Working Paper, Brown University, Department of Economics 2008-16
We first observe that two of Maskin's results do not extend beyond three players: we construct a four-player partition function with nonpositive externalities whose unique solution is inefficient, as well as a four-player characteristic function that has a unique efficient solution for each ordering of the players, but for which the payoff vector obtained by averaging these solutions over the different orderings does not coincide with the Shapley value. On the other hand, we reinforce Maskin's insight that externalities may play a crucial role in generating inefficiency. Many existing solutions on how to share profits assume or derive the property of efficiency. Yet we argue that players may have an interest to choose with whom to bargain. We illustrate how this may trigger inefficiency, especially in the presence of externalities, even if bargaining among any group of agents results in an efficient distribution of the surplus they can produce. We also provide some sufficient conditions for efficiency.