# Checking if a preference relation admits a utility function

Setting: We have two choices of goods $$(x_1,y_1)$$ and $$(x_2,y_2)$$ from the set of choices $$[-1,1]^2$$. Moreover, we have the following preference relation $$(x_1,y_1)\mathcal{R}(x_2,y_2)\iff |x_1|\geq|x_2|\>\>\text{or}\>\> |y_1|\geq|y_2|$$

Question: We have to check if there exists a utility function reprensation of this preference relation.

My attempt: So from what I have learned, we know that a preference relation admits a utility function representation if it is rational (reflexive, complete, transitive) and continuous. I have found that this preference relation is not transitive, but this does not mean that there does not exist a utility function representation, because the aforementioned statement is not an if and only if statement.

Moreover, I thought we could try to derive a contradiction from the fact that if there exists a utility function $$u$$ representation of the preference relation, then we have $$(x_1,y_1)\mathcal{R}(x_2,y_2)\iff u(x_1,y_1)\geq u(x_2,y_2)$$ I tried to use the fact that the relation is not transitive to derive a contradiction by using the statement above, but was unsuccessful.

Sadly, these are the two main theorems/propositions that I've learned to solve these problems.

Any help is appreciated!

The existence of a utility function $$u$$ implies transitivity. Let $$A$$, $$B$$ and $$C$$ be objects (pairs, in your example) for which $$ARB$$ and $$BRC$$. Then $$u(A) \ge u(B) \text{ and } u(B) \ge u(C)$$ so $$u(A) \ge u(C)$$ so $$ARC$$.