Vol. 50, No. 2, 1974

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Kernels of compound games with simple components

Nimrod Megiddo

Vol. 50 (1974), No. 2, 531–555

The kernel is a solution concept for a cooperative game. It reflects symmetry properties of the characteristic function and desirability relations over the set of the players. Given m games over disjoint sets of players and an m-person game, one defines a compound game over the union of the m disjoint sets. These m games are the components and the above m-person game is called the quotient. The quotient may be treated as a game played by representatives of the component games.

The kernel of the compound game is characterized fully. The compound kernel is, in fact, a composition of the components’ kernels by means of a distinguished subset of the imputation space of the quotient game. This subset depends also on the number of veto players in each component.

An effective formula for the compound kernel is given for compound simple games. This formula enables short cuts in the computations leading to the kernel of a decomposable game. The results are applied to compound majority games and a complete description of their kernels is given.

Mathematical Subject Classification
Primary: 90D12
Received: 28 September 1972
Revised: 20 March 1973
Published: 1 February 1974
Nimrod Megiddo