\section{Proper IP Scoring Rules}
\label{sec:imprecisescoringrules}

In this section, we introduce proper imprecise scoring rules, i.e., scores that incentivize the truthful reporting of an imprecise forecaster according to \Cref{def:truthfulness}. We start by focusing only on the elicitation of the imprecise forecaster without any communication from the DM. The following definition clarifies what it means for an imprecise scoring rule to be (strictly) proper, which naturally generalises \Cref{def:decision-making-with-imprecise-forecasts} from the forecaster's perspective. 
\begin{definition}
\label{def:propernessofimprecisescoringrule}
An imprecise scoring rule is (strictly) proper if for all credal sets $\cP,\cQ\subseteq\Delta(\cO)$, the forecaster with an imprecise belief $\cP\not\simeq\cQ$ (strictly) prefers $\cP$ over $\cQ$, i.e., $\cP\succeq_{\cP}\cQ$. The preference relation $\succeq_{\cP}$ is described by the (strict) dominance of $\bm{V}^{\cP}(\cP)$ over $\bm{V}^{\cP}(\cQ)$, i.e.,
\begin{align*}
     \mathbb{E}_p[s(\cP,o)]\geq \mathbb{E}_p[s(\cQ,o)] \quad \text{for all }p\in\cP,
\end{align*}
for strict dominance, at least one $\geq$ has to be strictly greater.
\end{definition}
We define strict properness of an imprecise scoring rule in \Cref{def:propernessofimprecisescoringrule} using dominance since it preserves the main idea behind strictly proper scoring rules in the precise setting, i.e. to incentivise the forecaster to be truthful. A strictly proper IP scoring rule incentivises the imprecise forecaster to be truthful according to \Cref{def:truthfulness}. 
\begin{theorem}
    \label{theorem:nosaneproperscoringrule}
    There does not exist a strictly proper imprecise scoring rule $s$. In addition, for a scoring rule $s$ to be proper it must be constant across all forecasts.
\end{theorem}
Similar impossibility results for imprecise forecasts have previously been reported in \citet{seidenfeld2012forecasting,mayo2015accuracy,schoenfield2017accuracy}. The implication of \Cref{theorem:nosaneproperscoringrule} is that under the current setup of an imprecise forecaster, any imprecise scoring rule satisfying properness in \Cref{def:propernessofimprecisescoringrule} has a constant score across all forecasts. 
%It provides no incentive to the forecaster (precise or imprecise) to invest effort in truthfully reporting their epistemic uncertainty, as they can obtain the same score by reporting the vacuous set of $\Delta(\cO)$. Consequently, we focus on studying the truthful elicitation of imprecise forecasts in the context of the decisions they induce. To this end, we introduce communication between the DM who consumes these forecasts to take action and the imprecise forecaster.
We observe in \Cref{sec:dmwithforecasts} that agents face possible indecision while making decision with the credal set $\cP$. As a result, we observe in \Cref{theorem:nosaneproperscoringrule} that it is not possible to design a scoring rule that incentivises the imprecise forecaster to report their imprecise belief $\cP$ honestly. Unlike in the precise setting, where the forecaster had a complete preference relation on plausible reports (see \Cref{def:strictlyproperscoringrule}), the epistemic uncertainty of the imprecise forecaster only allows for an incomplete preference relation $\succeq_{\cP}$ over plausible reports. 
Without further information, the imprecise forecaster cannot complete this incomplete preference relation. 