Optimising the decision threshold in a weighted voting system: The case of the IMF's Board of Governors
Dóra Gréta Petróczy
TL;DR
The study models the IMF Board as a weighted voting game with weights $w_i$ tied to member quotas and explores how the decision quota $q$ affects the alignment between weights and a priori voting power $\beta_i$ measured by the Banzhaf index. Using Binary Decision Diagrams to compute indices across $q$ in the range $[50\%,87\%]$, it evaluates alignment with $d_{\text{euc}}$, $d_{\text{man}}$, and $\omega(\beta,w)$, along with inequality metrics (Gini, HHI) and the PTA measure of decisiveness. The key finding is that a decision quota of about $58\%$ (and around $60\%$ under certain criteria) minimizes the disparity between weights and powers, with similar stabilization in inequality, implying threshold design is crucial alongside weights for fair representation. This quota-sensitive analysis provides a practical guideline for IMF governance reform, emphasizing that adjustments to the decision threshold can substantially influence member influence and equity.
Abstract
In a weighted majority voting game, the players' weights are determined based on the decision-maker's intentions. The weights are challenging to change in numerous cases, as they represent some desired disparity. However, the voting weights and the actual voting power do not necessarily coincide. Changing a decision threshold would offer some remedy. The International Monetary Fund (IMF) is one of the most important international organisations that uses a weighted voting system to make decisions. The voting weights in its Board of Governors depend on the quotas of the 191 member countries, which reflect their economic strengths to some extent. We analyse the connection between the decision threshold and the a priori voting power of the countries by calculating the Banzhaf indices for each threshold between 50% and 87\%. The difference between the quotas and voting powers is minimised if the decision threshold is 58% or 60%.