= [0, 0, 1], D = [\tfrac 1 3, \tfrac 1 3, \tfrac 1 3]\}$. A visual representation of this typespace and its convex hull can be found in figure 1 in the supplementary materials.

The allocation rule $\detallocs$ is also symmetric, in the sense that $\detalloci[1](\typei[1], \typei[2]) = \detalloci[2](\typei[2], \typei[1])$ for any $\typei[1], \typei[2] \in \dettypespace$. We summarize $\detalloci[1]$ with the diagram below. The rows are indexed by agent $1$’s own type $V_1$, and the columns by agent $2$’s type $V_2$: $$\begin{aligned}
\begin{blockarray}{c cccc}
 & A & B & C & D \\
\begin{block}{c(cccc)}
A & [1,1,0] & [2,0,2] & [3,0,3] & [4,0,4] \\ 
B & [0,1,1] & [2,2,0] & [3,3,0] & [4,4,0] \\ 
C & [1,0,1] & [0,2,2] & [0,3,3] & [0,4,4] \\ 
D & [0,1,1] & [2,2,0] & [3,3,0] & [4,4,0]  \\
\end{block}
\end{blockarray} \end{aligned}$$

The set of all feasible allocations is then ${\mathscr{F}}= \{(\detalloci[1](V_1, V_2), \detalloci[2](V_1, V_2) )\}_{V_1, V_2 \in \dettypespace}$. We hasten to point out that ${\mathscr{F}}$ is *not* the product between the two agents’ respective set of feasible allocations. For example, $[1, 1, 0, 1, 1, 0]$ is in ${\mathscr{F}}$ as it is $\detallocs(A, A)$, but $[1, 1, 0, 0, 1, 1]$ is not. 