Secure function evaluation (SFE) enables a group of players, by themselves, to evaluate a function on private inputs as securely as if a trusted third party had done it for them. A completely fair SFE is a protocol in which, conceptually, the function values are learned atomically. We provide a completely fair SFE protocol which is secure for any number of malicious players, using a novel combination of computational and physical channel assumptions. We also show how completely fair SFE has striking applications to game theory. In particular, it enables “cheaptalk” protocols that (a) achieve correlated-equilibrium payoffs in any game, (b) are the first protocols which provably give no additional power to any coalition of players, and (c) are exponentially more efficient than prior counterparts. Categories and Subject Descriptors: F.0 [Theory of Computation]: General General Terms: Theory, Security, Economics