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If I owned Hell and Texas I’d live in Hell and rent out Texas

See the thing about posturing is, it must be a credible threat in order to gain any negotiating power.

And if it's a credible threat, then it's a threat.

Forcing NATO into a corner isn't a smart play here, but Putin's not a military genius-- he's a bully and a thug.  So assigning reasonable strategic thinking to him doesn't make any sense.

Yeah, the problem with making a credible threat in order to get the other side to give up concessions...

...is what you do if they don't give up any concessions.

Do you then have to carry through on the threat you never wanted to actually carry through in the first place? Backing NATO into a corner is a smart play if you get them to give you something... If you don't, you've put yourself in a corner because not following through on the threat makes you look weak and you'll never get concessions in the future because they know your threat may not be something you're willing to follow through on.

Which is precisely why many of the NATO countries don't want to back down. 

We don't HAVE to do anything, but NATO by its very structure and nature is not an isolationist organization, and it's something the USA has put scores of years of investment into.  It's not easy for the US to dump its NATO allies.