This ain't a (crime)scene, it's an arms race

Gun violence has been in the news a lot recently, with incidents in Hamilton and Auckland among others, and predictably that has reignited the debate over the arming of police in New Zealand. However, we should resist the temptation. To see why, let's apply a little bit of game theory.

The arms race game is a variation of the prisoners' dilemma. For simplicity, let's assume that there are two players in our game, the police and the criminals. Both players have two strategies, to arm themselves or to disarm (or not arm). Both police and criminals are best off if they are armed and the other group is not, but if both are armed this escalates the potential for violent confrontation. Both players are making their decisions at the same time, making this a simultaneous game. The game is outlined in the payoff table below.

To find the Nash equilibrium in this game, we use the 'best response method'. To do this, we track: for each player, for each strategy, what is the best response of the other player. Where both players are selecting a best response, they are doing the best they can, given the choice of the other player (this is the definition of Nash equilibrium). In this game, the best responses are:

1. If the criminals choose to arm, the Police's best response is to arm (since -5 is a better payoff than -10) [we track the best responses with ticks, and not-best-responses with crosses; Note: I'm also tracking which payoffs I am comparing with numbers corresponding to the numbers in this list];
2. If the criminals choose to disarm, the Police's best response is to arm (since 10 is a better payoff than 5);
3. If the Police choose to arm, the criminals' best response is to arm (since -5 is a better payoff than -10); and
4. If the Police choose to arm, the criminals' best response is to arm (since 10 is a better payoff than 5).

Note that the Police's best response is always to choose to arm. This is their dominant strategy. Likewise, the criminals' best response is always to choose to arm, which makes it their dominant strategy as well. The single Nash equilibrium occurs where both players are playing a best response (where there are two ticks), which is where both the Police and criminals choose to arm. This leads to escalation of violence and is worse off for everyone than if both players chose to disarm. That demonstrates that the arms race is a type of prisoners' dilemma game (it's a dilemma because, when both players act in their own best interests, both are made worse off). If both police and criminals follow their dominant strategies, eventually we will end up in a situation like the U.S., with growing militarisation of police, which makes both police and citizens worse off.

How do we avoid this outcome? First, we need to recognise that the arms race game outlined above is not a one-shot game, it is a repeated game. That means that it is effectively played not once, but many times. In repeated games, the players are more likely to be able to work together to move away from the Nash equilibrium, and ensure a better outcome for all.

Let's assume for the sake of argument that the game is played weekly, and decisions are made simultaneously each week. Both police and criminals recognise that it is in their long-run interests to disarm. If they choose not to arm, and can trust the other player also not to arm, everyone benefits (or, rather, everyone faces a lower cost, since crime would continue, but it wouldn't be aggravated to the same extent by firearms).

Can each side really trust the other? That's the problem. In a repeated prisoners' dilemma game like this, each player can encourage the other to cooperate by using the tit-for-tat strategy. That strategy, identified by Robert Axelrod in the 1980s, works by initially cooperating (disarming), and then in each play of game after the first, you do whatever the other player did last time. So, if the criminals armed last week, the Police arm this week. And if the criminals disarmed last week, the Police disarm this week. Essentially, that means that each player punishes the other for not cooperating, by themselves choosing not to cooperate in the next play of the game. It also means that each player rewards the other for cooperating, by themselves choosing to cooperate in the next play of the game. However, the tit-for-tit strategy doesn't eliminate the incentive for either side to cheat. Can the Police trust the criminals not to arm, if they knew for sure that the Police would not arm?

A more severe form of punishment strategy is the grim strategy. This involves initially cooperating (like the tit-for-tat strategy), when the other player chooses not to cooperate, you switch to not cooperating and never cooperate again. You can see that this strategy essentially locks in the worst outcome in the game. And unfortunately, arming the police is a type of grim strategy if it is difficult to back out of once the decision is made (which, judging by the experience in the U.S., may well be the case).

We need another option. If the criminals are going to arm, and Police can't trust them not to, and we want to avoid the grim strategy, we end up in the outcome where Police (and society more generally) have the worst outcome, and criminals have the best. To encourage criminals not to arm, Police need to be armed at least some of the time. At least enough of the time that the long-run best outcome is for criminals not to arm all of the time. If criminals get a payoff of 10 for sure each week, they would surely arm. But if they get a payoff of 10 some weeks, but -5 in other weeks, then they might be better off on average to not arm.

The challenge in this approach is that the Police need to find the right balance. If criminals are arming more often (as appears to be the case right now), then Police need to arm a little bit more as a deterrent. This doesn't mean that having all Police routinely armed is the right solution, only that a little movement in that direction might help to restore an uneasy but preferable outcome.

Let's not finish on such a gloomy note. Time for a musical interlude, courtesy of Fall Out Boy (since I ripped off their song title for the title of this post, it seems only fair):

Crack cocaine and the gun violence equilibrium in the U.S.

In economics, we often recognise that there may be multiple equilibriums, and that even a relatively small shock may be enough to cause the economy to move from one equilibrium to another. Consider gun violence as an example. If gun violence is low, people feel safe and therefore don't feel the need to carry a gun for self-defence purposes. Therefore there exists a low gun violence equilibrium. [*] However, if some shock occurs, and gun violence increases, then people will feel less safe. They will be more likely to carry a gun for self-defence purposes, and therefore more likely to use a gun, perpetuating the level of gun violence. Therefore there also exists a high gun violence equilibrium. However, once a society is in a high gun violence equilibrium, it is going to be very difficult to reverse things.

What would cause society to move from a low gun violence equilibrium to a high gun violence equilibrium? A 2018 NBER Working Paper by William Evans (University of Notre Dame), Craig Garthwaite (Northwestern University), and Timothy Moore (Purdue University) provides evidence that the rise of the crack cocaine market in the U.S. in the 1990s caused a shift from a lower gun violence equilibrium to a higher gun violence equilibrium. Their argument is that:

...the daily experiences of young black males were fundamentally altered by the emergence of violent crack cocaine markets in the United States. We demonstrate that the diffusion of guns both as a part of, and in response to, these violent crack markets permanently changed the young black males’ rates of gun possession and their norms around carrying guns. The ramifications of these changes in the prevalence of gun possession among successive cohorts of young black males are felt to this day in the higher murder rates in this community.

They use city-level data from the largest 57 metropolitan areas (in 1980) on age-, sex- and race-specific murder rates, over the period:

...from eight years prior to the arrival of crack and 17 years after, for a total of 26 years for each city. As the earliest date of crack’s arrival is 1982 and the latest is 1994, our data set spans from 1974 through 2011.

Essentially, they use a difference-in-differences approach that compares the change in murder rate for young black males (aged 15-24 years) before and after the introduction of crack into their city, with the same change for black males aged 35 years and over. They find that:

...the emergence of crack cocaine markets is associated with an increase in the murder rate of young black males that peaks at 129 percent in the decade after these markets first emerge...

...17 years after crack markets arrived, the murder rates for young black males were 70 percent higher than they would have been had they followed the trends of older black males. 

They key figure from the paper is this one, which plots the change in murder rate for young black males:

The x-axis tracks years before (negative numbers) and after (positive numbers) the introduction of crack cocaine into the city. There is a clear and statistically significant increase in murders among young black males, compared with older black males. Evans et al. then go on to show that this likely arose from increases in gun violence, in three ways, by showing:

...in the six years after crack markets emerged, the share of all murders attributable to young black males increased by 75 percent. Seventeen years after crack markets emerged, young black males still accounted for a 45 percent greater share of all murders than they had in the years before the arrival of crack markets...

...these murders [between family members] increase markedly in the years after crack markets and remain elevated over the next sixteen years. This increase is driven entirely by murders involving guns, with no detectable change in the non-gun domestic violence murder rate over this time period...

...we further show that there is a strong correlation between ten-year changes in gun ownership and changes in the fraction of suicides involving guns among 15-19 year olds.

These results are all consistent with a story that the arrival of crack cocaine in a city increases murders primarily through an increase in gun-related violence. I liked this paper also because it gave a detailed account of the development of crack cocaine markets in the U.S. Here are the highlights:

In the early 1970s, much of the cocaine shipped to the U.S. originated in Chile. After the 1973 military coup by Augusto Pinochet in Chile that toppled the administration of Salvador Allende, Pinochet initiated a military crackdown on cocaine smuggling operations. Many smugglers moved to Colombia with the goal of using established marijuana smuggling routes as a way of getting cocaine to the United States...

As these organizations [the Colombian drug cartels] grew, an informal agreement was struck where the Medellin cartel would primarily control supply into Miami and Los Angeles, while Cali would concentrate its operations in New York...

The large-scale entrance of the Colombian cocaine cartels into Miami, New York and Los Angeles meant that by the early 1980s, these areas had relatively high cocaine supply leading to falling prices. Despite the downward pressure on prices, many low-income consumers remained priced out of the market...

Crack cocaine was an innovation that provided a safer way to smoke cocaine... This new product has two attractive properties. First, it produced an instant high, and its users could quickly become addicted. Second, an intense high could be produced with a minimal amount of cocaine, meaning that the profit-maximizing per-dose price was a fraction of the price per high for powder cocaine...

Crack was first introduced to the market by innovative retail organizations in New York, Miami and Los Angeles, which had a large supply of powder cocaine. It then spread from those cities...

The combination of a liquidity-constrained customer base and the short-lived high offered by the product meant many customers purchased multiple times a day... crack cocaine was sold in small doses, often in open-air drug markets where the dealer and the customer had no pre-existing contact to arrange that particular sale (though may have participated in a similarly anonymous sale at that location before)...

The lack of preexisting arrangements with buyers meant that geography was a key determinant of a crack dealer’s revenue...

The violence associated with establishing and defending a market from entry was a key reason for a substantial amount of drug-related violence.

If you need to fight a turf war to protect your market (or gain access to a market), guns are an efficient way to do so. Crack cocaine was a key driver that moved the U.S. from a lower gun violence equilibrium to a higher gun violence equilibrium.

[HT: Marginal Revolution, last year]

*****

[*] However, this equilibrium is very unstable. Readers who understand some game theory will probably recognise this as a form of the 'arms race' game, which is itself a type of prisoners' dilemma. Everyone would be safe(r) if no one carried a gun. However, if no one else carries a gun, you can be both safe and powerful by carrying a gun. So, there are incentives to carry a gun, regardless of whether everyone else is, or no one else is. The low gun violence 'equilibrium' is not actually a Nash equilibrium in this game. It is unstable, but may be kept in place by cultural norms against carrying guns, or high penalties for doing so.