The Edgeworth-Pareto Principle
in Decision Making
Vladimir D. Noghin
Saint-Petersburg State University
Russia
URL: www.apmath.spbu.ru/staff/noghin
dgmo-2006
Introduction
Since the 19 century, the Edgeworth-Pareto principle is an
effective tool for solving multicriteria problems.
A ‘naive’ version of this principle states that we should
make our choice within the set of Pareto optimal
alternatives.
There are practical situations when this principle does not
‘work’. In such cases selected alternatives are not
necessarily Pareto optimal.
In this connection, it is important to describe a class of
multicriteria choice problems for which the EdgeworthPareto principle is valid. It may be correctly done on the
basis of axiomatic justification.
Below an axiomatic approach is applied to separate
multicriteria choice problems for which the EdgeworthPareto principle may be successfully applied.
Content
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Historical aspects
Multicriteria problem
Multicriteria choice model
Axiomatic Edgeworth-Pareto principle
The principle for a fuzzy preference relation
The principle in terms of choice function
Extension
References
Francis Edgeworth
(1845-1926)
F. Edgeworth was a brilliant British
economist: almost the whole of his
literary output was addressed to his
fellow economists, taking the form of elegant technical
essays on taxation, monopoly and duopoly pricing, the pure
theory of international trade and the theory of index
numbers. He was introduced ‘indifference curves’, the ‘core’
of an exchange economy, and the so-called ‘Edgeworth box’
based on a concept of local Pareto optimality for two criteria.
Vilfredo Pareto
(1848-1923)
V. Pareto was a famous Italian
economist and sociologist. In his most
valuable work Manual in Political
Economy (1906) he presented the basis of modern economics
of welfare and introduced a concept of efficiency (‘Pareto
optimality’) in a local sense as a state that could not be
locally improved by any member of economics without
worsening of the state of at least one of the other members.
Famous Contributors to Multicriteria
Optimization Theory
• А. Wald (1939)
Dealt with a concept of maximal element of partially
ordered set (i.e. with Pareto optimality in a global sense)
• G. Birkgoff (1940)
Obtained a characterization of complete transitive binary
relation in terms of lexicographic order
• М. Slater (1950)
Introduced a concept of weakly efficient point and
derived optimality condition for these points in a saddlepoint form
• Т. Koopmans (1951)
Applied a concept of Pareto optimality to analyze
production and allocation problems
Famous Contributors to Multicriteria
Optimization Theory
• X. Kuhn – А. Tucker (1951)
Authors of the paper Nonlinear Programming where they
presented different optimality conditions for vectorvalued goal function under nonlinear constraints
• D. Gale – X. Kuhn – А. Tucker (1951)
Proposed duality theory for linear multiobjective
programming
• К. Arrow – E. Barankin – D. Blackwell (1953)
Proved that a Pareto set is dense in the set of all optimal
points of some linear scalarizing functions
• L. Hurwicz (1958)
Extended main results by X. Kuhn and A. Tucker to
general linear vector spaces
Famous Contributors to Multicriteria
Optimization Theory
• S. Karlin (1959)
Using linear scalarizing functions with nonnegative
coefficients, obtained a necessary condition for weakly
efficient points in convex case (this condition was
implicitly presented in the paper by M. Slater, 1950)
• Yu. Germeyer (1967)
Was the first who received optimality conditions for
weakly efficient points using
‘maxmin’ scalarizing
function. Later this condition was rediscovered several
times
• А. Geoffrion (1968)
Introduced a concept of proper efficient point and
established some optimality conditions for these points
• B. Peleg (1972)
Studied some topological properties of the Pareto set.
Multicriteria (Vector) Optimization
Is a generalization of scalar optimization theory. One
operates with the following two objects
• X is a set of feasible alternatives (points, vectors)
• f = (f1,...,fm) is a numerical vector-valued function,
defined on X
Main topics:
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Optimality conditions
Existence theorems
Duality theory
Numerical methods
Multicriteria Problem
〈 X, f 〉
On the page 5 of her book ‘Nonlinear Multiobjective
Optimization’ Prof. Kaisa Miettenen writes: a multicriteria
problem is to maximize on X all the objective
functions f1,...,fm simultaneously, assuming that there
does not exist a single solution that is optimal with respect to
every objective function.
Unfortunately, this passage does not explain what is a
solution of the problem, i.e. which alternatives
(vectors) must be chosen from X?
Meanwhile, in practice a major question is what are the ‘best’
solutions and how to find them?
Pareto Set
For maximization problem a Pareto set is defined by
Pf(X) = {x* ∈ X | does not exist x ∈ X such that f(x*) ≥ f(x)}
where a ≥ b means ai ≥ bi , i=1,2,..m, and a ≠ b.
Prof. Kaisa Miettenen on the page 13 of the mentioned book
declares: ‘Usually, we are interested in Pareto optimal
solutions and can forget all other solutions.’
Why?
She does not give an answer. So do almost all other authors.
Question
How to find out when selected solutions must be Pareto
optimal and when they may be non-Pareto optimal?
In other words, how to describe mathematically a class of
multicriteria problems in which just Pareto optimal
solutions are desirable?
In order to answer to this question we need to extend the
multicriteria model 〈X, f〉 and then to apply an
axiomatic approach.
Multicriteria Choice Model
〈 X, f, > 〉
consists of
• a set of feasible alternatives X
• a numerical vector-valued function f = (f1,...,fm)
• an asymmetric binary (preference) relation > of a Decision
Maker (DM) defined on X.
The relation > describes personal preferences of the DM, so
that
x1 > x2 means that the DM x1 prefers to x2.
Solution of
Multicriteria Choice Problem
Let X and f = (f1,...,fm) be fixed.
Possessing the reference relation >, the Decision Maker has
to select from X one or more alternatives which are the
‘best’ for him/her.
We will denote this set by Sel(X), Sel(X) ⊂ X, and call it a set
of selected alternatives.
To solve the multicriteria choice problem means to find Sel(X).
Axioms of Reasonable Choice
Axiom 1 (Pareto Axiom):
∀ x1 , x2 ∈ X :
f(x1) ≥ f(x2) ⇒ x2 ∉ Sel(X).
Axiom 2 (Axiom of exclusion of dominated
alternatives):
∀ x1 , x2 ∈ X :
x1 > x2 ⇒ x2 ∉ Sel(X).
These two axioms determine a ‘reasonable’ behavior of the
DM in decision making process.
Axiomatic Edgeworth-Pareto Principle
Theorem 1. Let Pareto Axiom be accepted. Then for any
Sel(X) satisfying Axiom 2 the inclusion
Sel(X) ⊂ Pf(X)
(1)
is valid.
This theorem says that a ‘reasonable’ DM makes his\her
choice only within Pareto optimal alternatives.
Remark. Theorem 1 is true for arbitrary nonempty set X as
well as for arbitrary numerical vector-valued function f.
Proof of Theorem 1
Let us introduce a set of non-dominated alternatives
Nf(X) ={x* ∈ X | does not exist x ∈ X such that x* > x}. (a)
First prove the inclusion
Sel(X) ⊂ Nf(X).
(b)
Assume the contrary: there exists x ∈ Sel(X) such that
x ∉ Nf(X). According to (a), we have x* > x for some x* ∈
X. Applying Axiom 2, we obtain x ∉ Sel(X). It contradicts the
initial assumption x ∈ Sel(X) .
Similarly, using Axiom 1 we can easily prove
Nf(X) ⊂ Pf(X).
(c)
The inequalities (b) − (c) imply (1). Q.E.D.
Geometric Illustration
X
Pf (X)
Sel(X)
Minimality Property of the Axioms
Theorem 2. If at least one of above two axioms is
ignored, then the inclusion (1) may be violated.
The proof contains two counterexamples, which are
omitted here.
Some Conclusions
• If we propose (or use) some numerical method to
compute definite Pareto optimal solution (or solutions)
as the ‘best’, then we must assume that both
‘reasonable’ axioms are satisfied. Otherwise (when at
least one of the axioms is ignored), the ‘best’ solution
may be non-Pareto optimal and it cannot be determined
by the proposed (used) method.
• Selecting some non-Pareto optimal solution as the ‘best’,
we reject at least one of two ‘reasonable’ axioms.
Why WE Usually Select Pareto
Optimal Solutions
Let us return to the page 13 of the book by Prof. Kaisa
Miettenen where she discusses a value of Pareto optimal
solution. Theorems 1-2 help us to conclude that
being a reasoning person, the DM usually makes
his\her choice according to two above
mentioned axioms. That is why usually we are
interested in Pareto optimal solutions and can
forget all other solutions.
But we should not forget that our choice may lay outside
the Pareto set if at least one of the axioms is unavailable
for us.
Fuzzy Sets and Fuzzy Relations
Let A be a nonempty set.
A fuzzy set X on A is defined by its membership function
λA(·) : A → [0, 1]. For every x ∈ A, the number λA(x) is
interpreted as the degree to which x is a member of X.
Standard set-theoretic operations were proposed for
fuzzy sets.
A fuzzy relation on A is defined by its membership
function μ(·,·) : A × A → [0, 1]. Here, μ(x,y) is
interpreted as the degree of confidence that the given
relation between x and y holds.
Fuzzy Multicriteria Choice Model
〈 X, f, > 〉
where
• X is a crisp set of feasible alternatives
• f = (f1,...,fm) is a numerical vector-valued function
• > is an asymmetric fuzzy preference relation with a
membership function μ(·,·).
Solution of the fuzzy multicriteria choice problem is a fuzzy
set of selected alternatives whose membership
function we will denote by λ(·).
Basic Axioms in Fuzzy Case
Axiom 1 (Fuzzy Pareto Axiom):
∀ x1 , x2 ∈ X :
f(x1) ≥ f(x2) ⇒ μ(x1,x2) = 1.
Axiom 2 (Axiom of exclusion of dominated
alternatives):
∀ x1 , x2 ∈ X :
μ(x1,x2) = μ* ∈ [0,1] ⇒ λ(x2) ≤ 1− μ*.
Edgeworth-Pareto Principle
(fuzzy case)
Theorem 3. Let Pareto Axiom be accepted. Then for any
λ(·) satisfying Axiom 2 the inequality
λ(x) ≤ λP(x)
∀x∈X
(2)
holds, where λP(·) is a membership function of Pareto set:
λP(x) =
{
λP(x) =1, if x ∈ Pf(X)
λP(x) = 0, if x ∉ Pf(X)
Choice Function
Let X be a nonempty set of alternatives. A class of all
nonempty subsets of X we will denote by X:
X = 2X\{∅}.
Definition. A single-valued mapping C defined on X that
assigns to every A ∈ X a certain set C(A) such that
C(A) ⊂ A is said to be a choice function.
Example. Let X = {a,b,c}. Then X = {a, b, c, {a,b}, {a,c},
{b,c}, {a,b,c}} and, for instance, C({a})={a}, C({b})={b},
C({c})={c}, C({a,b})={∅}, C({a,c})={a,c}, C({b,c})={b},
C({a,b,c})={a,c}.
Multicriteria Choice Model
〈 X, f, C 〉
where
• X is a set of feasible alternatives
• f = (f1,...,fm) is a numerical vector-valued function
• C is a choice function defined on X.
To solve this problem means to find C(X). This set consists of
chosen (selected) alternatives. Its cardinal number may be
greater or equal to 1.
In practice, usually, information on C is only partial. By this
reason, C(X) is unknown a priori.
Axioms of Reasonable Choice
Axiom 1 (Pareto Axiom in terms of choice function):
∀ x1 , x2 ∈ X :
f(x1) ≥ f(x2) ⇒ x2 ∉ C(X).
Axiom 2 (Axiom of Exclusion):
∀ x1 , x2 ∈ X :
C({x1,x2}) = {x1} ⇒ x2 ∉ C(X).
According to Axiom 2, if x1 is not selected from {x1,x2}, then
this alternative should not be selected from the whole X.
Edgeworth-Pareto Principle
(in terms of choice function)
Theorem 4. For any choice function C(X) satisfying above
two axioms it holds that
C(X) ⊂ Pf(X)
(3)
According to (3), Pf(X) can be considered as an upper
estimate for unknown C(X) (when above two axioms
are satisfied).
Extension
Extension of the presented results have been realized for
the following cases:
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Vector function f is not numerical
Set X is fuzzy
Choice function C is fuzzy
Some combinations of above points
References
1. Noghin V.D. A Logical justification of the Edgeworth-Pareto principle.
Comp. Mathematics and Math. Physics, 2002, V. 42, PP. 915-920.
2. Noghin V.D. The Edgeworth-Pareto principle and the relative
importance of criteria in the case of a fuzzy preference relation. Comp.
Mathematics and Math. Physics, 2003, V. 43, PP. 1604-1612.
3. Noghin V.D. A generalized Edgeworth-Pareto principle and the bounds
of its application, Economika i matem. metody, 2005, V. 41, No. 3, PP.
128-134 [in Russian].
4. Noghin V.D. Decision making in multicriteria environment: a
quantitative approach. 2nd ed., 2005, Fizmatlit, Moscow [in Russian].
5. Noghin V.D. The Edgeworth-Pareto principle in terms of choice
functions. Math. Social Sci., 2006, forthcoming.
6. Noghin V.D. The Edgeworth-Pareto principle in terms of a fuzzy
choice function. Comp. Mathematics and Math. Physics, 2006, V. 46,
PP. 554-562.
7. Noghin V.D., Volkova N.A. Evolution of the Edgeworth-Pareto
principle. Tavricheskii Vestnik Informatiki i Matematiki, 2006, No. 1, PP.
23-33 [in Russian].