Weak differential monotonicity and axiomatization of convex combinations of solutions to cooperative games

The \(\alpha \) $\alpha$ -egalitarian Shapley values are convex combinations of the Shapley value and the equal division value. For cooperative games that contain more than three players, Casajus and Yokote (Int J Game Theory 48:979–997, 2019) propose a characterization of the \(\alpha \) $\alpha$ -egalitarian Shapley values by efficiency, weak differential monotonicity, and the average dummy player property. However, it remains an open question whether the characterization holds true for games with three players or not. In this paper, we answer this question in the affirmative. Furthermore, we establish a new axiomatization of the \(\alpha \) $\alpha$ -equal surplus division values, which are convex combinations of the equal surplus division value and the equal division value.