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There are occasions where you basically just start a chapter with a short descriptive passage and go straight to dialogues, so in those situations I am not sure how to deal with acronyms in dialogues.

As Mike settled down for work, he noticed a strange note on the corner
of his desk. Curious, he grabbed the note and looked at it to see if
there was anything inscribed on it, but without warning Kate opened
the door, which startled him and compelled him to put the note inside
his jacket.

"Mike, we have an emergency!" she said.

"What is it?" he asked.

"The NDPI requested a meeting."

NDPI stands for National Directorate of Police Intelligence, but it wasn't mentioned yet, do I have to sort of add a chapter or dialogue prior to that where the full name is used? What if I don't want to do that? What are my options? I would like to choose the most popular one.

If you don't provide a hint, then readers will know only that somebody requested a meeting and that's considered an emergency. If you want to convey something about the nature of the organization (or emergency) you need to provide more of a hint, but that doesn't necessarily mean expanding the acronym. One way to do this is to have someone react to the reference.

For example:

"The NDPI requested a meeting."

"What? How did our police report become a national matter?"

Or:

Mike groaned. Dealing with the police was frustrating at any level, but when the national body got involved, long tedious meetings and forms filed in triplicate were just the beginning.

I typically would not expand a single acronym, and would not use too many. I would give hints, but perhaps I wouldn't do even that.

Consider the TV series "NCIS", you can go half a season without knowing what NCIS stands for. Something to do with the Navy, and Investigating. They have badges. The people they investigate groan and say "NCIS", or seem happy to see them, or ask "What's the Navy's interest here?". But the authors do not go out of their way to inform the audience what the acronym stands for. (Technically when you pronounce each letter, it is called "initialism". Acronyms are when the initials are pronounced as words; like "laser" or "NASA".)

Nor does it make any difference! You may be very proud of your acronym, but in the real world, nobody cares. We have the FBI, NSA, NSF, CIA. We have PHDs, MDs, RNs, some people do not know what those initials stand for either. Same for IBM, GM, the NYSE, the EU, LED. We have all kinds of "xxPD", like NYPD, LAPD, we get it without it ever being spelled out.

These initials just become the name, a label for an organization or a title, and IRL we stop thinking so much about what they stand for and just treat them like a name. So it can feel unnatural for characters to be reciting to themselves, or each other, the meanings of acronyms. Their reactions and thoughts can give oblique hints; If we hear the FBI wants to talk to us, our thought might be What do they want? I don't know anything about any crimes. Maybe about somebody I knew in school?

Remember, show, don't tell. Have your law-abiding characters treat the NDPI like cops (they sound like the FBI), with caution and deference and a little bit of fear. Have your openly law-breaking characters treat the NDPI with suspicion, resistance, and a dollop of anger, hatred and resentment. Have your secretly law-breaking character feel the latter and fake the former.

Have the NDPI act like cops, ask questions like cops, say stuff like cops. A good place to expand acronyms is in formal introductions.

As they entered the meeting room, a man in a crisp business suit sat at the conference table, reading a folder. He looked up, closing the folder and laying it on the table, standing up to greet them.

"Please, take a seat," he said, gesturing to the empty seats near him. "I am agent Malloy. Before we begin, In this particular circumstance I am required to inform you that I am an agent of the National Directorate of Police Intelligence, and lying to an NDPI agent about any matter related to a case is a national crime that may carry a prison sentence of up to five years."

Agent Malloy grinned. "That is never a pleasant way to start a conversation, but it has to be on the tape. I can't lie to the NDPI either."

So we are being recorded, Mike thought. And he can't lie to the NDPI, but he can lie to us. He worried about Kate, she was just reckless enough to get herself into serious trouble here, and he didn't even know what the NDPI wanted.

Something like that; from your POV character.

If the character would naturally use the acronym in dialog, you should write it that way. however, you should try to find a way to explain it, or at least give a hint, as soon as you can without harming the flow of the story. The suggestions in the answer by Monica Cellio seem like good ones. If appropriate, one might also quote a document which expands the acronym, and may official documents do. If the reader understands the general concept conveyed by the acronym, it is ok if the exact expansion or referent of the term is not explained.

In technical writing, as opposed to fiction, acronyms should usually be expanded on first use unless all readers of the document can be expected to know the term well. (Writing for web developers one might not expand "HTML". Writing for economists, one might not expand "GDP".)

Clarity is always good.

I feel it's important to convey the POV that the story is trying to convey. If the narrator is a limited third person POV, the information conveyed should match that perspective. If the story is told from an omniscient perspective, the acronym should be expanded within a paragraph or so of first being used.

I've read stories that did this pretty successfully with a few POV characters. For example, the acronym was first introduced to a character who was totally unfamiliar with it. In the next chapter, our POV switched to somebody knew that the acronym was at least that of a federal body. A few chapters later, we had a new POV character, who didn't think the acronym at all, because they knew what it was. As such, their opening paragraph spelled out that the big deal the book was having was the [full name of the organization] idiots had their Ts dotted and Is crossed again.

TL;DR: Stick with the POV you've chosen for the story. Go with what is consistent with that POV.

Ben Aaronovitch in his Rivers of London series often introduces an acronym without explaining it. Generally, the protagonist will introduce the acronym in his role as the narrator. Then he'll use that acronym with another character who won't understand it. Only then does he explain to that character, and therefore also to the audience, what the acronym means.

The stories are mostly set in London which has a history of unique and confusing speech patterns (think Cockney Rhyming Slang). Using language that others don't understand is a type of power and indicates which groups the characters do and don't belong to. Group A understands this set of language, group B understands that set of language, etc.

As the reader, when this happens you feel temporarily out of the loop, which I believe allows you to empathise with more than just the protagonist. He does also go on a lot of relevant short tangents, e.g. recounting part a of prior conversation to the audience during a current conversation. It's a very colloquial, informal, contemporary style which works for this series of books, but would be entirely unsuitable for other stories.

Don't leave the reader hanging too long. A short time between introducing a word/acronym/concept and explaining it is acceptable and engages the audience if not overdone. If done well, you certainly don't need to explain it straightaway.

It doesn't need to be prior but it should be in the same scene or context. In fact, this goes the other way around as well. You should not explain an acronym without using it in the same scene or context.

The reason is simply that unless the explanation and use are in the same scene or context, human memory cannot be trusted to connect the two.

Same scene is probably self explanatory. If the explanation and use are close enough for short term memory to handle and there is nothing in the scene to confuse the reader from making the connection, it will work.

Same context is probably what you want. Basically you make NDPI memorable, give people things to remember it by. Usually this would be by giving characters some sort of an emotional reaction readers can remember. Or by connecting NDPI to the plot. Anyway you want but the point is that readers think that being approached by NDPI is important, it matters, it makes a difference, it is memorable.

Downside is that you will need to deliver on that promise. If you make it look like NDPI has personal impact on characters, you'll need to show it. If you imply it matters to the plot, you'll need to develop that as part of the plot or sub-plot.

And whenever you do that that will be in the same context and you can deliver explanation of what the acronym means and readers will connect it properly.

What reading experience do you want to provide to your readers?

Are you writing from a viewpoint character familiar with a world that is foreign to your readers, and your readers have to slowly figure out that world in the course of the novel? Then do not explain NDPI and let your readers understand it from the following events.

Or are you explaining the world to your readers as you introduce it so they always have the relevant knowledge? Then explain the NDPI either in the narration:

"The NDPI requested a meeting." The National Directorate of Police Intelligence are in charge of [whatever].

Or in the thoughts or words of one of the characters:

I had to tell the National Directorate of Police Intelligence, I thought. "I'll call the NDPI," I told Mike.