Computing voting power in easy weighted voting games

Weighted voting games are ubiquitous mathematical models which are used in economics, political science, neuroscience, threshold logic, reliability theory and distributed systems. They model situations where agents with variable voting weight vote in favour of or against a decision. A coalition of agents is winning if and only if the sum of weights of the coalition exceeds or equals a specified quota. The Banzhaf index is a measure of voting power of an agent in a weighted voting game. It depends on the number of coalitions in which the agent is the difference in the coalition winning or losing. It is well known that computing Banzhaf indices in a weighted voting game is NP-hard. We give a comprehensive classification of weighted voting games which can be solved in polynomial time. Among other results, we provide a polynomial ($O(k{(\frac{n}{k})}^k)$) algorithm to compute the Banzhaf indices in weighted voting games in which the number of weight values is bounded by $k$.

💡 Research Summary

Weighted voting games (WVGs) are a fundamental model for collective decision‑making in many disciplines, from political science to distributed computing. In a WVG each player i carries a weight w_i and a coalition is winning if the sum of its members’ weights meets or exceeds a quota q. The Banzhaf index measures a player’s power by counting the coalitions in which the player is pivotal – i.e., the coalition switches from losing to winning when the player joins. Computing Banzhaf indices is known to be NP‑hard in the general case because it requires examining an exponential number of subsets.

The authors address this computational barrier by identifying structural restrictions that make the problem tractable. Their central contribution is a complete classification of “easy” subclasses of WVGs and a polynomial‑time algorithm for one of the most natural restrictions: the number of distinct weight values, denoted k, is bounded by a constant. When only k different weights appear, the n players can be partitioned into k groups, each group containing all players with the same weight. Let g_i be the size of group i and w_i the common weight. The problem then reduces to deciding how many players to pick from each group so that the total weight reaches the quota.

The algorithm builds a dynamic‑programming (DP) table DP