Summary: Can we always design a game form so that all its Bayesian equilibrium outcomes are either interim individually rational efficient or interim envy-free efficient in asymmetric information economies? We show that the answer is no. Given that such games exist in complete information economies, can this gap be filled by games that leak information endogenously without enlarging the equilibrium set? We adopt two approaches. First, we introduce a theory of `cheap play' as a natural model of such games. The second is an extreme approach that adopts an ex post viewpoint. Neither yields a possibility result. Information leakage does not recover efficiency.