each agent’s allocation are given by $\detallocs$ when fed by the corresponding types in $\dettypespaces$, and the remaining coordinates are filled in according to the opponent’s report. More specifically, $$\begin{gathered}
\forall V_1 \in \{A',B',C',D'\}, \\
\randalloci[1](V_1, A')=[\detalloci[1](\det(V_1), A), 0, 100, 100,100],\\
\randalloci[1](V_1, B')=[\detalloci[1](\det(V_1), B),100 , 0, 100, 100],\\
\randalloci[1](V_1, C')=[\detalloci[1](\det(V_1), C),100, 100, 0, 100], \\
\randalloci[1](V_1, D')=[\detalloci[1](\det(V_1), D), 100, 100, 100,0]\end{gathered}$$ For the other types, we have $$\begin{gathered}
    \forall V_1 \in \{E', F', G', H'\}, \forall V_2 \in \{A', B', C', D'\}, \\
    \randalloci[1](V_1, V_2) = \randalloci[1](C', V_2).\\
    \forall V_2 \in \{E', F', G', H'\}, \forall V_1 \in \randtypespace, \\
    \randalloci[1](V_1, V_2) = [0, 0, 0, 100, 100, 100, 100].\end{gathered}$$

Note that $\randallocs$ itself is deterministic. The difficulty we need to overcome in this section is that the *extension* of $\randallocs$ may be randomized, and we must show that any extension to $\operatorname{Conv}(\randtypespaces)$ cannot be DSIC.

The set of feasible allocations is ${\mathscr{F}}$ $= $ $\{\randallocs(V_1, V_2)\}_{V_1, V_2 \in \randtypespace}$. Again we emphasize that the set of feasible allocations is *not* a product space.

We first show that a subset of the payments are still “locked” as they were in the deterministic setting.

\[lem:randequal-pay\] For any allocation rule $\extallocs$ that 