André Casajus

• We provide a concise characterization of the class of positively weighted Shapley values by three properties, two standard properties, efficiency and marginality, and a relaxation of the balanced contributions property called the weak balanced contributions property. Balanced contributions: the amount one player gains or loses when another player leaves the game equals the amount the latter player gains or loses when the former player leaves the game. Weakly balanced contributions: the direction (sign) of the change of one player's payoff when another player leaves the game equals the direction (sign) of the change of the latter player's payoff when the former player leaves the game. Since the (symmetric) Shapley value is characterized by efficiency and the balanced contributions property and satisfies marginality, we pinpoint position of the Shapley value within the class of positively weighted Shapley values to obeying the balanced contributions property versus just obeying the weak balanced contributions property.