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DC, barely. Batman (assuming Bruce) and Captain America are pretty even. I give the slight edge to CA. Elektra in a slight edge over Shiva. But Nightwing and Canary are both outright better than Psylocke and Widow IMO.
In terms of hth skills, I think they are very close. Batman has mastered every form of fighting on earth, even ones lost in time, he did fairly good against 3boot Val, while fighting in the air and even surprised Val that he learned so quickly.
Captain America has been stated on panel to be a master of every known martial art. Batman has not. Batman has been stated on panel know 126 martial arts, which isn't the same as every known. Also Cap is physically superior on top of his skill edge so it isn't hard to figure out how that fight will end up landing.
Here they are from first to last in order of fighting skill:
1. Captain America / Elektra
2. " "
3. Shiva
4. Psylocke
5. Batman
6. Nightwing
7. Black Canary
8. Black Widow
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I would put Canary on the same tier as Nightwing, but otherwise agreed. I'd also put Psylocke on top with the other two because of the Ogun stuff, but I think she lacks the h2h feats to back that up.
Marvel win, because of Elektra and Captain America:
I think that on paper, overall, DC is more skilled, but Elektra, IMO, is the most skilled on the field.
And the clincher:
Than, you have Captain America. Skill wise, I put Batman over Cap, but Cap is metahuman: He doesn't fatigue (or do it slowly), and he's peak human in everything, but in strenght, where he's above human.
So, it's Batman with better moves but not that much vs Batman on steroids. And after one or two hour of fighting, even Batman will fall of fatigue. Cap's not.
I don't think Betsy is quite there yet with Cap and Elektra, but I could see bumping her up to or slightly above Shiva. Also I count Grayson Batman as Nightwing. Now, I would agree that Canary was above Dick when he was still Nightwing, but IMO he got a skill boost when he dawned the cowl.
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