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

TL;DR

This paper explores methods to find voting systems that closely match a desired power distribution, using exact and heuristic approaches for the Penrose-Banzhaf index in small voting bodies.

Contribution

It introduces enumeration and integer linear programming techniques for the inverse power index problem and compares their effectiveness to simple heuristics.

Findings

Exact methods outperform heuristics in accuracy.

Heuristics perform well but can be significantly improved.

Results inform termination criteria for local search algorithms.

Abstract

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.