A simple model of gun ownership and crime rates

Does gun ownership increase or reduce violent crime? I’m not sure whether a model for this already exists, but in this post, I attempt to model the causal effect of gun ownership on violent crimes in this post (I’ll share any relevant literature I find, if I find any, in subsequent posts). I’ll jump straight into the model setup, with explanations injected as I go along.

Model setup

  • We assume that a person makes the decision about whether or not to commit a violent crime based on two variables: their innate desire to commit the crime (\beta_{i} \sim U(0,1)) and their probability of success (\chi_{i}). We represent this with \kappa_{i} = 0.5(\beta_{i} + \chi_{i}).
  • Their probability of success depends on two factors: some innate ability to use force (a_{i} \sim U(0,1)) and the ability of the other party to use force (a_{-i} \sim U(0,1)). Their innate ability is known to them. The ability of the other party is not. We represent this with \chi_{i} = a_{i} - a_{-i} + 0.5, i.e. if you are better able to use force than the other party (a_{i} > a_{-i}), it is likely that you successfully commit the crime and subdue them with force, with your probability of success increasing linearly as the gap between your ability and the other party’s ability grows. If you are less able to use force than the other party (a_{i} < a_{-i}), the converse applies. If your ability is equal to that of the other party (a_{i} = a_{-i}), your probability of success is \chi_{i} = 0.5, i.e. you have a 50% chance of success and 50% chance of failure.
  • If \kappa_{i} crosses some threshold, \bar{\kappa}, 1 > \bar{\kappa} > 0.5, then the person will commit a crime. If it doesn’t, then they will not. We set the upper bound on \bar{\kappa} such that there will always be people who commit violent crimes (\bar{\kappa} < 1 = maximum \kappa). We set the lower bound on \bar{\kappa} such that there will always be people who deeply want to commit violent crimes (\beta_{i} = 1) but who are completely unable to (\chi_{i} = 0) and so \kappa_{i} = 0.5 < \bar{\kappa}, and these people do not commit violent crimes. Similarly, people who are very able to commit violent crimes (\chi_{i} = 0) but really don’t want to (\beta_{i} = 0) have \kappa_{i} = 0.5 < \bar{\kappa} and don’t commit violent crimes.

Baseline case: No guns

The individual decides on whether or not to commit a violent crime, i.e., he considers his desire and his probability of success. Not knowing the ability of the other party, he takes the expected ability of the other party (E[a_{-i}] = 0.5, since we are considering a uniform distribution over the interval [0,1]), i.e.

\kappa_{i} = 0.5E[\beta_{i} + a_{i} - a_{-i} + 0.5] = 0.5(\beta_{i} + a_{i})

If 0.5(\beta_{i} + a_{i}) \geq \bar{\kappa}, the individual attempts a crime (this doesn’t say anything about whether or not he succeeds, only that he attempts it).

Everyone has a gun

Now we consider the case in which guns are provided to everyone. What happens when a gun adds \alpha to each person’s ability to be violent? We note that this does not change the number of violent crimes committed, since

\chi_{i} = 0.5(a_{i} + \alpha - (a_{-i} + \alpha) + 1) = 0.5(a_{i} - a_{-i} + 0.5)

Let’s say that the value a gun adds to someone’s ability to be violent is increasing in \kappa_{i}, i.e. people who have decided to or have very high likelihood of deciding to commit a violent crime are more able to use a gun to their advantage than people who don’t want to commit a violent crime. Intuitively, this seems reasonable, because violent criminals have the advantage of surprise, and their victims, even when carrying guns, are unable to reach for their guns in time or don’t want to because they don’t want to risk angering their attackers and being shot. For simplicity, let’s say \alpha_{i} is a function that takes the form \gamma\kappa_{i} + c, where \gamma < 1, c > 0.

What we then have is

\kappa_{i} = 0.5E[\beta_{i} + (a_{i} + \gamma\kappa_{i} + c - (a_{-i} + \gamma\kappa_{-i} + c) + 0.5] = 0.5(\beta_{i} + a_{i} + \gamma\kappa_{i} - 0.5\gamma)

\kappa_{i} - \gamma \kappa_{i} = 0.5(\beta_{i} + a_{i} - 0.5\gamma)

\kappa_{i} = \frac{0.5(\beta_{i} + a_{i}) - 0.25\gamma}{1-\gamma}

What we see is that for sufficiently large \gamma (\gamma > 0.5), \frac{0.5(\beta_{i} + a_{i}) - 0.5}{1-\gamma} > 0.5(\beta_{i} + a_{i}), no matter what \beta_{i} and a_{i} are.

What is the implication of this?

Let’s take a \bar{\kappa} of 0.7 and \gamma of 0.5. I previously had

\kappa_{i} = 0.5(0.6 + 0.6) = 0.6 < \bar{\kappa}

Now, I have

\kappa_{i} = \frac{0.5(0.6 + 0.6) - 0.5(0.5)}{1-0.5} = 0.7 = \bar{\kappa}

So, given \gamma > 0.5, we will see an increase in the number of people committing violent crimes.

Ban on guns, but individuals can obtain them illegally

Next, we introduce a situation with a ban on guns but the ability to obtain weapons illegally. We assume the probability of an individual obtaining a gun is dependent on \beta_{i} (the intuition behind this is that the more dogged they are in their pursuit of a gun to act out their violent tendencies, the more likely it is that they get the gun), with some cutoff: \beta_{i} > \bar{\beta} > 0.5 means that the individual has a gun. What this means is that the average person does not have a gun, and the expectation that someone is carrying a gun is 1 - \bar{\beta}.

For an individual with \beta_{i} > \bar{\beta}, and with the same expectations as laid out before, it’s easy to observe that

\kappa_{i} = 0.5(\beta_{i} + a_{i} + \gamma\kappa_{i} - (1-\bar{\beta})(0.5\gamma+ \bar{\beta}c)

\kappa_{i} = \frac{0.5(\beta_{i} + a_{i} + \bar{\beta}c - 0.5\gamma(1-\bar{\beta}))}{1-\gamma}

Since 1 - \gamma < 1 and c > 0, \kappa_{i} here is obviously higher than in the case with no guns at all. It is also higher than \frac{0.5(\beta_{i} + a_{i}) - 0.25\gamma}{1-\gamma}, derived from the case where everyone carries a gun.

For an individual with \beta_{i} < \bar{\beta}, and with the same expectations as laid out before, we observe that

\kappa_{i} = 0.5(\beta_{i} + a_{i} - (1-\bar{\beta})(0.5\gamma + c))

The individual has a lower \kappa_{i} than in the case with no guns, as well as a lower \kappa_{i} than in the case with everyone carrying a gun.

What this tells us is that allowing everyone to carry a gun (relative to a situation in which we ban guns but there is still illegal gun ownership) decreases the probability that someone who has high pre-existing violent desire (\beta_{i} < \bar{\beta}) commits a crime, but increases the probability that someone who has lower pre-existing violent desire (\beta_{i} > \bar{\beta}) commits a crime. What this means for the overall violent crime rate depends on what \bar{\beta} is: the implication being that if there’s a very low proportion of people who manage to obtain guns in the presence of a gun ban, then we should ban guns to increase general safety.

I’ll stop here for now and just highlight some simplifying assumptions made in this model. There are, of course, assumptions made in this model. One is that the ability is exogenously determined, and independent of desire to be violent. Another important one is that the decision to commit a violent crime is made based on an expectation of the average ability of all victims to be violent and retaliate, where in reality, violent criminals are choosing their victims because these victims are perceived to be of lower ability to retaliate.

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