The EdgeworthThe Edgeworth--Pareto Principle Pareto Principle

in Decision Makingin Decision Making

Vladimir D. Noghin Vladimir D. Noghin

SaintSaint--Petersburg State University Petersburg State University RussiaRussia

URL: URL: www.apmath.spbu.ru/staff/noghinwww.apmath.spbu.ru/staff/noghin dgmodgmo--20062006

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 Edgeworth-Pareto 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 Edgeworth-Pareto principle may be successfully applied.

Content

• 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 saddle-point 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 vector-valued 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:

• 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 toevery 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

(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.

Sel(X) ⊂ Pf(X)

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 inclusionSel(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.

X

)X(Pf

)X(Sel

Geometric Illustration

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. KaisaMiettenen 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) =1, if x ∈ Pf(X)λP(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:

• Vector function f is not numerical• Set X is fuzzy• Choice function C is fuzzy• Some combinations of above points

References1. 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].