re: Who wins in a fight? Captain America or Batman?

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Batman. Bats would've already studied Cap and broke down ever element of his fighting style just on the 1/2 percent chance they would ever meet.

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Batman has a game plan for everyone. Batman has a gameplan to take out Robin and Alfred in case they turn. That's how planned out he is. Look at all the instances of Batman looking into the weaknesses of the JL.

Yes in a wrestling or boxing match that was sanctioned, Captain America would crush Batman.

If they were to fight any other way like Captain America getting his brain taken over by the Penguin or Batman got touched by Loki's spear, then in that scenario each would anticipate a fight with the other. Captain America will leap with out looking into any trap or scenario Batman plans out.

In any scenario in which Batman anticipates any foe (even many God-like powered villains) he wins.

I think Batman has this.
Unless America get the jump on him. But the Captain was always more direct into battle, which plays up the Batman's tactics.