Theorem 2, p. 17 — the equilibrium detection rate r*_{b1} is the same in every Γ(b1,b2) with 1 ≤ b2 < m*, and Γ(b1,1) equilibrium inspection strategies stay equilibrium strategies

Let $\mathcal G=(\mathcal V,\mathcal E,\{\mathcal C_i\})$ be a detection model and $b_1\in\mathbb N$ the number of detectors. Then:

1. there is a number $r^*_{b_1}\in[0,1]$ such that, for every $b_2$ with $1\le b_2<m^*$ and every Nash equilibrium $\sigma^*$ of $\Gamma(b_1,b_2)$,

2. every inspection strategy $\sigma^{1*}$ that is part of a Nash equilibrium of $\Gamma(b_1,1)$ is also part of a Nash equilibrium of every game $\Gamma(b_1,b_2)$ with $1\le b_2<m^*$: some attack strategy $\sigma^{2\prime}\in\Delta(\mathcal A_2)$ makes $(\sigma^{1*},\sigma^{2\prime})$ an equilibrium of $\Gamma(b_1,b_2)$.

The expected detection rate in equilibrium depends on the defender's resources only, as long as the attacker's resources stay below the maximum set packing size $m^*$; and the defender can compute an equilibrium inspection strategy as if the attacker had a single resource.

Formalization Note The number $r^*_{b_1}$ is chosen after $b_1$ and before $b_2$, as in (17). The hypothesis $b_2\ge 1$ is added: in $\Gamma(b_1,0)$ every profile is an equilibrium and the rate is $1$, whereas $r^*_{b_1}<1$ for $b_1<n^*$; the paper's base case is $\Gamma(b_1,1)$. The case $b_1\ge n^*$ is included, as printed; it relies on the convention $F(S,\emptyset)/|\emptyset|=1$ of the rate. If $m^*\le 1$ there is no admissible $b_2$ and both parts hold vacuously. The new attack strategy in part 2 lives on the action set of $\Gamma(b_1,b_2)$ and is not required to be related to the equilibrium attack strategy of $\Gamma(b_1,1)$.