\textbf{Generalized Gini social-evaluation welfare function} (GSF; \citet{weymark1981generalized, sim2021collaborative}) is a common class of aggregation functions that can smoothly interpolate between utilitarian and egalitarian aggregation of a utility profile $\{u^{(i)}(\cdot)\}_{i\in[n]}$:
\begin{align*}\label{eq:gini}
    \rho_{\lambda}[\{u^{(i)}(\cdot)\}_{i\in[n]}] &:= \textbf{w}^\top \phi(\{u^{(i)}(\cdot)\}_{i\in[n]}) \\\quad \text{s.t.} \quad w_i &:= \frac{(1-\lambda)+\lambda\frac{n-i}{n-1}}{\sum_{j=1}^n((1-\lambda)+\lambda\frac{n-j}{n-1})} \\ w_i &> 0 \\ 0 < &\lambda \leq 1
\end{align*} 
where $\textbf{w} := (w_i)_{i \in V}$ is a weight vector, $\textbf{1}$ is the one-vector, and $\phi$ is a sorting function that arranges the elements of the input vector in ascending order and returns the sorted vector and $\lambda\in[0,1]$ is a parameter that allows for smooth interpolation between utilitarian $(\lambda=1)$ and egalitarian $(\lambda\rightarrow 0)$ aggregations. 
\begin{assumption}
    \label{assumption:strongly-concave-u} The decision maker's utility function 
    $u:\cA\times \cO\rightarrow\mathbb{R}$ is strictly conncave in $\cA$, i.e. 
    \begin{align*}
        \forall o \in \cO \quad \forall a,a'\in\cA \quad \forall &\mu\in[0,1] \\
        u(\mu a+ (1-\mu)a',o)&> \mu u(a,o) + (1-\mu)u(a',o)
    \end{align*}
\end{assumption}
\begin{lemma}(Known)
    \label{lemma:scalarization-full-pareto-front}
    Given assumption~\ref{assumption:strongly-concave-u} i.e., the utility function $u:\cA\times\cO\rightarrow\cR$ is strictly concave in $\cA$. The Pareto front of the actions of the Decision-Maker is fully characterized by 
    scalarization.    
\end{lemma}
\begin{proof}   
    The Pareto set of actions for a decision-maker who obtains a forecast $\cP$ can be represented by 
    \begin{align*}
        \cA^{\cP}_{\text{PF}}=\Bigg\{a\in\cA\text{ }|\text{ }\not\exists a'\in\cA \text{ s.t. }a'\succ_{\cP} a'\Bigg\}
    \end{align*}
    where $\succeq_{\cP}$ is the partial order induced by expected utilities of all distributions in the forecast. We define the Pareto front by considering the set of all aggregation rules, i.e.  $\bm{\rho}$ as 
    \begin{align*}
        \cA^{\cP}_{\bm{\rho}-\text{PF}}=\Bigg\{a^*_{\rho,\cP}\text{ }|\text{ }\exists\text{ }\rho\in\rho_D \quad a^*_{\rho,\cP}=arg\max_{a\in\cA}\rho(\{\mathbb{E}_p[u(a,o)]\}_{p\in\cP})\Bigg\}
    \end{align*}
    It is straightforward to see that $\cA^{\cP}_{\bm{\rho}-\text{PF}}\subseteq\cA^{\cP}_{\text{PF}}$ when $\bm{\rho}$ contains only Pareto efficient rules. 
    
    contains all the 
Correspondingly we consider the set of actions that can be achieved with scalarization of objectives i.e.
\begin{align*}
    \cA^{\cP}_{\text{Scalar}}=\Bigg\{a^*_{w,\cP}\text{ }|\text{ }\exists\text{ }w\in\Delta^{|ext(\cP)|} \quad a^*_{w,\cP}=arg\max_{a\in\cA}\sum_{p\in\cP}w_p\mathbb{E}_p[u(a,o)]\Bigg\}
\end{align*}
\end{proof}
\begin{proof}
From Proposition ~\ref{prop:tailoredscoringrule} we know that for any Pareto efficient aggregation rule $\rho$, the corresponding aggregation rule $s_\rho$ is proper. Which means that for all rules in the set of desirable and pareto efficient aggregation rules, for an imprecise forecaster with belief $\cP\subseteq\Delta(\cO)$, following is true
\begin{align*}
    \forall \rho\in \rho_{PE} \quad V_\rho^\cP(\cP)\geq V_\rho^{\cP}(\cQ) &&(s_\rho \text{ is proper for Pareto Efficient }\rho)
\end{align*}
Therefore, for any $q\in\Delta(\rho_{PE})$ we can say that, 
\begin{align*}
    \mathbb{E}_{\rho\sim q}[V_\rho^\cP(\cP)]&\geq \mathbb{E}_{\rho\sim q}[V_\rho^{\cP}(\cQ)]\\
    V_q^{\cP}(\cP)\geq V_q^(\cQ)
\end{align*}
We now use the fact that $\rho_D$ is desirable, which means that if for any arbitrary $\cQ\subseteq\Delta(\cO)$ following holds true, 
\begin{align*}
    \forall \rho \in\rho_D \quad \rho(\{\mathbb{E}_{p}[s_\rho(\cP,o)]\}_{p\in\cP})=\rho(\{\mathbb{E}_{p}[s_\rho(\cQ,o)]\}_{p\in\cP}) 
\end{align*}
Essesntially it says that for all $\rho\in\rho_D$ $V_\rho^\cP(\cP)= V_\rho^{\cP}(\cQ)$. From desirability it implies that, 
\begin{align*} \forall \cP \in \Delta (\cO ),\ \forall \rho \in \rho_{PE}:\
   \forall p\in\cP \quad  \mathbb{E}_{p}[s_\rho(\cP,o)]=\mathbb{E}_{p}[s_\rho(\cQ,o)]
\end{align*}
Given $s_\rho$ is a tailored scoring rule, 
\begin{align}
   \forall p\in\cP \quad  \mathbb{E}_{p}[s_\rho(\cP,o)]=\mathbb{E}_{p}[s_\rho(\cQ,o)]\\
   \implies \quad \forall p\in\cP \quad  \mathbb{E}_{p}[u(a^*_{\rho,\cP},o)]=\mathbb{E}_{p}[u(a^*_{\rho,\cQ},o)]\\
\end{align}

Case 1: Assume $\rho$ is dictatorial


Since $\rho\in\rho_D$ is dictatorial, we can say that, 
\begin{align*}
    \forall p\in\cP \quad  \mathbb{E}_{p}[u(a^*_{\rho,\cP},o)]=\mathbb{E}_{p}[u(a^*_{\rho,\cQ},o)]\\
   \implies \quad \forall p\in\cP \quad  \mathbb{E}_{p}[u(a^*_p,o)]=\mathbb{E}_{p}[u(a^*_{\rho,\cP},o)]=\mathbb{E}_{p}[u(a^*_{\rho,\cQ},o)] && \rho \text{ is dictatorship}\\
   \implies a^*_p= a^*_{\rho,\cP}=a^*_{\rho,\cQ}\\
\end{align*}
Since for $s_\rho$ to be strictly proper for precise distributions we need that $a^*_p$ is a unique maximizer. This implies that $p\in\cP$ and $p\in\cQ$, i.e. dictatorship must pick the same distribution from both the credal sets. And we do this for all the distributions, it implies that $P\simeq Q$. 

Therefore if for all $\rho\in\rho_D$, $V_\rho^\cP(\cP)= V_\rho^{\cP}(\cQ)$, then $\cP\simeq \cQ$. Therefore, $s_\rho$ is strictly proper. 
\end{proof}
\section{Additonal Lemmas}
\subsection{Pareto Fronts}
\begin{lemma}
    \label{appendix:lemma-pareto-fronts}
    Convexification of the utility profile does not change the Pareto front of actions, i.e. for a utility profile $U:=\{u^{(i)}(\cdot)\}_{i\in [n]}$, where $u^{(i)}:\cX\rightarrow\mathbb{R}$ is a function from some input space $cX$ to a $\mathbb{R}$.  Then, 
    \begin{align*}
        \text{Pareto front}(U) = \text{Pareto front}(Conv(U))  
    \end{align*}
    where $conv(U)=\Bigg\{u|\text{ }\exists\text{ }w\in\Delta^n\quad \text{s.t.} \quad u:=\sum^{n}_{i=1}w_iu^{(i)}(\cdot)\Bigg\}$
\end{lemma}
\begin{proof}
We begin by showing that $\text{Pareto front}(U) \subseteq \text{Pareto front}(Conv(U))$. Which means that there exists $x\in \cX$ such that $x\in \text{Pareto front}(U)$, implies that $x\in \text{Pareto front}(conv(U))$. 
\begin{align*}
    x \in \text{Pareto front}(U) &\implies \not \exists \quad x' \in \cX \quad s.t. \quad x'\succ x\\
    &\implies \not \exists \quad x' \in \cX \quad s.t. \quad \forall\text{ }i \in [n] \quad u^{(i)}(x')\geq u^{(i)}(x) \text{ and } \exists\text{ }i \in [n] \quad u^{(i)}(x')> u^{(i)}(x)\\
    &\implies \not \exists \quad x' \in \cX \quad s.t. \quad \forall\text{ }w\in\Delta^n \quad \sum^{n}_{i=1}w_iu^{(i)}(x')\geq \sum^{n}_{i=1}w_iu^{(i)}(x)\\
    &\quad\quad\quad\quad\quad\hspace{ 10em }\text{ and } \exists\text{ }i \in [n]\sum^{n}_{i=1}w_iu^{(i)}(x')> \sum^{n}_{i=1}w_iu^{(i)}(x)\\
    &\implies x \in \text{Pareto front}(conv(U))\\
\end{align*}
Now we show the reverse direction that $\text{Pareto front}(Conv(U))\subseteq\text{Pareto front}(U)$, meaning that there exists $x\in \cX$ such that $x\in \text{Pareto front}(conv(U))$, implies that $x\in \text{Pareto front}(U)$. 
\begin{align*}
    x \in \text{Pareto front}(conv(U)) &\implies \not \exists \quad x' \in \cX \quad s.t. \quad x'\succ x\\
    &\implies \not \exists \quad x' \in \cX \quad s.t. \quad \forall u\in Conv(U) \quad u(x')\geq u(x) \text{ and } \exists\text{ } u\in conv(U) \quad u(x')> u(x)\\
    &\implies \not \exists \quad x' \in \cX \quad s.t. \quad \forall\text{ }w\in\Delta^n \quad \sum^{n}_{i=1}w_iu^{(i)}(x')\geq \sum^{n}_{i=1}w_iu^{(i)}(x)\\
    &\quad\quad\quad\quad\quad\hspace{ 10em }\text{ and } \exists\text{ }i \in [n]\sum^{n}_{i=1}w_iu^{(i)}(x')> \sum^{n}_{i=1}w_iu^{(i)}(x)\\
    &\implies \not \exists \quad x' \in \cX \quad s.t. \quad \forall\text{ }i \in [n] \quad u^{(i)}(x')\geq u^{(i)}(x) \text{ and } \exists\text{ }i \in [n] \quad u^{(i)}(x')> u^{(i)}(x)\\
    && \hspace{-7em}(\{\delta_i\}_{i\in[n]}\subset \Delta^n)\\
    &\implies x \in \text{Pareto front}(U)\\
\end{align*}
\end{proof}
\begin{corollary}
A corollary of Lemma ~\ref{appendix:lemma-pareto-fronts} is that for any two utility profiles $U_i:=\{u^{(i)}(\cdot)\}_{i\in [n]}$ and $U_j:=\{u^{(j)}(\cdot)\}_{j\in [m]}$, following holds 
\begin{align*}
   conv(U_i)=Conv(U_j) \implies \text{Pareto front}(U_i)=\text{Pareto front}(U_j)
\end{align*}
\end{corollary}
\begin{proof}
    Let us assume that $conv(U_i)=Conv(U_j)$, then $\text{Pareto front}(Conv(U_i))=\text{Pareto front}(Conv(U_j))$, since for same utility profiles the Pareto fronts are the same. Then with direct application of Lemma~\ref{appendix:lemma-pareto-fronts}, $\text{Pareto front}(U_i)=\text{Pareto front}(U_j)$.
\end{proof}
\subsubsection{When does the same Pareto Front imply the same convex set of utilities?}
In general, it is not always the case that a Pareto front is unique. Often two different convex-hull of utilities can obtain the same Pareto front. In other words, since the Pareto fronts contain only the non-dominated elements when two Pareto fronts of the utilities are the same, the partial order between the dominated elements can differ. Then is it enough to say that two sets of utilities are the same up to the convex hull if they induce the same partial order of actions? Not quite. Consider this counter-example, a space of inputs $\cX:=\{x,y,w,z\}$. 

Consider a set of utilities $U_i=\{u_i^{k}\}_{k\in[n]}$ and $U_j=\{u_j^{k}\}_{k\in[n]}$. We assume $n=2$. 

\begin{table}[h]
    \centering
    \begin{tabular}{c|c|c|c|c}
     & x & y & w & z\\
       $u_i^{(1)}$ & 0.9 & 1 & 0.2 & 0.1 \\
       $u_i^{(2)}$  & 1 & 0.9 & 0.1 & 0.2 \\
       \hline 
      $u_j^{(1)}$ & 0.9 & 1 & 0.8 & 0.95 \\
      $u_j^{(2)}$ & 1 & 0.9 & 0.95 & 0.8 \\
    \end{tabular}
    \caption{Utilities $U_i$ and $U_j$ are different but induce same partial order i.e. $\{(x,y), (w,z)\}$}
    \label{tab:my_label} 
\end{table}
This shows that it is not even enough to compare two partial orders on a set of actions to identify if a set of utilities is the same upto convex combinations. We need stronger assumputions that the space of actions is \emph{rich} enough that any difference in the set of utilities upto the convex hull reflects in the partial orders of the input.  