Continuous Preferences Upper Contour Set is Open

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 [edit]

Given a relation on pairs of elements of set X {\displaystyle X}

X 2 {\displaystyle \succcurlyeq ~\subseteq ~X^{2}}

and an element x {\displaystyle x} of X {\displaystyle X}

x X {\displaystyle x\in X}

The upper contour set of x {\displaystyle x} is the set of all y {\displaystyle y} that are related to x {\displaystyle x} :

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

The lower contour set of x {\displaystyle x} is the set of all y {\displaystyle y} such that x {\displaystyle x} is related to them:

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

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

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

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

{ y ( x y ) ¬ ( y x ) } {\displaystyle \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

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

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

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

and the strict lower contour set of x {\displaystyle x} is

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

Contour sets of a function [edit]

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

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

Examples [edit]

Arithmetic [edit]

Consider a real number x {\displaystyle x} , and the relation {\displaystyle \geq } . Then

Consider, more generally, the relation

( a b ) [ f ( a ) f ( b ) ] {\displaystyle (a\succcurlyeq b)~\Leftarrow ~[f(a)\geq f(b)]}

Then

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

( a b ) [ f ( a ) f ( b ) ] {\displaystyle (a\succcurlyeq b)~\Leftarrow ~[f(a)\leq f(b)]}

though such definitions would tend to confound ready understanding.

In the case of a real-valued function f ( ) {\displaystyle 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 b ) [ f ( a ) f ( b ) ] {\displaystyle (a\succcurlyeq b)~\Leftarrow ~[f(a)\geq f(b)]}

Note that the arguments to f ( ) {\displaystyle f()} might be vectors, and that the notation used might instead be

[ ( a 1 , a 2 , ) ( b 1 , b 2 , ) ] [ f ( a 1 , a 2 , ) f ( b 1 , b 2 , ) ] {\displaystyle [(a_{1},a_{2},\ldots )\succcurlyeq (b_{1},b_{2},\ldots )]~\Leftarrow ~[f(a_{1},a_{2},\ldots )\geq f(b_{1},b_{2},\ldots )]}

Economics [edit]

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

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

Such preferences might be captured by a utility function u ( ) {\displaystyle u()} , in which case

Complementarity [edit]

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

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

See also [edit]

  • Epigraph
  • Hypograph

References [edit]

  1. ^ a b Robert P. Gilles (1996). Economic Exchange and Social Organization: The Edgeworthian Foundations of General Equilibrium Theory. Springer. p. 35. ISBN9780792342007.

Bibliography [edit]

  • 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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Source: https://en.wikipedia.org/wiki/Contour_set

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