A value for bi-cooperative games

Bi-cooperative games have been introduced by Bilbao et al as a generalization of TU cooperative games, where each player can participate positively to the game, negatively, or do not participate. In this paper, we propose a definition of a share of the wealth obtained by some players after they decided on their participation to the game. It turns out that the cost allocation rule does not look for a given player to her contribution at the opposite action to the one she chooses. The relevance of the value is discussed on several examples.