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According to a textbook, a monotone increasing but nonlinear transformation of a utility function might not represent the same preferences. Why is it so?

An example of such preference would be appreciated.

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    $\begingroup$ Maybe I am underthinking this, but the transformation has to be strictly monotone increasing. I think it is simple to see why, as only the strictly monotone function would keep the ordering of any preference relation intact. But you probably meant something else? $\endgroup$ – IMA Oct 21 '19 at 9:09
  • $\begingroup$ Thank you. I somehow thought that monotone increasing the same as strictly monotone increasing. $\endgroup$ – Aqqqq Oct 21 '19 at 9:27
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One reason I could think of is regarding convexity of the function (and hence risk aversion). For example, $u(x) = x$ is risk neutral, but $v(x) = \ln(u(x)) = \ln(x)$ is risk averse.

See also this.

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