According to a textbook, in the context of uncertainty (e.g. in lottery), if the preference is complete, a monotone increasing but nonlinear transformation of a utility function would not represent the same preferences. Why is it so?
An example of such preference would be appreciated.
Consider lotteries over $\{x,y,z\}$. Let $u(x)=0, u(y)=\frac{1}{2}, u(z)=1$. Consider the nonlinear transformation f(t)=t^2. Let $v:=f\circ u$, so $v(x)=0,v(y)=\frac{1}{4}, v(z)=1$.
Consider two lotteries, $P=(0,1,0)$ and $Q=(\frac{1}{2},0,\frac{1}{2})$.
$$E_P[u]=\frac{1}{2}=E_Q[u]$$
$$E_P[v]=\frac{1}{4}<\frac{1}{2}=E_Q[v]$$
In general, let $\succeq$ be a preference over lotteries $\Delta(X)$. Let $U$ be a utility function of $\succeq$ that has the EU form, so $U(P)=E_P[u]$ for some $u$. Take any increasing transformation $f$ and define $V(P):=f(U(P))$ then $V$ will also be a utility function for $\succeq$, that is $V(P)\geq V(Q)\iff P\succeq Q$. However, unless $f$ is a positive affine transformation, that is $f(x)=Ax+B$ where $A>0$ then $V$ will not be an expected utility function. That is, there will not exists a $v$ such that $V(p)=E_p[v]$.
