SELF–DUALITY IN GROUP DECISION
LAVOSLAV ČAKLOVIĆ, ZAGREB
A BSTRACT. The self-ranking of the group of decision makers is considered. Each decision maker, a member of the group, gives his individual preferences on the set of all group
members. Apart from that, their weights (importance) are calculated (reconstructed) as a
fixed point of some non-linear function Φ which represents a group consensus. Uniqueness
of the fixed point of is proved under some mild assumptions the size of the group. In that
case the fixed point can be calculated as the limit point of the sequence Φn (ξ), where ξ
is some initial ranking. The theory is ilustrated on the example of re-evaluation of a state
budget and some numerical experiments are provided for the case when this assumption is
not satisfied.
An`existence´ and uniqueness of the fixed point is also proved under the assumption that
1 ∈
/ σ Φ′X (ξ) for all ξ ∈ Aff(Σ), where σ denotes the spectrum of a linear operator
and Σ the standard simplex in Rn .
1. I NTRODUCTION
In self-ranking we aggregate individual ’opinions’ of the group members into the group
opinion about the same group, but without external evaluator. Any aggregation needs a
priori given weights of the group members but here they are not known. In this article
we suggest a mathemathical model for calculating those weights from the individual preference graphs given by each group member. Possible applications are: re-evaluation of
the state budget (see the example on the page 7), self-ranking of the group of decision
makers before any kind of group decision, ranking the efficient units in Data Envelopment
Analysis (DEA) and many others.
It is well known that DEA is not capable to distinguish efficient DMUs among themselves. It evaluates the relative efficiency of DMUs but does not allow for the ranking of
efficient units themselves. This approach can be used to rank only efficient DMUs using
cross-efficiency matrix, see Čaklović, Hunjak [6] or to rank all DMUs directly avoiding
the linear programming technique, see Čaklović [5].
There are some cases where the selfranking can be done using Analytical Hierarchy
Process (AHP) developed by Saaty [9], but those cases are limited with necessity of having
the full data set, missing data are not allowed. Our approach is based on a graph theoretic
approach developed by the author [4].
The paper is organized as follows: the section 2 describes the basic notions of Potential
Method and aggregation procedure in multicriteria and group decision context. In the
section 3 the main theorem about the existence and uniqueness of the fixed point is stated
and proved. Its generalization is given as well. A real world example is given in the section
4.1 and numerical test for selfish potential is done in section 5. And finally, the program
code written in Mathematica which was used to generate the numerical results is done in
the last section.
Key words and phrases. group decision making, self ranking, graph theory, preference flow, potential method,
multi-criteria decision problem.
1
2
LAVOSLAV ČAKLOVIĆ, ZAGREB
2. P OTENTIAL
METHOD , A BRIEF DESCRIPTION
2.1. Preference Graph. A preference graph is a digraph G = (V, A) where V is
the set of nodes and A is the set of arcs of G. We say that node a is more preferred
than node b, in notation a < b, if there is an arc (a, b) outgoing from b and ingoing
to a. A preference flow is a non-negative real function F defined on the set of
arcs. The value Fα on arc α is an intensity of the preference on some scale. For the arc
α = (a, b), Fα = 0 means that the decision maker is indifferent to the pair {a, b}. In that
case orientation of the arc is arbitrary. Two nodes are adjancent in the graph if and only if
they are compared.
An example of a preference graph in a voting procedure was considered by [7, Condorcet, 1785]. He defined the social preference flow as
FC (u, v) := N (u, v) − N (v, u)
where N (u, v) denote the number of voters choosing u over v. We say that the node u is
socially preferred to v if FC (u, v) ≥ 0. It is easy to see that Condorcet’s winner
exists if the Condorcet’s graph has no positive cycle. Potential Method allways gives the
answer, see the formula (2), but the interpretation is left to the decision maker, see the
notion of the measure of inconsistency, section 2.2. Another example of the preference
graph is shown in the Table 4.4. This is the individual preference graph given by ARMY
ministry from the example in the Section 4.1 on the page 7.
The incidence matrix A = (aα,v ) of the graph is m × n matrix, m = Card A,
n = Card V where

 −1, if α leaves node v
1, if α enters node v
aα,v =

0, otherwise.
X1



A=


−1
1 0
−1
0 1
0 −1 1
−1
0 0
0 −1 0
0
0
0
1
1






α4 , F4
α1 , F1
α2 , F2
X4
α5 , F5
X2
α3 , F3
X3
TABLE 2.1. The preference graph and its incidence matrix.
We shall write aij where i is the index of i-th arc and j is the index of j-th node.
The vector space Rm is called the arcs space and the vector space Rn is called the
vertex space (of the preference graph). The incidence matrix generates the orthogonal
decomposition
N (Aτ ) ⊕ R(A) = Rm
where R(A) is the column space of the matrix A and N (Aτ ) is the null-space of the matrix
Aτ . N (Aτ ) is called cycle space because it is generated by all cycles of the graph.
SELF–DUALITY IN GROUP DECISION
3
The preference flow F can be considered as an element if the arcs space and we identify
it by the column vector of length m. From the computational point of view it is practical
to denote, for an arc α ∈ A, F−α = −Fα .
A preference flow is consistent if there is no component of the flow in the cyclespace of the graph. According to the definition, F is consistent if the sum of algebraic
components of the flow along each cycle is equal to zero. Evidently,
Theorem 1. Let F denotes a preference flow. The following statements are equivalent:
(i)
(ii)
(iii)
(iv)
F is consistent.
F is a linear combination of columns of the incidence matrix A of the graph.
There exists X ∈ Rn such that AX = F .
Scalar product y τ F = 0 for each cycle y, i.e. F is orthogonal to the cycle space.
We shall examine the consistency of a given flow F by solving the equation
AX = F .
(1)
A solution of the equation AX = F , if it exists, is called the potential of F . Evidently,
X is not unique because the constant column 1 τ = [ 1 · · · 1 ]τ is an element of the kernel
N (A). The consistent flow is a potential difference because each row of AX is of the form
Xi − Xj for some i, j ∈ {1, . . . , n}.
2.2. Inconsistent Flow. In practice, while performing pairwise comparisons, the decision
maker does not give, in general, a flow which is consistent. In that case the best approximation of the flow by the column space of the incidence matrix should be calculated. Such
X is the solution of the normal equation of (1) i.e.
(2)
Aτ AX = Aτ F ,
and, for uniqueness we put extra condition
X
(3)
X(v) = 0.
v∈V
Measure of inconsistency of the flow F , in notation inc(F ), is defined as the
angle between F and the column space of the incidence matrix, i.e. the angle between F
and AX. Evidently, inc(F ) = 0 if and only if F is consistent. The practice suggest to
accept the inconsistency bellow 12o .
2.3. Aggregation of Flows in MCDM. The procedure of making a consensus graph
(V, A) and consensus flow is the following. Each criterion Ci , i ∈ {1, . . . , k} generates its own preference graph (V, Ai ) and its own preference flow Fi . Let wi denote the
weight of i-th criterion. First, for the given pair α = (u, v) of alternatives we calculate
(4)
Fα :=
k
X
wi Fi (α)
i=1
±α∈Ai
where the item wi Fi (α) is taken into account if and only if α ∈ Ai or −α ∈ Ai . If
this sum is non-negative, then we include α in the set A of arcs of the consensus graph,
and we put F (α) := Fα . If it is negative, we define −α = (v, u) as an arc in A and
F (−α) := −Fα . The flow F becomes non-negative. If Fα is not defined then u and v are
not adjacent in the consensus graph.
4
LAVOSLAV ČAKLOVIĆ, ZAGREB
2.3.1. Hierarchical decision structure. In a hierarchical decision structure each node, except the nodes on the last level, is a parent node for its children (leaves) from some other
level. A parent node may be considered as a criterion for evaluation of its children. The
only restriction is that children of the parent should be in the same level. The parents of
a node can be from different levels. Restriction, made by the conservation law, is that the
sum of the weights of nodes in some level set should be the sum of the weights of their
parents.
PM calculates the weights of nodes in some level in the following way. First, the weight
of the goal is set to be 1. For a particular level which is not yet ranked, the aggregation of
flows is done over the set of all parents. A consensus graph need not be connected. For
each connected component C the potential XC is calculated first and after that the weights
wC using the formula
wC = kC ·
(5)
aXC
kaXC k1
where k·k1 represents l1 -norm and kC is the sum of weights of the parents for that particular
component C. The exponential function X 7→ aX is defined by the components and a is
a positive constant. Currently, we use the value a = 2, but the user may change it, if
necessary. The process is repeated until the bottom level of the hierarchy is ranked.
2.4. Group decision. Let G = {1, 2, . . . , n} denotes the group of decision makers. The
aim of the group decision is to synthesize individual judgments x1 , . . . , xn of some entity
into a ‘group opinion’ f (x1 , . . . , xn ). This operation, let us denote it by ◦, should be an
associative and continuous operation of ‘individual opinions’ fi (xi ), i = 1, . . . , n. This
means
f (x1 , . . . , xn ) = f1 (x1 ) ◦ f2 (x2 ) ◦ · · · ◦ fn (xn ).
If all group members have equal importance, then f1 = · · · = fn and the operation should
be commutative, too. In [1, 1983] it is proved that under the natural conditions:
f
f (x, x, . . . , x) = x
1
1
1 1
=
, ,...,
x1 x2
xn
f (x1 , x2 , . . . , xn )
f has the form
(6)
n
h
X
i
1
ψ0 (log xk )
f (x1 , x2 , . . . , xn ) = exp ψ0−1
n
k=1
where ψ0 is an arbitrary continuous monotonous odd function. We will give one generalization of the formula (6) in the case of missing data, and apply it to the self ranking
problem of the group, i.e. when the weights of the group members are not known a priori.
Let us describe the connection of PM and the formula (6). If we suppose that all individual preference graphs are complete and xk = aFk (α) for the fixed arc α ∈ A. Then the
formula (6) becomes
n
X
1 ψ0 log aFk (α) .
ψ0 log aF (α) =
n
k=1
SELF–DUALITY IN GROUP DECISION
If moreover, we define ψ0 (t) =
t
log a
5
then ψ0 (log at ) = t and
F (α) =
n
X
1
Fk (α)
n
k=1
which is exactly the formula (4) when all individual graphs are complete.
In a group decision, each group member can have its own hierarchy under one condition:
that the set of alternatives, i.e. the bottom level of the hierarchy, should be the same.
The consensus graph for the whole group is calculated using the formula (4). Even this
condition can be dropped, a consensus graph then grows while adding each new individual
preference graph. The only reasonable condition is that the composite preference graph
should be connected.
3. S ELF -R ANKING
Self-ranking means that a group of decision makers ranks themselves. A typical example is re-evaluation of the state budget given bellow. The first step in self-ranking is to fix
some initial ranking w0 and calculate w1 . Repeating this process we get the sequence
(7)
w0 7→ w1 7→ · · · 7→ wn 7→ · · · .
The question is whether this sequence has a limit. We shall see that this is a good approach
and, under some condition on the consensus graph, this limit is the unique fixed point of
some non-linear function.
P
For the beginning let Σ = {ξ ∈ Rn | ξ ≥ 0,
ξi = 1} denote a standard simplex in
Rn and let us suppose that each decision maker i ∈ G gives the preference flow Fi on the
set G with respect to certain criteria. If ξ ∈ Σ is the given group ranking and FG is the
consensus flow, then it should satisfy the equation
X
ξi Fi = FG .
i∈G
Because of the linearity, the same relation should take place for potentials Xi and XG of
the consensus flow i.e.
X
ξi Xi = XG .
i∈G
If X denotes the matrix with columns Xi , i = 1, . . . , n, then, the above equation may be
written as Xξ = XG . The function
aXξ
,
kaXξ k1
ΦX : Σ → Σ is now well-defined and the group ranking should be the fixed point of Φ,
i.e. it should satisfy the equation
ΦX : ξ 7→
(8)
ξ = ΦX (ξ).
The existence of the fixed point is a consequence of the Brouwer’s fixed point theorem.
The uniqueness of the fixed point is assured under some mild conditions on the number of
nodes and the constant a.
Theorem 2. Let us suppose that
(9)
2 ln akXk∞ < 1
Then, ΦX is a contraction and for each ξ ∈ Σ the sequence (ΦX )n (ξ) converges to the
unique fixed point ξ0 ∈ Σ of ΦX .
6
LAVOSLAV ČAKLOVIĆ, ZAGREB
Proof. A short calculation gives that the derivative of ΦX in point ξ ∈ Σ is given by
(10)
Φ′X (ξ)η = ln a D(ΦX (ξ))Xη − ΦX (ξ)hΦX (ξ), Xηi , ∀η, hη, 1 i = 0.
where D(x) denotes a diagonal matrix with vector x on the main diagonal, and h· , ·i stands
for the euclidean scalar product. The norm of Φ′X (ξ) is bounded above by 2 ln a · kXk∞
which proves the theorem.
The condition (9) can be satisfied either by taking a closer to 1, or by multiplying
the composite flow FG by a small positive number t. This is the same as replacing the
matrix X by tX and taking t small enough to ensure that ΦtX becomes contraction. The
corresponding group ranking of tX may be taken as the desired ranking.
The following generalization of the previous theorem gives another view on the uniqueness. Let Aff(Σ) denote an affine manifold generated by Σ.
Theorem 3. Let us suppose that 1 ∈
/ σ Φ′X (ξ) for all ξ ∈ Aff(Σ), where σ denotes the
spectrum of a linear operator. Then, ΦX : Σ → Σ has a unique fixed point.
Proof. 1st step. Some properties of the function ΦX .
Let us extend a the function Φ on the affine manifold Aff(Σ) by the same formula. From
the equation (10) it easily follows that the tangent space on Σ is invariant on Φ′ (ξ), ∀ξ ∈
Aff(Σ). The proof is straightforward, because
hΦ′X (ξ)η, 1 i = hD(ΦX (ξ))Xη, 1 i − hΦX (ξ), 1 ihΦX (ξ), Xηi
= hXη, D(ΦX (ξ))1 i − hΦX (ξ), Xηi
= hXη, ΦX (ξ)i − hΦX (ξ), Xηi = 0,
for all η ⊥ 1 . Let us consider the real function
φ(ξ) := kΦX (ξ) − ξk2 .
Evidently, the minimum point of φ is the fixed point of ΦX , and vice verse. By the inverse function theorem the function ΦX (ξ) − ξ is locally invertible, because 1 is not the
eigenvalue of it’s derivative. The consequence is that the equation ΦX (ξ) = ξ has isolated
solutions, and φ has isolated critical points. There are finitely many of them because of the
compactness of Σ.
Let us suppose that φ has at least two minimum points. Then,
2nd step. φ has a critical point which is not the minimum point.
We are in the situation where the Mountain Pass theorem from the Critical Point Theory
can be applied, [10, Struwe, Theorem 10.3], to obtain the third critical point that is neither
a local, nor a global minimum point.
To apply the Mountain Pass theorem here, one should first check that φ satisfies the
compactness condition of Palais–Smale:
if ξn is a sequence such that
φ(ξn ) is bounded and kφ′ (ξn )k → 0
then it has a convergent sub sequence.
This is evident by the fact that φ is coercive in the sense that (∀M > 0)(∃R > 0) such that
φ(ξ) > M whenever kξk > R.
SELF–DUALITY IN GROUP DECISION
7
3rd step. The critical point of φ is the minimum point of φ.
Let ξ denote this third critical point. Then,
hΦX (ξ) − ξ, (Φ′X (ξ) − I)ηi = 0,
for all η such that hη, 1 i = 0.
Φ′X (ξ)
Because of the surjectivity of
− I we conclude that ΦX (ξ) − ξ = 0 and ξ is the
minimum point of φ which is exactly what we wanted to prove.
The Theorem 2 is the consequence of the Theorem 3 because the condition (9) implies
the premise of the Theorem 3. The question is whether the sequence ΦnX (ξ) is convergent
or not for each ξ ∈ Σ.
Conjecture 1. We believe that the supposition in the theorem 3 may berelaxed. Instead of
1∈
/ σ Φ′X (ξ) for all ξ ∈ Aff(Σ) we may assume only 1 ∈
/ σ Φ′X (ξ) for all ξ ∈ Σ.
We believe that this can be proved either by showing that ΦX cannot have a critical
point outside Σ or considering the simplex as the Hilbert space in the sense of [2].
4. E XAMPLES
4.1. Re-evaluation of the State Budget. The government of some state decided to reevaluate the state budget. The prime minister wants to do that transparently and he asked
the ministry of transport, industry, tourism, army and science to give their own preferences
concernig financial necessities.
The group of decision makers here is G = {TRANSP, IND, TOURISM , ARMY, SCIENCE }.
In Table 4.3 the preference flow is given for each ministry. From the table we see that
TOURISM gives priority to TRANSP in comparison TRANSP – ARMY and the strength of this
preference is 4. At the same time IND prefers ARMY with strength 2 while ARMY and
SCIENCE do not give any preference in comparison TRANSP – ARMY . None of the individual preference flow is complete.
The fixed point of the function ΦX (a = 2) is calculated and given in the Table 4.2.
According to this result the prime minister can decide how to distribute the financial resources. The same model can also be used for harmonizing functions of a company to see
wether some of them are proportionally big or small with respect to some function.
Ministry
weight potential
IND
0.230
0.210
SCIENCE
0.206
0.051
TRANSP
0.201
0.017
TOURISM
0.189
−0.076
ARMY
0.173
−0.202
TABLE 4.2. The fixed point in the state budget example.
4.2. The Importance of Process Standards in Education of Math. Here are the five
process standards in mathematics as seen by the National Council of Teachers of Mathematics (USA) in the Curriculum Focal Points released in 2006. Those five process are:
problem solving, reasoning, communication, connections, and representations.
For instance, to improve communication students:
• Organize and consolidate their mathematical thinking through communication;
8
LAVOSLAV ČAKLOVIĆ, ZAGREB
TOURISM
ARMY
IND
TRANSP – ARMY
−4
IND – TOURISM
−2
−3
−2
2
1
3
2
TRANSP – IND
−1
2
TRANSP – TOURISM
−2
ARMY – TRANSP
TRANSP
−1
0
ARMY – TOURISM
3
2
−3
−2
0
2
0
−2
−2
SCIENCE – TOURISM
2
3
1
IND – ARMY
SCIENCE – TRANSP
SCIENCE
−1
2
−1
1
TRANSP – SCIENCE
−2
SCIENCE – ARMY
2
−1
SCIENCE – IND
3
TABLE 4.3. Individual preferences given by ARMY ministry.
TRANSPORT
2
0
2
TOURISM
INDUSTRY
3
2
3
0
SCIENCE
ARMY
TABLE 4.4. Preference graph for ARMY.
• Communicate their mathematical thinking coherently and clearly to peers, teachers, and others;
• Analyze and evaluate the mathematical thinking and strategies of others;
• Use the language of mathematics to express mathematical ideas precisely.
To become skilfull in representations students should:
• Create and use representations to organize, record, and communicate mathematical ideas;
SELF–DUALITY IN GROUP DECISION
9
• Select, apply, and translate among mathematical representations to solve problems;
• Use representations to model and interpret physical, social, and mathematical
phenomena.
From the point of view of reasoning may be that the communication is more important
than representation but from the point of view of problem solving communication seems
to be less important than representation. Analysing carefully the preferences among the
pairs of processes from the point of view of each process, as the criterion, we obtain five
preference graphs; one for each process. The fix point of the consensus function ΦX gives
the importance of the processes in education. The preference graph given by reasoning,
for example, can be obtained as the consensus over some other criteria in: cultural, economical, ethical or social context.
The reader can experiment on his own with the preferences from the example pointing his browser to http://decision.math.hr/examples/curriculum/. The
result of self-ranking the group of processes is given in the Table 4.5.
process weight potential
PROBLEM SOLVING
REASONING
COMMUNICATION
CONNECTIONS
REPRESENTATIONS
TABLE 4.5. The importance of five processes in education.
5. S ELFISH P OTENTIALS
For some special preference flows we can measure the speed of convergence in each
step in the process (7). This is the ’symmetric’ case which we are going to describe. Let
us introduce some notions first.
We say that potential Xi is selfish potential with parameter ε if
(11)
Xiτ = ε · · · M · · · ε
where (n − 1)ε + M = 0, ε < M and M is the i-th component of Xi . A selfish rank
is the rank vector obtained from the selfish potential by formula the (5).
The following statement is useful.
Proposition 4. Let Xi , i = 1, . . . , n are given potentials such that {aXi } are linearly
independent. Then, there exists a column stochastic matrix Z = [Z1 , . . . , Zn ], where
Zi ∈ Σ are selfish ranks, such that (∀ξ ∈ Σ) (∃η ∈ Σ) with the property
ΦX (ξ) = Zη.
Specialy, if the potentials Xi , i = 1, . . ., n are selfish potentials, then
Zi =
aXi
, i = 1, . . ., n.
||aXi ||1
10
LAVOSLAV ČAKLOVIĆ, ZAGREB
Proof.
aXξ
i
=
=
(12)
n
Y
k=1
n
Y
aξk Xk
(aXk )i
k=1
≥ min aXk
k
i
i
ξk
=: εi
If we denote ǫ = (ε1 , . . ., εn ) and ei the i-th vector of the standard base in Rn then,
−hei , aX − ǫi ≤ 0, i = 1, . . ., n,
which implies that
aX − ǫ ∈ Rn+ .
(13)
Let us consider a half-line γi = {ǫ + tei | t ≥ 0} and a hyperplane H generated by
{aX1 , . . . , aXn }. They have one and only only point in common, let us denote it xi , and
let ti satisfies
xi =: ǫ + ti ei , ti > 0.
(14)
Evidently,
ǫ ∈ cone (x1 , . . ., xn ).
(15)
On the other side, by arithmetic-geometric mean, we have that
aXξ
i
≤
m
X
ξk aXk
i=1
i
, i = 1, . . ., n,
and the consequence is that aXξ and ǫ are elements of the same half-space generated by
H. This implies
aXξ ∈ cone (ǫ, x1 , . . ., xn )
and, because of (15)
aXξ ∈ cone (x1 , . . ., xn ).
If we denote
xi
, i = 1, . . ., n
kxi k
then, because of the previous formula, ΦX (ξ) is a convex combination of z1 , . . ., zn i.e.
Zi =
ΦX (ξ) = Zη
for some η ∈ Σ.
Corollary 5. Let us suppose that Xi , i = 1, . . ., n are selfish potentials with parameter ε.
Denote:
mε := (n − 1)aε + aM , t :=
aM − aε
,
mε
Then,
(i) ΦX (Zξ) = ΦtX (ξ), ∀ξ ∈ Σ.
(ii) ΦX (ΦtX (ξ)) = Φτ (t)X (η), for some η ∈ Σ
(iii) Φn+1
X (ξ) = Φτ n (1)X (η), for some η ∈ Σ.
τ (t) :=
at(M−ε) − 1
n − 1 + at(M−ε)
SELF–DUALITY IN GROUP DECISION
11
Proof. Because of (11)
aXi
and
ΦX (Xi ) =

aε
 .. 
 . 
 M

=
a 
 . 
 .. 
aε

aXi
=: Zi .
(n − 1)aε + aM
Because of Lemma 4 we can calculate ΦX (ΦX (ξ)) = ΦX (Zλ) for some λ ∈ Σ. Let us
calculate XZ first.
X
Xj (Zi )j
XZi =
j
=
X
Xj (Zi )j + Xi (Zi )i
j6=i
1 X
1
1
X j aε +
X i aM ±
X i aε
mε
mε
mε
j6=i
!
1
1 X
X j aε +
Xi (aM − aε ),
=
mε
m
ε
j
| {z }
=
=0
which implies
XZ = [XZ1 , . . . , XZn ]
aM − aε
[X1 , . . . , Xn ]
mε
aM − aε
X
=
mε
=
and
aXZλ = atXλ , ΦX (Zλ) = ΦtX (λ),
which proves (i).
To prove (ii) we use the same idea as in the proof of (i). Obviously, because ΦtX (ξ) ∈
Σ, we can write
atXξ
ΦtX (ξ) = tXξ = Z t η, for some η ∈ Σ,
|a |1
where the i-th column of the stochastic matrix Z t is
Zit =
Let us calculate XZit .
[atε . . . atM . . . atε ]τ
.
a(n−1)tε + atM
12
LAVOSLAV ČAKLOVIĆ, ZAGREB
XZit =
X
Xj (Zit )j + Xi (Zit )i
j6=i
1
1
1 X
Xj atε +
Xi atM ±
Xi atε
mtε
mtε
mtε
j6=i
!
X
1
1
=
Xj atε +
Xi (atM − atε ),
mtε
m
tε
j
| {z }
=
=0
tM
=
a − atε
Xi ,
(n − 1)atε + atM
which implies
XZ t = τ (t)X,
and
ΦX (ΦtX (ξ)) = ΦX (Z t η)
t
aXZ
= XZ t
ka
k
aτ (t)Xη
kaτ (t)Xη k
= Φaτ (t) X (η)
=
for some η ∈ Σ, which proves (ii). The proof of (iii) may be done by induction using (ii).
Remark 1. The geometrical meaning of τ (t) is the following. If e denotes one vertex of Σ,
ξ0 = n1 1 and et = (1 − t)ξ0 + te then
ΦX (et ) = eτ (t)
which shows that τ (t) measures the distance of ΦX (et ) from ξ0 . This claim is evident
from the following calculation:
aXet = a(1−t)Xξ0 ◦ atXe
∼ [atε . . . atM . . . atε ]τ
∼ [1 . . . at(M−ε) . . . 1]τ
(where ◦ denotes Hadamard’s product and ∼ denotes proportionality),
at(M−ε) − 1
1
ξ0 +
e
t(M−ε)
n−1+a
n − 1 + at(M−ε)
= (1 − τ (t))ξ0 + τ (t)e.
ΦX (et ) =
Theorem 6. Let us suppose that
(16)
|ε| ln a < 1.
1
n 1 and
n
ΦX (ξ)
Then ΦX : Σ → Σ has a unique fixed point ξ0 :=
ξ0 = lim
n→+∞
it can be calculated as a limit
SELF–DUALITY IN GROUP DECISION
13
2
1.5
1
0.5
-1
-0.5
0.5
1
1.5
2
-0.5
-1
F IGURE 1. Graph of τ (t) for τ ′ (0) > 1.
where ξ ∈ Σ is arbitrary.
Proof. From the Corollary 5 (iii) we have that for each n ∈ N there exists ηn ∈ Σ such
that
Φn+1
X (ξ) = Φτ n (1)X (ηn ).
A careful analysis of the function
τ (t) =
at(M−ε) − 1
n − 1 + at(M−ε)
(Figure 1) gives that τ n (1) → 0 if τ ′ (0) < 1. Easy calculation gives
τ ′ (t) = n(M − ε) ln a
at(M−ε)
(n − 1 + at(M−ε) )2
and
τ ′ (0) =
M −ε
ln a = |ε| ln a < 1
n
which proves the theorem.
5.1. Numerical experiment. The group has four elements G = {A, B, C, D} and the
preference flow is given in the Table 5.6 bellow. For the value a = 2 and the error 10−13
the fixed point is given in the last row obtained after 78 iterations.
In the Table 5.7 are given limit points of the sequence ΦX (ξ), where the initial point is
ξ0 = [0.2, 0.4, 0.2, 0.2]τ , for different values of the parameter a. The preference flow is the
same one as above. Calculation was done using Mathematica 5.0. It should be emphasized
that the limit point is
Φ(limn τ n (1))X (0, 1, 0, 0).
For the value a = 2.378414230005443 the derivative τ ′ (0) is slightly greater than 1
and the second component of the fixed point has the value of t-intercept of the intersection
of the graph of τ (t) and the diagonal τ = t.
In Table 5.8 we give a list of fixed points for some initial points (a = 4). The number
M is any number greater than m, mi .
14
LAVOSLAV ČAKLOVIĆ, ZAGREB
a=2
A
B
C
D
F(A,B)
4
−4
0
0
F(A,C)
4
0
−4
0
F(A,D)
4
0
0
−4
F(C,B)
0
−4
4
0
F(D,B)
0
−4
0
4
F(D,C)
0
0
−4
4
0.2
0.4
0.2
0.2
init. point
fixed point 0.25 0.25 0.25 0.25
TABLE 5.6. Fix point for a = 2.
A
B
C
D
0.2
0.4
0.2
0.2
iter.
initial point
a=2
78
a = 2.378414230005442 10
8
0.250 0.250 0.250 0.250
0.2
0.4
0.2
0.2
a = 2.378414230005443 405
0.057 0.828 0.057 0.057
a = 2.5
58
0.041 0.877 0.041 0.041
a=5
13
0.002 0.995 0.002 0.002
a=9
9
0.000 1.000 0.000 0.000
TABLE 5.7. Limit points for different values of a.
a=4
initial point
limit point
initial point
limit point
initial point
limit point
initial point
limit point
A
B
C
D
M
M
M
M
0.250 0.250 0.250 0.250
m
M
M
M
0.097 0.301 0.301 0.301
m1
m2
M
M
0.048 0.048 0.452 0.452
m1
m2
m3
M
0.004 0.004 0.004 0.987
TABLE 5.8. Limit points for a = 4.
A DDENDUM
For the sake of completeness we give the program code written in Mathematica which
was used to obtain the above numerical results.
SELF–DUALITY IN GROUP DECISION
15
(* filename: potmetfix.nb *)
(* input data *)
X={{3,-1,-1,-1},{-1,3,-1,-1},{-1,-1,3,-1},{-1,-1,-1,3}};
(* functions definitions *)
F[x_] := Catch[ n=0; Map [(
n +=Abs[ #] )&,x] ; Throw[n//N] ];
finito[e_,j_,x_, mError_] :=
(
If[ e < mError, Print["w_",j,"
(*Print["error = ",e//N];*)
Break[]
]
);
= ",x];
Iterate [b_,x_] :=
(
y =x . X;
y=b^y//N;
y= 1/F[y] * y//N;
Return[y//N];
);
fixedPoint[base_,wInit_,jMax_,mErr_] :=
(
w=1/F[wInit] * wInit//N;
For[
j=1, j\[LessEqual]jMax, j++,
x=w;
w = Catch[ Iterate[base//N,w] ];
error = F[x-w];
If [ mErr > 0, finito[ error, j, w,mErr] ,
If [ j \[Equal] jMax, Print["w_",j," = ",w]
];
)
]
];
fixedPoint[2,{1,2,1,1},500,N[10^(-13)]];
R EFERENCES
[1] J. Aczel and T. L. Saaty, Procedures for synthesizing Ratio Judgements, J. Math. Psychology, Vol. 27, No.
1, 1983.
[2] Aichison, J. The statistical Analysis of compositional data, The Blackburn Press, Caldwell, 2003.
[3] J. A. Bobdy, U. S. R. Murty: Graph Theory and Applications, Macmillan, London, 1976.
[4] Čaklović L. A Graph approach to MCDM, in preparation
[5] Čaklović, L., An IO–Modification of Potential Method, Electronic Notes in Discrete Mathematics, Elsevier,
33 (2009), 131–138
[6] Čaklović, L. and Hunjak T., Measurement of DMU–efficiency, submitted
16
LAVOSLAV ČAKLOVIĆ, ZAGREB
[7] Condorcet, Marquis de, Essai sur l’application de l’analyse à la probabilité des décisions rendues à la
probabilité des voix, Paris, de l’imprimerie royale, 1785.
[8] Cooper, W. W., Seiford, L. M. and K. Tone, Data Envelopment Analysis, Kluwer, 2002
[9] Saaty T. L. (1996) The Analytic Hierarchy Process, RWS Publications, Pittsburgh
[10] M. Struwe, Variational Methods, Second ed., Springer 1996.