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Should governments pay hostage ransom demands?

The recent killing of journalist James Foley has reignited the debate over paying hostage ransom demands. The US, British and many other governments have a clear policy of not paying ransom demands. Does that policy make sense?
        The policy has a clear rationale in game theory. To see why consider the very stylized game below. The game begins with the kidnappers deciding whether or not to kidnap. If they kidnap then the hostage's representatives have to decide whether or not to pay the ransom. The numbers give the payoffs to the hostage takers and the hostage's representatives. If the hostage takers do not kidnap then payoffs are 0. If they kidnap and the ransom is paid 100 is transferred from the hostage's representatives to the hostages. If they kidnap and the ransom is not paid then the hostage takers pay some small cost of 10 while the hostage's representatives pay a big cost of 200.  
 
 
         The payoffs in the game are clearly somewhat arbitrary. This, though, doesn't matter too much - it is the relative ordering of payoffs that matter. Specifically, the hostage's representatives have an incentive to pay the ransom because -100 is better than -200. Predicting this, the hostage takers have an incentive to kidnap because 100 is better than 0. There is a unique sub-game perfect Nash equilibrium of 'kidnap, pay ransom'. This equilibrium looks like bad news.
        Fortunately, there is another Nash equilibrium. Suppose the hostage's representatives are expected to not pay the ransom. Then the hostage takers should not kidnap because 0 is better than -10. So, 'don't kidnap, don't pay ransom' is also an equilibrium. In a one shot-game this equilibrium is hard to motivate because the hostage takers should be able to predict that the ransom will be paid. In other words, the threat to not pay the ransom is not credible. If, however, the game is repeated many times then the representatives have an incentive to build up a credible reputation for not paying the ransom.  
        So governments commitment to not paying any ransom is aimed at making the 'don't kidnap, don't pay ransom' equilibrium a reality. And this is the best possible outcome for everyone, bar the hostage takers. There is, however, a fundamental flaw in this logic: According to the equilibrium there should not be any kidnapping. But James Foley, and many others, are kidnapped. Does that mean the policy is not working?  
         One thing the story so far neglects is heterogeneity. In the game above the hostage takers would rather not kidnap than kidnap and receive no ransom. This is surely the right ordering for most potential instances of kidnap. Suppose, however, that the hostage takers would rather kidnap and receive no ransom than not kidnap. Then we get a game something like that below. In this game the hostage takers have an incentive to kidnap no matter what. And this is arguably more representative of the threat posed by the Islamic State in Syria and Iraq.
 
  
 
        In this second game there is a unique equilibrium of 'kidnap, pay ransom'. The hostage's representatives gain nothing from a threat to not pay the ransom. Does this undermine a government policy of never paying hostage takers? We'll the problem here is that the government would need to pay the ransom in some cases and credibly not commit to paying the ransom in others. That seems way too subtle a policy to be realistic. Such policy would also change the incentives of hostage takers. In particular, they would have an incentive to 'burn bridges' in order to create something like the second game above. And it would also change the incentives of potential hostage victims. In particular, there is the moral hazard problem of individuals taking excessive risk.
       That kidnapping remains despite a credible commitment to not pay any ransom demands is not, therefore, evidence that government policy is not working. Things would be a lot, lot worse if governments were to start paying up.  




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