Analytic models for active shooter incidents with civilian resistance

This is the situation where a student or teacher finds himself in a classroom when a campus shooting occurs. In this case, we also assume the following:

1. (i)

   There are $N$ civilians in the closed arena, where $N$ is a positive integer. The active shooter is not considered as a civilian.

2. (ii)

   The shooter is already at the only door of the closed arena. There is no other place to exit the room. Therefore, the civilian of focus can either fight or hide. A civilian who does not fight will be killed by the shooter with equal probability. The shooter will walk around until every civilian is dead.

3. (iii)

   The shooter is equipped with a rifle, and his killing rate is $m$ people per minute where $m$ is a positive integer and $N=T_{1}m$ for some positive integer $T_{1}$. He carries plenty of magazines that are far more than enough to kill all the $N$ civilians. His rifle will not malfunction during the shooting. He plans to kill as many civilians as possible.

4. (iv)

   If an unarmed civilian fights back, his chance of success will be $p_{1}$. If an armed civilian fights back, his probability of success will be $p_{2}$, where $p_{2}>p_{1}>0$.

To avoid repetitions in our presentation, we will use $p$ to denote the survival probability of the civilian of focus, where $p=p_{1}$ if he is unarmed and $p=p_{2}$ if he is armed.

It is clear from our Assumption (iii) that $T_{1}$ is the number of minutes the shooter needs to kill all the civilians in the arena. Thus, $T_{1}$ plays a vital role in our analysis. In particular, we need to consider two cases: (I) $T_{1}>T_{2}$ and (II) $T_{1}\leq T_{2}$. When $T_{1}>T_{2}$, the authority will arrive before the shooter can kill all the civilians in the arena; hence, every civilian should stand a nonzero chance of survival anytime. However, if $T_{1}\leq T_{2}$, the shooter will murder all the civilians before the authority arrives; as a result, the chance of survival of each civilian should be zero after $T_{1}$ minutes. Our analysis below confirms this intuition.

Table 1 provides the civilian’s survival probabilities over time, where the $i$-th cell of the “fight” row is his survival probability when he hides for the first $(i-1)$-th minutes and starts to fight back at the beginning of the $i$-th minute. Table 1 offers some insight into civilian resistance during an active shooter incident. First, if the civilian decides to hide, his chance of survival decreases over time until the authority arrives. The same can be said for the civilian if he plans to fight back. Since $N=T_{1}m$ and $T_{2}<T_{1}$, we know that $1-T_{2}/m>0$. That is, if the authority is able to arrive quickly, a civilian will have a positive chance of survival if he decides to hide. This means the authority response time is critical in an active shooter event. For parents who are considering new schools for their children, they might want to take this into consideration. Everything else being equal, a school that is close to a police station or staffed with an armed guard should be chosen over the one that is not. Also, some schools in rural areas are usually far away from the authority, they should consider adding an armed guard if they do not have one yet.

Next, if $p>(1-m/N)$, then the civilian should fight back at the inception of the shooting. More generally, if $p[1-(k-1)m/N]>(1-km/N)$ for some $1\leq k\leq T_{2}$ but $p[1-(j-1)m/N]<(1-jm/N)$ for $1\leq j\leq k-1$, then a civilian should fight back during the $k$-th minute; otherwise, “hide” will be a better option. For an unarmed civilian (i.e., $p=p_{1}$), this analytically confirms the prediction made in Sporrer (2020) that there are some desperate situations where the best option of an unarmed civilian is to fight the shooter. This shows that hiding is not always the only rational option for an unarmed civilian. On the other hand, if $p[1-(k-1)m/N]<(1-km/N)$ for all $1\leq k\leq T_{2}$, or, equivalently, $p<[N-km]/[N-(k-1)m]=1-m/[N-(k-1)m]$ for all $1\leq k\leq T_{2}$, then a civilian should hide all the way through. Note that $1-m/[N-(k-1)m]$ is decreasing in $k$; therefore, the condition $p<1-m/[N-(k-1)m]$ for all $1\leq k\leq T_{2}$ is equivalent to the condition $p<1-m/(N-m)$.

Moreover, if $p_{2}>1-m/[N-(k-1)m]$ for some $1\leq k\leq T_{2}$, then $p_{2}>\max\{p_{1},1-m/[N-(k-1)m]\}$, showing that carrying a firearm may increase a civilian’s chance of survival in some cases. However, carrying a firearm does not always increase one’s chance of survival. For example, if $p_{2}<1-m/[N-(k-1)m]$ for all $1\leq k\leq T_{2}$, then a civilian would better hide throughout even if he is armed.

Finally, it is possible that $p_{2}>1-m/[N-(k-1)m]$ but $p_{1}<1-m/[N-(k-1)m]$ for some $1\leq k\leq T_{2}$. In this case, carrying a firearm increases the odds that “fight” is the best option.

By similar analysis as in the case where $T_{1}>T_{2}$, we can obtain the civilian’s survival probabilities over time that are given by Table 2.

A few remarks in order. First, if the civilian wants to hide and wait for the authority, his chance of survival will decrease over time and become zero before the authority arrives. The same can be said for the civilian if he plans to fight back. That is, if the authority is unable to arrive quickly (e.g., a campus shooting in a rural area), the civilian will stand no chance of surviving in a closed arena shooting after a certain time point, regardless of whether he decides to hide and wait for the authority.

Second, if $p>(1-m/N)$, then a civilian should fight back right away. More generally, if $p[1-(k-1)m/N]>(1-km/N)$ for some $1\leq k\leq T_{1}$ but $p[1-(j-1)m/N]<(1-jm/N)$ for all $1\leq j\leq k-1$, then a civilian should fight back during the $k$-th minute; otherwise, he had better to hide. Again, the aforementioned prediction made in Sporrer (2020) is confirmed here analytically. It is worth underlining that even if $p[1-(k-1)m/N]<(1-km/N)$ for all $1\leq k\leq T_{1}-1$, or, equivalently, $p<1-m/[N-(k-1)m]$ for all $1\leq k\leq T_{1}-1$, the civilian should fight in the $T_{1}$-th minute because his chance of survival is zero if he hides during the $T_{1}$-th minute.

We see again that, if $p_{2}>1-m/[N-(k-1)m]=\max\{p_{1},1-m/[N-(k-1)m]\}$ for some $1\leq k\leq T_{2}$, then carrying a firearm will increase a civilian’s chance of survival. Once again, this does not imply that carrying a firearm will necessarily increase one’s chance of survival because it is possible that $p_{2}<1-m/[N-(k-1)m]$ for all $1\leq k\leq T_{1}$. Hence, an armed civilian should not always choose to fight. If $p_{2}>1-m/[N-(k-1)m]$ but $p_{1}<1-m/[N-(k-1)m]$ for some $1\leq k\leq T_{1}$, then carrying a firearm will increase the chance of civilian resistance when everything else being equal.

A typical example of this is when a student or teacher finds himself in the hallway when a campus shooting happens. Besides the common assumptions in Section 2, we also assume the following:

1. (1)

   There are $M$ civilians in the hallway, where $M$ is a positive integer.

2. (2)

   There are $K$ separate closed arenas close to each civilian. The shooter will search these $K$ closed arenas one by one to look for his victims. The shooter will not re-enter a closed arena he has searched. Once the shooter enters a closed arena, he will not move to another closed arena until he kills all civilians in the current arena.

3. (3)

   Each civilian can either run or hide in one of the $K$ closed arenas. In addition, the civilian of focus can also fight.

4. (4)

   If an unarmed civilian fights, his chance of success will be $\tilde{p}_{1}$. If an armed civilian fights back, his probability of success will be $\tilde{p}_{2}$, where $\tilde{p}_{2}>\tilde{p}_{1}>0$. Note that $\tilde{p}_{1}$ may or may not equal to $p_{1}$. Similarly, $\tilde{p}_{2}$ may or may not equal to $p_{2}$. We will use $\tilde{p}$ to denote the survival probability of a civilian, where $\tilde{p}=\tilde{p}_{1}$ if he is unarmed and $\tilde{p}=\tilde{p}_{2}$ if he is armed.

5. (5)

   If a civilian runs, he can safely exit the area with a probability of $p_{r}>0$.

6. (6)

   If a civilian decides to hide in one of the $K$ nearby closed arenas, he will succeed with a probability $p_{h}>0$.

7. (7)

   The hallway interaction ends within a minute.

8. (8)

   The authority will not arrive during hallway interaction.

9. (9)

   There will be $N$ civilians in each of the $K$ closed arenas once the hallway interaction ends, where $N<M/K$.

The decision of the civilian of focus during hallway interaction (i.e., the first minute) is easy. If $p_{h}>\max\{p_{r},\tilde{p}\}$, then he should hide into a closed arena; if $p_{r}>\max\{p_{h},\tilde{p}\}$, then he should run; if $\tilde{p}>\max\{p_{r},p_{h}\}$, then he should fight back.

At the beginning of the second minute, the shooter may enter the closed arena where the civilian of focus is hiding with a probability $1/K$. If this happens, then the results in Section 2.1.1 will apply from the second minute to the $\max\{T_{1}+1,T_{2}+1\}$-th minute. However, if the shooter enters an arena other than where the civilian of focus is hiding at the beginning of the second minute, which happens with a probability $1-/K^{2}$, then the civilian of focus will hide from the second minute until the $(T_{1}+1)$-th minute. In this case, if $T_{2}\leq T_{1}+1$, the civilian of focus will surely survive. However, if $T_{2}>T_{1}+1$, then at the beginning of the $(T_{1}+2)$-th minute we will be in the same situation as the one we face at the beginning of the second minute, except now $K$ is being replaced by $K-1$. Then, the shooter might enter the arena where the civilian is hiding with a probability $1/(K-1)$, in which case the results in Section 2.1.1 will apply again from the $(T_{1}+2)$-th minute until the $\max\{2T_{1}+1,T_{1}+T_{2}+1\}$-th minute. The analysis for the rest of the time until the $(KT_{1}+1)$-th minute is similar. The larger $K$ is, the longer the whole event will last and hence the more likely the civilian of focus will have a nonzero chance of survival as time passes.

In this case, the incident will not last beyond time $T_{2}$. Note that if a civilian chooses to run, he can either exit the area or get killed in a minute. If he fights back, he will either subdue the shooter with a probability $p$ or get killed immediately. But if he hides, he can still choose to fight, hide, or run in the next minute. Employing a similar argument as in Section 2.1.1, we can obtain the survival probabilities of the civilian of focus over time. These probabilities are summarized in . For example, the civilian is still in the area at the beginning of the second minute if and only if he chooses to hide in the first minute. This happens with a probability $1-m/(N-e)$ because, according to our assumption, none of the $e$ civilian who exit the open arena is shot during the first minute. It follows that there are $N-(m+e)$ civilians left in the arena at the end of the first minute. If he chooses to hide again at the beginning of the second minute, then his chance of surviving during the second minute would be

Note that the assumptions on $e,m$, and $N$ imply that all probabilities in Table 3 are positive. Based on Table 3, we can make almost the same observations as for the case $T_{1}>T_{2}$ in Section 2.1.1. For the sake of brevity, they will not be repeated here. The same will be done for the remainder of the paper unless new insight arises. Readers will see that Observations (i)–(vi) at the end of Section 1 will hold in all cases.

In this case, the event will end at time $T_{1}$ if not before. Similar to the case where $T_{1}>T_{2}$, we can obtain the survival probabilities of the civilian over time; see Table 4.

Similar to the case where $T_{1}>T_{2}$, it is easy to see that all probabilities in Table 4 are positive. In addition, all survival probabilities after the $(T_{1}-1)$-th minute in the “hide” row are zeros because $N=T_{1}(e+m)$, i.e., all civilians will have either exited the arena or been killed at the end of the $T_{1}$-th minute. Again, we can make almost the same observations as in Section 2.1.2. For example, if the civilian of focus chooses to hide during the first $T_{1}-1$ minutes, then he should either fight back or run during the $T_{1}$-th minute.

Note that if a civilian is still in the arena and alive at the beginning of Phase 2, he must have survived Phase 0 and Phase 1 and his action in Phase 1 must be “hide”. Thus, the probability that the civilian is alive in the arena at the beginning of Phase 2, denoted as $p_{s}$, is given by

Following the same line of reasoning as in Section 2.2, we can obtain a civilian’s probabilities of survival over time during Phase 2. Table 5 summarizes these probabilities. We again make similar observations as in Section 1.2.1 and Section 2.2.1.

Similar to the case where $T_{1}<T_{2}$, we can derive all the survival probabilities in Table 6. Observations in this case are completely analogous to those in Section 1.2.2 and Section 2.2.2.

Besides the common assumptions for Section 3, we also take the following assumptions in this section:

1. (a)

   There are $N$ civilians in the closed arena, where $N$ is a positive integer. The active shooter is not considered as a civilian.

2. (b)

   The shooter is already at the only door of the closed arena. There is no other place to exit the room. Therefore, a civilian of focus can either fight or hide. A civilian who does not fight will be killed by the shooter with equal probability. The shooter will walk around until every civilian is dead.

3. (c)

   The shooter is equipped with a rifle, and his killing rate is $m$ people per minute, where $m$ is a positive integer, and $K_{i}-K_{i-1}<m$, and $N=T_{1}m$ for some positive integer $T_{1}$. He carries plenty of magazines that are far more than enough to kill all the $N$ civilians. His rifle will not malfunction during the shooting. He plans to kill as many civilians as possible.

4. (d)

   When an armed civilian fights back, he will kill the shooter with a probability $p_{2}$. At the same time, he may also mistakenly kill another civilian with a probability of $p_{f}$, where $p_{f}=cp_{2}$ for some $0<c<1$.

5. (e)

   If an unarmed civilian fights back, he will subdue the shooter with a probability $p_{1}$, where $p_{2}>p_{1}>0$.

Let us look at the $i$-th minute. First, the probability that no civilian is mistakenly killed when an armed civilian fights back equals $(1-p_{f})^{j_{K_{i}}}$. Next, the event that an armed civilian fights back and survives the shooting is the same as the event that the shooter is killed by an armed civilian or the remaining $j_{K_{i}}-1$ armed civilians who follow him and, at the same, he is not mistakenly killed by the other $j_{K_{i}}-1$ armed civilians. The probability that the shooter is killed by one of the $j_{K_{i}}$ armed civilians is $1-(1-p_{2})^{j_{K_{i}}}$. The probability that an armed civilian is not killed by another armed civilian when he fights back is $[1-p_{f}/(N_{i}-1)]^{j_{K_{i}}-1}$ where $N_{i}=N-(i-1)m$. Hence,

where $K_{i}\in\{1,2,\ldots,N_{i}\}$.

Proposition 1 shows that more armed civilians do not necessarily increase the survival probability of a civilian of focus. Quite the opposite, when there are “too many” armed civilians, the survival probability of a civilian of focus starts to decrease due to the increased likelihood of “friendly fire”. An interesting practical implication of this is that the scenario where everyone carries a firearm might be worse than the scenario where only a few of them has a firearm.

For $i=1,\ldots,\max\{T_{1},T_{2}\}$, we put $p_{2i}=[1-(1-p_{2})^{j_{K_{i}}}][1-p_{f}/(N_{i}-1)]^{j_{K_{i}}-1}$ if a civilian is armed and $p_{2i}=p_{1}$ if he is unarmed. Similar to our analysis in Section 2.1, we can derive the survival probabilities of a civilian over times for both $T_{1}>T_{2}$ and $T_{1}\leq T_{2}$. We summarize the results in Table 7 and Table 8 respectively. The respective observations for both cases are similar to those in Section 2.1.1. For example, in the case where $T_{1}>T_{1}$, if $p_{2T_{i}}[1-(i-1)m/N]>(1-im/N)$ for some $1\leq i\leq T_{2}$ but $p_{2j}[1-(j-1)m/N]<(1-jm/N)$ for $1\leq j\leq i-1$, then a civilian of focus should fight back during the $i$-th minute; otherwise, “hide” will be a better option. On the other hand, if $p_{2i}[1-(i-1)m/N]<(1-im/N)$ for all $1\leq i\leq T_{2}$, or, equivalently, $p_{2i}<[N-im]/[N-(i-1)m]=1-m/[N-(i-1)m]$ for all $1\leq i\leq T_{2}$, then a civilian of focus should hide.

As for the hallway interaction, the assumptions, analysis, and observations are similar to those in Section 2.1.2 except that $p_{2}$ is replaced by $p_{21}$. It is easy to see again that the survival probability of a civilian of focus, as a function of the number of armed civilians, first increases and then decreases.