Equilibrium Coalition Structures in Three-Player Symmetric Games

In symmetric games with externalities across coalitions, we investigate how three players form coalitions using two solutions: ${\mathbf{n}}^{\ast }$, which is a focal prediction of coalition structure in a class of noncooperative coalitional bargaining games, and equilibrium binding agreements, which represent the cooperative blocking approach. We find that the coarsest equilibrium coalition structure (based on the latter notion) is never finer than ${\mathbf{n}}^{\ast }$, and we provide a sufficient and necessary condition for these two solutions to generate the same coalition structure (i.e., the two solutions coincide if and only if the first coalition to form in ${\mathbf{n}}^{\ast }$ is not a two-player coalition or a particular condition about average coalitional worths is satisfied). In symmetric games with more than three players, we demonstrate through a series of examples that any relationship between these two solutions is possible. We also discuss symmetric games with positive externalities or equal division in which these two solutions coincide.