  Fortification-interdiction games are tri-level adversarial games where two
opponents act in succession to protect, disrupt and simply use an
infrastructure for a specific purpose. Many such games have been formulated and
tackled in the literature through specific algorithmic methods, however very
few investigations exist on the completeness of such fortification problems in
order to locate them rigorously in the polynomial hierarchy. We clarify the
completeness status of several well-known fortification problems, such as the
Tri-level Interdiction Knapsack Problem with unit fortification and attack
weights, the Max-flow Interdiction Problem and Shortest Path Interdiction
Problem with Fortification, the Multi-level Critical Node Problem with unit
weights, as well as a well-studied electric grid defence planning problem. For
all of these problems, we prove their completeness either for the $\Sigma^p_2$
or the $\Sigma^p_3$ class of the polynomial hierarchy. We also prove that the
Multi-level Fortification-Interdiction Knapsack Problem with an arbitrary
number of protection and interdiction rounds and unit fortification and attack
weights is complete for any level of the polynomial hierarchy, therefore
providing a useful basis for further attempts at proving the completeness of
protection-interdiction games at any level of said hierarchy.
