Coalition-proof Nash allocation in a barter game with multiple indivisible goods

We consider a barter market with multiple types of indivisible goods. Konishi et al. [J. Math. Econ. 35 (2001) 1-15] showed that the core of this market may be empty. We define a normal form game for this market and consider its equilibrium outcomes. Assuming separable strict preferences, we show that the game has a unique Coalition-proof Nash allocation. This Coalition-proof Nash allocation is precisely the unique commodity-wise competitive allocation of the market, and it may be inefficient. The commodity-wise competitive allocation is implementable in Coalition-proof Nash equilibria but not in Strong Nash equilibria.