Defending yourself defends others

Mar 10 JDN 2458553

There’s a meme going around the feminist community that is very well-intentioned, but dangerously misguided. I first encountered it as a tweet, though it may have originated elsewhere:

If you’re promoting changes to women’s behaviour to “prevent” rape, you’re really saying “make sure he rapes the other girl”.

The good intention here is that we need to stop blaming victims. Victim-blaming is ubiquitous, and especially common and harmful in the case of sexual assault. If someone assaults you—or robs you, or abuses you—it is never your fault.

But I fear that there is a baby being thrown out with this bathwater: While failing to defend yourself doesn’t make it your fault, being able to defend yourself can still make you safer.

And, just as importantly, it can make others safer too. The game theory behind that is the subject of this post.

For purposes of the theory, it doesn’t matter what the crime is. So let’s set aside the intense emotional implications of sexual assault and suppose the crime is grand theft auto.

Some cars are defended—they have a LoJack system installed that will allow them to be recovered and the thieves to be prosecuted. (Don’t suppose it’s a car alarm; those don’t work.)

Other cars are not defended—once stolen, they may not be recovered.

There are two cases to consider: Defense that is visible, and defense that is invisible.

Let’s start by assuming that the defense is visible: When choosing which car to try to steal, the thieves can intentionally pick one that doesn’t have a LoJack installed. (This doesn’t work well for car theft, but it’s worth considering for the general question of self-defense. The kind of clothes you wear, the way you carry yourself, how many people are with you, and overall just how big and strong you look are visible signs of a capacity for self-defense.)

In that case, the game is one of perfect information: First each car owner chooses whether or not to install a LoJack at some cost L (in real life, about $700), and then thieves see which cars are equipped and then choose which car to steal.

Let’s say the probability of a car theft being recovered and prosecuted if it’s defended is p, and the probability of it being recovered if it’s not defended is q; p > q. In the real world, about half of stolen cars are recovered—but over 90% of LoJack-equipped vehicles are recovered, so p = 0.9 and q = 0.5.

Then let’s say the cost of being caught and prosecuted is C. This is presumably quite high: If you get convicted, you could spend time in prison. But maybe the car will be recovered and the thief won’t be convicted. Let’s ballpark that at about $30,000.

Finally, the value of successfully stealing a car is V. The average price of a used car in the US is about $20,000, so V is probably close to that.

If no cars are defended, what will the thieves choose? Assuming they are risk-neutral (car thieves don’t seem like very risk averse folks, in general), the expected benefit of stealing a car is V – q C. With the parameters above, that’s (20000)-(0.5)(30000) = $5,000. The thieves will choose a car at random and steal it.

If some cars are defended and some are not, what will the thieves choose? They will avoid the defended cars and steal one of the undefended cars.

But what if all cars are defended? Now the expected benefit is V – p C, which is (20000)-(0.9)(30000) = -$7,000. The thieves will not steal any cars at all. (This is actually the unique subgame-perfect equilibrium: Everyone installs a LoJack and no cars get stolen. Of course, that assumes perfect rationality.)

Yet that isn’t so impressive; everyone defending themselves results in everyone being defended? That sounds tautological. Expecting everyone to successfully defend themselves all the time sounds quite unreasonable. This might be what people have in mind when they say things like the quote above: It’s impossible for everyone to be defended always.

But it turns out that we don’t actually need that. Things get a lot more interesting when we assume that self-defense can be invisible. It would be very hard to know whether a car has a LoJack installed without actually opening it up, and there are many other ways to defend yourself that are not so visible—such as knowing techniques of martial arts or using a self-defense phone app.

Now the game has imperfect information. The thieves don’t know whether you have chosen to defend your car or not.

We need to add a couple more parameters. First is the number of cars per thief n. Then we need the proportion of cars that are defended. Let’s call it d. Then with probability d a given car is defended, and with probability 1-d it is not.

The expected value of stealing a car for the thieves is now this: V – p d C – q (1-d) C. If this is positive, they will steal a car; if it is negative, they will not.

Knowing this, should you install a LoJack? Remember that it costs you L to do so.

What’s the probability your car will be stolen? If they are stealing cars at all, the probability of your car being one stolen is 1/n. If that happens, you will have an expected loss of (1-p)V if you have a LoJack, or (1-q)V if you don’t. The difference between those is (p-q)V.

So your expected benefit of having a LoJack is (p-q)V/n – L. With the parameters above, that comes to: (0.9-0.5)(20000)/n – (700) = 8000/n – 700. So if there are no more than 11 cars per thief, this is positive and you should buy a LoJack. If there are 12 or more cars per thief, you’re better off taking your chances.

This only applies if the thieves are willing to steal at all. And then the interesting question is whether V – p d C – q (1-d) C is positive. For these parameters, that’s (20000) – (0.9)(30000)d – (0.5)(30000) + (0.5)(30000)d = 5000 – 12000 d. Notice that if we substitute in d=0 we get back $5,000, and at d=1 we get back -$7,000, just as before. There is a critical value of d at which the thieves aren’t sure whether to try or not: d* = 5/12 = 0.42.

Assuming that a given car is worth defending if it would be stolen (n <= 11), the equilibrium is actually when precisely d* of the cars are defended and 1-d* are not. Any less than this, and there is an undefended car that would be worth defending. Any more than this, and the thieves aren’t going to try to steal anything, so why bother defending?

Of course this is a very stylized model: In particular, we assumed that all cars are equally valuable and equally easy to steal, which is surely not true in real life.

Yet this model is still enough to make the most important point: Since presumably we do not value the welfare of the car thieves, it could happen that people choosing on their own would not defend their cars, but society as a whole would be better off if they did.

The net loss to society from a stolen car is (1-q)V if the car was not defended, or (1-p)V if it was. But if the thieves don’t steal any cars at all, the net loss to society is zero. The cost of defending a proportion d* of all cars is n d* L.

So if we are currently at d = 0, society is currently losing (1-q)V. We could eliminate this cost entirely by paying n d* L to defend a sufficient number of cars. Suppose n = 30. Then this total cost is (30)(5/12)(700) = $8,750. The loss from cars being stolen was (0.5)(20000) = $10,000. So it would be worth it, from society’s perspective, to randomly install LoJack systems in 42% of cars.

But for any given car owner, it would not be worth it; the expected benefit is 8000/30 – 700 = -$433. (I guess we could ask how much you’re willing to pay for “peace of mind”.)

Where does the extra benefit go? To all the other car owners. By defending your car, you are raising d and thereby lowering the expected payoff for a car thief. There is a positive externality; this is a public good. You get some of that benefit yourself, but others also share in that benefit.

This brings me at last to the core message of this post:

Self-defense is a public good.

The better each person defends themselves, the riskier it becomes for criminals to try to victimize anyone. Never feel guilty for trying to defend yourself; you are defending everyone else at the same time. In fact, you should consider taking actions to defend yourself even when you aren’t sure it’s worth it for you personally: That positive externality may be large enough to make your actions worthwhile for society as a whole.

Again, this does not mean we should blame victims when they are unable to defend themselves. Self-defense is easier for some people than others, and everyone is bound to slip up on occasion. (Also, eternal vigilance can quickly shade over into paranoia.) It is always the perpetrator’s fault.

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