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?
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D.E.M. implies an implausible [to the audience] save; and there is no value to that. The minute the audience finds a story development implausible they lose interest; this is the point when they turn to the back to see the final page number so they can compute how much more time they must waste to see if the sub-plots are resolved better, or if they should just give up.
As mentioned in another answer; the DEM of the Greeks were not implausible to the audience: They believed in Gods and were seen as a time-saving device; as in "this is how it would all work out". Kind of like the "much later" post-script of some movies (e.g. When Harry Met Sally; with the two of them elderly and clearly married happily ever after).
The best equivalent to DEM that can still be used is seeding the implausible solution in the opening act, while you have the benefit of the audience's willing suspension of disbelief about anything. You can introduce magic, interstellar travel, telekinesis, telepathy, whatever: You can also demonstrate real life implausibilities: show a six year old winning a junior archery championship, or an eight year old successfully solving differential equations to the amazement of their PhD tutor, or using their phenomenally photographic memory at the behest of a parent that takes it completely for granted.
Now to some extent, these kinds of things force the story to be about those things; but you can also take a few notches off "perfection" to establish an unusual ability or character or coincidence that can serve where you thought the DEM had to be.
If the story demands something implausible; establish it very early in the story where it will be accepted as part of the foundation of the fiction. Once the first Act (the setup) is over the author has spent all their credit for implausibility.
If more is needed, rewrite the first Act: But of course it cannot go on forever, and there are limits to how much implausibility the audience will tolerate, even in the First Act.
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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.
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