fiction - From a writing standpoint, what is the value of Deus Ex Machina?

Deus ex machina is considered by TV Tropes to be a sub-trope to asspulling, and thus, often considered a bad writing trope.

Now, unlike most other bad writing tropes, that have their roots in fanfiction, it has a nifty Latin name and dates back to Greek drama, (Deus Ex Machina, i.e.: god from machine, i.e: we lower some random dude into the scene with a crane) so it must have some sort of value.

What value does this trope have from a writing standpoint?

Answer

It really comes down to merit. The heart of every story is moral, it is about the character making a choice about values, and the reader has a basic desire to see virtue rewarded and vice punished. If your Deus Ex Machina prevents your hero from having to make the great sacrifice that proves their moral worth, that is profoundly unsatisfactory to the reader. If the hero does make, or at very least fully commits to making the great sacrifice, then they have proven themselves worthy and it is then satisfying that some external force intervenes and saves them.

Thus when Luke has committed himself to the attack on the Death Star and has accepted his almost certain death, it is satisfying, and not a cheat, when Han turns up at the last moment and clears the tie fighters out of the way. On the other hand, if Han turns up before Luke commits to his desperate attack and say, "Hey guys, I just won this Death Star Neutralizing Ray in a poker game. You can all go home." That would not be satisfying because Luke's moral choice and commitment to sacrifice have not been made. The solution to the problem is not merited.

And, really, merit is everything in this. A complete logical, completely foreshadowed, resolution that involves no sort of sacrifice of difficult choice is unsatisfying, no matter how logical. A merited resolution, however unlikely, is satisfactory.