Abstract: Dan calls for more Concealed Weapons and has a qualitative mathematical model that gives a sense of why that could be very rational if your goal is to prevent mass public shootings.

Most of my friends will find it bizarre if not bat-guano insane that I don't agree with their call for more gun control after the Sandy Hook shootings. In fact, I think the biggest issue in mass shootings is our poor mental health care situation. However, when it comes to gun politics, I think that mass shootings are a problem whose answer is more guns, or at least more concealed carry. As a nation we own plenty of firearms, but in general law-abiding citizens do not have access to them in sufficient quantities at sufficiently large varieties of locations. This is not just a principled position about "what's right" though, it's an actual model-based opinion about how we can prevent horrible mass murders.

Preventing mass murders, not gun control, should be our goal. There are those who simply want gun control laws for their own sake, I believe those people are putting the lives of children and citizens in general second, even if they don't realize it. Here's why:

### A simple model of attacker-defender probabilities:

It's well known from research that concealed carry holders (CCW) are much less likely to commit crimes than average. In fact, they tend to be extremely responsible people, and especially when it comes to violent crime. So we can rule out the idea that more CCW means more overall crime (the research is in, we've done the experiment in 40 or so states, there is no big change before and after CCW shall-issue laws).

However, you could argue that there is no big change, so CCW has little effect. Let's see why that might be true, and what we might expect to change that, and why mass-shootings in particular are a good candidate for prevention by CCW.

Let $p$ be the probability that someone who is attacked is carrying a concealed weapon (this varies according to the general level of CCW in the population, but also according to the circumstances of the attack. Federal law makes it illegal to carry a gun near a school for example!). Then if $N$ is the number of people involved in the attack (both those targeted, and those standing nearby who might be able to help), the probability that an attacker will encounter zero armed defenders is $(1-p)^N$ and the probability that he will encounter exactly one armed defender is $np(1-p)^{(N-1)}$.

Statistics show that if an attacker encounters even one armed defender, the fact that they are concealed and not immediately identifiable puts the attacker at considerable peril. There have been quite a few mass shootings foiled by a single armed defender. However, let's suppose that what we really need is 2 armed defenders to turn the tables dramatically against an attacker. You can't simultaneously shoot two armed defenders so being outnumbered and non-identifiable, this circumstance will tend to quickly end mass shootings.

Let's see how the probability of "success" for the attacker varies with $N$ the number of potential victims, and $p$ the probability that a victim will be armed:

Armed mass attackers want large N, and small p. The "Agile Attacker" theory is that the attacker can wait and choose his location to minimize p. This is likely one major reason why attackers choose schools where children are guaranteed not to shoot back, and the adults have been banned by federal law from carrying weapons.

As you can see if all locations had equal levels of CCW in the vicinity of as little as 5% of the population then attackers would be dramatically prevented from succeeding in large shooting sprees. Remember $N$ here is not just the number of people who fall victim, but also the number of "bystanders" who might be able to  help. Typical mass shootings involve around 20 to 30 people killed, which means that the number of people who might reasonably be defenders is usually well over 50 and probably frequently exceeds 100.

One serious problem, though, is the existence of places where CCW is banned. These public places are essentially "designated victim zones" where the probability of success for the attacker is nearly 1 (see graph at p ~ 0).

Until we have essentially all locations protected by 3-5% of the population willing to undergo CCW background checks and training, and carrying in almost every location where people can be found in large quantities in public, the agile attacker can seek out the unprotected.

But widespread CCW at the 5% level is easily achieved. First of all, there will be no opposition from  "the gun lobby" (consisting primarily of people who have intuited the mathematical results above). Second of all, we already have many states where CCW is in the range of 3% or so. However, we still have restrictions on certain places where CCW is not allowed. This gives the agile attacker plenty of niches to carry out horrible deeds. Eliminating those niches by repealing laws that ban CCW in certain locations, and enforcing a strengthening of armed security in the remaining locations where CCW is still banned is the right approach. Since no-one wants to go to school in a locked-down prison-like environment, I particularly believe that CCW in schools needs to be allowed and even encouraged. We already entrust our children's safety to teachers and administrators, and we should give those teachers and administrators who are willing the opportunity to save our children in the event of horrible attacks.

4 Responses leave one →
1. December 17, 2012

Some issues brought up by friends on my Facebook list:

This model takes into account the probability that the attacker will face armed resistance, but doesn't take into account the effectiveness of said armed resistance. I believe that a careful study of the history of mass shootings will show that armed resistance by even 1 and especially by 2 or more defenders is quite successful. It is however quite dangerous, but so is passive waiting. Basically the situation is bad no matter what, but armed defenders certainly don't seem to make things worse, and there is data out there which suggests that they probably make things dramatically better. However I am not a criminologist, and I don't have a reference for what I would call a "high quality" study of the effectiveness of armed defenders in mass shootings so I'm not going to post links here. I invite you to try to find what you'd call a "high quality" study of effectiveness.

Also, I want to emphasize that Mental Illness is the first and foremost problem. I talk about guns and CCW only because it's the larger conversation on my FB feed and also I felt that I had a somewhat unique quantitative perspective to add to that conversation. If we had dramatically better mental health care, I believe we could significantly reduce the rate at which these things occur and simultaneously solve a lot of other societal problems caused by mental illness.

2. December 17, 2012

Another fair comment is that this model treats individuals as independent. In most public locations that is probably a good assumption. I don't consult with other people who are going to a mall or the same office building as to whether today is a good day to bring my camera for example, so if cameras were suddenly necessary you'd go around asking people if they had a camera and you'd find that a binomial random variable worked well to describe the random fluctuations in the number of cameras you find... similarly for CCW under sort of "average" conditions.

That all goes out the window when you have restrictions on CCW, so at schools, churches, and places with posted No CCW signs, everyone who is law abiding CCW holder leaves their gun somewhere else. similarly at a off-duty police convention you might expect that either a large number of people are carrying, >> 2 or if there is a restriction on carry in the convention hall, then 0 are carrying. That kind of correlation thing can happen but is probably relatively less important practically than the 'binomial random sample' model.

3. December 17, 2012

Wikipedia http://en.wikipedia.org/wiki/List_of_countries_by_firearm-related_death_rate claims that suicides and accidents dominate gun deaths (that is, outweigh homicides). If this is true, public safety suffers if gun ownership rises. Naively, a large increase in gun ownership would somewhat increase suicide rates and vastly increase accidents.

• December 17, 2012

Actually I don't advocate a substantial increase in firearms ownership, though for effect I sort of imply it in the title. I really think we have plenty of firearms, what I am advocating is that those people ready and willing to pass background checks and training etc required for a CCW be given that opportunity to posses their firearms in public in a uniform way. Probably the worst outcomes we can get would come from having a ton of firearms "out there" but also having a lot of laws that prevent law abiding citizens from using them for defense. That's pretty much the situation we have to a greater or lesser extent depending on which state you live in and which locations, such as schools and churches etc you are discussing.