Coalition-proof nash equilibria and the core in three-player games

We study 3-person noncooperative games of coalition formation where the underlying situation is represented by a game in coalitional form without side payments. We look at coalition-proof Nash equilibria and we show that if the underlying game is balanced (in the sense of Scarf), then, except for indifferences, the grand coalition forms, and the payoff is in the core. If the underlying game has an empty core, then only a two-player coalition can form, and the payoff to its members is given by the respective coordinates of a unique "outside-options" vector. If the underlying game is not balanced but has a nonempty core, then either one of the two mentioned cases may hold.