This is a slimmed down analysis, and non-technical in character, as to how to think about certain issues with North Korea in terms of game and nuclear deterrence theory.
As to games and nuclear deterrence theory, it would seem to the author that the theory of general sum transferable theory has a place, since money exists, but, that would depend on the players, and so each case is sui generis, a thing to itself.
That right there is a major relaxation of what rigor one could claim for any result, as it depends on how one models the players as individual actors, as in North Korea.
Then again, it would seem obvious that this sort of approach is how people actually make decisions, informed by models certainly, but also bringing things to the table with them at the meta-level of assumptions, maybe even just liking one country idiosyncratically. Some people like a France, some like Germany more, and it really has nothing to do with France or Germany, but who you are.
In any event, proceeding to the issues at hand with North Korea, there exists a market for protection with nuclear weapons that we call the nuclear deterrence market.
One can provide the nuclear deterrence oneself, or one can ally with another party for both transferable utility considerations, i.e. money invested in your currency and in factories and in economic access in general, and for other non-transferable, i.e. non-monetary, considerations as to physcial presence for example on the Asian and European landmasses to contain certain powers.
As to the issue of provision, clearly there are economies of scale in nuclear weapons, since if one has a fleet of Ohio class submarines, or Typhoons, and possibly Xia class, although the Chinese curiously can't see to get that to work well, just one of those things, then one can provide deterrence on a global basis, an economy of scale, since one doesn't have to alter weapons platforms to provide nuclear deterrence.
In this market for nuclear deterrence, there is then the question of entry into the market, e.g. North Korea is the potential entrant, and we are the incumbent whose life would be more simple without them doing that.
Here we have in game theory a question of "type" of player, in which to be momentarily non-seriously serious, there is the question of the "nutbomb" type, beyond that of "hard" or "soft."
There is good news here possibly in the non-detonation of a North Korean nuclear weapon, in that the "nutbomb" type always fights no matter the payoff to be discussed below.
As to the payoff, if you think about it as a two move game in the extensive form, North Korea goes first, and choses to observe the payoffs in the nodes, which are in effect chosen by the incumbent in the sense of chosing to "fight" or "not fight" with siganals as to probability of fighting possible, known as "mixed strategies," although of course in the literature some have always objected to the "Two Face" model this implies as to launching a war on a coin toss. It is probably best to think of such a thing as more of a "wishful thinking" or more positively "glass is half full" as states go through time to the endnodes of the game.
So in the case of North Korea, the incumbent, us, has to make a choice about what it estimates the world will look like if North Korea becomes a fully fleshed out nuclear armed state, by estimating its type in part, and then examining what set of signals it is willing to send, up to and including an Osirak type raid, as Israel did in 1981 with Iraq, notably one of the few cases in which nuclear entry was prevented.
As to data, Brazil and the ABC played around with that idea, but never took many steps down that road, although it would seem that in certain circles in Brazil, there is a hedging to what the world would look like if America declined and did not provide extended deterrence. The other main cases of nuclear non-entry were clients of the United States or the Soviet Union, with troops stationed there at that point in time and or subject to de facto or de jure extended deterrence guarantees, save possibly for Israel.
In any event, the real question then is:"Do you think that the costs of sending the signals to deter entry, up to and including the use of force, are worth it in terms of what's your estimated payoff of having that particular entrant in the nuclear deterrence market,"where here North Korea doesn't to the author appear promising.
If you go back in time to the Cheonan incident, and let's ignore that only save for crisis behavior revealing certain things, which is of course why sometimes people start crises in the first place, North Korea used very curious language as to its nuclear weapons capabilities not being deterrence in the usual sense used in popular quarters, although it is consistent with warfighting doctrines.
As to the nature of nuclear deterrence, the more popular view is "Minimal deterrence through Assured Destruction."
Crudely put, if I can burn down enough of your cities, you won't do that to me, and we all live in a room with guns happily enough. And people do that in the domestic world, and it is not thought to remarkable.
On the other hand, there was always a school of thought associated with professional militaries rather often that didn't like minimal deterrence with Assured Destruction, as there is no theory of victory in that; one just kills everyone on a Trident, and then has a Budweiser to celebrate. or cry.
Supposedly the British on their subs have a secret envelope that says one of two things;
"We have been destroyed, kill them all, and may God bless the Her Majesty and the United Kingdom."
or, "Sorry chaps, we're pretty much all dead here. No point in killing a bunch of bloody Russians and Chinese on a bluff that got called. Didn't pull that inside straight, and have a beer, you punters."
Or something like that, and only the Prime Minister knows which it is, and possibly one wonders the monarch too. Or so one reads, if people don't tell too many tales out of that school, for good reason.
But as to warfigthing theories of nuclear weapons, that was the North Korean language with Cheonan, in the sense that in the West we call compellence. I was taking a class as a refresher and validator to me at Harvard, and pointed this out to Tom Nichols, who also teaches at the Naval War College, in terms of:
"Now you are going to say that they aren't talking sense, because they mean compellence, not deterrence, but does that not prove that generalized theories of deterrence may run into grave cross-cultural barriers as to meta-level questions of what the purpose of nuclear weapons is, i.e. to merely prevent war, or to win one?"
"Yep." And implicitly, "We kind of just have to hope for the best, and frankly, I have met a lot of people in the military who think like you, and they scare me, like you do."
And then at the end of the class he said "Thanks for being a really good sport Don," and I was cool with that.
I think that is a fair statement, and I have no issue with Dr. Nichols, because he could be right about one thing for sure.
He could well be right that warfighting notions of nuclear deterrence are dangerous, although, he could be wrong too if there existed a state that had those notions in the sense of calculating what to do in the endnodes of a Stackelberg entry to nuclear deterrence game, and had revealed that in the Cheonan and recently by threatening the South Korean president, if I suppose one could also argue that was to get the talk of regime change in the North off of the table. I think not personaly, but I could be wrong.
I don't think there's anything that can be done about that either, but to just look in your heart, and really ask where your motive is coming from, and that could well not be good, but of evil, if that's unpopular to say in the materialist world we live in, and note, evil might well be the hardest to detect of all, if one believes in such things of course, and many do not, and they could I suppose be correct that people claiming personal knowledge are delusional, dangerous, or both, and there is a history of that.
So, in conclusion, games and nuclear deterrence theory are how to look at the ongoing situation with the North as to whether it tests a nuclear weapon, although the longer that it waits, that is a monotonicaly declining possiblity, and after May Day, one would think not, although then there remains the issue of what to do about North Korea's subrosa nuclear weapons program, in which the same considerations would apply as to type and such, which as has been argued elsewhere would also apply at least potentially to Iran.
But if they keep threatening, it would be zero surprise if that doggie bit us, with WMD, which is why backing down that little mutt, if in private, is a rather serious affair indeed.