The Solidarity Value as a Probabilistic Solidarity Value

Abstract

Our paper focuses on two evolutions in the Shapley value: on the one hand, Radzik and Nowak’s solidarity value (in short s-value) allows the needy to receive solidarity help from privileged players; on the other hand, Weber’s concept of probabilistic value provides a setting where the probability of joining a coalition depends both on the player being picked to join and on the coalition being constituted, unlike for the Shapley value which assumes equal chance to all players still waiting aside. In a context where the solidarity of Nowak and Radzik is the guiding policy for profit sharing, what is the meaning of this Weber’s idea of probabilistic value? We come out with a large class of values, called Probabilistic solidarity values ( ps-values ), that comprises the s-value. Our two main results are a characterization of Radzik-Nowak’s solidarity value as a specific Probabilistic solidarity value and a characterization of symmetric ps-values which exhibit a rational intuitive property found in literature on the per-capita productivity.