A voting with absenteeism game is deﬁned as a pair (G;r) where G is an n-player (monotonic) simple game and r is an n-vector for which r i is the probability that player i attends a vote. We deﬁne a power index for such games, called the absentee index. We axiomatize the absentee index and provide a multilinear extension formula for it. Using this analysis we re-derive Myerson’s (1977, 1980) “balanced contributions” property for the Shapley-Shubik power index. In fact, we derive a formula which quantitatively gives the amount of the ‘balanced contributions” in terms of the coeﬀicients of the multilinear extension of the game. Finally, we deﬁne the notion of substitutes and complements in simple games. We compare these concepts with the familiar concepts of dummy player, veto player, and master player.
Quint, Thomas and Shubik, Martin, "Absenteeism, Substitutes, and Complements in Simple Games" (2003). Cowles Foundation Discussion Papers. 1723.