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Hostage Situations

A hostage situation is a pretty common scenario (at least in movies). I often find myself questioning the actions of those involved. So I was wondering, what is the right decision in a hostage situation?

The captor is holding a single hostage, and demands something in exchange for release of the hostage. For the sake of simplicity, let’s say it is a million dollars. The negotiator has the money, and is negotiating with the captor.

The simplest set of possibilities is this. It’s very similar to the prisoner’s dilemma. Both players (captor and negotiator) can either cooperate or defect.

1. Both cooperate: Captor gets the money, hands over the hostage and escapes. Everybody wins.
2. Both defect: Negotiator doesn’t pay the captor, and the captor doesn’t hand over the hostage. The situation is the same as before.
3. Cooperate/defect: Captor hands over the hostage, but isn’t paid. Captor loses.
4. Defect/cooperate: Captor is paid, but doesn’t hand over the hostages. Negotiator loses.

In a single play, the only smart strategy is to defect. But in an iterated game, other strategies prevail.

This alone is not very interesting in itself, because it is identical to the prisoner’s dilemma. But things start to get more interesting once complications are added.

Complication 1: Killing

The captor may at any point kill the hostage. This is very bad for the negotiator, but ensures that the captor will not receive payment. Also, there is the possibility that the captor will be arrested.

At this layer, the captor should not threaten to kill the hostage. Rather, he should threaten to keep the hostage captive until the money is paid. If the hostage is killed, the captor loses all bargaining power and both parties lose.

Complication 2: Unequal Values

There is also the fact that the hostage’s life and the million dollars are not equally valued. It is possible that if the captor kills the hostage, even if he loses his million dollars, then the loss of the negotiator is a good deal greater. Since the negotiator is trying to maximize her own needs regardless of what the captor gets (at least theoretically), she will not risk the captor being killed. If the pays the million dollars but the captor still refuses to give up the hostage, this is still better for the negotiator than if the hostage were killed. But even if the captor loses the million dollars by killing the hostage, this is no better or worse than if the negotiator refused to pay the million dollars — the money is lost either way. So the negotiator has a much stronger incentive to keep the hostage alive.

Complication 3: Arrest

If the captor still has the hostage, then he is safe from arrest because he can use the hostage as leverage. But if he kills the hostage, then he can be arrested — he has lost his leverage. He cares more about his own life than the negotiators care about that of the hostage, so he has the strongest incentive yet to keep the hostage alive. The negotiator knows that a fail to pay could in result in the killing of the hostage followed by the arrest of the captor, which is even worse for the captor. It would seem that the captor no longer has an incentive to kill the hostage, which means that the negotiator no longer has an incentive to pay.

But this is not so. Remember that neither side cares if the other loses, only if they themselves win. If the captor assumes that he will be arrested if he kills the hostage, then he has no reason to kill the hostage — even if the negotiator tries to arrest him while he still has the hostage. He goes to prison either way. He no longer has an incentive not to kill the hostage, which means that the negotiator can’t expect him not to.

Despite these complications, the best way to try to maximize one’s own gain is still to defect. The captor has no incentive to release the hostage after being paid, because if he does he may be arrested. Therefore, the negotiator has no incentive to pay the money. Therefore, the captor has no incentive not to kill the hostage. Therefore, the negotiator has no incentive not to arrest the captor. Everybody loses.

This all changes, of course, in iterated hostage situations and in multiple hostage situations.

Notice that this is all just speculation; there are plenty of other options one could include in the scenario. Although I find it fascinating, I don’t know much about game theory, so I may be wrong about this reasoning.

How do other factors change the situation? Is there any way to ensure that everyone gets what they want? Discuss.

Categories: Math
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  1. February 25, 2015 at 6:02 am

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