Game Theory Question

Here is one that my friend Paul and I were talking about:
Two Players
10 Rounds
Each round, Player 1 is given five dollars. He can elect to give Player 2 as much of the five dollars as he wants. For every dollar that 1 gives, 2 gets three bucks. Player 2 can then give back as much of the money as he wants. What is the equilibrium in this game? We used basic backwards induction and figured that Player 1 would keep the $5 every round. We aren't certain on this, however, and Paul says that he's read that our equilibrium doesn't hold in tests of this game.