The main assumption in cooperative game theory is that the grand coalition formula_1 will form. The challenge is then to allocate the payoff formula_53 among the players in some fair way. (This assumption is not restrictive, because even if players split off and form smaller coalitions, we can apply solution concepts to the subgames defined by whatever coalitions actually form.) A "solution concept" is a vector formula_54 that represents the allocation to each player. Researchers have proposed different solution concepts based on different notions of fairness. Some properties to look for in a solution concept include: