\section{Preliminaries}
\label{sec:preliminaries}
This section introduces proper scoring rules, imprecise probabilities (IP), and credal sets. We begin by establishing the notation. Let $(\cO,\cF)$ be a measurable space where $\cO$ is a finite, discrete, non-empty set of possible outcomes (or states of nature) and $\cF$ is a corresponding sigma-algebra.  Let $O:\cO\rightarrow\mathbb{R}$ be a random variable associated with p.m.f. $p:\cO\rightarrow[0,1]$ on outcome $o\in\cO$. The probability simplex $\Delta(\cO)$ denotes the set of all probability distributions on $\cO$. Our framework involves two agents: a forecaster and a decision-maker~(DM), each with an associated utility function $u:\cX\times\cO\rightarrow\mathbb{R}$, where $\cX$ represents the decision space relevant to the agent's utility. 
Since we often refer to specific outcomes $o\in\mathcal{O}$, we will use $O$ and $o$ interchangeability. Thus, for some $x\in\mathcal{X}$, the agent's expected utility $\mathbb{E}_{O\sim p}[u(x,O)]$ is expressed as $\mathbb{E}_{o\sim p}[u(x,o)]$. For a set $\cP$, $co(\cP)$ corresponds to the convex hull and $\ext(\cP)$ to its extreme points.

\subsection{Precise Scoring Rules}
\label{section:proper-scoring-rule-preliminary}

Scoring rules incentivize a forecaster to truthfully report their probability assessments of an uncertain event~\citep{winkler1967quantification,brier1950verification}.  Specifically, a scoring rule $s:\Delta(\cO)\times \cO \rightarrow \mathbb{R}$ assigns a score of $s(q,o)$ to a forecaster with a forecast $q\in\Delta(\cO)$ when an outcome $o$ happens.
\begin{definition}
    A forecaster is precise if their true belief can be expressed as a probability distribution $p\in\Delta(\cO)$.
\end{definition}
Since classical proper scoring rules focus on truthful reporting and evaluation of \emph{precise} forecasts, we refer to them as precise scoring rules. To discourage a forecaster from making overly confident predictions, e.g., $q(o)=0$. We introduce $\textit{regular}$ precise scoring rule, i.e. $s(q,o)\in\mathbb{R}$ for all $o\in\cO$ and $s(q,o)=-\infty$ only if $q(o)=0$. 
\begin{definition}[Expected utility of the forecaster] Precise scoring rules implicitly assume that the forecaster is an expected utility-maximising agent. Therefore, for a forecaster with true belief $p$, the  utility of reporting forecast $q$ is
\begin{align}
\label{eq:expected-score-utility}
    u_{p}(q)=\mathbb{E}_{o\sim p}[s(q,o)].
\end{align}
\end{definition}
We now define a subclass of regular precise scoring rules, known as \emph{strictly} proper precise scoring rules that incentivize truthful reporting of the forecaster's belief.
\begin{definition}[Strictly Proper Precise Scoring Rule]
\label{def:strictlyproperscoringrule}
A scoring rule $s:\Delta(\cO)\times \cO\rightarrow \mathbb{R}\cup\{-\infty\}$ is strictly proper if the forecaster's true belief $p\in\Delta(\cO)$ uniquely maximizes their expected utility, i.e., for all $p,q\in\Delta(\cO)$ s.t. $q\neq p$,
\begin{align}
    \mathbb{E}_{o\sim p}[s(p,o)] > \mathbb{E}_{o\sim p}[s(q,o)].
\end{align}
\end{definition}
Some examples of strictly proper precise scoring rules are, logarithmic scoring rule $s(q,o)=a_o+b\text{ }\log(q(o))$ and quadratic scoring rule $s(q,o)=a_o+b(2q(o)-\mathbb{E}_{o\sim q}[q(o)])$ with $b\in\mathbb{R}_{+}$ and $a_o\in\mathbb{R}$ as arbitrary parameters. Proper precise scoring rules are closely related to convexity and can be characterized using convex functions as shown in ~\citet{mccarthy1956measures,savage1971elicitation,gneiting2007strictly}.

\begin{theorem}[\citealt{gneiting2007strictly}]
~\label{theorem:gneiting}
A regular scoring rule $s$ is (strictly) proper if and only if 
\begin{align}
    s(q,o) = G(q) - \sum_{o\in\cO} G'(q) dq(o) + G'(q)(o)
\end{align}
where $G:\Delta(\cO)\rightarrow \mathbb{R}$ is a (strictly) convex function and $G'(q)$ is a subgradient of $G$ at point $q$ and $G'(q)(o)$ is the value of gradient at outcome $o$.
\end{theorem}
An implication of Theorem~\ref{theorem:gneiting} is that with this characterisation of the scoring rule $s$, we can interpret $G$ as the corresponding maximum expected score ~\citep{frongillo2014general}.
The derivation of $G$ as the expected score is included in  Appendix \ref{remark:gneiting} for completeness. 

%%%%
\subsection{IP and Credal Sets}
\label{subsection:Imprecise-probability-and-credal-sets}

Standard probability theory assigns a unique numerical value to each event, whereas \emph{imprecise probabilities} (IP) allows a range of plausible values to represent uncertainty in the presence of limited or ambiguous information. One common approach to modelling such uncertainty is via \emph{credal sets}. 
Given a subset $\cP\subseteq\Delta(\cO)$ of the plausible probability distributions, a credal set is defined as a closed and convex combination of $\cP$.
The assumption of convexity and closedness allows for rational decision-making~\citep{gajdos2004decision,troffaes2007decision} and satisfies axioms such as coherence~\citep{definetti1974theory,walley1991statistical}. While $\cP$ directly specifies the plausible beliefs about the state of nature, $\co(\cP)$ denotes the uncertainty inferred by a rational agent~\citep{walley1991statistical, augustin2014introduction}.
