Construct utility function for a risk-averse agent I am trying to construct utility function for an agent who can be risk-seeking or risk-averse. We have an agent $i$ who has an ideal point $x$ in a policy space $X = [0,1]$. There is a policy (option) given by $a \in X$. Can we construct the agent $i$'s utility function as a distance between his ideal point and $a$ as $u_i (a) = -(x - a)^\gamma$, where $\gamma &gt; 0$ represents $i$'s attitude towards risk, such as $\gamma &gt; 1$ means $i$ is risk-averse and $\gamma &lt; 1$ means that $i$ is risk-seeking?
If $-|x-a|$ represents the monetary payoff associated with the policy choice $a$, then $u(a) =-\left(|x-a|^\gamma\right)$ is risk averse for $\gamma &gt; 1$, and risk loving for $0&lt;\gamma &lt; 1$.