EzekielRaiden
Follower of the Way
Not really, other than total number of combat rounds per rest (short or long.) I mean, in theory weapon die has an effect because crits are "roll 2x the dice," but this effect is very small due to Champions only getting a crit in 10% of cases where they would not otherwise have. Likewise for fighting style, and the only style which benefits is Great Weapon Fighting, because it affects the dice output. All others affect something other than the dice (hit bonus or static damage) which means being unaffected by crits. That is, if you go from a 1d8 one-handed weapon to a 2d6 greatsword, the difference in damage is 2.5 from damage dice (going from 4.5 to 7 average), and GWF bumps that up to 8.33... Meaning that, on average, per swing, fighting style is only adding 0.1×3.833... = 0.3833... extra points of damage per swing. Sure, it adds up over the course of many combat rounds. But it's gotta be a lot of combat rounds to close the gap, regardless of fighting style.I'm no expert in CharOp but i believe it depends of a few factors like weapon, fighting style, number of combat per adventuring day, lenght of combat etc...
I've crunched the numbers on this one. Except at very high level (15+), you need at absolute minimum 20 rounds of combat between short rests in order for the Champion to catch up to the Battle Master. You'd need even more to catch up to something like a Paladin. This either means having extremely long combats (e.g. two 10-round combats or three 7-round combats per short rest) or having extremely frequent ones (e.g. 5-7 combats again per short rest.)
It's a simple function of dividing the average damage bonus from BM dice by the average damage bonus per swing from Champion, and then dividing that result by the Fighter's total attacks per Attack. (This is because (Damage)/(Damage/attack) = # attacks, so if we divide by attacks/Attack we get # combat rounds.) Of course in doing this I did make some simplifying assumptions, but most of those assumptions specifically favor the Champion, not the Battle Master. E.g. it is generally more powerful to increase you hit rating rather than your damage dice due to the existence of Great Weapon Master, so this is a floor for what the Champion must reach, not a ceiling. I also ignored any benefits the BM might get from her natural 5% crit chance, because BMs should try to have a maneuver available to spend on any crits they may get, further pushing up the average.
The Champion extra damage from crits is laughably small at any level actually likely to be played.
Edit: Note that the above numbers DO assume a Champion wielding a greatsword with GWF. The numbers would be slightly less than doubled (about 80% larger) if I had instead used a 1d8 one-handed weapon.