Marginalist and efficient values for TU games (Q1303881)

Single-valued solutions (values) of cooperative games in the characteristic function form are considered. The author develops an axiomatic approach. For \(n\)-person games with \(n\geq 3\) the following theorems are proved. Theorem 1 (1). A marginalist, efficient, and monotonic value possessing the null-player property is additive, covariant, and random-ordered. Theorem 1 (2). A marginalist, efficient, and continuous value possessing the null-player property is additive and covariant. Theorem 2. The only set of marginalist, efficient, and monotonic values possessing the null-player property is the set of random-ordered values. Theorem 3. The only set of marginalist, efficient, monotonic, and covariant values is the set of random-ordered values. Theorem 4. All the axioms in the hypothesis of both Theorem 2 and Theorem 3 are independent. The proof of Theorem 4 is provided by appropriate examples, including the case of \(n=2\). Two lemmas are proved to provide the proofs of theorems 1 and 3. A good historical survey is given which shows the directions of a generalization of the Shapley value.