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 ${\displaystyle Y={\big \{}y_{1},y_{2},\ldots ,y_{n}{\big \}}}$. The set ${\displaystyle S\left(Y\right)}$ is the smallest set of simple polynomials such that:

• ${\displaystyle y_{i}\in S\left(Y\right),i=1,\ldots ,n}$;
• ${\displaystyle Y_{1},Y_{2}\subset Y}$ such that ${\displaystyle Y_{1}\cap Y_{2}=\varnothing ,G_{1}\in S\left(Y_{1}\right)}$ and ${\displaystyle G_{2}\in S\left(Y_{2}\right)}$, then ${\displaystyle G_{1}+G_{2}\,}$ and ${\displaystyle G_{1}G_{2}\,}$ are in ${\displaystyle S\left(Y\right)}$.

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 ${\displaystyle \times }$ 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:

• Additive: ${\displaystyle A+P+U\,}$;
• Distributive: ${\displaystyle \left(A+P\right)U\,}$; plus 2 others obtained by interchanging A, P and U;
• Dual distributive: ${\displaystyle AP+U\,}$ plus 2 others as per above;
• Multiplicative: ${\displaystyle APU\,}$.

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 ${\displaystyle A={\big \{}a,b,c,\ldots {\big \}}}$, ${\displaystyle P={\big \{}p,q,r,\ldots {\big \}}}$ and ${\displaystyle U={\big \{}u,v,w,\ldots {\big \}}}$ be non-empty and disjoint sets. Let " ${\displaystyle \succsim }$ " be a simple order. Krantz et al. (1971) argued the quadruple ${\displaystyle Z=\langle A,P,U,\succsim \rangle }$ is a polynomial conjoint system if and only if the following axioms hold.

• WEAK ORDER.
• SINGLE CANCELLATION. The relation " ${\displaystyle \succsim }$ " satisfies single cancellation upon A whenever ${\displaystyle \left(a,p,u\right)\succsim \left(b,p,u\right)}$ if and only if ${\displaystyle \left(a,q,v\right)\succsim \left(b,q,v\right)}$ holds for all ${\displaystyle a,b\in A;p,q\in P}$ and ${\displaystyle u,v\in U}$. Single cancellation upon P and U is similarly defined.
• DOUBLE CANCELLATION. The relation " ${\displaystyle \succsim }$ " upon ${\displaystyle A\times P}$ satisfies double cancellation if and only if for all ${\displaystyle a,b,c\in A}$ and ${\displaystyle p,q,r\in P}$, ${\displaystyle \left(a,q,u\right)\succsim \left(b,p,u\right)}$ and ${\displaystyle \left(b,r,u\right)\succsim \left(c,q,u\right)}$ therefore ${\displaystyle \left(a,r,u\right)\succsim \left(c,p,u\right)}$ is true for all ${\displaystyle u\in U}$. The condition holds similarly upon ${\displaystyle A\times U}$ and ${\displaystyle U\times P}$.
• JOINT SINGLE CANCELLATION. The relation " ${\displaystyle \succsim }$ " upon ${\displaystyle A\times P}$ satisfies joint single cancellation such that ${\displaystyle \left(a,p,u\right)\succsim \left(b,q,u\right)}$ if and only if ${\displaystyle \left(a,p,v\right)\succsim \left(b,q,v\right)}$ is true for all ${\displaystyle a,b\in A;p,q\in P}$ and ${\displaystyle u,v\in U}$. Joint independence is similarly defined for ${\displaystyle A\times U}$ and ${\displaystyle U\times P}$.
• DISTRIBUTIVE CANCELLATION. Distributive cancellation holds upon ${\displaystyle A\times P\times U}$ if and only if ${\displaystyle \left(a,p,u\right)\succsim \left(c,r,v\right)}$, ${\displaystyle \left(b,q,u\right)\succsim \left(d,s,v\right)}$ and ${\displaystyle \left(d,r,v\right)\succsim \left(b,p,u\right)}$ implies ${\displaystyle \left(a,q,u\right)\succsim \left(c,s,v\right)}$ is true for all ${\displaystyle a,b,c,d\in A;p,q,r,s\in P}$ and ${\displaystyle u,v\in U}$.
• DUAL DISTRIBUTIVE CANCELLATION. Dual distributive cancellation holds upon ${\displaystyle A\times P\times U}$ if and only if

${\displaystyle \left(a,r,w\right)\succsim \left(c,s,v\right)}$, ${\displaystyle \left(d,p,u\right)\succsim \left(b,t,x\right)}$, ${\displaystyle \left(d,r,x\right)\succsim \left(e,s,u\right)}$ and ${\displaystyle \left(c,t,y\right)\succsim \left(d,q,y\right)}$ implies ${\displaystyle \left(a,p,v\right)\succsim \left(b,q,w\right)}$ is true for all ${\displaystyle a,b,c,d,e\in A;p,q,r,s,t\in P}$ and ${\displaystyle u,v,w,x,y\in U}$.

• SOLVABILITY. The relation " ${\displaystyle \succsim }$ " upon ${\displaystyle A\times P\times U}$ is solvable if and only if for all ${\displaystyle a,b\in A;p,q\in P}$ and ${\displaystyle u,v\in U}$, there exists ${\displaystyle c\in A;r\in P}$ and ${\displaystyle w\in U}$ such that ${\displaystyle a\sim \left(b,q,w\right)\sim \left(b,r,v\right)\sim \left(c,q,v\right)}$.
• ARCHIMEDEAN CONDITION.

Representation theorems

The quadruple ${\displaystyle Z=\langle A,P,U,\succsim \rangle }$ falls into one class of three variable simple polynomials by virtue of the joint single cancellation axiom.