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?

  • 1
    $\begingroup$ What are multi-dimensional single-peaked preferences? $\endgroup$ Commented Sep 27, 2020 at 11:19
  • $\begingroup$ By multi-dimensional single peakedness, I was referring to Bossert and Peters' paper titled "Single-peaked choice." $\endgroup$
    – Polime
    Commented Sep 30, 2020 at 11:16

1 Answer 1


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.

  • $\begingroup$ Did you mean $r\mapsto q_r$ for the injection from $\mathbb R_{++}$ to $\mathbb Q$? $\endgroup$
    – Herr K.
    Commented Sep 28, 2020 at 4:36
  • $\begingroup$ @HerrK. Yes; thank you for pointing it out. $\endgroup$ Commented Sep 28, 2020 at 7:15

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