This isn't an easy question because there are so many possibilities. The main thing to worry about when choosing a background for your actor's headshot is to think about whether or not there are distracting elements present.
Ideally, you want a background that draws little attention to itself, and there are only two things that lead to this result: a flat white background or a dark background.
Having a very dark background (not black) is good because dark colors don't demand attention. The brightest part should be the actor's face so that attention is immediately drawn there.
On the other hand, a flat white background is great because flat white is looked at as non-existent, meaning it feels as if there is no background. Therefore, the viewer's attention will be fully drawn to the actor in the photo. Just make sure the background isn't too bright that it "spills" onto the actor.
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