\frac{2}{3} V_1 + \frac{1}{3} D, \detalloci[1](V_1, V_2') \rangle$ for the allocation. In order for $\typei[1]$ to have no incentive to misreport $V_1^*$, we must have $$\begin{gathered}
\langle \frac{2}{3} V_1 + \frac{1}{3} D, \detalloci[1](V_1, V_2') \rangle - \payi[1](\typei[1], V_2) 
\\ 
\geq \langle \frac{2}{3} V_1^* + \frac{1}{3} D, \detalloci[1](V_1^*, V_2) \rangle - \payi[1](V_1^*, V_2) 
\label{eq:t1-V1}\end{gathered}$$

On the other hand, in order for type $V_1$ not to have incentive for deviating to $\typei[1]$, we have $$\begin{gathered}
    \langle V_1, \detalloci[1](V_1, V_2) \rangle - \payi[1](V_1, V_2) 
    \\
    = \langle V_1^*, \detalloci[1](V_1^*, V_2) \rangle - \payi[1](V_1^*, V_2) 
    \\
    \geq \langle V_1, \detalloci[1](V_1, V_2') \rangle - \payi[1](\typei[1], V_2);
    \label{eq:V1-t1}
    \end{gathered}$$ where for the equality we used the fact that the value obtained by reporting truthfully is the same for every type in $\{A, B, C\}$ given a fixed type of the opponent, and that $\payi[1](V_1, V_2) = \payi[1](V_1^{*}, V_2)$ by Lemma \[lem:equal-pay\].

Similarly, in order for type $D$ not to have incentive for deviating to $\typei[1]$, we have $$\begin{gathered}
    \langle D, \detalloci[1](D, V_2) \rangle - \payi[1](D, V_2) 
    \\
    = \langle D, \detalloci[1](V_1^*, V_2) \rangle - \payi[1](V_1^*, V_2) 
    \\
    \geq \langle D, \detalloci[1](V_1, V_2') \rangle - \payi[1](\typei[1], V_2),
    \label{eq:D-t1}\end{gathered}$$ where for the equality we used the fact that 