Utility representation of single peaked preferences Is it true that a single-peaked preference (with the peak at some finite point) over the set of real numbers, always has a utility representation ??
If yes, can you please hint towards the proof or references.
Is this result generalizable to multi-dimensional single-peakedness?
No. Basically, you can encode a form of lexicographic preferences, probably the most familiar example of non-representable preferences, as single-peaked preferences on $\mathbb{R}$.
Define $\succeq$ so that $x\succeq y$ exactly if either $|x|&lt;|y|$ or $|x|=|y|$ and $x\leq y$. Basically, the closer to the peak of $0$ a number is, the better, and in case of a tie, the number to the left of $0$ is better.
Suppose for the sake of contradiction that there is a utility representation $v:\mathbb{R}\to \mathbb{R}$ of $\succeq$. For each $r\in\mathbb{R}_{++}$ (the strictly positive numbers), let $q_r$ be a rational number in the interval $\big(v(r),v(-r)\big)$. Since for $r\neq r'$, $\big(v(r),v(-r)\big)\cap\big(v(r'),v(-r')\big)=\emptyset$, we have an injection $r\mapsto q_r$ from $\mathbb{R}_{++}$ to $\mathbb{Q}$, which is impossible since $\mathbb{R}_{++}$ is uncountable and $\mathbb{Q}$ is countable.