Assume the following hypothetical scenario:

There are only three relevant nations, two large ones (A, B) both with nuclear weapons sufficient to destroy everyone (MAD) and a smaller one without nuclear weapons (C).

Further suppose that one of the large nations (A) is determined to conquer C (for whatever reason, but it's a strong one). A may or may not be a democracy but in any case there is a very strong will to invade and capture all of C. The army of A shall further be capable of doing that against the will of C.

B is actually on very friendly terms with C and would like for C to remain as it is. B is on much less friendly terms with A, surely less so than with C. People living in B will probably strongly feel for people in C and therefore B would likely try to defend C. Let's further assume that B and C together are more capable in a military sense than A.

Now I wonder if the mutually assured destruction of A and B actually helps A in conquering C or rather hinders A in doing that or is largely neutral in that regard?

Mutually assured destruction means that A and B should hardly go to full scale war with each other, because that would mean the end of the world. In game theory that is described as a Nash equilibrium.

But what does it mean in this scenario here and how does mutually assured destruction change the equation?

Without nuclear weapons: B would defend C, they would be stronger than A, so A would likely be defeated or decide not to attack in the first place.

With nuclear weapons, there are more (contradicting) possibilities:

• A can tell B to not interfere in its conquest of C or else A would consider that an attack on itself and retaliate. A would basically virtually extend its borders to include C. MAD would mean that B could not attack A, A could successfully conquer C.
• Or B could tell A that it considers an attack on C an attack on itself and is ready to retaliate. B would basically virtually extends its "protection zone" to include C. MAD would mean that A would not attack C in order to avoid total destruction.
• Or both A and B both include C in their MAD and nobody knows what happens.

I'm totally confused by all that game theory and that is where I cannot get to any further. I just want to know if MAD means that small, third, non-nuclear weaponized countries are easy prey, even if (bonus question) in principle they could be part of a defense alliance? Would the situation change if B and C would be in a defensive alliance?

My guess is that it's still like a game of chicken in the end, and whoever hangs on to his life less wins. But what confuses me is that basically anything can happen with some reasoning (see above).

This question is very similar to Does a policy of mutually assured destruction favor rogue states? but in this case here I do not assume that A is a rogue state (just somewhat aggressive and wanting to conquer C otherwise equal to B and there are no madmen involved, just rational actors with specific interests) and it's a specific scenario where a third minor nation is to be invaded. So more specific answers might be possible.

I'm not interested in any special cases here, A could be the US, Russia, China (with more nuclear weapons), C could be Ukraine, Poland, Taiwan, Mexico, ...(or any other small country) and B could be the rest of the world combined in each case. If the outcome depends on specific circumstances just explain to me which they are.