Weakly balanced contributions and solutions for cooperative games

We explore a relaxation of the balanced contributions property for solutions for TU games that requires the direction (sign) of one player's change of payoffs when another player leaves the game to equal the direction (sign) of the latter player's change of payoffs when the former leaves the game. There exists a large class of solutions that satisfy both efficiency and this weak balanced contributions property. The Shapley value is the unique solution that also obeys weak differential marginality.