Referring this question.

There's an answer that answers my exact question, and there's a different answer that helps me win/gain in the game, but considers slightly altered initial conditions.

Replayability (and question re-usability)-wise altered conditions are reasonably common and
based on answer to related question I'd rather accept a win-the-game answer.

So, should I accept 'formally correct' answer, a beneficial answer to slightly generalized question or re-word the question title to be more general?

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Accept the answer that answers your exact question as it is the answer to the question you're asking.

From what I gather you're suggesting that this other answer would also answer the question but isn't as specific to your problem. The idea behind picking an answer as "the answer" is that it solves the solves the problem being asked not every potentially related problem.

Now I understand the concern of what happens to the more generalized answer and people potentially not seeing at as "the answer" but what's nice about Stack Exchange sites is the community can indicate that something, although it isn't the answer to the question is still useful information pertaining to the problem via upvotes. Sure it may not be "the answer" but any user who has the same problem as you did and stumbles upon this question will still be able to see the solution to your problem as well as the generalized advice too.

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Its upto YOU to choose which answer to accept.

Personally, with 2 or more useful answers I would:

  1. Upvote all of them
  2. Accept the one that actually answers my current question and is also widely applicable to understanding similar scenarios
  3. Comment on other posts that answer my posted question (not those that are useful & I upvoted, but only helpful for related but not-quite-same scenarios) thanking them that I found it useful too.

there's a different answer that helps me win/gain in the game, but considers slightly altered initial conditions

It seems from your description that this different answer doesn't answer your actual question. If so, then just upvote it but accept the other one. If it did answer your question but was more general, then I'd prefer accepting that one.

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