Contour set

In mathematics, contour sets generalize and formalize the everyday notions of

everything superior to something
everything superior or equivalent to something
everything inferior to something
everything inferior or equivalent to something.

Formal definitions

Given a relation on pairs of elements of set $X$

\succcurlyeq~\subseteq~X^2

and an element $x$ of $X$

x\in X

The upper contour set of $x$ is the set of all $y$ that are related to $x$ :

\left\{ y~\backepsilon~y\succcurlyeq x\right\}

The lower contour set of $x$ is the set of all $y$ such that $x$ is related to them:

\left\{ y~\backepsilon~x\succcurlyeq y\right\}

The strict upper contour set of $x$ is the set of all $y$ that are related to $x$ without $x$ being in this way related to any of them:

\left\{ y~\backepsilon~(y\succcurlyeq x)\land\lnot(x\succcurlyeq y)\right\}

The strict lower contour set of $x$ is the set of all $y$ such that $x$ is related to them without any of them being in this way related to $x$ :

\left\{ y~\backepsilon~(x\succcurlyeq y)\land\lnot(y\succcurlyeq x)\right\}

The formal expressions of the last two may be simplified if we have defined

\succ~=~\left\{ \left(a,b\right)~\backepsilon~\left(a\succcurlyeq b\right)\land\lnot(b\succcurlyeq a)\right\}

so that $a$ is related to $b$ but $b$ is not related to $a$ , in which case the strict upper contour set of $x$ is

\left\{ y~\backepsilon~y\succ x\right\}

and the strict lower contour set of $x$ is

\left\{ y~\backepsilon~x\succ y\right\}

Contour sets of a function

In the case of a function $f()$ considered in terms of relation $\triangleright$ , reference to the contour sets of the function is implicitly to the contour sets of the implied relation

(a\succcurlyeq b)~\Leftarrow~[f(a)\triangleright f(b)]

Examples

Arithmetic

Consider a real number $x$ , and the relation $\geq$ . Then

the upper contour set of $x$ would be the set of numbers that were greater than or equal to $x$ ,
the strict upper contour set of $x$ would be the set of numbers that were greater than $x$ ,
the lower contour set of $x$ would be the set of numbers that were less than or equal to $x$ , and
the strict lower contour set of $x$ would be the set of numbers that were less than $x$ .

Consider, more generally, the relation

(a\succcurlyeq b)~\Leftarrow~[f(a)\ge f(b)]

Then

the upper contour set of $x$ would be the set of all $y$ such that $f(y)\ge f(x)$ ,
the strict upper contour set of $x$ would be the set of all $y$ such that $f(y)>f(x)$ ,
the lower contour set of $x$ would be the set of all $y$ such that $f(x)\ge f(y)$ , and
the strict lower contour set of $x$ would be the set of all $y$ such that $f(x)>f(y)$ .

It would be technically possible to define contour sets in terms of the relation

(a\succcurlyeq b)~\Leftarrow~[f(a)\le f(b)]

though such definitions would tend to confound ready understanding.

In the case of a real-valued function $f()$ (whose arguments might or might not be themselves real numbers), reference to the contour sets of the function is implicitly to the contour sets of the relation

(a\succcurlyeq b)~\Leftarrow~[f(a)\ge f(b)]

Note that the arguments to $f()$ might be vectors, and that the notation used might instead be

[(a_1 ,a_2 ,\ldots)\succcurlyeq(b_1 ,b_2 ,\ldots)]~\Leftarrow~[f(a_1 ,a_2 ,\ldots)\ge f(b_1 ,b_2 ,\ldots)]

Economic

In economics, the set $X$ could be interpreted as a set of goods and services or of possible outcomes, the relation $\succ$ as strict preference, and the relationship $\succcurlyeq$ as weak preference. Then

the upper contour set, or better set,^[1] of $x$ would be the set of all goods, services, or outcomes that were at least as desired as $x$ ,
the strict upper contour set of $x$ would be the set of all goods, services, or outcomes that were more desired than $x$ ,
the lower contour set, or worse set,^[1] of $x$ would be the set of all goods, services, or outcomes that were no more desired than $x$ , and
the strict lower contour set of $x$ would be the set of all goods, services, or outcomes that were less desired than $x$ .

Such preferences might be captured by a utility function $u()$ , in which case

the upper contour set of $x$ would be the set of all $y$ such that $u(y)\ge u(x)$ ,
the strict upper contour set of $x$ would be the set of all $y$ such that $u(y)>u(x)$ ,
the lower contour set of $x$ would be the set of all $y$ such that $u(x)\ge u(y)$ , and
the strict lower contour set of $x$ would be the set of all $y$ such that $u(x)>u(y)$ .

Complementarity

On the assumption that $\succcurlyeq$ is a total ordering of $X$ , the complement of the upper contour set is the strict lower contour set.

X^2\backslash\left\{ y~\backepsilon~y\succcurlyeq x\right\}=\left\{ y~\backepsilon~x\succ y\right\}

X^2\backslash\left\{ y~\backepsilon~x\succ y\right\}=\left\{ y~\backepsilon~y\succcurlyeq x\right\}

and the complement of the strict upper contour set is the lower contour set.

X^2\backslash\left\{ y~\backepsilon~y\succ x\right\}=\left\{ y~\backepsilon~x\succcurlyeq y\right\}

X^2\backslash\left\{ y~\backepsilon~x\succcurlyeq y\right\}=\left\{ y~\backepsilon~y\succ x\right\}

References

1 2 Robert P. Gilles (1996). Economic Exchange and Social Organization: The Edgeworthian Foundations of General Equilibrium Theory. Springer. p. 35.

Bibliography

Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green, Microeconomic Theory (LCC HB172.M6247 1995), p43. ISBN 0-19-507340-1 (cloth) ISBN 0-19-510268-1 (paper)

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