COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 1447

Absenteeism, Substitutes, and Complements in Simple Games

Tom Quint and Martin Shubik

December 2003

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.

Keywords: Simple game, Shapley-Shubik power index, Absenteeism, Multilinear extension, Balanced contributions, Substitute, Complement

JEL Classification: C7, C71, D72
AMS Classification: 91A23, 91B12