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The Fighting Initiate feat from Tasha's Cauldron of Everything allows anybody proficient with a martial weapon to pick up a Fighting Style option from the fighter class.

For someone who does not already have access to the Fighting Style and wants to maximize their weapon damage from a one-handed melee weapon, would it be better to pick this feat for Dueling, or to just take an Ability Score Increase for the relevant ability?

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In short, yes, but...

Just looking at the damage numbers, the ASI will obviously give you a +1,
and the feat – given that you fulfill the requirements – a +2.

In that sense, the fighting style wins.
But the ASI has more to offer than just a simple +1 to damage.

First of all, the +1 modifier will of course help for all other rolls with that ability. How useful that is will depend on what it is, and how important that is is up to you, but this has little to do with damage.

More importantly, the ASI will also give you a +1 to hit, and this is where the calculations get a bit more complicated. But overall, a +1 to hit is about a 5% increase to hit. But there are more advanced calculations. And unless the bonuses on the character makes him/her hit everything, an average increase of hit by 5%, would mean an average increase in damage applied by 5%.

How much 5% damage is, well that also depends on the build and level. But at 20 damage, that should even out (5% of 20 being 1).
Looking at this damage graph that means somewhere around level 10 and beyond this should be worth it – or somewhere between lvl 5 and 20 depending on the build.

In the end, it comes down to how much bonus you have to hit already, how high AC enemies you face, and most importantly, if you aren't already at level 20 in your main damage attribute – because then you've hit a limit which the fighting style will help you go beyond. On top of that, you could potentially change the fighting style if you hit another ASI, should the need for a different style arise.

In Summary

Pick ASI if: you miss as expected (about 25-50% of the time) and expect an average damage larger than 20.
This will give you better return on damage on average, plus a bonus to saves/checks/skills.

Pick Dueling if: you never miss, do little damage each attack and don't care about the bonus stats or already hit 20 in your ability.

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    \$\begingroup\$ Another pick dueling if: you have ruled out all the much better ways of causing more damage, such as multiclassing. \$\endgroup\$
    – SeriousBri
    Commented Nov 29, 2022 at 8:10
  • \$\begingroup\$ @SeriousBri Figured it was outside the scope of the question, but definitely worth pointing out. \$\endgroup\$ Commented Nov 29, 2022 at 11:34

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