Shapley values seem to be the standard answer to "how should a coalition split the rewards of their cooperation", but I'm curious about alternatives.
The standard characterization of Shapley values says that Shapley values are the unique coalition payments which satisfy a bunch of properties. Three of them (efficiency, symmetry, and null player) seem pretty necessary for any "reasonable" or "practical" coalition payment rule, but the last one (linearity) does not.
If I didn't care for linearity (or its close synonyms, additivity and aggregation):
- What sorts of payment rules become available?
- What other properties of Shapley values are maintained?
- What other properties would produce a uniquely characterized payment rule?
In that case, you may be interested in the pre-kernel/pre-nucleolus. These solutions do not satisfy neither linearity nor additivity nor aggregation. The pre-nucleolus is a single-valued solution like the Shapley value that satisfies in addition covariance, anonymity, and the reduced game property.
For more information, have a look into the book of B. Peleg, and P. Sudhölter (2007), Introduction to Theory of Cooperative Games. Springer Publisher.
Book Info here