Alternative representation of semivalues, the inverse problem and coalitional rationality

The concept of semivalue of a transferable utility game has been introduced by Dubey, Neyman and Weber as weighted sum of marginal contributions. Later, Puente has introduced a particular class of semivalues, called binomial semivalues, where weights are obtained through a recursive procedure. In the present paper, we extend Puente's procedure to obtain an equivalent representation of semivalues that turns out to be useful to solve the inverse problem and the question of coalitional rationality.