Linear symmetric rankings for TU-games

We define ranking as an equivalence relation on the set of power indices and study those that have a linear and symmetric representative. Moreover, we classify—or parametrize—those rankings that reward “positive” payoffs for “positive” participation. It is shown that these are in 1-1 correspondence with the points of the standard simplex. Moreover, this correspondence is convex. Finally, we contrast this classification with Saari–Sieberg’s approach via “positive” semi-values.