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

Heuristic and exact solutions to the inverse power index problem for small voting bodies