Cooperative Game - Solution Concepts

Solution Concepts

The main assumption in cooperative game theory is that the grand coalition will form. The challenge is then to allocate the payoff 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 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:

• Efficiency: The payoff vector exactly splits the total value: .
• Individual rationality: No player receives less than what he could get on his own: .
• Existence: The solution concept exists for any game .
• Uniqueness: The solution concept is unique for any game .
• Computational ease: The solution concept can be calculated efficiently (i.e. in polynomial time with respect to the number of players .)
• Symmetry: The solution concept allocates equal payments to symmetric players, . Two players, are symmetric if ; that is, we can exchange one player for the other in any coalition that contains only one of the players and not change the payoff.
• Additivity: The allocation to a player in a sum of two games is the sum of the allocations to the player in each individual game. Mathematically, if and are games, the game simply assigns to any coalition the sum of the payoffs the coalition would get in the two individual games. An additive solution concept assigns to every player in the sum of what he would receive in and .
• Zero Allocation to Null Players: The allocation to a null player is zero. A null player satisfies . In economic terms, a null player's marginal value to any coalition that does not contain him is zero.

An efficient payoff vector is called a pre-imputation, and an individually rational pre-imputation is called an imputation. Most solution concepts are imputations.