I managed to find this. There is a noticeable tendency for proposers to keep more of the money if they have earned it. This is especially pronounced in the Dictator Game, but also exists in the Ultimatum Game.

Although I can't recall where I got it from, and Google is failing me, I'm pretty sure there's a body of experimental evidence along these lines, showing that the second player is overwhelmingly more likely to accept an unfair split if the roles are designated in a way you describe.