The North Korea missile crisis as a coalition game
Alexander Levakov, Senior Research Fellow, Ph.D
December, 2017
1 Current situation
“In 2017, North Korea conducted a series of missile and nuclear tests that demonstrated the country’s ability to launch ballistic missiles beyond its immediate region and suggested that North Korea’s nuclear weapons capability was developing at a faster rate than had been assessed by the U.S. intelligence community.This, coupled with a regular joint U.S.–South Korea military exercise undertaken in August 2017, as well as U.S. threats, raised international tensions in the region and beyond.”1
2 Game theory
Current crisis is a classical game theory 2 case where two or more players are pursuing long term goals and applying set of strategies coincided with payoffs.
3 Coalitions and players
In this case (North Korea missile crisis) we have got two groups of players i.e. coalitions: USA, South Korea, Japan and China, North Korea, Russia. The point is that the first coalition (USA, South Korea, Japan) is explicit while the second one (China, North Korea, Russia) is tacit. Both coalitions are stable in the current situation i.e. no one of players can leave coalition.
3.1 Goals of coalitions
China and Russia support North Korea as a big challenge for the USA military buildup in the region 3. China is the leader of this coalition as a big trade partner and investor of the USA economy4. The goal of China is to preserve and strengthen its leadership and negate USA influence in South Asia and Pacific region (ACEAN). The goal of Russia is to lift up sanctions by USA. The goal of North Korea is to gain official status of regional nuclear power, dismount US and South Korea military buildup, lift up sanctions and rule out dependence from China. Neither China, nor Russia are interested in nuclear North Korea both as current partner or future rival, too. Both China and Russia are not interested in united Korea, no matter communist or noncommunist. This is the main and principal contradiction within coalition.
The USA supports South Korea and Japan by THAAD and PATRIOT installations, military drills and nuclear weapons arsenal. USA is the leader of this coalition. The goal of the USA is to dismount North Korea nuclear and missile programs 5 and to overthrow Kim as North Korea leader. The goal of South Korea and Japan is to dismount nuclear weapons and missile program of North Korea. Japan is not interested in united Korea as a trade competetitor while USA and Soth Korea are interested ruther. This is the main and principal contradiction within coalition.
USA has to keep balance between its two main treasury securities holders in terms of dollars: China ($1,200,500,000,000) and Japan ($1,101,700,000,000). While Russia ($105,400,000,000) and South Korea ($95,000,000,000) have very little influence on USA in terms of treasury securities 6.
3.2 Strategies of coalitions
China doesn’t want to increase it’s nuclear arsenal understanding useless of further escalations with the USA 7, 8, 9, 10. China helps or doesn’t prevent North Korea missile program in order to gain future payoffs as a mediator with the USA11. China and Russia are blocking USA sanctions as long as possible 12, 13.
USA can’t wage the war against North Korea due to the potential losses in South Korea and China 14. USA strengthens economical and political sanctions on North Korea leaving the room for contacts with Kim 15. USA and its allies in coalition deploy and strengthen missile defense both in the region and in the mainland USA demonstrating the will and power for military actions in case of war16. While the room for diplomacy is decresing the military option becomes more realistic in spite of consequences for all countries engaged in conflict. Thus USA and its allies are trying to make China and Russia more dependent from North Korea position on ICBM program i.e. make both countries more reluctant to its continuation 17.
4 Coalitional model18
In a coalitional model, we focus on what groups of players can achieve rather than on what individual players can do. A stability requirement for a coalitional game is that the outcome be immune to deviations by groups of players, i.e., no subset of players can unilaterally improve their outcome. A strong criterion is that all players in the group are strictly better off. A weaker criterion is that the outcome improves for at least one player in the group and makes it no worse for all others.
We now consider a simple version of a coalitional game, namely a coalitional game with transferable payoff (transferable payoff means that there is no restriction on how the total payoff may be divided among the members of a group).
4.1 Rules for stable sharing
We have a finite set \(N\) of players and a function \(v:2^N \to R \geq 0\). So, \(v\) assigns a non-negative number to every \(S \in N\). We can think of \(v(S)\) as the total payoff available to the members of set \(S\). A natural question that arises is how to share the value \(v(N)\) among all the players so that there is no incentive to deviate.
The core is a solution concept for coalitional games, analogous to Nash equilibria for non-cooperative games. For a coalitional game with transferable payoff, a cost sharing is in the core if no coalition can obtain a payoff which is better than the sum of the members’ current payoffs.
We note that a cost sharing \(x\) has \(x_i \geq 0: \forall i \in N\) and \(\sum_{i\in N} x_i = v(N)\). Then, the condition for a cost sharing \(x\) to be in the core is that \(\forall S \subseteq N, \sum_{i\in S} x_i \geq v(S)\). Equivalently, there is no set \(S\) and payoff vector \(y\) with \(\sum_{i\in S} {y_i = v(S)}\) for which \({y_i} > {x_i} : \forall i \in {S}\).
US options in the coalition game
5 Historical case analogue
In 2007 Issac G. Gipson published his thesis “THE EFFECTIVENESS OF THE U.S. MISSILE DEFENSE CAPABILITIES AS A DETERRENT TO THE NORTH KOREAN MISSILE THREAT” applying game theory to the subject i.e. current US-NK missile crisis .
The United States enhances the current missile defense capabilities through its continued development and testing, while North Korea accepts the benefit of the Six Party Talks, but secretly continues to develop and test ballistic missiles (2,2).19, 20
As we can see nothing has changed since that time except ICBM and nuclear tests made by North Korea. 
6 Conclusions
1.Both coalitions have explicit strategies and payoffs in the game.
2.Neither player can leave coalition.
3.The solution of the game depends on rationality of all players 21.
4.North Korea has to change its ICBM program for the space one, leaving the room for negotiations with USA under the pressure from China.
5.USA has to deploy Aegis Ashore and THAAD installations together with F-35 jets in the region making Japan the main military rival for China.
6.There is no explicit military solution for this game.
7 Note
It’s a personal point of view on the current situation.
8 References
https://www.theguardian.com/world/2017/dec/07/north-korea-ready-direct-talks-us-sergei-lavrov↩
https://www.statista.com/statistics/246420/major-foreign-holders-of-us-treasury-debt/↩
https://www.statista.com/statistics/246420/major-foreign-holders-of-us-treasury-debt/↩
https://www.theguardian.com/world/2017/dec/12/china-refugee-camps-border-north-korea↩
http://edition.cnn.com/2017/12/10/asia/north-korea-united-nations/index.html↩
https://www.theguardian.com/world/2017/dec/08/pentagon-believed-it-would-undoubtedly-win-war-with-north-korea-in-1994↩
https://www.theguardian.com/us-news/2017/dec/12/us-ready-for-talks-with-north-korea-without-preconditions-tillerson-says↩
EVA TARDOS, “Algorithmic game theory”↩
Issac G. Gipson, THE EFFECTIVENESS OF THE U.S. MISSILE DEFENSE CAPABILITIES AS A DETERRENT TO THE NORTH KOREAN MISSILE THREAT, December 2007, NAVAL POSTGRADUATE SCHOOL↩