Axiomatization of some power indices in voting games with abstention

Power measures are used to quantify the influence of members of a democratic institution. We consider voting games with abstention or (3,2) games, which are decision-making processes in which voting options include yes, no and abstention. The power indices that we study are based on the notions of minimal and shift minimal winning tripartitions. We define and characterize the Deegan–Packel and shift Deegan–Packel power indices in the class of (3,2) games. Furthermore, owing to the parameterization result obtained by Freixas et al. (Discret Appl Math 255:21–39, 2019), we provide computational formulae of these indices in the class of I-complete (3,2) games. These formulae allow us to determine the power of each player in a game, regardless of the number of minimal and/or shift minimal winning tripartitions of the game.