Microeconomics - Expected Utility Theory - Piecewise utility index, certainty equivalence, etc.
Suppose random variable $X\sim F$. 


Certainty equivalent of the lottery $X$ is the constant $c$ that solves: 


  $u(c) = \mathbb{E}(u(X))$

Because $u(x) = \min(\sqrt{x}, \sqrt{M})$ is concave, by Jensen's inequality:


  $u(c) = \mathbb{E}(u(X)) \leq u(\mathbb{E}(X)) $

Agent is risk averse because $u$ is concave.
If agent prefers lottery $X\sim F$ to lottery $Y\sim G$, it is not necessary that $F$ FOSD $G$. Consider the following two lotteries:



  $X = \frac{M}{2}$ with probability 1.
  
  $Y = M$ with probability $\frac{1}{2}$ and $Y = 0$ with probability $\frac{1}{2}$. 


Since $u$ is concave, lottery $X$ is preferred over lottery $Y$. This because $X$ gives the expected value of the lottery $Y$ with probability 1. Also, $F$ does not FOSD $G$ because $F\left(\frac{M}{2}\right) = 1 &gt; \frac{1}{2} = G\left(\frac{M}{2}\right) $.