This is important for the proof.

$\detallocs$ is DSIC implementable.

Let the payment be 0 for both agents and all type profiles. As the allocation and payment rules are both symmetric, consider either agent $i$. If $\typei[-i] = A$, the maximum value agent $i$ could get, when her type is $A$, $B$, or $C$, is 1, attained with truthful bidding. For $\typei = D$, the four allocations all give the same value $\tfrac 2 3$. Similar arguments hold when $\typei[-i]$ is $B$, $C$ or $D$.

\[thm:det\] There exists no deterministic DSIC extension of $\detallocs$.

Before proving Theorem \[thm:det\], we make several preparatory observations.

A key difficulty with multi-dimensional preferences is the the lack of a payment identity à la Myerson [@Myerson81]. In order to argue that any extension of an allocation rule is not DSIC, one has to either check the many cyclic (or weak) monotonicity conditions, or show that no payment rule can support the extension in a DSIC mechanism. We designed $\dettypespaces$ and $\detallocs$ carefully so that the allocations “lock” the payment rules.

\[lem:equal-pay\] For any allocation rule $\extallocs$ that is an extension of $\detallocs$, if $\extallocs$ can be implemented by a DSIC mechanism with payment rule $\pays$, 