
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.
Cooperative games, closure spaces, cooperative structure, restricted games, Shapley and Banzhaf values
Cooperative games, closure spaces, cooperative structure, restricted games, Shapley and Banzhaf values
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