Cooperative Games in Game Theory
The Core solution is a solution concept in cooperative game theory that identifies the set of allocations that cannot be improved by any coalition of players. It is a concept that measures the stability of a game. In other words, it defines the allocation of payoffs that is the most secure against deviations by subgroups of players.
The core solution can be defined as the set of feasible payoff vectors, such that no coalition of players can improve their payoff by leaving the grand coalition and forming a new coalition among themselves. In a game with transferable utility, the core is non-empty if and only if the game is balanced, meaning that the total worth of the grand coalition is divided equally among all its members.
For instance, consider a cooperative game with three players. Suppose that the worth of the grand coalition is 9. If the players could form a coalition among themselves and get a total worth of 10, then the allocation is not in the core. If, on the other hand, the players could only form a coalition and get a total worth of 8, then the allocation is in the core, as no coalition can improve upon it.
The core solution has many applications, including in the study of market power and oligopoly. It can also be used to determine the minimal amount of compensation required to keep all players in the grand coalition.
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