# How to resolve a hostage crisis: Part I

Suppose an archvillain, with a few lackeys, comes into a bank lookin’ for loot. Knowing that he has the potential to get more out of the bank than merely what’s in the vaults, and knowing that it’s not so simple to just drive away, they decide to hold the innocent civilians in the bank hostage. Unfortunately, Batman is away in Shangri-La for now, so he can’t help you. Spiderman is busy trying to learn how to fly. And the other superheroes just suck. So your only option is to engage in a standoff, with an elite SWAT team standing outside, ready to storm the building if possible/necessary. Meanwhile, the criminals are inside, threatening to kill their hostages unless they get \$100 million.

Suppose, for now, that the criminals are completely rational, and this is known to the SWAT team as well. Let’s start with the case where there is only one hostage, and the SWAT team cares about this dude so much that his life is worth \$100 million to save. Nonetheless, if the SWAT team refuses to pay the ransom, what are the criminals to do? If they shoot the guy, then there’s nothing to prevent the SWAT team from storming the bank and shooting them dead, or at the very least capturing them, resulting in a long prison term for murder (on top of robbing the bank). This extra penalty makes it worse than not killing the hostage. Thus the SWAT team has no incentive to give in to their demands, as the threat of killing the hostage is not credible – the criminals are worse off if they do so.

Now let’s see what happens if there are more hostages. We can use induction to see that the hostage-takers face this predicament, no matter how many hostages are involved. Suppose that, once there are N hostages left, the SWAT team has no reason to give in. We then look at the case with N+1 hostages. If the hostage-takers kill the spare, they are left in a situation in which they automatically lose – they won’t get their ransom, and they’re stuck there until they are forced to surrender. Thus they won’t want to kill the N+1th hostage. We therefore end up with no dead hostages, no matter how menacing the threat.

I know what you’re thinking now: “Yeah right. Why don’t you try this strategy and see what happens? If you do, the hostages will end up DEAD.” And you’re probably right. So where have I gone wrong?

Uncertainty in the number of hostages doesn’t help much here, because the number of hostages will be bounded. We can therefore apply the same induction reasoning as before to the uncertain number of hostages, and once the criminals have claimed to kill the upper bound of the number of people that could possibly have been in the bank, they’re still dead meat. So we’ll have to find some other reason.

The key to my original argument was that the criminals are rational. Obviously, that’s not always going to be the case. After all, most rational, normal people do not go around robbing banks. So, chances are pretty good that the people robbing the bank aren’t quite rational, and will kill the hostages even though it ends up hurting them. In this case, it might be worth paying up to prevent the deaths of innocents. We’ll analyze this in part 2, which I will post in a week.

### 3 Comments on “How to resolve a hostage crisis: Part I”

1. Ben F says:

Nice work, Jeff. I’ve been following from afar for a while. Your article suggests the idea that it would be rational for the hostage takers to make the SWAT team think that they’re not rational; killing one of many hostages is actually the perfect way of doing that, given your analysis.

2. Tune in for Part II

3. Konstantinos says:

Reblogged this on Between numbers and commented:
Too cool, so here is a reblog of it!