Consistency and the core in games with externalities
The starting point in the author's considerations is a generalization of the classical transferable utility games, called \textit{games with externalities}. Their structure is the following: Let \(N\) be a finite \textit{player set}. A \textit{coalition} \(S\) is any subset of \(N\). Let \(\Pi(S)\) be the set of all partitions of a coalition \(S\). For a coalition \(S\) and a partition \({\mathcal P}\in \Pi(N\setminus S)\), an \textit{embedded coalition} is defined as the pair \((S,{\mathcal P})\). Let \(EC(N)\) be the set of all embedded coalitions of \(N\). A \textit{partition function form game} is a pair \((N,v)\) where a \textit{partition function} \(v\) is a function that assigns a real number to each embedded coalition, that is, \(v: EC(N) \rightarrow \mathbb{R}\). Let \(\Gamma\) be a subset of the set of all partition function form games. Now a \textit{solution on} \(\Gamma\) is defined as any function \(\sigma\) that associates a subset \(\sigma(N,v)\) of \(F(N,v)\) with every game \((N,v)\in \Gamma\), where \(F(N,v)\) is the set of \textit{payoff vectors} of the form \(F(N,v) = \{x\in \mathbb{R}^N | \sum_{j\in N}x_j \leq v(N,\{\emptyset\})\}\). In the context of games with externalities, the author considers some version of a reduced game called the \textit{complement-reduced game}, and notions of a \textit{subset consistency}, \textit{complement-consistency}, and of a \(\psi\)-\textit{core}. In terms of the first three notions, he gets several results describing different properties of a solution \(\sigma\) on \(\Gamma\) and of the \(\psi\)-\textit{core}. The paper also contains a wide discussion about other versions of reduced games, and about several related results from the literature.
scientific article; zbMATH DE number 6858290