Self-duality in decision process arise when alternatives are also criteria
for themselves. A typical example of self-duality is a selfassessment of a
group of decision makers. Each one of them create his own
preference graph over the set of group members including himself. Aggregation
of those graphs, a group preference graph, serves for calculating the group ranks.
In real life situations our goals are frequently unambiguous or even unknown.
Instead of them we are ruled by some vague principles and habits that even leads
to a conflict. Generaly, we have two possibilities:
reconsider the importance of our principles and goals,
introduce/reject new principles and repeat the process.
We shall analyze carefully the first case. From the mathematical point of
view this leads to the fixed point problem from nonlinear analysis.
If we change the importance/weights w of the goals, this directly influences
the weights of actions. Problem is that we do not know the weights
of our goals. Let us follow the mental process of reconsidering the goals
from the point of view of actions. For each action there are some goals that
support that actions more than the others. . .
. . . which means that each action can make its own preference
graph over the set of goals, and indirectly, each goal,
using the set of actions, defines preferences on the set of
Hierarchy in the figure below designs exactly this situation.
Figure: Hierarchy of selfranking problem.
For the hypothetical
weights w of the goals we now calculate the new weights of the same
goals, let us denote them ɸ(w). Repeating the process:
w -> ɸ(w) -> ɸ2(w) -> ... -> ɸn(w) -> ...
we get the infinite sequence of weights. It can be proved now that this
sequence has the a fixed point λ i.e. the point that satisfies equation
ɸ(λ) = λ
and moreover, this point is unique. We interpret it as the weights of the
goals that were unknown at the beginning of the process.
A simple numerical example
Use the following links to see input data or start the iterative process.
Please notify that fixed point of iterative process doesn't depend
upon the starting ranking of the goals. Fixed point λ is determined
by interior relations between the nodes in the hierarchy i.e. in
preference graphs. In this example we have
λ(G1) = 0.337,
λ(G2) = 0.228
λ(G3) = 0.434.
Notify that one iteration gives very good aproximation of the fixed point.
This is a consequence of the fact that convergence of iterative process is