Polynomial conjoint measurement
Encyclopedia
Polynomial conjoint measurement is an extension of the theory of conjoint measurement
to three or more attributes. It was initially developed by the mathematical psychologists David Krantz (1968) and Amos Tversky
(1967). The theory was given a comprehensive mathematical exposition in the first volume of Foundations of Measurement (Krantz, Luce, Suppes & Tversky, 1971), which Krantz and Tversky wrote in collaboration with the mathematical psychologist R. Duncan Luce
and philosopher Patrick Suppes
. Krantz & Tversky (1971) also published a non-technical paper on polynomial conjoint measurement for behavioural scientists in the journal Psychological Review.
As with the theory of conjoint measurement, the significance of polynomial conjoint measurement lies in the quantification of natural attributes in the absence of concatenation operations. Polynomial conjoint measurement differs from the two attribute case discovered by Luce & Tukey (1964) in that more complex composition rules are involved.
Let
. The set 
is the smallest set of simple polynomials such that:
Informally, the schema argues:
a) single attributes are simple polynomials;
b) if G1 and G2 are simple polynomials that are disjoint (i.e. have no attributes in common), then G1 + G2 and G1
G2 are simple polynomials; and
c) no polynomials are simple except as given by a) and b).
Let A, P and U be single disjoint attributes. From Krantz’s (1968) schema it follows that four classes of simple polynomials in three variables exist which contain a total of eight simple polynomials:
Krantz’s (1968) schema can be used to construct simple polynomials of greater numbers of attributes. For example, if D is a single variable disjoint to A, B, and C then three classes of simple polynomials in four variables are A + B + C + D, D + (B + AC) and D + ABC. This procedure can be employed for any finite number of variables. A simple test is that a simple polynomial can be ‘split’ into either a product or sum of two smaller, disjoint simple polynomials. These polynomials can be further ‘split’ until single variables are obtained. An expression not amenable to ‘splitting’ in this manner is not a simple polynomial (e.g. AB + BC + AC (Krantz & Tversky, 1971)).
, 
and 
be non-empty and disjoint sets. Let " 
" be a simple order. Krantz et al. (1971) argued the quadruple 
is a polynomial conjoint system if and only if the following axioms hold.
, 
, 
and 
implies 
is true for all 
and 
.
falls into one class of three variable simple polynomials by virtue of the joint single cancellation axiom.
Theory of conjoint measurement
The theory of conjoint measurement is a general, formal theory of continuous quantity. It was independently discovered by the French economist Gerard Debreu and by the American mathematical psychologist R...
to three or more attributes. It was initially developed by the mathematical psychologists David Krantz (1968) and Amos Tversky
Amos Tversky
Amos Nathan Tversky, was a cognitive and mathematical psychologist, a pioneer of cognitive science, a longtime collaborator of Daniel Kahneman, and a key figure in the discovery of systematic human cognitive bias and handling of risk. Much of his early work concerned the foundations of measurement...
(1967). The theory was given a comprehensive mathematical exposition in the first volume of Foundations of Measurement (Krantz, Luce, Suppes & Tversky, 1971), which Krantz and Tversky wrote in collaboration with the mathematical psychologist R. Duncan Luce
R. Duncan Luce
Robert Duncan Luce is the Distinguished Research Professor of Cognitive Science at the University of California, Irvine.Luce received a B.S. in Aeronautical Engineering from the Massachusetts Institute of Technology in 1945, and PhD in Mathematics from the same university in 1950...
and philosopher Patrick Suppes
Patrick Suppes
Patrick Colonel Suppes is an American philosopher who has made significant contributions to philosophy of science, the theory of measurement, the foundations of quantum mechanics, decision theory, psychology, and educational technology...
. Krantz & Tversky (1971) also published a non-technical paper on polynomial conjoint measurement for behavioural scientists in the journal Psychological Review.
As with the theory of conjoint measurement, the significance of polynomial conjoint measurement lies in the quantification of natural attributes in the absence of concatenation operations. Polynomial conjoint measurement differs from the two attribute case discovered by Luce & Tukey (1964) in that more complex composition rules are involved.
Krantz's (1968) schema
Most scientific theories involve more than just two attributes; and thus the two variable case of conjoint measurement has rather limited scope. Moreover, contrary to the theory of n - component conjoint measurement, many attributes are non-additive compositions of other attributes (Krantz, et al., 1971). Krantz (1968) proposed a general schema to ascertain the sufficient set of cancellation axioms for a class of polynomial combination rules he called simple polynomials. The formal definition of this schema given by Krantz, et al., (1971, p.328) is as follows.Let
. The set is the smallest set of simple polynomials such that:
-
; -
such that 
and 
, then 
and 
are in 
.
Informally, the schema argues:
a) single attributes are simple polynomials;
b) if G1 and G2 are simple polynomials that are disjoint (i.e. have no attributes in common), then G1 + G2 and G1
G2 are simple polynomials; andc) no polynomials are simple except as given by a) and b).
Let A, P and U be single disjoint attributes. From Krantz’s (1968) schema it follows that four classes of simple polynomials in three variables exist which contain a total of eight simple polynomials:
- Additive:
; - Distributive:
; plus 2 others obtained by interchanging A, P and U; - Dual distributive:
plus 2 others as per above; - Multiplicative:
.
Krantz’s (1968) schema can be used to construct simple polynomials of greater numbers of attributes. For example, if D is a single variable disjoint to A, B, and C then three classes of simple polynomials in four variables are A + B + C + D, D + (B + AC) and D + ABC. This procedure can be employed for any finite number of variables. A simple test is that a simple polynomial can be ‘split’ into either a product or sum of two smaller, disjoint simple polynomials. These polynomials can be further ‘split’ until single variables are obtained. An expression not amenable to ‘splitting’ in this manner is not a simple polynomial (e.g. AB + BC + AC (Krantz & Tversky, 1971)).
Axioms
Let, and be non-empty and disjoint sets. Let " " be a simple order. Krantz et al. (1971) argued the quadruple is a polynomial conjoint system if and only if the following axioms hold.- WEAK ORDER.
- SINGLE CANCELLATION. The relation "
" satisfies single cancellation upon A whenever 
if and only if 
holds for all 
and 
. Single cancellation upon P and U is similarly defined.
- DOUBLE CANCELLATION. The relation "
" upon 
satisfies double cancellation if and only if for all 
and 
, 
and 
therefore 
is true for all 
. The condition holds similarly upon 
and 
.
- JOINT SINGLE CANCELLATION. The relation "
" upon 
satisfies joint single cancellation such that 
if and only if 
is true for all 
and 
. Joint independence is similarly defined for 
and 
.
- DISTRIBUTIVE CANCELLATION. Distributive cancellation holds upon
if and only if 
, 
and 
implies 
is true for all 
and 
.
- DUAL DISTRIBUTIVE CANCELLATION. Dual distributive cancellation holds upon
if and only if
, , and implies is true for all and .- SOLVABILITY. The relation "
" upon 
is solvable if and only if for all 
and 
, there exists 
and 
such that 
.
- ARCHIMEDEAN CONDITION.
Representation theorems
The quadruple falls into one class of three variable simple polynomials by virtue of the joint single cancellation axiom.