the hunger games - How could the Capitol use nuclear weapons against District 13 provided that "Mutual Assured Destruction" doctrine was in place?

In The Hunger Games, Capitol nuked District 13 during the first rebellion of Panem. But, we later learned that District 13 also had nukes, but they didn't use it because it would have meant nobody would have survived.

Now, the question is: How could Capitol even think of using nukes provided Mutual Assured Destruction was in place?

Mutual assured destruction is a doctrine of military strategy and national security policy in which a full-scale use of nuclear weapons by two or more opposing sides would cause the complete annihilation of both the attacker and the defender. It is based on the theory of deterrence, which holds that the threat of using strong weapons against the enemy prevents the enemy's use of those same weapons.

In case Capitol didn't know District 13 had nukes, District 13 could always showed it off to ensure Mutual Assured Destruction was in place.

Answer

I think you're missing the point. The idea of Mutually Assured Destruction is precisely why the Capitol didn't use nukes in the first war.

During the Dark Days, the rebels in 13 wrested control from the government forces, trained their nuclear missiles on the Capitol, and then struck a bargain: They would play dead in exchange for being left alone. The Capitol had another nuclear arsenal out west, but it couldn't attack 13 without certain retaliation. It was forced to accept 13's deal. The Capitol demolished the visible remains of the district and cut off all access from the outside. Perhaps the Capitol's leaders thought that, without help, 13 would die off on its own. It almost did a few times, but it always managed to pull through due to strict sharing of resources, strenuous discipline, and constant vigilance against any further attacks from the Capitol.

_{-- Mockingjay, chapter 2 (emphasis mine)}

Mutually Assured Destruction ("if you attack us, we both die") is the exact concept being referred to here. The Capitol didn't nuke Thirteen because of MAD - that's why Thirteen was allowed to survive. In short, the answer to your question is:

How could the Capitol use nuclear weapons against District 13 provided that “Mutual Assured Destruction” doctrine was in place?

They didn't, and precisely for this reason.