  We study the complexity of the following problem: Given two weighted voting
games G' and G'' that each contain a player p, in which of these games is p's
power index value higher? We study this problem with respect to both the
Shapley-Shubik power index [SS54] and the Banzhaf power index [Ban65,DS79]. Our
main result is that for both of these power indices the problem is complete for
probabilistic polynomial time (i.e., is PP-complete). We apply our results to
partially resolve some recently proposed problems regarding the complexity of
weighted voting games. We also study the complexity of the raw Shapley-Shubik
power index. Deng and Papadimitriou [DP94] showed that the raw Shapley-Shubik
power index is #P-metric-complete. We strengthen this by showing that the raw
Shapley-Shubik power index is many-one complete for #P. And our strengthening
cannot possibly be further improved to parsimonious completeness, since we
observe that, in contrast with the raw Banzhaf power index, the raw
Shapley-Shubik power index is not #P-parsimonious-complete.
