possible.

Non-existence of randomized extensions
--------------------------------------

We need a more convoluted construction and a more careful argument to prove the impossibility of extensions that are possibly randomized. We build on $\dettypespaces$ and $\detallocs$ to construct $\randtypespaces$ and $\randallocs$ and prove Theorem \[thm:rand\].

We first raise types in $\dettypespace$ to a space of seven dimensions. Define $A' = [1, 0, 0, 0, 0, 0, 0]$, $B' = [0, 1, 0, 0, 0, 0, 0]$, $C' = [0, 0, 1, 0, 0, 0, 0]$ and $D' = \frac 1 3 (A' + B' + C')$. For ease of notation, we define a mapping $\det: \{A', B', C', D'\} \to \dettypespace$, with $\det(A') = A, \det(B') = B, \det(C') = C$ and $\det(D') = D$. We also introduce four new types, $E' = [0, 0, 0, 1, 0, 0, 0]$, $F'=[0,0,0,0,1,0,0]$, $G'=[0,0,0,0,0,1,0]$ and $H'=[0,0,0,0,0,0,1]$. Define $\randtypespace = \{A', B', C', D', E', F', G'\}$, and $\randtypespaces = \randtypespace \times \randtypespace$.

We now define $\randallocs$, which is again symmetric, in the sense that $\randalloci[1](V_1, V_2) = \randalloci[2](V_2, V_1)$ for every $V_1, V_2 \in \randtypespace$. We therefore only describe $\randalloci[1]$. When both agents report types in $\{A', B', C', D'\}$, the first three coordinates of 