According to the Sharpie company, a permanent marker is permanent if it contains dyes or pigments, adheres to most surfaces and it water resistant. That last requirement pretty much rules out trying to remove permanent marker stains from rubber by scrubbing them with soap and water. Fortunately, there are ways to do it that are just as easy, so the face of your child's doll doesn't have to remain permanently disfigured.
Among the solvents you can use to remove marker stains are:
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Procedure for Removing Stains
Apply a liberal amount of rubbing alcohol, acetone or commercial stain remover to a clean cloth. Use nail polish remover in place of acetone, as long as the label indicates it is acetone-based. You can also use hair spray -- it contains alcohol as a solvent.
Test the solvent -- especially acetone -- on an inconspicuous part of the surface you're cleaning before using it. It may dull the finish or even dissolve some rubber and plastic amalgams.
Rub the marker stain vigorously. It may take more than one application of solvent; change the cloth if it gets discolored.