Absenteeism, Substitutes, and Complements in Simple Games

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 ri 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.