# 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?

• What are multi-dimensional single-peaked preferences? Commented Sep 27, 2020 at 11:19
• By multi-dimensional single peakedness, I was referring to Bossert and Peters' paper titled "Single-peaked choice." Commented Sep 30, 2020 at 11:16

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 $$|x|<|y|$$ 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.
• Did you mean $r\mapsto q_r$ for the injection from $\mathbb R_{++}$ to $\mathbb Q$? Commented Sep 28, 2020 at 4:36