Cooperative Games in Game Theory
In cooperative game theory, a coalition refers to a group of players who agree to work together to achieve a common goal. The formation of coalitions is a key feature of many cooperative games.
When forming a coalition, it is important for the members to decide how to divide the payoff among themselves. There are different ways to do this, depending on the specific game and the goals of the players. One common approach is to use the Shapley value, which we will discuss in more detail in a later lesson.
Another approach is to use bargaining solutions, such as the Nash bargaining solution. This solution provides a way for the members of a coalition to divide the payoff in a way that is fair and efficient. The Nash bargaining solution is based on the idea that the members of the coalition will bargain over the distribution of the payoff, and each member will have a bargaining power that reflects their contribution to the coalition.
In some cases, the members of a coalition may not be able to agree on how to divide the payoff. This is known as a bargaining failure. In such situations, it may be necessary to use an external mechanism, such as a mediator, to help the members reach an agreement. Alternatively, a game may have a unique solution that does not require bargaining, such as the core solution or the nucleolus.
Overall, coalition formation and payoff distribution are important concepts in cooperative game theory because they allow players to work together to achieve a common goal. By understanding these concepts, we can better analyze and design cooperative games that are fair and efficient.
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