Here's the payoff matrix. Both sides can do one of three things: strike first, retaliate with an equally-destructive second strike, or sit on their hands. The fourth option, make friends and play nice, is a different game altogether.
Not all alternatives make sense. For example, both sides can't retaliate, and it's unlikely that both sides could exactly synchronise a first strike. Those cells are left blank.
Note that the payoff matrix format best suits a game where both players make their move at the same time. However, nuclear war is typically a sequential game which, as we shall see, has consequences.
Note that the scores indicate that if both parties strike, the result is armageddon, with payoff minus infinity for both parties.
If you strike and your opponent does not, they are obliterated and you have won. As you handle the fallout, you can reassure yourself that at least your enemy no longer exists: minus infinity to them and a modest +1 to you.
If you both do nothing, then you each get a score of zero, reflecting your continuing insecurity.
The first point to note is that the "do nothing strategy", while pareto-optimal (i.e. welfare maximising by most people's reckoning), is not a Nash equilibrium. Each side can improve its payoff by striking first, on the assumption that the other side will not retaliate because it doesn't increase its payoff (-∞) by so doing. In either case it's annihilated.
The scoring method is explained here (Nash equilibria in a payoff matrix).
However, recall that this is a sequential game, so the appropriate model is a "subgame perfect Nash equilibrium". To compute this we have to eliminate 'non-credible moves'. To prevent the first strike option being the Nash equilibrium it's necessary to make not-retaliating non-credible.
Hence the onus on all politicians with fingers on the button to solemnly state that in the event of a nuclear attack there is not the faintest chance that they would do anything else but retaliate at once. There: (-∞, -∞) guaranteed (!).
Looks like sticking with the insecurity is the best bet after all.
Next: MAD and Iran