The arrival of a large number of immigrants in the city has led to a zero-sum scenario.

Have I used 'zero-sum' correctly and naturally? Is this okay?

The arrival of a large number of immigrants in the city has led to a zero-sum scenario. There are countless applicants for each job opening, and migrants also face a zero-sum competition when trying to secure school spots for their children.

No. That is not how it's used. It basically refers to a situation in which there is no net gain for any of the entities involved. Look it up in the dictionary, and then revise your sentences and repost them below. Note also that "competition" in the sense you've used it (generic - not a particular event) is not countable.

I disagree. "Zero sum" refers to a situation where if I win, you lose.

I don't think it is used naturally here, and it doesn't add anything to the sentence. They are in competition for limited spots. It's inherently zero sum, and it doesn't need to be said.

SoothingDave said:

I disagree. "Zero sum" refers to a situation where if I win, you lose.

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Perhaps this is another AmE-vs-the-rest difference.
See below, for example.

Yes, that's the definition. You said something different, that there is no net gain for any of the entities involved. The winners win. That's a gain.

Competition for limited spots in a school is zero sum. Competition for a job opening is zero sum.

I don't understand your point, and I'm not all that interested to be honest with you.

SoothingDave said:

You said something different, ... The winners win. That's a gain.

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I did not say that.

I think the OP can see the definition and how it does not fit with either of his sentences.

We're both using the same definition. Can you give an example of something that you think is "zero sum?"

Aren't the example sentences in the link I posted above enough for you?

Is there no way I can use 'zero-sum' in that context in a natural way?

In mathematical game theory zero -sum is a technical term meaning that for each winner there is exactly one loser. In everyday parlance I supoose its usage is more fuzzy or flexible. Nevertheless I'd advise you to use a different term @alpacinou.

alpacinou said:

Is there no way I can use 'zero-sum' in that context in a natural way?

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What are you trying to do?

Are you trying to learn the word 'zero sum' by making up your own context for it? Or do you have an idea in mind that you're trying to express?

probus said:

In everyday parlance I supoose its usage is more fuzzy or flexible.

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Well, it has become quite common recently outside of economics—enough that people are frequently using it incorrectly.

jutfrank said:

What are you trying to do?

Are you trying to learn the word 'zero sum' by making up your own context for it? Or do you have an idea in mind that you're trying to express?

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A bit of both. At the minute, I'm trying to see if it works in that context.

alpacinou said:

A bit of both.

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That could be a problem. Stick to one or the other.

alpacinou said:

At the minute, I'm trying to see if it works in that context.

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We've already said it doesn't.

Applications for jobs and places in school are about competition for a limited number of places available based on merit, which is a no-brainer. In a zero-sum situation, one party gains at the expense of another. I don't see how the two can be linked.