Equal support from others for unproductive players: efficient and linear values that satisfy the equal treatment and weak null player out properties for cooperative games

In cooperative games with transferable utilities defined on a variable set of players, we characterize the family of values satisfying efficiency, linearity, the equal treatment property, and the weak null player out property. The last property weakens the usual null player out property, and together with efficiency, it is interpreted as considering equal support from others for null players. Together with the fact that efficiency, linearity, and the equal treatment property characterize the Shapley value along with the usual null player out property, our result reveals how weakening the null player out property can expand the possibilities of solutions. The characterized family contains well-known values in the literature, such as the Shapley value, equal division value, equal surplus division value, and the egalitarian non-separable contributions value, etc. In addition, each value in the characterized family is determined by an infinite sequence of real numbers. Furthermore, the equal treatment property in our characterization can be replaced by the balanced contributions property for symmetric players. Comparing this result with the existing one also shows that how weakening the balanced contributions property more can expand the possibilities of solutions.