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Paths and consistency in additive cost sharing. (English) Zbl 1098.91012
Summary: We provide a direct proof of a representation theorem for additive cost sharing methods as sums of path methods. Also, by directly considering the paths that generate some common additive cost sharing methods (Aumann-Shapley, Shapley-Shubik, and Serial Cost) we show that they are consistent. These results follow directly from a simple sufficient condition for consistency: being generated by an associative path. We also introduce a new axiom, dummy consistency, which is quite mild. Using this, we show that the Aumann-Shapley and Serial Cost methods are the unique (additive) consistent extension of their restriction on all two agent problems, while the Shapley-Shubik method has multiple consistent extensions but a unique anonymous scale invariant one.

MSC:
91A12 Cooperative games
91B32 Resource and cost allocation (including fair division, apportionment, etc.)
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