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 $\gamma > 0$ represents $i$'s attitude towards risk, such as $\gamma > 1$ means $i$ is risk-averse and $\gamma < 1$ means that $i$ is risk-seeking?
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$\begingroup$ What do you mean by risk here? Usually, risk aversion is defined for lotteries over monetary outcomes. $\endgroup$ – Michael Greinecker Jul 19 '18 at 22:41
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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 $\gamma > 1$, and risk loving for $0<\gamma < 1$.
