Upper and lower bounds of the kernel and nucleolus (Q1105498)

The author investigates the classical n-person monotonic game (N,v). Let \(m(i)=\min \{v(S\cup \{i\})-v(S):\) \(i\not\in S\), \(S\neq \emptyset \}\); \(z_ i=v(N)-v(N-\{i\})+n^{-1}(v(N)-\sum_{j\in N}(v(N)-v(N-\{\) \(j\})\))); \(L=\{x:\) \(x_ i\geq m(i)\), \(i\in N\}\); \(U=\{x:\) \(x_ i\leq m(i)\), \(i\in N\}.\) Many interesting results are proved about the relationships between the sets L, U, C (core), K (kernel) and N (nucleolus). For instance, it is proved that \(K\subset L\cup int(C)\), i.e., the vector \(\{\) m(i)\(\}\) is a lower bound of the kernel outside the interior of the core; \(U\subset C\); if \(z_ t\leq m(i)\), then \(K\cap U=\{z\}\); if \(z_ t\leq m(i)\), then \(N=z\), i.e., in this case the vector \(\{\) m(i)\(\}\) is an upper bound of the nucleolus.