Is it Preference relation?

Question

I need to check if the relation $ \succeq ( \space \succeq \space \subset X × X, \space X=VB[0,1] ) $ define as below $$ f \succeq g \Longleftrightarrow Var(f+g) \geq Varf $$ is preference relation. If it is preferance relation then is it represented by any utility function ?

I have no idea how to slove it with good explanation, i want to understand how i schoud deal with it.

Answer 1

A relation is called – mostly in economics terminology – preference relation if is reflexive, transitive, and dichotomous.

Az $R$ relation on an $X$ set is called

• reflexive, if for all $x \in X$: $(x,x) \in R$;
• transitive, if for all $x,y,z \in X$ if $(x,y) \in R$ and $(y,z) \in R$ then $(x,z) \in R$;
• dichotomous or total, if for all $x,y \in R$ $(x,y) \in R$ or $(y,x) \in R$ (possibly both) holds.

So to answer the first question we have to check the properties.

• reflexivity: $\operatorname{Var}(f+f)=\operatorname{Var}(2f) = 4\operatorname{Var}(f) \geq \operatorname{Var}(f)$
• etc $\dots$ I let you the rest! Check it!

To the second part of the problem you have to check that the relation is continous of not. $\succeq$ is countinous on $X$ is for all $x,y,z \in X$ the sets $\{ \alpha \in [0,1] \mid \alpha x+(1-\alpha)y \succeq z \}$ and $\{ \alpha \in [0,1] \mid z \succeq \alpha x+(1-\alpha)y \}$ sets are closed. You just have to check it.