  Power indices are mappings that quantify the influence of the members of a
voting body on collective decisions a priori. Their nonlinearity and
discontinuity makes it difficult to compute inverse images, i.e., to determine
a voting system which induces a power distribution as close as possible to a
desired one. This paper considers approximations and exact solutions to this
inverse problem for the Penrose-Banzhaf index, which are obtained by
enumeration and integer linear programming techniques. They are compared to the
results of three simple solution heuristics. The heuristics perform well in
absolute terms but can be improved upon very considerably in relative terms.
The findings complement known asymptotic results for large voting bodies and
may improve termination criteria for local search algorithms.
