Group Decision Making

Y. İlker TOPCU, Ph.D.

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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 majorityEach 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 ballotOn the first ballot a candidate can’t be elected unless he

obtains an absolute majority of the votes castThe 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 voteEach voter has one vote

• Multiple voteEach 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 voteEach 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 voteEach voter has as many votes as the number of

seats to be filled

Voters can cast more than one vote for candidates

• List systemsVoter chooses between lists of candidates• Highest average (d’Hondt’s rule)• Greatest remainder

Election of two or more members cont.

• Approval votingEach 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 86,000

B 56,000

C 38,000

D 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

/2 /3

A 86,000 43,000 43,000 28,667 28,667 3B 56,000 56,000 28,000 28,000 28,000 1C 38,000 38,000 38,000 38,000 19,000 1D 20,000 20,000 20,000 20,000 20,000 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

List Votes Seats Remainder SeatsA 86.000 2 6.000 2B 56.000 1 16.000 1C 38.000 0 38.000 1D 20.000 0 20.000 1

200,000 / 5= 40,000

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 V2 V3 V4 V5 V6 V7 V8 V9 V10 V111 A A A B B B B C C C D2 C C C A A A A A A A A3 D D D C C C C D D D C4 B B B D D D D B B B B

Voters

Case 2 of Dodgson

• Contradiction in absolute majority: Candidate A and B

Order of

Preference V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V111 B B B B B B A A A A A2 A A A A A A C C C D D3 C C C D D D D D D C C4 D D D C C C B B B B B

Voters

Case 3 of Dodgson

• Contradiction in absolute majority, the second ballot : Elimination of candidate A

Order of

Preference V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V111 B B B C C C C D D A A2 A A A A A A A A A B D3 D C D B B B D C B D C4 C D C D D D B B C C B

Voters

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: a P c P b32 votes: b P c P a27 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 votesa P c: 38 votes; c P a 62 votes c P b P ab 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

32 votes: b P c P a

27 votes: c P b P a

3 votes: c P a P b

a: 38 votesb: 32 votes

c: 27+3=30 votesSimple Majority

Absolute majority is 51 votes: c is eliminatedThe second ballot is a simple plurality ballot(Suppose preferential ranks are not changed) a: 41 votes

b: 59 votes

Second ballot

Disadvantages of Preferential Voting

• Committee would have a circular preference among the alternatives: would not be able to arrive at a transitive ranking23 votes: a P b P c17 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 1 2 3 4V1 A B C DV2 D A B CV3 B C D A

Order of preference

WinnerBP D AP B AP C ADP A BP D BP C BAP B DP A CP D CAP B AP C DP A 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)

‘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

yA\{x}

b P a P c

  a b c fC

a - 55 41 41

b 45 - 69 45

c 59 31 - 31

Borda’s Function

• The candidates are ranked in the order of the values of fB

fB(x) = #(i: x Pi y)yA

b P a P c

  a b c fB

a - 55 41 96

b 45 - 69 114

c 59 31 - 90

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 = 96b: 2 * ( 28 + 3 ) + 1 * ( 38 + 14 ) + 0 * 17 = 114c: 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

A1 = A = {a, b, c}

fB(a) = 96

fB(b) =114

fB(c) = 90

j

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: a P b P c

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).

b P a P c

  a b c change

a - 55/45 41/59 9b 45/55 - 69/31 5c 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

Eigenvector Function

• First construct the pairwise comparison matrix D:

• Then find the eigenvector of D

b P a P c

  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 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 0.307 0.4575 0.1772 0.314b 0.2512 0.3743 0.5677 0.398c 0.4418 0.1682 0.2551 0.288

1 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

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

Social Choice Functions Ranking

Condercet’s Function b P a P c

Borda’s Function b P a P c

Dodgson’s Function b P a P c

Nanson’s Function a P b P c

Eigenvector Function b P a P c

Kemeny’s function

• fK= max <E, L> E = M-MT

b P a P c Fk (bPaPc) = -10 -18 -10 +38 -18 +38 = 20

a P b P cFk (aPbPc) = 10 -18 +10 +38 -18 +38 = 60

M a b ca 0 55 41b 45 0 69c 59 31 1

E a b ca 0 10 -18b -10 0 38c 18 -38 0

L a b c

a 0 -1 1b 1 0 1

c -1 -1 0

L a b c

a 0 1 1

b -1 0 1

c -1 -1 0

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 VotesA 150.000B 95.000C 76.000D 47.000E 32.000

Total 400.000

Example – Social Choice Functions

a P b P c 23b P c P a 17

b P a P c 2

c P a P b 10

c P b P a 8

60

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?