North Korea, nuclear missiles, and credible threats of extortion

Eric Rasmussen wrote an excellent post recently about North Korea's nuclear threat. I thought this would be topical to cover here, given that we discussed game theory in ECON100 this week and Rasmussen's post makes use of sequential games. Rasmussen writes:

Besides defense, though, the North Korean military does have another purpose: to make money...

The army can also make money by extortion.  North Korea’s army is too weak to engage in plunder by conquest, but it is strong enough to engage in demanding nuisance fees.   Would it be worth $20 billion per year to South Korea to avoid Seoul being bombarded? North Korea could be like the Barbary Pirates of 1800, who were enough of a nuisance to extract a goodly amount of their revenue as protection money, but so poor that that same amount was trivial to the European countries that paid it. The   United States,  having more principle and less monetary calculation, ironically, than the aristocratic Europeans, proved problematic on the shores of Tripoli and eventually France ended the game by conquering Algeria. However, the Barbary pirates did have a good run for their money.

The problem is making the threat to bombard Seoul credible.  The threat is credible if South Korea invades the North. If a South Korean invasion begins,  and  Kaesong is about to be occupied, North Korea has nothing to lose by destroying Seoul. If South Korea purposely bypasses Kaesong to avoid triggering that response and heads straight for Pyongyang, the Kim regime would see its demise and, again, might as well destroy Seoul and get a bit of revenge. Either way, the threat of bombardment is credible.

On the other hand, if North Korea simply says it will shell Seoul unless $20 billion is deposited in a certain Swiss bank account, that threat is not credible. If South Korea refuses, and North Korea shells Seoul, South Korea will respond by destroying the guns. Once the guns are gone,   South can conquer   North without fear of retaliation. North Korea will have almost literally “shot its wad.” South Korea may have lost 100,000 people, but that is small comfort for the Kim regime if it loses power. Thus, looking ahead, South can see that North will not retaliate and its $20 billion demand can be safely refused.

The game that Rasmussen describes is laid out in the figure below. The payoffs in the figure are (North Korea, South Korea). We can solve sequential games like this using backward induction - that is, we start with the last decision and work our way backwards. So, in this case the final decision is South Korea's - whether to Bomb Pyongyang, or not. If South Korea bombs Pyongyang, their payoff is -95, compared with -100 for not bombing Pyongyang. So, South Korea will bomb Pyongyang (because -95 is better than -100). Now, working backwards one step, we can work out whether North Korea will bombard Seoul. North Korea knows that South Korea will bomb Pyongyang if they bombard Seoul, so North Korea's payoffs are -200 if they bombard Seoul, or -1 if they don't. So, North Korea will choose not to bombard Seoul (because -1 is better than -200). Their threat to bombard Seoul if South Korea doesn't pay them $20 billion is not credible - North Korea wouldn't follow through on the threat.

Next, we can work out whether South Korea will choose to pay the $20 billion demand. If South Korea pays the $20 billion their payoff is -20, but if they don't pay their payoff is 1 (since North Korea will choose not to bombard Seoul). South Korea will choose not to pay the $20 billion (because 1 is better than -20). Finally, we can work out whether North Korea will threaten Seoul. If they issue the threat, their payoff is -1 (since South Korea will choose not to pay, and then North Korea will choose not to bombard Seoul), but if they don't issue the threat their payoff is 0. So, North Korea will not threaten Seoul (because 0 is better than -1). The subgame perfect Nash equilibrium is that North Korea doesn't threaten Seoul (and South Korea doesn't need to do anything in response, because the game ends right there). Note that Rasmussen has tracked all of the best responses as arrows in the figure.

Rasmussen then goes on to describe how the game would change if North Korea develops a nuclear arsenal:

...nukes are good for extortion in themselves and a good backup for artillery. Imagine now that Kim has nuclear missiles pointed at Seoul. This changes the game... There is now a new move at the end, Nuke Seoul or Not. Many of the arrows change, though, because that last move changes everything.

The game with nuclear weapons is in the figure below. Again, we can solve this game with backward induction. Now the final decision in the game is North Korea's - whether to Nuke Seoul, or not. If North Korea nukes Seoul, their payoff is -180, compared with -200 for not nuking Seoul. So, North Korea will nuke Seoul (because -180 is better than -200). Now, working backwards one step, we can work out whether South Korea will bomb Pyongyang. This time, if South Korea bombs Pyongyang their payoff is -195 (since North Korea will nuke Seoul), but their payoff is -100 if they don't bomb Pyongyang. So, South Korea will not bomb Pyongyang (because -100 is better than -195). Next, we can work out whether North Korea will bombard Seoul. North Korea knows that South Korea will not bomb Pyongyang if they bombard Seoul, so North Korea's payoffs are 2 if they bombard Seoul, or -1 if they don't. So, North Korea will now choose to bombard Seoul (because 2 is better than -1). If North Korea has nuclear weapons, notice that their threat to bombard Seoul is now credible - they will follow through on it.

Next, we can work out whether South Korea will choose to pay the $20 billion demand. If South Korea pay the $20 billion their payoff is -20, but if they don't pay their payoff is now -100 (since North Korea will choose to bombard Seoul, and then South Korea will choose not to bomb Pyongyang because Seoul would then get nuked). South Korea will choose to pay the $20 billion (because -20 is better than -100). Finally, we can work out whether North Korea will threaten Seoul. If they issue the threat, their payoff is 20 (since South Korea will choose to pay them), but if they don't issue the threat their payoff is 0. So, North Korea will now threaten Seoul (because 20 is better than 0). The subgame perfect Nash equilibrium is that North Korea threatens Seoul, and South Korea pays the $20 billion.

So, that provides one more reason (if any were needed) why South Korea (and its allies) should be working hard to prevent North Korea from developing nuclear weapons. Because North Korea could then use them for extortion.

[HT: Marginal Revolution]