necessarily the same set of evaluation criteria Prior to the GDM process, each group member must have performed her/his own assessment of preferences. The output is a vector of normalized and cardinal ranking, a vector of ordinal ranking, or a vector of outranking relations performed on the alternatives. Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 6
Content-Oriented Approaches Implicit Multiattribute Evaluation (Social Choice Theory) Explicit Multiattribute Evaluation Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 7
SOCIAL CHOICE THEORY Voting Social Choice Function Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 8
Voting Methods Nonranked Voting System Preferential Voting System Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 9
Nonranked Voting System One member elected from two candidates One member elected from many candidates Election of two or more members Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 10
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 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 11
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 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 12
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 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 13
Election of two or more members ctd. 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 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 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 14
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 86,000 B 56,000 C 38,000 D 20,000 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 15
Solution with Highest average (d Hondt s) method 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 43,000 28,667 28,667 3 B 56,000 56,000 28,000 28,000 28,000 1 C 38,000 38,000 38,000 38,000 19,000 1 D 20,000 20,000 20,000 20,000 20,000 0 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 16
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 Votes Seats Remainder Seats A 86.000 2 6.000 2 B 56.000 1 16.000 1 C 38.000 0 38.000 1 D 20.000 0 20.000 1 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 17
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) Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 18
Case 1 of Dodgson Contradiction in simple majority: Candidate A and B Order of preference V1 V2 V3 V4 V5 Voters 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 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 19
Case 2 of Dodgson Contradiction in absolute majority: Candidate A and B Order of Preference V1 V2 V3 V4 V5 Voters 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 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 20
Case 3 of Dodgson Contradiction in absolute majority, the second ballot: Elimination of candidate A Order of Preference V1 V2 V3 V4 V5 Voters 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 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 21
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:xp i y) > #(i:yp i x) Weak Simple Majority xry: #(i:xp i y) > #(i:yp i x) Tie xiy: #(i:xp i y) = #(i:yp i x) Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 22
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 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 23
EXAMPLE Suppose the 100 voters preferential judgments are as follows: 38 votes: a P c P b 32 votes: b P c P a 27 votes: c P b P a 3 votes: c P a P b 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 c P b P a b P c: 32 votes; c P b 68 votes c is condorcet winner Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 24
Advantages of Preferential Voting Simple Majority Second ballot If nonranked voting is utilized for the example: 38 votes: a P c P b 32 votes: b P c P a 27 votes: c P b P a 3 votes: c P a P b Absolute majority is 51 votes: c is eliminated The second ballot is a simple plurality ballot Suppose preferential ranks are not changed a: 38 votes b: 32 votes c: 27+3=30 votes a: 41 votes b: 59 votes Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 25
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) Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 26
Disadvantages of Preferential Voting ctd. Aggregate judgments can be incompatible Order of preference Voters 1 2 3 4 V1 A B C D V2 D A B C V3 B C D A Winner BP D AP B AP C A DP A BP D BP C B AP B DP A CP D C AP B AP C DP A D Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 27
Social Choice Functions Condorcet s function Borda s function Copeland s function Nanson s function Dodgson s function Eigenvector function Kemeny s function Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 28
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 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 29
Condercet s Function The candidates are ranked in the order of the values of f C f C (x) = min #(i: x P i y) y A\{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 fc a - 55 41 41 b 45-69 45 c 59 31-31 b P a P c Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 30
Borda s Function The candidates are ranked in the order of the values of f B f B (x) = #(i: x P i y) y A a b c fb a - 55 41 96 b 45-69 114 c 59 31-90 b P a P c Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 31
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 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 32
Copeland s Function The candidates are ranked in the order of the values of f CP f CP (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 f CP (x) = #(y: y A x P y) - #(y: y A y P x) #(i: a P i b) = 55 > #(i: b P i a) = 45 a P b #(i: a P i c) = 41 < #(i: c P i a) = 59 c P a #(i: b P i c) = 69 > #(i: c P i b) = 31 b P c f CP (a) = 1-1 = 0, f CP (b) = 1-1 = 0, f CP (c) = 1-1 = 0 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 33
Copeland s Function (another example) 38 votes: a P c P b 32 votes: b P c P a 27 votes: c P b P a 3 votes: c P a P b Judgments of simple majority: b P a, c P a and c P b f CP (a) = 0 2 = 2; f CP (b) = 1 1 = 0; f CP (c) = 2 0 = 2 The ranking of alternatives: c P b P a Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 34
Nanson s Function Let A 1 = A and for each j > 1 let A j+1 = A j \ {x A j : f B (x) < f B (y) for all y A j, and f B (x) < f B (y) for some y A j } where f B (x) is the Borda score Then f N (x) = lim A j gives the winning candidate A 1 = A = {a, b, c} f B (a) = 96 f B (b) =114 f B (c) = 90 j Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 35
Nanson s Function ctd. Candidate c is eliminated as s/he has the lowest score: A 2 = {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 f B (a) = 55 f B (b) = 45 Candidate b is eliminated and candidate a is the winner: a P b P c Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 36
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 c change a - 55/45 41/59 b 45/55-69/31 c 59/41 31/69-9 5 19 b P a P c Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 37
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) X 1 X 2. X m X 1 1 n 12 / n 21 n 1m / n m1 X 2 n 21 / n 12 1 n 2m / n m2 X m n m1 / n 1m n m2 / n 2m 1 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 38
Eigenvector Function First construct the pairwise comparison matrix A: Then find the eigenvector of A: a b c a 1 1.2222 0.6949 b 0.8182 1 2.2258 c 1.439 0.4493 1 sum 3.2572 2.6715 3.9207 a b c a 1 55/45 41/59 b 45/55 1 69/31 c 59/41 31/69 1 a b c a 0.307 0.4575 0.1772 0.314 b 0.2512 0.3743 0.5677 0.398 c 0.4418 0.1682 0.2551 0.288 1 1 1 b P a P c Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 39
Which one to choose? The most appropriate compromise or consensus ranking should be defined according to Kemeny s function Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 40
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-M t f K = max <E, L> where <E, L> is the inner product of E and L Inner (dot) product of vectors [1, 3, 5] and [4, 2, 1]: (1)(4) + (3)(-2) + (-5)(-1) = 3 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 41
Example Social Choice Functions Condercet s Function Borda s Function Dodgson s Function Nanson s Function Eigenvector Function Ranking b P a P c b P a P c b P a P c a P b P c b P a P c Evaluate two rankings according to Kemeny s function: b P a P c a P b P c Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 42
f K = max <E, L> E = M-M T M a b c a 0 55 41 b 45 0 69 c 59 31 1 E a b c a 0 10-18 b -10 0 38 c 18-38 0 L a b c a 0-1 1 b 1 0 1 c -1-1 0 b P a P c F k (bpapc) = -10-18 -10+38-18+38 = 20 a P b P c L a b c a 0 1 1 b -1 0 1 c -1-1 0 F k (apbpc) = 10-18 +10+38-18+38 = 60 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 43
Additional 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 parliament, how many deputies should each party get? Parties Votes A 150.000 B 95.000 C 76.000 D 47.000 E 32.000 Total 400.000 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 44
Additional 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? a P b P c 23 b P c P a 17 b P a P c 2 c P a P b 10 c P b P a 8 60 Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 45