André Casajus

• An extension operator assigns to any TU game its extension, a mapping that assigns a worth to any non-negative resource vector for the players. It satisfies three properties: linearity in the game, homogeneity of extensions, and the extension property. The latter requires the indicator vector of any coalition to be assigned the worth generated by this coalition in the underlying TU game. Algaba et al. (2004, Theor Decis 56, 229-238) advocate the Lovász extension (Lovász, 1983, Mathematical Programming: The State of the Art, Springer, 235-256) as a natural extension operator. We show that it is the unique extension operator that satisfies two desirable properties. Resources of players outside a carrier of the TU game do not affect the worth generated. For monotonic TU games, extensions are monotonic. Further, we discuss generalizations of the Lovász extension using CES production functions.