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):