GROUP DECISION MAKING UNDER MULTIPLE CRITERIA 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.

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 ­ 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 (exhaustive ) ­ 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 - is shown to be the only non 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.

­ : ����� ����� / ����� ���� ­ : 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.

The preferential vote was first proposed by the Chevalier de Borda, in a paper that he read before the Academie Royale des Sciences in Paris in 1770.

This voting system was regularly used in Britain for pparliamentary for more than thirty years until its abolition in 1950.

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... Preferential Voting System

The problem of aggregating individual preferences to form a group or social choice has been the subject of much discussion and controversy. In 1785, Condorcet discovered the "paradox of voting": the fact that social choice processes based on a principle of can give rise to nontransitive (cyclical) ranking among candidates. Preferential Voting System Two-alternative case

SIMPLE MAJORITY DECISION RULE The votes are counted and the individual preferences are aggregated

Strict Simple Majority xPy: #(i:xPiy) > #(i:yPix)

Weak Simple Majority xRy: #(i:xPiy) ³ #(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 – Existance of Condorcet winner

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 Advantages of Preferential Voting

If nonranked voting is utilized for the previous example: Simple majority

38 votes: a P c P b a: 38 votes 32 votes: b P c P a b: 32 votes 27 votes: c P b P a 3 votes: c P a P b c: 27+3=30 votes

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) Þ Intransitivity (paradox of voting) Social Choice Functions

Condorcet’s function Borda’s function Copeland’s function Nanson’s function Dodgson’s function Eigenvector function Kemeny’s function

Single Transferable Vote Coombs Method 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 {y Î A(x) | # (x Pi y) }

‘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 b P a P c c 59 31 - 31 Sum-of-Ranks (Borda) Rule

The candidates are ranked in the order of the values of fB fB(x) = å #(i: x Pi y) yÎA

a b c fB a - 55 41 96 b 45 - 69 114 b P a P c c 59 31 - 90 Sum-of-Ranks (Borda) Rule (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 Nanson’s Function

The Nanson method is a recursive elimination method based on Borda scores. In the first step one calculates each candidate’s Borda score. At the end of the first step the candidates whose Borda scores do not exceed the average Borda score of the candidates in this step are eliminated from all ballots In the second step a revised Borda score is computed for the remaining candidates. The elimination process is continued in this way until one candidate is left.

A1 = A = {a, b, c} fB(a) = 96 fB(b) =114 fB(c) = 90 Average Borda Score: 100 à a and c eliminated; b wins Nanson’s Function – Example 2

A1 = {a, b, c, d, e}

2 votes: a b c d e 2 votes: d b e c a 2 votes: c a d e b 2 votes: e d a c b 1 vote: b c e a d

fB (a) = 19 fB (b) = 16 fB (c) = 19 fB (d) = 20 fB (e) = 16

Average Borda Score: 18 à b, and e are eliminated Nanson’s Function – Example 2 (Cont.)

A1= A1 ∖{b, e} = {a, c, d} 2 votes: a c d 2 votes: d c a 2 votes: c a d 2 votes: d a c 1 vote: c a d

fB (a) = 9 fB (c) = 10 fB (d) = 8

Average Borda Score: 9 à a, and d are eliminated, c wins 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: yÎA Ù x P y) - #(y: yÎA Ù y P x)

#(i: a Pi b) = 55 > #(i: b Pi a) = 45 Þ ‘a P b’

#(i: a Pi c) = 41 < #(i: c Pi a) = 59 Þ ‘c P a’

#(i: b Pi c) = 69 > #(i: c Pi b) = 31 Þ ‘b P c’ fCP(a) = 1 - 1 = 0, fCP(b) = 1 - 1 = 0, fCP(c) = 1 - 1 = 0 Copeland’s Function (Example 2)

A

B C

f (A) = 3-1 =2 D E CP fCP(B) = 2-2 =0 fCP(C) = 1-3 =-2 fCP(D) = 2-2 =0 fCP(E) = 2-2 =0

A is the winner! 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 9 b 45/55 - 69/31 b P a P c 5 c 59/41 31/69 - 19 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

…

Xm nm1 / n1m nm2 / n2m 1 How to find the vector of weight (eigenvector)?

Crudest way Sum the elements in each row and normalize by dividing each sum by the total of all the sums Better Take the sum of the elements in each column and form the reciprocals of these sums. To normalize so that these numbers add to unity, divide each reciprocal by the sum of the reciprocals Good Divide the elements of each column by the sum of that column(i.e. Normalize that column) and then add the elemets in each resulting row and divide this sum by the number of elements in the row. This is a process of averaging over normalized columns Good Multiply the n elements in each row and take the nth root. Normalize the resulting numbers Eigenvector Function

First construct the pairwise comparison matrix D: a b c Weights by the other a 1 55/45 41/59 good method a 0.947 b 45/55 1 69/31 b 1.221 c 59/41 31/69 1 c 0.865

Then find the eigenvector of D

a b c a b c a 0.307 0.4575 0.1772 a 1 1.2222 0.6949 0.314 b 0.2512 0.3743 0.5677 b 0.8182 1 2.2258 0.398 c 1.439 0.4493 1 c 0.4418 0.1682 0.2551 0.288 sum 3.2572 2.6715 3.9207 1 1 1

b P a P c Kemeny’s function

Aims to find the most appropriate compromise or consensus ranking Evaluates different ranking alternatives.

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 ­ where is the (ordinary inner product of E and L) Kemeny’s function

Evaluate two rankings according to Kemeny’s function: ­ b P a P c ­ a P b P c Kemeny’s function

T ­ fK= max E = M-M

M a b c E a b c a 0 55 41 a 0 10 -18 b 45 0 69 b -10 0 38 c 59 31 1 c 18 -38 0

L a b c a 0 -1 1 b P a P c F (bPaPc) = -10 -18 - b 1 0 1 k 10 +38 -18 +38 = 20 c -1 -1 0

L a b c a 0 1 1 F (aPbPc) = 10 -18 a P b P c k ü b -1 0 1 +10 +38 -18 +38 = 60 c -1 -1 0 Project Groups

8 students have enrolled in the course!

You will constitute groups of 2-3 students.

We will have 3 groups. One group will make a presentation in each presentation day. Project Groups

The first group’s presentation is approaching! ­ Topic: Implicit Multiattribute Evaluation ­ I will give some of their papers next week, ­ They have to search recent papers and find 2 – 3 appropriate papers.

Please constitute your groups before next week’s course Next week

Topics: Arrow's conditions (STV) The Coombs method

Read “STV transfer rules”, the the file on ninova. (no submission needed)

Homework 2: Please read the following paper and submit one page summary:

­ Nurmi, H. (2018). Voting theory: cui bono. Politeia, 91, 106-121. http://politeia.ru/files/articles/rus/Politeia-2018-4(91)-106-121.pdf