Group Decision Making

Report
Group Decision Making
Y. İlker TOPCU, Ph.D.
www.ilkertopcu.net www.ilkertopcu.org www.ilkertopcu.info
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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 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)
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
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)
yA
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: yA  x P y) - #(y: yA  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 \ {xAj: fB(x) < fB(y) for all yAj,
and fB(x) < fB(y) for some yAj}
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

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