= Solution
With agent variables, each <constraint domain> has <cardinality> $2^m$. Their total number of listed domain values is $n2^m$, while the <Cartesian product> of the domains has
$$
\boxed{(2^m)^n=2^{mn}}
$$
complete assignments before constraints.
With item variables, each <constraint domain> has <cardinality> $n$. Their total number of listed domain values is $mn$, while their <Cartesian product> has
$$
\boxed{n^m}
$$
complete assignments. These are raw domain-space sizes. The larger agent-variable space includes overlaps and missing items; imposing the allocation constraints reduces it to the same $n^m$ actual complete allocations represented directly by item variables.
Back to article page
