The Shapley transfer value without zero weights
Let N be a finite set of players, and let \({\mathcal N}:=2^ N\setminus \{\emptyset \}\) be the family of nonempty coalitions. Denote the unit simplex \(\{\) \(\lambda\in {\mathfrak R}^ N_+|\) \(\sum_{i\in N}\lambda_ i=1\}\) by \(\Delta^ N\), and its relative interior by icr \(\Delta\) \({}^ N\). For each \(S\in {\mathcal N}\), denote by \({\mathfrak R}^ S\) the subspace \(\{\) \(x\in {\mathfrak R}^ N|\) \(x_ i=0\), \(\forall i\in N\setminus S\}\). Define a non-sidepayment game as a correspondence (set- valued map) \(V: {\mathcal N}\to {\mathfrak R}^ N\) such that V(S)\(\subset {\mathfrak R}^ S\) for every \(S\in {\mathcal N}\). Given a non-sidepayment game V, one can construct a sidepayment game \(v_{\lambda}: {\mathcal N}\to {\mathfrak R}\) by \(v_{\lambda}(S):=\sup \{\lambda \cdot u|\) \(u\in V(S)\}\), for each \(\lambda \in \Delta^ N\). Let \(\phi\) \(v\in {\mathfrak R}^ N\) be the Shapley value of a sidepayment game v. A utility allocation \(\Phi\) (V)\(\in {\mathfrak R}^ N\) is called a \(\lambda\)-transfer value of a non- sidepayment game V, if there exists \(\lambda^*\in \Delta^ N\) for which \((\lambda^*_ i\Phi_ i(V))_{i\in N}=\phi v_{\lambda^*}\), \(\Phi\) (V)\(\in V(N)\). The first result of this paper establishes conditions on a non-sidepayment game V, under which there exists a \(\lambda\)-transfer value of V such that its associated \(\lambda^*\) is in icr \(\Delta\) \({}^ N\). A non-sidepayment game V is called cardinal- convex, if \(V(S)+V(T)\subset V(S\cup T)+V(S\cap T)\) for every S,T\(\in {\mathcal N}\). The second result of the paper is that a \(\lambda\)-transfer value of a cardinal-convex game, for which the associated \(\lambda^*\) is in icr \(\Delta\) \({}^ N\), is in the core of V. The third result of this paper axiomatizes the \(\lambda\)-transfer value correspondence, \(V\mapsto \{\Phi (V)\}\). See a related, independent work of \textit{R. J. Aumann} [Econometrica 53, 599-612 (1985), Exercise 2 (pp. 148-149)], in regard to the third result. See also a related, independent work of the reviewer [''Game theory for economic analysis'' (1983; Zbl 0522.90104)] in regard to the second result.
scientific article; zbMATH DE number 3910176
scientific article