
\subsection{Aggregation Functions}
\label{subsection:aggregation}
To resolve indecision arising from epistemic uncertainty in the credal set $\cP$, the DM exercises a subjective choice through aggregation function $\rho$ to make $\succeq_{\cP}$ complete. The DM communicates the choice of $\rho$ to the forecaster \emph{prior} to elicitation, and the elicited credal set then informs downstream decisions for the DM. The resulting utility can be shared with the forecaster as an incentive.
%The DM decides how the forecaster and DM make a subjective choice to resolve indecision due to epistemic uncertainty from the credal set $\cP$. To this end, the DM employs an aggregation function $\rho$ that aggregates utilities into a single utility. Thus completing the incomplete preference relation $\succeq_{\cP}$ with subjective choice. $\rho$ is communicated \emph{prior} to elicitation by the DM to the forecaster. The elicited credal set is then used by the DM for downstream decision-making and the utility obtained by the DM can be shared with the forecaster as an incentive. 

\begin{definition}[Aggregation Function]\label{def:aggregate}
     For a credal set $\cQ$ an aggregation function \(\rho:{(\mathbb{R}^{\cX})}^{|\cQ|}\rightarrow\mathbb{R}^{\cX}\) combines multiple utilities via a positive linear combination, i.e., for any $x\in\cX$:
    $$\rho[\{\mathbb{E}_{q}[u(x,o)]\}_{q\in\cQ}] = \int_{q\in\cQ}\bm{w}(q) \mathbb{E}_q[u(x,o)]dq$$ 
    where $\bm{w}(q) \in \RR^{|\cQ|}_{\geq 0}$ for all $q\in\cQ$ depends on the expected utilities $\{\mathbb{E}_{q}[u(x,o)]\}_{q\in\cQ}$.
\end{definition}
We focus on linear aggregations because, according to \citet{harsanyi1955cardinal}, this class of aggregation rules uniquely satisfies both VNM axioms \citep{von2007theory} and Bayes Optimality \citep{brown1981complete}. Many popular aggregation functions such as utilitarian and egalitarian rules can be expressed with linear aggregations as they characterise relative utilitarianism \citep{dhillon1999relative}. 

For the utilitarian and egalitarian rules, the decision-making process from an agent's perspective is picking an $x\in\cX$. Either for the DM an $a\in\cA$ or for the forecaster a $\cQ\subseteq\Delta(\cO)$. Illustrating this from the DM's perspective, the utilitarian rule corresponds to the linear combination $\rho[\{\mathbb{E}_q[u(a,o)]\}_{q\in\cQ}] = \nicefrac{1}{|\cQ|}\sum_{q\in \cQ}\mathbb{E}_q[u(a,o)]$, whereas the egalitarian rule corresponds to
$\rho[\{\mathbb{E}_q[u(a,o)]\}_{q\in\cQ}] = \min_{q\in \cQ}\mathbb{E}_q[u(a,o)]$. Here, the weights $\bm{w}$ can be interpreted as $\bm{w}(q)=\nicefrac{1}{|\cQ|}$ and a one-hot vector, respectively. A VNM-rational DM (see Eq. \eqref{eq:forecaster-aggregation} for forecaster) uses $\rho$ to obtain the best action  $a^*_{\cQ,\rho}$: 
\begin{equation}
\label{eq:decision-maker-opt}
    a^*_{\cQ,\rho} = \arg\max_{a\in\cA}\; \rho[\{\mathbb{E}_{q}[u(a,o)]\}_{q\in\cQ}].
\end{equation}
Given the incomplete preference relation from a credal set $\cQ$, i.e., $\succeq_{\cQ}:=\{\succeq_q\}_{q\in\cQ}$, the aggregation rule $\rho$ allows us to define the corresponding complete preference relation $\succeq_{\rho[\cQ]}$, representing the aggregated utility from Equation~\eqref{eq:decision-maker-opt}. By abuse of notation, $\succeq_{\rho[\cQ]}$ represents the aggregation of utilities rather than the credal set.



\textbf{Axiomatisation of $\rho$:} When interpreting imprecise forecasts as a “collective” report of precise forecasters, a social choice perspective naturally emerges for the downstream DM. Although non-intuitive, this perspective applies even to a single-agent imprecise forecaster. Following \citet{arrow1950difficulty}, we outline three desirable properties of any aggregation rule $\rho$: Pareto Efficiency (PE), Independence from Irrelevant Alternatives (IIA), and Non-Dictatorship (ND).  
\begin{definition}[Pareto Efficiency]
\label{def:paretoefficiency}
    An aggregation rule $\rho$ is Pareto Efficient iff for all $x,x'\in\cX$,
    \begin{equation*}
       x\succeq_{\cQ} x' \implies x\succeq_{\rho[\cQ]} x'.
    \end{equation*}
    %where $\succeq_{\{u^{(i)}(\cdot)\}_{i\in[n]}}$ represents the dominance of input $x$ over $x'$, i.e., $u^{(i)}(x)> u^{(i)}(x')$ for all $i\in[n]$. And $\succeq_{\rho[\{u^{(i)}(\cdot)\}_{i\in[n]}}$ is the preference relation obtained post aggregation. 
\end{definition}
From the DM's perspective, $\cX=\cA$, and as a result, a Pareto efficient $\rho$ will respect the inherent partial order $\succeq_{\cQ}$ over actions which DM could infer from the reported credal set $\cQ$. Therefore, the DM can be assured that application of $\rho$ only resolves indecision and similarly for the forecaster when choosing the best report. Additionally, from the forecaster's perspective, a non-PE $\rho$ can distort recommendations of their forecast $\cQ$ of an action $a$ over $a'$. 
The aggregation rule that violates PE may result in the payment/score that misaligns with the forecaster's report.

\begin{definition}[IIA]
\label{def:IIA-scoringrule}
An aggregation function $\rho$ is considered IIA if preferences between $x,y\in \cX$, i.e., $x\succeq_{\rho[\cQ]}y$ or $y\succeq_{\rho[\cQ]}x$ is independent of whether any other $z$ is in $\cX$.
\end{definition}
Although cryptic, IIA is desirable to the DM. From the DM's perspective, $\cX=\cA$. Imagine a scenario where there exists a $z\in \cA$ such that both $x,y\in\cA$ dominate $z$ w.r.t. the partial order $\succeq_{\cQ}$, implying that $z$ is irrelevant to the DM under forecasts $\cQ$. However, if $\rho$ violates IIA, the post-aggregation preference $\succeq_{\rho[\cQ]}$ between $x$ and $y$ can be influenced by the presence or absence of $z$. This makes the DM vulnerable to strategic manipulation regarding the best action to take by adding or removing $z$, which in turn creates uncertainty for the forecaster about their own incentives.
%Although cryptic, IIA is desirable to the DM. Given that we know from the DM's perspective $\cX=\cA$, imagine a scenario where, there exists a $z\in \cA$ such that both $x,y\in\cA$ dominate $z$ w.r.t. the partial order $\succeq_{\phi(\cP)}$ implying that $z$ is irrelevant to the DM under forecasts $\cP$. However in case of a $\rho$ that violates IIA, the preference $\succeq_{\rho[\phi(\cP)]}$ between $x$ and $y$ can be influenced by $z$. In other words, a non-IIA satisfying $\rho$ can result in different best action when DM first uses partial order $\succeq_{\phi(\cP)}$ to identify the Pareto set of actions and then use $\rho$ to find best action within the Pareto set vs using $\rho$ to find the best action from the entire space $\cA$.
\begin{definition}[Non-Dictatorship]
\label{def:non-dictatorship}
    An aggregation rule $\rho$ is said to be non-dictatorial if for a profile of preferences $\succeq_{\cQ}:=\{\succeq_q\}_{q\in\cQ}$ there does not exist $q\in\cQ$ (dictator) such that for all $x,y\in \cX$, $x\succeq_{q} y$ implies $x\succeq_{\rho[\cQ]}y$.
\end{definition}
From the downstream decision-making perspective for a DM, non-dictatorship is optional. However, when DM wants to communicate the aggregation rule $\rho$ to the forecaster and wishes to truthfully elicit their true belief, non-dictatorship becomes crucial. Given a dictatorial $\rho$, the forecaster can manipulate the DM by strategically reporting the dictator $q$. We discuss this more formally in \Cref{appendix:dictatorships}.   
%%%
\subsection{Proper IP scores with aggregation}
The DM communicates the aggregation function $\rho$ to the forecaster and incentivises them using an IP scoring rule. This communication helps resolve the forecaster's epistemic uncertainty, parameterizing the IP scoring rule as $s_\rho:2^{\Delta(\cO)}\times\cO\rightarrow\mathbb{R}$. The forecaster reports $\cQ\in 2^{\Delta(\cO)}$ and receives a score of $s_\rho(\cQ,o)$ when outcome $o\in\cO$ occurs. Unlike prior IP scoring rules, the forecaster can now use $\rho$ to resolve indecision and complete the preference relation over $2^{\Delta(\cO)}$. This is evident from the expected utility of the forecaster with belief $\cP$ when reporting $\cQ\subseteq\Delta(\cO)$:
%The DM communicates the aggregation function $\rho$ to the forecaster and incentivises the forecaster with an IP scoring rule. The aggregation rule will also guide the forecaster in resolving their epistemic uncertainty and therefore, it parameterises the IP scoring rule as follows $s_\rho:2^{\Delta(\cO)}\times\cO\rightarrow\mathbb{R}$. 
%The forecaster can then report their forecast $\cQ\in 2^{\Delta(\cO)}$ and they obtain a score of $s_\rho(\cQ,o)$ for it when the outcome $o\in\cO$ occurs. The crucial difference from the setting of IP scoring rules is that now the forecaster can use $\rho$ from the DM to resolve their indecision and complete the preference relation on the space of reports $2^{\Delta(\cO)}$. this is evident from The expected utility of the forecaster with imprecise belief $\cP$ when reporting $\cQ\subseteq\Delta(\cO)$ is
\begin{equation}
\label{eq:forecaster-aggregation}
    V^{\cP}_{\rho}(\cQ):=\rho[\bm{V}^{\cP}(\cQ)]=\rho[\{\mathbb{E}_{p}[s_\rho(\cQ,o)]\}_{p\in\cP}].
\end{equation}
\begin{comment}
Since any choice of $\rho$ effectively performs a linear combination of $\bm{V}^{\cP}(\cQ)$, it implies that $V^{\cP}_{\rho}(\cQ)\in\text{im}(\bm{V}^{\cP}(\cQ))$. Therefore there exists a $p_{\cQ}\in\cP$ such that we can equivalently write
\begin{equation}
\label{eq:p-effective-definition}
    V^{\cP}_{\rho}(\cQ)=\mathbb{E}_{p_{\cQ}}[s_\rho(\cQ,o)]
\end{equation}
We subscript $p_{\cQ}$ with $\cQ$ since the linear combination weights used by $\rho$ depend on $\{\mathbb{E}_{p}[s_\rho(\cQ,o)]\}_{p\in\cP}$. And therefore, $p_\cQ$ indirectly depends on $Q$. Leveraging the alternative characterisation of $V^{\cP}_{\rho}(\cQ)$ in terms of $p_{\cQ}\in\cP$ in Equation~\ref{eq:p-effective-definition} we now define regular, proper and strictly proper imprecise decision scoring rules in the same spirit as precise scoring rules. 
\end{comment}
Since an imprecise decision scoring rule $s_\rho$ is simply a parameterised IP scoring rule, its regularity is defined exactly as in \Cref{sec:imprecisescoringrules}. We extend (strict) properness for IP scoring rules from \Cref{def:strictlyproperscoringrule} to aggregation as follows.
\begin{definition}
    \label{def:properscoringimprecisedecisionrule}
    A regular IP scoring rule $s_\rho$ for an aggregation function $\rho$ is proper if, for all $\cP\subseteq\Delta(\cO)$ and all $\cQ\not\simeq\cP$, $V^{\cP}_\rho(\cP) \geq V^{\cP}_\rho(\cQ)$. The IP scoring rule $s_\rho$ is strictly proper if and only if at least one of the inequalities is strict.
\end{definition}
Notably, strictness in \Cref{def:properscoringimprecisedecisionrule} adheres to the notion of truthfulness defined in \Cref{def:truthfulness}. Since DM needs to evaluate the forecaster, we employ the class of scoring rules that accommodate a DM in evaluating a forecast, called tailored scoring rules \citep{dawid2007geometry,richmond2008scoring,johnstone2011tailored}, following the ideas of business sharing proposed in \citet{savage1971elicitation}. We now define them in the context of aggregation functions for imprecise forecasts.
\begin{definition}[Tailored Scoring Rules] 
\label{def:tailoredscoringrule}
An IP scoring rule $s$ is tailored for a DM with utility function $u$ and aggregation function $\rho$, if for any $k,c\in\mathbb{R}_{\geq 0}$, the score is defined as 
    \begin{equation*}
        s_{\rho}(\cQ,o)=k u(a^*_{\cQ,\rho},o) + c .
    \end{equation*}
\end{definition}
In \Cref{def:tailoredscoringrule}, $k$ can be referred to as the business share obtained by the forecaster in the utility of the DM and $c$ is the fixed fee of the forecaster. Next, we show that the class of tailored scoring rules is proper for any choice of $\rho$,
\begin{proposition}
\label{prop:tailoredscoringrule}
    A tailored scoring rule $s_\rho$ is proper with respect to \Cref{def:properscoringimprecisedecisionrule} for any aggregation rule $\rho$.
\end{proposition}
While necessary, the properness of scoring rules is easy to satisfy (see \Cref{theorem:nosaneproperscoringrule}). For example, a constant scoring rule is always  proper. We therefore characterise strict properness of $s_\rho$ for imprecise forecasts.
\begin{lemma}
Let $s_\rho$ be a tailored scoring rule. Then, the following holds:
\begin{enumerate}
\label{lemma:strictness-for-precise-distributions}
    \item $s_\rho$ is strictly proper for \textbf{precise distributions} if and only if $a^*_{q} := \arg\max_{a\in\cA}\mathbb{E}_{q}[u(a,o)]$ is a unique maximiser for all $q\in\Delta(\cO)$.
    \label{theorem:aggregation-strictness-impossibility}
    \item $s_\rho$ is not strictly proper, i.e., does not satisfy \Cref{def:properscoringimprecisedecisionrule}, for any Pareto efficient $\rho$.
\end{enumerate}
\end{lemma}
\Cref{lemma:strictness-for-precise-distributions} ensures the existence of non-constant proper IP scoring rules. Beyond this positive result, we observe that Pareto efficiency leads to the impossibility of truthful elicitation under \Cref{def:truthfulness}. Although $s_\rho$ in \Cref{lemma:strictness-for-precise-distributions} is not strictly proper for imprecise forecasts, it remains practical to implement while being proper for all forecasts and strictly proper for precise ones. We speculate that the properties of $s_\rho$ are optimal for deterministic scoring rules, given the prior impossibility of any real-valued strictly proper IP scoring rules \citep{seidenfeld2012forecasting}. To explore this further, we investigate whether allowing randomisation in the choice of aggregation rule can enable truthful elicitation.

% \Alan{There is only one subsection (4.1) in section 4. Consider restructuring it because there is no need to set things into subsection if there is only one subsection.}

