A recursive measure of voting power that satisfies reasonable postulates

The classical measures of voting power are based on players' decisiveness or full causal efficacy in vote configurations or divisions. We design an alternative, recursive measure departing from this classical approach. We motivate the measure via an axiomatic characterisation based on reasonable axioms and by offering two complementary interpretations of its meaning: first, we interpret the measure to represent, not the player's probability of being decisive in a voting structure, but its expected probability of being decisive in a uniform random walk from a vote configuration in the subset lattice (through which we represent the voting structure); and, second, we interpret it as representing a player's expected efficacy, thereby incorporating the notion of partial and not just full causal efficacy. We shore up our measure by demonstrating that it satisfies a set of postulates any reasonable voting measure should satisfy, namely, the iso-invariance, dummy, dominance, donation, minimum-power bloc, and quarrel postulates.