Static Spatial Modeling of Gun Ownership with Varying Degrees of Segregation

This is an academic paper for the Computational Economics course at Davidson College, with Dr. Shyam Gouri Suresh. This is a group project with Brooke Whitcomb, Sreylin Touch, Satyajit Banerjee. Written on 28th September 2021.

  1. Introduction 

US citizens have yet to come to a consensus on whether or how to regulate guns despite the high prevalence of gun violence. It is a polarizing topic, with some people advocating for government intervention and some people advocating against it. Gun ownership is argued to be the fundamental right for citizens of the United States; it is the Second Amendment: the right to bear arms for lawful purposes such as self-defense. 

Assuming that guns are owned primarily for self-defense purposes, we test the influence of people’s individual decisions to purchase a gun on aggregate ownership outcomes. An individual’s choice to buy a gun is based on personal optimization in response to the actions of the people surrounding them but has aggregate effects that are not intended nor considered by the individual. An individual may be compelled to buy a gun if someone else in their neighborhood has one, or many people in their neighborhood have one to feel safe. But their decision may also lead to more people in their neighborhood buying guns, and so on till everyone owns a gun. Because gun-prevalence is linked to violence, an individual’s choice to buy a gun to increase safety may counter-intuitively have negative aggregate safety effects. 

Using a static special model, we study this emergent phenomenon under different levels of segregation and differing extent of the sphere of an individual neighborhood. We study neighborhoods with a high degree of segregation and the level of gun ownership, and we also test how the size of someone’s neighborhood (how many people in their region they consider a neighbor) affects their decision to buy a gun holding preferences constant. Linking gun ownership prevalence with segregation and regional effects has implications for urban planning and local/national gun control policies. 

  1. Description

We use a Moore neighborhood, island lattice to simulate a neighborhood. Each cell of the lattice represents an agent in a neighborhood. Each agent is either in race 1 or race 0. We compare the likelihood of agents owning a gun in an integrated neighborhood to a segregated neighborhood using asynchronous updating. Agents of both races are randomly and normally distributed. Later in our work, we look at extended Moore neighborhoods to account for the sphere of influence condition. 

We use a 100×100 grid with 10,000 agents. We set the initial race of each agent by dividing the board into four quadrants and choosing the level of segregation between 0.5 and 1. When the segregation level is 0.5, each agent in every quadrant has a 50% chance of being either race 1 or race 0, which, on average, makes a perfectly integrated neighborhood. When the level is anything greater, the likelihood of an agent in a quadrant is skewed symmetrically in opposite quadrants to be dominated by one or the other race (see Figure 1 and Figure 2 below). 

Figure 1. Perfectly Integrated Neighborhood (0.5)  Figure 2. Segregated Neighborhood (0.9)

Once, the race is set for that level of segregation, the initial gun ownership of each agent is determined by the percentage of neighbors of the same race as the agent. If less than 25% of the neighbors of a particular agent look like they do i.e. they are primarily surrounded by the other race, then the agent starts with a gun. We assign initial gun ownership by this rule rather than randomizing the process because we assume that agents only own a gun when they want to increase their sense of safety and ignore other factors that might lead agents to purchase guns e.g. hunting sport preference, growing up around guns, etc. We hypothesize that more integrated neighborhoods might start with more guns than segregated neighborhoods because, in segregated neighborhoods, agents are more likely to be surrounded by people who look like them. 

We define someone’s sphere of influence by the number of people surrounding them they consider when making a decision to buy a gun. The smallest sphere of influence is 1 in which an agent only considers the eight closest neighbors, and the largest sphere of influence we study is 10 in which an agent considers the people surrounding them by a width of 10, so 440 neighbors total at most.

Agents asynchronously decide to either buy or sell their gun depending on the agent’s neighborhood’s gun situation relative to the entire neighborhood’s. Thus, after the initial setup, agents optimize regardless of race. We designed this rule ignoring race factors because peer effects and fear effects work in the same direction. In other words, an agent might buy a gun because all the people that look like them have guns due to peer effects, and also an agent might buy a gun if unlike people near them have guns because of fear effects. We anchor an agent’s decision to buy or sell a gun based on the current portion of gun owners in the entire neighborhood compared to their sphere of influence (i.e. based on the agents surrounding them). An agent sells their gun if the current gun ownership in their sphere (P) is less than the current gun ownership of the entire neighborhood (g0), and an agent buys a gun when the current gun ownership rate in the agent’s sphere is greater than two times the gun ownership rate of the entire neighborhood. See below. 

(1)      buy = P ( g > 2g0)

(2)      sell = P (g <=g0) 

Notice, we make it slightly more difficult to buy a gun than sell a gun to account for the associated costs or barriers to buying a gun such as getting a license. Similarly, you buying a gun is strictly greater than to account for the case whereg0= 0 so that agents do not “spontaneously ” buy a gun even when no one around them is buying a gun. We assume that people have a general sense of what is going on in their greater area and use that as a reference to compare their personal situation.

We test each combination of segregation levels from 0.5 to 0.9 incrementing by 0.1 and extent of sphere of influence from 1-10 incrementing by 1. We run the simulation 100 times for each combination for 2 generations (each agent makes 2 decisions asynchronously). In total, we ran 5000 simulations, and collected the starting portion of gun owners (race effects) and ending portion, and then calculated the change in gun ownership (peer effects). 

  1. Results

Segregation

As expected, increasing levels of segregation leads to an increased likelihood of initially owning a gun when people optimize based on the race of their neighbors alone. However, interestingly, this proposition only holds true at the lowest level of the sphere of influence, which is 1, despite holding the preference to own a gun at the 25% threshold. We believe this result occurs because as an agent expands their sphere of influence, the portion of neighbors who are of the same race as that agent approaches 50%. This conclusion is implicitly derived from the fact that the entire neighborhood has almost exactly half of each race. 

For neighborhoods with large spheres of influence, the segregated ones had the smallest change in gun ownership. Whereas the most integrated neighborhoods had more drastic increases in gun ownership even though these neighborhoods started with the lowest levels of gun ownership. This result may seem counter-intuitive, but actually has strong implications for reference-point bias. Because the threshold to buy a gun is dependent on the current percentage of people who own a gun (in the entire neighborhood) is relatively small, then a small number of people in your neighborhoods can be significant enough to influence you to buy a gun. So even though the neighborhood started with very few people owning guns for racial reasons, those few agents influenced others in their neighborhood to buy a gun and the effects keep spreading. 

National vs Local 

Segregation and Sphere of Influence 

We adopt the idea of the Sphere of Influence to try to capture the effect of regional factors. This is to see how the rate of gun ownership changes at varying degrees of segregation and sphere of influence. At low levels of segregation (0.5-0.7), the larger the sphere of influence, the more likely an agent is to buy a gun. Whereas for high levels of segregation, while the likelihood to buy a gun still increases over the course of a couple of generations, it is a smaller magnitude when the spheres of influence are larger than smaller. We believe this occurs because of a clustering effect that we discuss later in this paper. 

Table 1 shows average changes in gun ownership (%) of 100 trials for each set of conditions. There are a few important observations in the table. As the degree of locality increases (going from 1 to 10), the effect of segregation on gun ownership increases. In another word, when agents account for national factors or their sense of locality is wider while holding their preferences constant, the effect of degree of segregation on the rate of gun ownership increases quite significantly. More segregated neighborhoods’ change in gun ownership approaches zero as locality increases. Whereas for integrated neighborhoods, the change in gun ownership increases.

Table 1: Average of Change in Gun Ownership (%)

  Extent of Sphere of Influence
  12345678910
Level of Segregation 0.5-6.0%14.3%19.3%13.6%14.8%17.0%19.0%19.7%20.7%21.1%
0.6-4.6%11.8%18.3%19.8%18.8%16.9%18.6%19.8%20.6%21.5%
0.7-0.4%2.4%10.2%13.9%16.3%17.7%18.9%19.3%19.8%20.2%
0.86.3%1.3%-1.0%-2.1%-2.4%-2.5%-2.2%-1.8%-1.2%-0.4%
0.97.5%7.6%6.7%5.0%4.1%3.0%3.0%2.8%2.5%3.0%
Notes: Level of Segregation (0.5=Integrated, 0.9=Segregated), Extent of Sphere of Influence (1=Local, 10=Global), Positive sign of the % indicates an increase in the likelihood of owning a gun, and a negative sign indicates a decrease. 

Notice in Figure 1 as the sphere of influence widens, clusters of gun owners form at all levels of segregation. In segregated models, clusters form within racial groups separated by the borders of the ghettos. We expected the most guns to be owned near the borders of racial clusters, however, the opposite occurred. The reason is that racial clusters arise when a particular agent happens to be in a neighborhood dominated by another race and therefore starts with a gun, and there are more likely to be agents predominately surrounded by a different race in a segregated neighborhood than in integrated neighborhoods. The gun-free borders seem to grow out (see 0.8-7 and 0.8-9 in figure 1) and the concentration of guns grows in the deeper parts of the segregated regions. 

In integrated neighborhoods, the clusters form along the edges where people have fewer neighbors, and therefore each neighbor has a heavier influence relative to the agents in the center of the entire neighborhood. In other words, in a third-degree neighborhood, someone in the middle has 48 neighbors, so if two neighbors have a gun the portion of people is 4%, while someone on the edge has 7 neighbors and if two neighbors have a gun the portion of gun owners is 28.5%. In this case, the edge case would buy a gun, but the agent in the middle wouldn’t. 

Figure 1. Illustration of Gun ownership at different degrees of segregations and locality

Note: for each simulation, the left figure is the starting map of gun owners (black) and the right is the final map of gun ownership after 2 generations (each agent makes 2 decisions randomly)

  1. Conclusions

In our model, each race has identical ability and costs to buy a gun. We could extend this model such that one group has higher costs to purchasing a gun. This variation is applicable to study the effect when one race is disproportionately incarcerated, and therefore would have a harder time obtaining a gun license. Under asymmetric costs, we would expect gun ownership to decrease overall because each agent is less likely to have a neighbor with a gun, and in segregated neighborhoods, we would expect the gun ownership of the incarcerated/high-cost race group to own fewer guns. Although you may also argue that if someone has broken the law, they may be more likely to obtain a gun by illegal means and have potentially lower costs.  

Another version we would have explored was a neighborhood with a mix of larger and smaller spheres of influence amongst the agents. This would model the effects of having more worldly-oriented and locally-oriented agents. Based on our results, these two groups would opposingly optimize, the more worldly people are less likely to sell their gun and the local people are more likely to buy a gun. 

  1. References

Mitchell, Melanie. “Cellular Automata, Life, and the Universe.” Complexity: A Guided Tour , 2009, pp. 145–159. 

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