Cooperative game theorists propose the following attractive process: (1) capture the abstract value of each possible coalition of individuals, (2) write down some principles, or axioms, on how to distribute the value, and then, (3) find a set of allocations that satisfy the principles. The Shapley value has received much attention – but it is just one solution concept, satisfying one set of principles, in one class of games. In our work, we contrast the Shapley value with solution concepts in the class of multi-choice games (MC-games), which extends the class of transferrable-utility games to cases where features or parameters have levels of participation. These MC-game solutions are model agnostic, and unique to their own set of axioms, just like the Shapley value. We demonstrate in a simple setting in the context of polynomial regression that three MC-game solution concepts can outperform the Shapley value. Our work offers a general algorithm for constructing any MC-game framework with polynomial time complexity in the number of parameter levels, and an application of this algorithm that is transparent, and can be used to generalise local explanation frameworks such as SHapley Additive exPlanations (SHAP).
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