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In cooperative game theory, the payoffs for the grand coalition can be distributed in a number of ways. Each of those ways is a solution concept. Well-known examples of solution concepts include the core, the Shapley value, the nucleolus, and the kernel. Each of them are useful in their own way because they each satisfy certain important requirements.

I wonder which other solution concepts are interesting and why they are as such. Can you think of examples of solution concepts which I have not named that are studied? Why are they important?

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There are many many solution concepts, and google can help you in finding some of them. Two that are missing from your list are the bargaining set and the Banzhaf value. Note that some solution concepts are point solutions, while others are set solutions.

This answers your first question. Regarding the second question: they are important because they are mathematically interesting. The significance of many mathematical objects is their mathematical beauty and the challenges their study poses. This is the same with solution concepts. True, one can argue that in specific cases of sharing a pie, the axiomatization of the various point solution concepts can help in identifying the right way to split the pie among the participants. This may include sharing power after elections, and sharing income by different operators of a networks. However, it is usually difficult to argue why one set of axioms better fits a specific problem than another set of axioms, and participants will often prefer the solution concept that benefits them. Some may also argue that set solution concepts help in identifying outcomes that will not be realized. Since each set solution concept has a different prediction, which one should we adhere to?

Thus, for me the relevance of this theory is its mathematical beauty.

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