Gun Control, Part 2: Shall we play a game?


In my last post we undertook a cursory analysis of data related to firearm ownership and homicide rates across various jurisdictions. We concluded that strict gun-control laws and reduced firearm ownership are correlated with increased variance in the total homicide rate:

Homicide vs. Firearms 4

In other words, the worst-case scenarios are worse under strict gun-control laws and lower rates of firearm ownership.

Mere correlation, however, does not necessarily imply causation. We need a hypothesis that explains how gun control can enable more murders to be committed, then we need to test that hypothesis empirically before we reject it or fail to reject it.

But first, you and I need to rob a bank. For educational purposes.

You suggest that we knock off a local branch of an unnamed bailed-out bank holding company that recently froze the accounts of a legal firearms manufacturer. I insist that no weapons or threats of violence are needed to rob a bank these days, we’re peaceful, enlightened criminals who have no need for such things. You go inside, walk up to the teller, demand money, and leave. I drive the getaway car. Electric, of course. We stash the loot in an abandoned building in Detroit and lay low in our hideout until the whole thing blows over.

We’re up late one night watching Current TV. Suddenly, sirens wail! We look at each other in a panic, aware that the jig may be up. We reaffirm our loyalty to one another, we pinky-swear that we will never confess to our crime or implicate one another, no matter what the cost. You start to cry. I steel my resolve and pick up a pointy stick, ready for battle.

The cops burst through the door. I raise my weapon and ask to see a warrant. They taser me. You chortle a bit when you see me flailing about on the floor. I can’t feel my legs. You get handcuffed, I get hog-tied. This is the last time we see each other. They put us into separate cars and haul us off to the Graybar Hotel. We’re booked and held separately overnight.

Late the next morning a jail officer leads me to a drab, windowless room and motions for me to sit in a rather flimsy and uncomfortable chair. The jailer exits and another man who looks like Barney Frank waddles through the door and plops his corpulence into the chair across from me. He leans forward and squints for a moment through thick glasses.

“We know everything,” he intones somewhat nasally, his fetid breath inducing a wave of nausea. I gag and try to cover it by clearing my throat.

“Do you?”

“Of course!” he retorts sharply. “Your partner sang like a canary.” He manages a wry smile.

I put on my very best poker face. I’ve been practicing regularly in anticipation of a moment like this. After an awkward silence, Barney’s doppelganger turns a bit redder in the face, inhales sharply and bellows:

“You have two options! Confess everything, and you’ll get five years. Or keep quiet, and you’ll get twenty years. Either way, we own you!”

He produces a cassette tape player and hits the record button. I am momentarily startled by the reappearance of such an ancient technological artifact, then I close my eyes to concentrate and mentally draw the following matrix:

Game theory 1

As a result of some serendipitous timing and your convincing work at the bank, we made off with a total of $1 million that we have agreed to split 50-50. So if you and I cooperate by honoring our pinky-sworn agreement to not rat each other out, each of us will receive $500,000 once the prosecutor realizes he doesn’t have enough evidence to convict us.

But if I honor our agreement and you defect, I go to jail for 20 years (for the purposes of the exercise, I value my freedom at $100,000 per year) and you probably get to cut a deal with the prosecutor to walk away with only probation after you testify against me. You’ll say the whole thing was my idea, that I hid the money and you have no idea where it is. And once I go to jail, you’ll pick up the $1 million payout at that abandoned building in Detroit. Given the way you laughed at me last night while I was being shocked into oblivion, how do I know that this wasn’t your plan all along? You sly devil!

It’s clear that you’re better off if you defect, and I now realize that if you do, I’m much worse off if I don’t defect. We both go to jail for five years and the bank will get its money back, but that’s preferable to me doing twenty while you live it up with all that cheddar.

You’re probably being put through this same ordeal in the room next door. We’re locked in a distributed Battle of the Wits. I bet your interrogator looks more like Princess Buttercup than mine does.

This little anecdote represents a specific instance of a game that economists call Prisoner’s Dilemma. Generally, you and I would both be better off if we cooperated, but we each have an incentive to cheat. And if there is an incentive for you to cheat, the rational thing for me to do is defect, and vice versa. We call this rationally-optimal state a Nash equilibrium (named after the mathematician, not the point guard).

It turns out that Prisoner’s Dilemma can help us model all sorts of interactions in which the players have a choice to either collaborate or defect: Advertising, the use of performance-enhancing drugs in sports, OPEC oil production quotas, etc.

One of the most famous applications of game theory was the military doctrine of strategy known as mutually assured destruction that defined how the Cold War was carried out. The United States and the Soviet Union each had enough nuclear-armed missiles to destroy the other several times over. We cooperated by not firing them at each other. If we had defected by launching our missiles at the Soviets, they also would have defected by launching their missiles at us. We’d both be annihilated, which is the worst possible state, so neither of us launched the first strike.

But if we had disarmed unilaterally, mutual destruction would no longer be assured. There would be no reciprocal penalty for the Soviets should they defect and launch the first strike, and vice-versa. The ironic reality is that two nuclear-armed superpower rivals are safer than one nuclear-armed superpower who could strike with impunity with no threat of reciprocal strikes.

Let’s use game theory to test gun control scenarios.

You are walking down the street with $100 in your pocket. I am sitting on the corner with nothing in my pocket. As you approach, I have a strategic decision to make:

  • I can cooperate by letting you pass, or
  • I can defect by attempting to rob you of your $100.

Being a rational thug, I quickly create the following payout matrix in my mind:

Game theory 2

If I choose to cooperate, you keep the $100 and I get nothing, which seems like a pretty good deal for you but a lousy deal for me. If I defect and decide to rob you and you cooperate, I get the $100, and you keep nothing. If I attempt the robbery, you could decide to defect by running away or putting up a fight. I size you up and figure that I have a 90% chance of winning either a fight or a foot race, which makes my expected payoff 90% x $100 = $90. Therefore you have a 10% chance of keeping the $100 plus a 90% chance of me breaking your nose and sending you to the ER, which makes your expected payoff 10% x $100 + 90% x -$1000 = -$890 if you decide to defect.

My worst-case scenario if I defect is that I get $90. My best-case scenario if I cooperate is that I get nothing. So, rationally, I step up and demand that you hand over your money. When I do, you instantly calculate the same payoff matrix and decide that while $0 is worse than $100,  -$890 is a lot worse than $0. So you cooperate. The Nash equilibrium here is that I defect and you cooperate, and as a thug this equilibrium pleases me immensely.

But I have a problem: There are people who I can’t intimidate into cooperating. I can segment my “market” and target only the easy prey: Women, smaller men, people walking alone at night. But these targets get wise to my strategy and start changing their behavior to reduce their vulnerability. They walk in groups to improve their odds of escape or winning a fight against me. They cross the street when they see me up ahead. They don’t go out after dark when it’s harder to see me and there are fewer Good Samaritans to rescue them.

I need to change my strategy to adapt. I could partner with a few of my friends to improve the odds of success against stronger victims or groups of people, but we’d be a bit conspicuous sitting around waiting for someone to rob. And I’d have to split the loot among the group, which I don’t like. And being thugs like me, they can’t be trusted.

Here’s a thought: I could use a weapon. I could present the weapon as I make my demands. People are conditioned to fear weapons, almost every time they see one in the media it is associated with something bad happening to a good person. And if I’m armed, I’ll be able to take on stronger individuals and even small groups of people, increasing the size of my target market, my expected profit per transaction, and my win rate! Here is the updated payout matrix:

Game theory 3

Let’s assume you have a 1% chance of being brave/stupid/fast enough to run from someone who is able to threaten you with a deadly weapon. And while I have no intention of actually using the weapon on you, it’s important that you understand that you don’t know what my intentions are. All you know is that if I use the weapon on you, your very negative payout represents a reasonable risk of immediate death.

Clearly, the armed-robbery business is much better than the unarmed-robbery business. For the thug, at least. And things don’t have to stop at mere robbery. A few early successes can induce grandiose delusions or narcissism as the thug realizes the power he wields over his victims. The thrill of power may lead him to act out other, more perverse fantasies to increase his “payout” at the expense of his victim, who is rationally willing to settle for any outcome marginally better than death.

But let’s say that you, the victim-in-waiting, have studied game theory a bit as well. You understand that the Nash equilibrium above is bad news for you. It sets you up to be either a victim or a hermit, and neither of those is any way to pursue life, liberty, and property, not to mention happiness.

What would happen if you exercised your natural right to defend yourself? You happen to live in one of the 49 states that issue concealed carry permits (the one holdout, Illinois, was recently ordered by a Federal court to come up with a concealed-carry permit program within 6 months), so you obtain one and carry a firearm legally. Here’s how the payout matrix changes:

Game theory 4

This changes everything. The thug knows that if he attempts to rob you, armed or otherwise, you are entitled to defend yourself. If he picks the wrong target, he’ll get two rounds to his center-of-mass and one between the eyes. That’s a very bad ending for any rational being, even a thug. So in a scenario where there is a chance that the victim may be armed, the Nash equilibrium is for both parties to cooperate (the upper-left cell).

That’s a lot of words to illustrate what happens in this short video. Two armed tough guys attempt to knock off an internet café in Florida. They begin rounding up the patrons to separate them from the possessions (and who knows what else), when an armed 70-year-old rains (lead) on their parade. Spoiler alert: People get shot (though you wouldn’t know it from the video), nobody dies, and you may laugh out loud when you see the replay of Thug 1 running over Thug 2 as both cowards try to squeeze through the exit door at precisely the same moment:

If you think all this game-theory nonsense is the result of me twisting microeconomics to fit a uniquely modern problem with crime and violence, you’re wrong. Take a look at this excerpt from Cesare Beccaria’s Essay on Crimes and Punishments, as quoted in Thomas Jefferson’s “Legal Commonplace Book”:

Laws that forbid the carrying of arms…disarm only those who are neither inclined nor determined to commit crimes. Such laws make things worse for the assaulted and better for the assailants; they serve rather to encourage than prevent homicides, for an unarmed man may be attacked with greater confidence than an armed one.

So, can correlation imply causation? Yes indeed, provided there is a hypothesis that can be tested and validated empirically. I think the game-theory hypothesis is quite compelling. Based on the evidence I’ve studied, I even think it’s correct. But I can’t in good conscience recommend that we carry out a randomized double-blind controlled study when the lives of innocent people are at stake. We need to look at the variance that already exists in crime rates and firearm ownership, there is a mountain of evidence that we can use if we are willing to do the science rather than acquiesce to political demagoguery.

7 thoughts on “Gun Control, Part 2: Shall we play a game?

  1. Just for fun, what if you only posted homicide rates vs firearm ownership across OECD countries? I’d be curious to see such an apples-to-apples correlation.

    1. To compare “firearm-related death rate” vs. total homicide rates? That’s an interesting idea. I’ll see what I can come up with.

  2. Doesn’t this lead to an arms race? and/or Mutually assured destruction? Could the first guy just start wearing body armour, the next guy getting an assault rifle so on and so forth?

  3. You fail to factor in the fact that in a robbery there is more than is at least three parties involved the robber the robbed and the law. Robbing someone with a weapon raises the risks for both the robbed and the robber(in terms of legal repercussion). Also you fail to see the inherent danger of the MAD theory: It requires near perfect communication of intention between the parties. In fact I feel that your last payout matrix argues shows that even with a firearm I have still haven’t improved my defenses, I’ve only Increase my offense. Now while some might say that a good offense is the best defense, that fact remains that when he robs me the robber can’t tell whether or not I’m armed, so that point becomes moot. Not to mention that if I am armed and the robber does choose to rob me he still has the ‘drop’ on me, unless I’m some Clint Eastwood type sharpshooter. Lastly I would like to mention that for those who are afraid of being robber by scoundrels on the street the best thing you can do is to use Batesian mimicry. Simply by projecting to the robber that your are indeed some one who is likely to be armed. e.g: wear an NRA Shirt, a cowboy hat and other tough guy paraphernalia; the robber will sense that you might be a threat an move on to more greener pastures.

    1. Thanks for reading and for your comments. For practical purposes the (low) likelihood of the law punishing a mugger can be factored into the payoff matrix. It’s also noteworthy that in an armed confrontation there is no mutually-assured destruction as you imply. My reference to MAD was more of a parenthetical of well-known applications of game theory. In reality, MAD does not apply to an armed-robbery scenario because most shootings are not fatal, i.e. mutual destruction is hardly “assured”: http://appliedrationality.blogspot.com/2013/04/the-increasing-chances-of-being-shot.html

      Furthermore, Prisoner’s Dilemma (MAD or otherwise) does not require symmetric information as you assert. In fact, Robert Axelrod’s excellent book The Evolution of Cooperation http://www.amazon.com/The-Evolution-Cooperation-Revised-Edition/dp/0465005640 provides ample empirical evidence (as well as some useful real-world anecdotes) to prove that reciprocity is an optimal strategy under asymmetric information.

      Finally, I agree that signaling can be an effective deterrent as you describe, but only in a jurisdiction that allows concealed carry for law-abiding citizens; otherwise all the signaling devices you describe communicate nothing about what might happen to the bad guy.

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