It's entirely possible Legolas decided to forego Shot On The Run, but with Point Blank Shot, Rapid Shot, and however many other feats he possesses, this should be a two, three round encounter at most for the elf. /Nerd
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Though because of the no look shot, Barton obviously has a +8 compound bow, blind fighting, a high BAB, and the third party feat that gives a Power Attack like ability on ranged attacks.
Legolas also blew some feats on Two Weapon fighting, making him less effective in that sense. But he has too high of an AC
Clint could beat Optimus Prime with an arrow up the spark chamber.
Hawkeye wins
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Legolas: massive speed and agility advantage, advantage in H2H and melee combat, and he has feats from the Hobbit that go from borderline impossible to defying the laws of physics (river barrel chase scene and final fight with Bolg).
Agree with Darth Thor, Hawkeye gets the top accuracy award. His no look Chitauri speeder bike shot and the USB port shot were great, and, especially, the SHIELD helicarrier shot was insane. Legolas also had stuff like the long shot to the orc's skull in the Mines of Moria in Fellowship, the Warg rider shots in Two Towers, and the no look whilse jumping lateral head shot in Desolation, though.
Hawkeye might be a more effective force on the battlefield but Legolas just counters him.
I'm not sure they can both dodge succesfully and aim correctly at the same time. So I figure they both miss the first arrow. Then Legolas simply "reloads" faster. Now if Hawkeye had a pillar or some cover and so did Legolas I think Hawkeye would win. But open space is a win for Legolas.