Closure spaces and restricted games (Q1298732)

Consider a cooperative game \((N,v)\) with transferable utility, where \(N\) is finite. A game with a cooperation structure is the game \((N,v)\) together with a cooperation structure \( P^* \) which associates with each coalition \(S\subset N,\) a partition \( P^*(S)\) of \(S.\) The author considers the restricted game \((N,v^ {P^*}) \) where \[ v^{ P^*}= \sum _{S_{i}\in P^*(S)}v(S_i),\quad S\subseteq N \] as defined by \textit{R. J. Weber} in the following paper: Aumann, R. J. et al., Handbook of Game Theory II, 1285-1303 (1994; Zbl 0925.90093). The cooperation structure considered by the author satisfies additional properties and is a closure system. For such a cooperation structure, the author studies the restricted game and develops the relation between the dividends of Harsanyi in the restricted game and the worth function in the original game. A method of computing the Shapley and Banzhaf values of the closure restricted games is also obtained.