Just to note that, when you think of what's gone on in Libya and Cote d'Ivoire, the logical consequence of the combination of:
1) There are situations which would warrant our intervention everywhere
2) Of course we cannot intervene everywhere
3) We can do whatever we like.
Now, let us put this in the words of a model. The fact that western countries cannot intervene everywhere they would like to comes from something like a budget constraint, say they have W to invest in war.
Those countries would like to intervene in all countries where the atrocities committed by the dictator, rebellion forces, etc... are above a certain level. Assume, once again, that you can measure this thing by a number: the western countries want to intervene if the level of atrocities is above a level X. Probably more practically, assume that X is the level above which the UN actually intervenes(i.e. China and Russia abstain, western countries except Germany all vote to intervene)
If you have multiple conflicts, humanitarian catastrophes, etc..., there will not be enough money to cover all the things the western countries want to do. So there's a choice to make. How do you do it?
One idea is to raise the level of X above which you intervene. You will intervene if the level of atrocities is above Y>X. Now, this means that any dictator aware of that for instance, can raise the level of atrocities.
Another possible strategy is to randomize over all conflicts with atrocities above X. This might be good, because dictators below Y might not want to reach Y, if the marginal gain in raising the level of atrocities is higher than the marginal increase in the probability of intervention. The drawback is that because of the budget constraint, there is also a positive probability that the countries will not intervene in areas where the level of atrocities is really high.
That is, in my opinion, tremendously interesting. One extension of this would be to look at the impact of uncertainty, what if the dictator and the western countries observe the variables with noise? What are the impacts? One could think, for instance, that western countries are more risk averse than old dictators at the end of their term. Think of the consequences of the war in Iraq, or the fact that in a democracy it might be harder to go to war.