Coalition-proof Nash equilibria and cores in a strategic pure exchange game of bads

We present an example of a strategic game in which the major solutions allowing coalitions always exist. Specifically, we consider the pure exchange game due to Scarf [Scarf, H.E., 1971. On the existence of a cooperative solution for a general class of n-person games. Journal of Economic Theory 3, 169–181] with all the commodities being replaced by bads. It is shown that the coalition-proof Nash equilibrium describes the behavior with every player dumping all the initial bads onto just one next player unilaterally. On the other hand, the α-core also exists and coincides with the β-core without any convexity assumptions. If, in particular, the bads are of one type, any coalition-proof Nash equilibrium reduces to a strong Nash equilibrium; and every player retaining all the initial bads is in the α-core if and only if the distribution of the initial bads is moderate without too ‘big’ players.