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This question and the accepted answer are of the opinion that granting an extra feat to all PCs at level 1 does not affect game balance.

Would granting every PC the choice of a feat or a cantrip (as opposed to a feat only) be unbalancing?

I recognize that in some cases characters get racial cantrips, in which case, this house-rule extra cantrip would be in addition to racially granted ones.

Jack
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No.

Since the Feat Magic Initiate grants 2 cantrips, giving the choice of gaining a single cantrip instead would be worse than granting a Feat.

Destruktor
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    Excellent observation. With my mod, a character could take a free house-rule cantrip at level 1, then Magic Initiate instead of an ASI, and with the potential for racial and class cantrips, end up cantripy as heck. That would be alright. – Jack Dec 05 '18 at 17:23
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    @Jack If you are worried about a character with many cantrips you may be interested in checking this question – Sdjz Dec 05 '18 at 17:26
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    While true, this doesn't really answer the question. "Would it unbalance the game?" is the question, not "would anyone pick a cantrip over a feat?". – GreySage Dec 05 '18 at 18:20
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    @Jack You're missing the point... Given the choice between "a feat" and "a cantrip", nobody in their right mind would take the cantrip, because there is a feat that allows a more than just that. The existence of Magic Initiate indicates that a feat is more valuable than a single cantrip. Your game would become slightly unbalanced in favor of the people who took the feat. – T.J.L. Dec 05 '18 at 18:21
  • @Sdjz That's a good link. I'm good with all PCs having one more cantrip than they would otherwise, I'm just looking for what I might not have considered in terms of something that would somehow take something special some class or subclass or race or subrace. – Jack Dec 05 '18 at 18:23
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    @T.J.L. I see your point. Why take one cantrip, when you can have two? I suppose the only reason would be to take a cantrip at level one and then Magic Initiate as a feat at the right opportunity and get three. But good point. – Jack Dec 05 '18 at 18:26
  • @Jack I doubt anybody is going to be that desperate for that many cantrips and not just be a spell caster. I mean... I've made a character with a cantrip for every kind of damage, but it's really not that good a build. – T.J.L. Dec 05 '18 at 18:31
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    It might be good to edit some of what @T.J.L. is saying into your answer if you agree. There's good stuff there that could improve what you have currently. Also, I do think answering the title question is also a good idea. – Rubiksmoose Dec 05 '18 at 18:52
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    I feel like this isn't actually an answer to the question. just a problem with the question that should have really been a comment. – goodguy5 Dec 05 '18 at 19:28
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    One way I can see the single cantrip being a good alternative is if the player can choose the ability to use for casting instead of being restricted the casting ability of the chosen class for Magic Initiate. – Paul Pearce Dec 05 '18 at 20:34
  • @goodguy5 The answer to the question is "no", and thus proven ny a simple example. Doubtless there may be some edge case, but there is no simple way to evaluate how unbalanced a single cantrip gained at high level vs another feat or ASI.. – Destruktor Dec 06 '18 at 00:30
  • It might be worth clarifying the header, as there's another yes/no question in the body of the question that's phrased slightly differently. – V2Blast Dec 06 '18 at 01:35
  • @T.J.L. I think you're right. That makes it pointless to offer a cantrip in addition. – Jack Dec 06 '18 at 10:06