\section{Elicitation Procedure for Operator}
In an Imprecise Learning setup, a model operator receives a choice space $\Lambda$ and a corresponding augmented Hypothesis $h\in\mathcal{H}_\Lambda$. For final model deployment, the model operator has to decide on the choice of $\lambda\in\Lambda$. It is possible that the model operator may not be able to determine an exact $\lambda$ that reflects their precise risk behaviour. Thus this situation will require elicitation of the model operator's risk disposition.  
In previous works, under institutional separation when decision makers are aware of their choice of $\lambda$ it implies that they can solve the following optimisation task
\begin{equation}
    \lambda^* = arg\max_{\lambda\in\lambda} u(\lambda)
    \label{eq:bo-task}
\end{equation}
where $u:\Lambda\rightarrow\mathbb{R}_{+}$ is the utility of a choice $\lambda$. Thus by solving Eq ~\ref{eq:bo-task} decision makers can obtain a $\lambda^*$ which can be provided to model developers at the time of deployment in the case of Institutional Separation as in Imprecise Learning. In the case of Precise Learning, it can be interpreted as $\lambda^*$ being provided before training so that the right objective can be optimised.

However, these setups implicitly assume that (1) the operators know the full space of judgment parameters $\lambda\in\Lambda$ considered by the model designers. (2) They understand apriori to deploying a judgment parameter and its consequences, i.e. have access to a utility function $u:\Lambda\rightarrow\mathbb{R}$. In real-world scenarios, operators need support to translate their level of risk aversion into an appropriate choice of $\lambda$. To this end, we assume that the operators do not have any utility function on space of judgments and neither do they possess any prior belief about the judgement parameters.  
\subsection{Elicitation Setup}
In our elicitation setup, operators initially receive an $x\in\cX$ from nature which is obtained from an unknown distribution $\mathbb{P}_{t}\in\Delta(\cX)$. Note, that initially nature only reveals the $x$ and with-holds the $y$, which is revealed to the operator much later. We place no assumptions on how nature generates $y$ given an $x$, it can be stochastic or even adversarial.  
We assume that the operator possesses a belief over how $x$ generates $y$ in the form of a conditional distribution $q(Y|X)$. As rational agents operators use their belief to select the optimal action from the space of actions $\cA$. The abstract notion of actions generalises several setups from the real world. For example, when $\cA=\cY$, actions correspond to predictions $y\in\cY$, or they can correspond to other decisions such as treatment recommendations i.e. $\cA=T$. The operator obtains utility $u:\cA\times\cY\rightarrow\mathbb{R}$ upon selecting action $a\in\cA$ when the outcome $y\in\cY$. We assume that the utility function is common knowledge to both, operators and model designers.

To elicit the operator, model designers ask the operator to recommend an action policy $\pi:\cX\rightarrow\cA$ from the space of policies $\Pi$. As a rational agent, operators compute the expected utility of each policy as
\begin{equation}
    EU_{op}(\pi)=\mathbb{E}_{x\sim \mathbb{P}_t}\Bigg[\mathbb{E}_{q(Y|X=x)}[u(\pi_{op}(x),y)]\Bigg]
\end{equation} 
and report the expected utility maximizing policy $\pi_{op}:= arg\max_{\pi\in\Pi} EU_{op}(\pi)$. 
Which in practice can be learned by the operator with a set of samples $\{x_i\}_{i=1}^n$ as 
\begin{equation}
    \hat{\pi}_{op}:= arg\max_{\pi\in\Pi} \frac{1}{n}\Bigg[\mathbb{E}_{q(Y|X=x_i)}[u(\pi(x_i),y)]\Bigg]
\end{equation}
\subsection{The elicitation Contract}
As part of the elicitation, the model designer incentivizes the operators to truthfully report their maximum expected utility $EU_{op}^{max}$ via a utility transfer function $t:\cX\times\cY\times \Pi\rightarrow\mathbb{R}$ describing the transfer of utility from the model designers to operators given outcomes from nature $(x,y)$ and the action $a$ chosen by the operator.   
\begin{proposition}
\label{prop:payments}
    A utility transfer function $t:\cX\times\cY\times \Pi\rightarrow\mathbb{R}$ insures the operators iff
    \begin{equation}
    \label{eq:utility_transfer}
    t(y,x. \pi) = g(x, \pi) - u(y,\pi(x))
    \end{equation}
     Here $g:\cX\times\Pi\rightarrow\mathbb{R}$ is the guaranteed return to the operator based on their action policy and $x\in\cX$.
\end{proposition}

Additionally, transfers of the form in Eq~\ref{eq:utility_transfer} protect against strategic manipulations by the operators such as performativity for appropriate choices of $g$ (See Appendix~\ref{sec:peformitivity-appendix}).