Is it rational to own a firearm for self protection?

Consider the following graphic:


This graphic is a representation of the costs and benefits of gun ownership as a function of environment and ability.  Costs and benefits are simplified to ‘probability of causing my death’ and ‘probability of preventing my death’, respectively.  This does not account for all (or even most) of the costs and benefits of gun ownership, and the values on the axes on the graph are for illustration, and do not accurately represent probabilities of harm specifically or generally.  Nevertheless, the graph can be used to understand the circumstances in which gun possession could be considered rational and irrational.  This is done by dividing the probability on the y-axis by the probability on the x-axis; if the resultant ratio is above 1, then gun ownership is rational.  Otherwise it is not.

As one would expect, for a child, gun ownership is usually a bad idea since their lack of skill and maturity is more likely to result in them harming themselves than using the firearm for self protection. At the other extreme, trained soldiers in war zones are probably more likely to survive when in possession of a firearm.  Nevertheless, an untrained soldier is still worse off than a trained soldier since the probability of accidentally dying at the hands of his own gun is higher than his trained contemporaries.

The more general question for the public is where am I on this graph?  Am I more likely to die or prevent death when in possession of a gun?  To some degree, this question can be answered with a little bit of data and some trivial arithmetic.

Let’s start with the annual homicide rate in the US.  It’s somewhere around 5 per 100,000 per year.  One of the arguments in favour of firearm ownership is that individual risk of homicide is lowered by possessing a firearm (moving up the y-axis of my graph).  It is hard to know what value to assign this reduction in homicide risk, but John Lott has suggested that concealed handgun laws reduce murders by 8%.  This seems a reasonable number to start with, but I’ll be generous and round it up to 10%, and use that as a working value.  So I am assuming that by possessing a firearm, one can reduce their risk of homicide by 0.5 per 100,000.

Next, we must determine the rate of accidental death from a firearm, the rate of firearm suicide, and the rate of firearms being turned on the person wielding them.  There are between 500 and 700 accidental gun deaths per year in the US, which gives us a rate of between 0.16 and 0.22 per 100,000.  I’ll choose the lower of these two values to work with.  The rate of firearm suicide is roughly 6 per 100 000.  However we can’t use this rate in our calculation since in the absence of a gun, people may commit suicide using other means.  So I am going to arbitrarily drop this value to 0.6 per 100,000—this is an estimate of firearm suicide that wouldn’t have otherwise occurred even if firearms weren’t available.  Even harder is to know the rate of deaths in which a person’s firearm is turned on them in a confrontation.  I don’t have an estimate of the rate of death from this type, so let’s just call this value an unknown, X.  We do know that X can’t exceed 5 (since this is the homicide rate) and that X it is probably a fairly small value.

So the ratio of probabilities, r,  is calculated as

r = 0.5 / (0.16 + 0.6 + X).

This formula tells us whether or not the gun ownership is rational in a population over the long term.  The numerator is the net survival benefit of firearm ownership (y-axis).  The denominator is the net cost (x-axis).  The formula might have been very different if we were to include externalized costs of gun ownership on survival, but let’s keep it simple.  If r is above 1, then owning a firearm is rational, if below 1, owning a firearm is irrational.

Given the assumptions thus far, owning a gun does not seem rational for the population as a whole.   Even if we set X=0, r is still less than 1.  But this does not take into account variations in environments and contexts of gun ownership, which seem to explain important differences in suicide and homicide rates.  The suicide and homicide rates in the US vary considerably with ethnicity.  Based on US Department of Justice Statistics, the homicide rate for white Americans is around 3 per 100,000, and for African Americans around 17 per 100,000.  Suicide rates in the white population are about double that of the African American population.  So now we have two additional ratios to consider:

rwhites = 0.3 / (0.16 + 0.7 + X), and

rAfrican Americans = 1.7 / (0.16 + 0.35 + X).

Based on these data gun ownership is probably not rational in the white population (rwhites is less than 1).  On the other hand, for African Americans it may be very rational to own a gun (rAfrican Americans is greater than 1).

If we stratify suicide rates and homicide rates by gender, the results are also interesting:

rwomen= 0.2 / (0.16 + 0.3 + X), and

rmen = 0.68 / (0.16 + 0.9 + X).

From these calculations, gun ownership is probably irrational for women and men, but more irrational for women than men.  I suspect if we stratified by income, race and sex simultaneously, an even clearer picture would result: gun ownership is clearly irrational rich white women, and clearly rational for poor African American men.

If this all seems rather cynical, well it should.  While I’ve framed this thought experiment in terms of a simple calculation of rationality, it merely adopts the controversial logic behind the argument that firearms reduce an individual’s risk of being a victim of violence.  This is the kind of statement that can be analyzed empirically, and given the data available to me, the logic does not seem to hold universally, but is very much dependent on circumstance, and in particular, the social environment in which one lives.

Even if some of the parameters I’ve used  are imprecise, there are some important things to take away from this exercise.

First, as the homicide rate goes down, it becomes less and less rational to own a gun, and in many US states (and in most OECD countries) gun ownership is probably universally irrational when it comes to self-preservation.  People own guns for reasons other than defense against murder, and that has not been taken into account here, but nor has the fact that gun ownership promotes a culture that many people find distasteful, and creates negative social externalities that can have wide reaching effects. In any case, in a debate on the question of survival alone, gun-control advocates may be able to make the superior appeal to logic and common sense.  Second, the demographic inequalities in the US suggest a perverse race paradox in gun ownership.  White people in the U.S. are more likely to own guns than African Americans, yet are less likely to be killed in a homicide whether or not they own a gun.

I am not so naive to think that any evidence I present will change the minds of the pro-gun advocates, or even that what I’ve written here is really new.  However, I do favour trying to understand decisions by examining whether or not they are logically coherent and rational, since at least we can then better understand the true motives for favouring one position over another.  In this case, one has to question the real motives of gun advocates–if guns don’t make most of us safer, then why frame them as a tool of self defense?


Thanks to William Bland, who pointed out the importance of stratifying suicide rates by gender and ethnicity.