\usetikzlibrary{shapes.geometric, arrows}
\begin{figure}[h]
\centering
\begin{tikzpicture}[node distance=2cm, thick, >=stealth]
    % Define block styles
    
    % Case I
    \node (case1) [] {\textbf{Case I}};
    \node (process1) [below=0.5cm of case1] {$a\succeq_{\cP}a'$ implies $a\succeq_{\cQ}a'$};
    \node (end1) [below=0.1cm of process1] {(\Cref{lemma:additional-subset-q-preference})};
    % Case II
    \node (case2) [right of=case1, xshift=5cm] {\textbf{Case II}};
    \node (decision) [below=0.5cm of case2] {$(a\not\succeq_{\cP}a')\land(a'\not\succeq_{\cP}a)$};
    \node (branch1) [below left of=decision, xshift=-1cm] {Case II.1};
    \node (branchend1) [below=0.05cm of branch1] {$a\succeq_{\cQ} a':=a\succeq_{\rho[\cP]}a'$};
    \node (branch2) [below right of=decision, xshift=1cm] {Case II.2};
    \node (branchend2) [below=0.05cm of branch2] {$(a\not\succeq_{\cQ}a')\land(a'\not\succeq_{\cQ}a)$};
    \draw[->] (decision) --  (branch1);
    \draw[->] (decision) --  (branch2);
\end{tikzpicture}
\caption{In \textbf{Case I} when actions are comparable in $\succeq_{\cP}$ the \Cref{lemma:additional-subset-q-preference} dictates their order to be the same for partial preference induced by $\succeq_{\cQ}$. However, in \textbf{Case II} when the actions are incomparable w.r.t $\succeq_{\cP}$ either their order in $\succeq_{\cQ}$ must be set to aggregated order of $\cP$ i.e. $\succeq_{\rho[\cP]}$ or they are left untouched, i.e. incomparable w.r.t $\succeq_{\cQ}$}
\end{figure}