Eigenvectors and Decision Making

• Analytic Hierarchy Process (AHP) • Positive Symmetrically Reciprocal Matrices

(PSRMs) and Transitive PSRMs • Estimation methods -- How close? Close to what?

Close in what sense?

Analytic Hierarchy Process

“In the 1970’s the Egyptian government asked Tom Saaty, a pioneering mathematician with a fistful of awards, to help clarify the Middle East conflict. The Egyptians needed a coherent, analytical way to assess the pros and cons of their less than cozy strategic relationship with the Soviet Union. Saaty, a Wharton professor with a background in arms-control research, tackled the question with game theory.”

Analytic Hierarchy Process

“The Egyptian government was pleased with his work (and eventually did ask the Russians to leave), but Saaty himself wasn’t satisfied with the process. He felt his conclusion was incomplete – that important but intangible information was left out of the final equation because game theory was too rigid. ‘I couldn’t use it to solve a real-life problem.’”

Analytic Hierarchy Process

• Xerox (Software: • Boeing Expert • IBM Choice)• Northrop Grumman• US Steel• Corning• Governments of U.S., South Africa, Canada, and

Indonesia

Analytic Hierarchy Process

Decision:

Choose between a

Toyota, Honda, or Citation (Chevrolet)

Analytic Hierarchy Process

One set of criteria: cost, dependability, size, and aestheticsFirst we want to assign numbers to these four criteria that

represent their relative importance w1, w2, w3, w4 .

Cost Depen Size Aesth Cost 1 2 3 3Dependability 1/2 1 3 3Size 1/3 1/3 1 1/2Aesthetics 1/3 1/3 2 1

This matrix has maximum eigenvalue approximately 4.08 and eigenvector is (.44, .31, .10, .15).

Analytic Hierarchy Process

A n x n matrix

x n x 1

λ scaler

Ax = λx

(λI-A)x = 0

det(λI-A) = 0

Analytic Hierarchy Process

15.

1.

31.

44.

08.4

15.

1.

31.

44.

123/13/1

2/113/13/1

3312/1

3321

Generate reciprocal matrices for each criterion, comparing the three types of cars pairwise on each criterion as shown below.

 For Dependability:

Toy Hon Cit Weights  Toy 1 1 3 .43

Hon 1 1 3 .43Cit 1/3 1/3 1 .14

 For Cost:

Toy Hon Cit Weights 

Toy 1 1 2 .40Hon 1 1 1 .35Cit 1/2 1 1 .25

 

 For Size:

Toy Hon Cit Weights 

Toy 1 3 1 .43Hon 1 1 3 .14Cit 1/3 1/3 1 .43

 For Aesthetics:

Toy Hon Cit Weights 

Toy 1 4 3 .63Hon 1/4 1 2 .22Cit 1/3 1/2 1 .15

The weight for each criterion is then multiplied by the weight for each car within that criterion, and added across criteria as shown below:

  Cost Depen Size Aesth Composite Priorities (.44) (.31) (.10) (.15)

 Toy .40 .43 .43 .63 .45Hon .35 .43 .14 .33 .33Cit .25 .14 .43 .15 .22

Positive Symmetrically Reciprocal Matrices

n x n matrix A 

aij > 0 

aji = 1/aij 

Transitive or consistent: aik = aijajk

wi/wj = wi/wk x wk/wj

(The entries aij represent the ratio wi/wj.)

Cost Depen Size Aesth

Cost w1/w1 w1/w2 w1/w3 w1/w4

Dependability w2/w1, w2/w2 w2/w3 w2/w4

Size w3/w1 w3/w2 w3/w3 w3/w4

Aesthetics w4/w1 w4/w2 w4/w3 w4/w4

 

w1/w1 w1/w2 w1/w3 w1/w4 w1 w1

w2/w1, w2/w2 w2/w3 w2/w4 w2 = 4 w2

w3/w1 w3/w2 w3/w3 w3/w4 w3 w3

w4/w1 w4/w2 w4/w3 w4/w4 w4 w4

Positive Symmetrically Reciprocal Matrices

Theorem (Perron-Frobenius): (Let A be non-negative and irreducible.) (1) A has a real positive simple (not multiple) eigenvalue λmax (Perron root) which is not exceeded in modulus by any other eigenvalue of A.

(2) The eigenvector of A corresponding to the eigenvalue λmax has positive components and is essentially unique (to within multiplication by a constant).

Theorem: A positive, reciprocal matrix is consistent if and only if λmax = n. (Saaty. J of Mathematical Psychology, 1977).

Positive Symmetrically Reciprocal Matrices

Theorem:

Let R be an n x n positive reciprocal matrix with entries

1/S < rij < S, 1 < i, j < n, for some S >1

and let λmax denote the largest eigenvalue of R in modulus, which is known to be real and positive. Then

n < λmax < 1 + ½(n-1)(S + 1/S),

the lower and upper bound being reached if and only if R is supertransitive or maximally intransitive, respectively.

(Aupetit and Genest – European Journal of Operational Research, 1993):

Positive Symmetrically Reciprocal Matrices

Theorem:A transitive matrix is SR and has rank one. (Farkas, Rozsa, Stubnya, Linear Algebra Appl., 1999.)

Close?

15.

1.

31.

44.

08.4

15.

1.

31.

44.

123/13/1

2/113/13/1

3312/1

3321

Close?

• Saaty: Limit of normalized row sums of powers of the matrix.

• μ =(λmax-n)/(n-1)

• If we consider inconsistencies in the powers of A, generated by cycles of length 1, 2, 3, … then the right eigenvector represents the dominance of one alternative over all others through these cycles. (Mathematics Magazine, 1987.)

Aee

eAT

k

klim

Close?

• Logarithmic Least Square Method

• Least Squares

• Another kind of eigenvector in another metric, the “max eigenvector” minimizes relative error (Elsner & vanden Driesscle, Linear Algebra and Its Applications, 2004).

2

,

lnln

n

ji j

iij w

wa

2

,

n

ji j

iij w

wa

Finally

Why do I care?

What am I doing with these matrices and eigenvectors?