Reconstructing a simple game from a uniparametric family of allocations

Several relationships between simple games and a particular type of solutions for cooperative games are studied in this paper. These solutions belong to the set of semivalues and they are related to a unique parameter that explicitly provides their weighting coefficients. Through the allocations offered by this family of solutions, so-called binomial semivalues, and also from their respective potentials, some characteristics of the simple games can be recovered. The paper analyzes the capacity of binomial semivalues to summarize the structure of simple games, and, moreover, a property of separation among simple games is given. Copyright Sociedad de Estadística e Investigación Operativa 2013