items $l$ and $r$ that can be critical according to ordering . For item $l$ we have $$\begin{aligned}
\label{comL}
l := \min \{ j: \sum\limits_{i=1}^{j} w_{i} \geq C_l\}. \end{aligned}$$ All items $1, \dots, (l-1)$ cannot in fact be critical even without the leader’s interdiction. For the last item $r$, we first compute the maximum weight of the follower that can be interdicted by the leader (similarly as in [@CaCarLoWo16]) by solving the following problem (denoted by $LW$).\
$$\begin{aligned}
\text{max}\quad & \sum\limits_{i=1}^{n} w_{i}x_{i}  \label{eq:ObjWmax}\\
    \text{subject to}\quad
    & \sum\limits_{i=1}^{n} v_{i}x_{i} \leq C_u \label{eq:capLWmax}\\
& x_{i} \in\{0,1\} \qquad i= 1,\dots,n \label{eq:varDefXWmax}\end{aligned}$$ Item $r$ is defined as $$\begin{aligned}
\label{comR}
r := \min \{ j: \sum\limits_{i=1}^{j} w_{i} \geq C_l + z(LW)\}. \end{aligned}$$ Since from we have$\sum\limits_{i=1}^{r} w_{i}(1 -x_i) \geq C_l$ for any leader’s strategy, all items from $(r+1)$ to $n$ cannot be critical.

### Building models $CRIT_2(c)$ {#BuildMod}

For each candidate critical item $c \in  [l,r]$, we formulate model $CRIT_2(c)$ by constructing a proper set $\mathcal{F}(\pi, x, k)$ as follows. Consider the subsets involving items in the interval $[c - \delta, c + \delta]$. Even for small value of $\delta$, the number of subsets can be very large. Hence, in order to limit the number 