  We study incentive-compatible mechanisms that maximize the Nash Social
Welfare. Since traditional incentive-compatible mechanisms cannot maximize the
Nash Social Welfare even approximately, we propose changing the traditional
model. Inspired by a widely used charging method (e.g., royalties, a lawyer
that charges some percentage of possible future compensation), we suggest
charging the players some percentage of their value of the outcome. We call
this model the \emph{percentage fee} model.
  We show that there is a mechanism that maximizes exactly the Nash Social
Welfare in every setting with non-negative valuations. Moreover, we prove an
analog of Roberts theorem that essentially says that if the valuations are
non-negative, then the only implementable social choice functions are those
that maximize weighted variants of the Nash Social Welfare. We develop
polynomial time incentive compatible approximation algorithms for the Nash
Social Welfare with subadditive valuations and prove some hardness results.
