• Nuclear Proliferation and the “Hierarchy of Hungry Cannibals” Game

    Last week I posed a game called “Hierarchy of Hungry Cannibals.” The game is easy to grasp but takes too many words to state to repeat it here. This Monday I analyzed the game. In this post I attempt to relate it to a real world issue. Because the game is so structured and contrived it isn’t easy to find a good application in real life. Sometimes games are just games. However, I’m going to try anyway. The application is nuclear proliferation.

    Instead of ten cannibals on an island we have ten countries in the world (or any other even number greater than two). Like the cannibals, name the countries 1, 2, 3, …, 10. Country 1 is the dominant superpower, 2 its chief rival but less powerful than 1, 3 the chief rival but less powerful than 2, and so on up to 10. Each country becomes interested in acquiring nuclear weapons only if its more powerful rival has them. But each country is more interested in preventing its next lower rival from acquiring nuclear weapons than in acquiring them itself.

    This set up is almost identical to that of the hierarchy of hungry cannibals just with different words (“countries” for “cannibals”, “acquiring nuclear weapons” for “eating”, and so forth). So the solution is almost the same. There is one key difference. In the case of the cannibals, if number 1 passes on eating Meathead then he has no one to eat and therefore cannot eat. In the case of nuclear proliferation, country 1 can acquire nuclear weapons any time it likes. It doesn’t give up the option. If it figures out that country 2 will acquire nuclear weapons despite what it does, then country 1 might as well acquire them too.

    Using backward induction in the same way used on to find the solution to the hierarchy of hungry cannibals game, we find that country 2 will indeed acquire nuclear weapons without risk of country 3 doing so. Thus, country 1 might as well too. Therefore, the preferred strategies of the countries are:

    1. Acquire
    2. Acquire
    3. Don’t acquire
    4. Acquire
    5. Don’t acquire
    6. Acquire
    7. Don’t acquire
    8. Acquire
    9. Don’t acquire
    10. Acquire

    This is just like the cannibals case with “acquire” in place of “eat” with one difference. While country 1 would seem to have a preferred strategy of “don’t acquire” in order to prevent country 2 from acquiring, we know that country 2 will acquire them anyway because it is risk free for that country to do so. Country 3 cannot acquire them or else the next country–number 4–will (same goes for every odd country other than 1). Therefore, country 1 might as well acquire nuclear weapons.

    Thus, the final solution is:

    1. Acquire
    2. Acquire
    3. Don’t acquire
    4. Don’t acquire
    5. Don’t acquire
    6. Don’t acquire
    7. Don’t acquire
    8. Don’t acquire
    9. Don’t acquire
    10. Don’t acquire

    The two dominant countries have nuclear weapons, but nuclear weapons are prevented from proliferating to other countries. Once upon a time that was, in fact, the state of the world. But this correspondence is a bit of luck (or, to be honest, I rigged it this way). The solution would be different if we started with an odd number of countries. If you’ve read this far you can likely deduce that for yourself.

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