Brinkmanship doesn't get explored all that much in games. Precipice is almost certainly derived from Balance of Power, a political strategy simulation from 1985 about the Cold War powers and their winner-armageddons-all staring contest. It was impressively original, complex, and completely unplayable. Happily, Precipice has kept most of what made that game interesting to begin with, and dumped the baggage.
You take control of the USA or USSR during the big tiff, taking turns to invest political capital into attempts to sway the relations of other nations round to your ideology - or at least paying it enough lipservice that they're effectively Finlandized. Precipice is far simpler than Balance Of Power, and all the better for it. It can be grasped in the space of half an hour. The main things you'll be doing are making diplomatic overtures to non-hostile nations, and destabilising or even outright attacking hostile ones, with the goal of what the winners would call regime change and everyone else would call installing your puppet.
You can play it surprisingly nice, thanks to the trade system. Sealing a trade deal is a solid way to gradually win a nation over for little cost, and gains you resources each turn. These can be further exchanged for more trade deals, or saved up and generously donated to nations that have a crisis of some kind, which can even win over staunch friends of the other superpower if you play it right. Being bold pays off, but so can taking the moral high ground as well. Any time you or the enemy are caught doing underhanded things like arming dissidents, or something you can't disguise like invading, the other side can challenge your actions. Doing so kicks off the thing that also defined Balance of Power best: the face-offs.
Whoever backs down first loses face with some of the neutral, and even friendly powers. If both sides refuse, the stakes raise, again and again, until - oops! - you obliterated all human life. Unlike Shadow President, there's no cool/chilling animation and body count followed by a slow collapse of the world order as everyone aligns against you. It's just over. But hey, if you'd let Vietnam turn to communism the whole region would have followed, right? Right.
Precipice isn't difficult, and I suspect it would be much more fun with two players going head to head. But it's a good, stripped down game of bluffing and second guessing as you place and root out spies, and try to balance the bonus of turning whole blocs vs the potential to drive a wedge between your enemy and their mates. It's a fun time watching the USA screw up so badly that even Canada and Bootlick Island turn against them, or divesting the USSR of the Warsaw Pact. Sure it's unlikely, but this is more of a fun board game than a simulation. The main downside is that after a handful of games you'll find a strategy that almost always works, and the only variation is a question of degrees. Maybe grab a friend and see who blinks first.