This type of game theory mostly only deals with the principal actors and their direct reactions, though. What's often more important is what everybody else in the world thinks of the actions. A nuclear exchange is not the end of the world, and what that world looks like afterwards will be shaped by what people think of your actions. That's mostly what I was addressing. In the event of a nuclear attack by a 'rouge state', i.e. one that doesn't have enough weapons and delivery systems for a major attack, the world will be looking very closely indeed at what our response is.

A response that is not sufficiently swift and terrifying could embolden our enemies/rivals. It could also make countries allied to us and trusting the protection of our nuclear umbrella doubt the protection of that umbrella. They might seek to ally with our rival countries instead, or build their own nuclear deterrent. Consider Japan, for example. China is their rival, and a nuclear power. Japan has no nuclear weapons of their own, as they trust us to back them if China threatens them with nuclear weapons. If they have reason to doubt that our backing is not sufficient to deter China, they might seek to build their own nuclear arsenal instead.

Of course, a response that is seen as excessively brutal and/or unconsidered could potentially cause diplomatic issues as well. Countries unfriendly to us might become more worried about what we might do, and might seek a better arsenal of their own, or closer alliances with nuclear powers opposed to us. Countries friendly to us or more allied to us might become worried about what we might to do them, or about being associated with our actions, and might become more distant.

In reality, there will probably be some of both reactions, no matter what we actually did.

I'm inclined to be rather skeptical of nuclear pre-emption because of the second. Not quite 100% against, but the situation would have to be very clearly defined as it being a necessity. In the case of North Korea, in the world as it is today, I don't see it being necessary because of Chinese influence. China is clearly okay with having North Korea be a thorn in our side, but they very much don't want to let North Korea do anything that might provoke us into either a conventional or nuclear attack against them. They don't want American troops or American nuclear weapons detonations right on their border. They have a lot of influence with North Korea, being pretty much the only thing between them and starving and freezing in the dark.

I also have a nice essay around here somewhere describing how your stereotypical crazy dictator tends to get very sane very quickly when faced with the prospect of nuclear war. China, India, and Pakistan have toned down the crazy considerably since becoming nuclear powers.

Note that the last 2 paragraphs treat the world as it is today - where any actor launching a nuclear attack could very reasonably fear a swift and devastating response. This is exactly what a nuclear attack by another country followed by a response perceived as weak could change. It might, for example, turn nuclear pre-emption into a necessity instead of an unfavorable option. If we couldn't count on Chinese and North Korean fear to deter an attack, we might have to actually launch a pre-emptive attack to stop theirs.