Correlation and inequality in weighted majority voting games

In a weighted majority voting game, the weights of the players are determined based on some socio-economic parameter. A number of measures have been proposed to measure the voting powers of the different players. A basic question in this area is to what extent does the variation in the voting powers reflect the variation in the weights? The voting powers depend on the winning threshold. So, a second question is what is the appropriate value of the winning threshold? In this work, we propose two simple ideas to address these and related questions in a quantifiable manner. The first idea is to use Pearson's Correlation Coefficient between the weight vector and the power profile to measure the similarity between weight and power. The second idea is to use standard inequality measures to quantify the inequality in the weight vector as well as in the power profile. These two ideas answer the first question. Both the weight-power similarity and inequality scores of voting power profiles depend on the value of the winning threshold. For situations of practical interest, it turns out that it is possible to choose a value of the winning threshold which maximises the similarity score and the also minimises the difference in the inequality scores of the weight vector and the power profile. This provides an answer to the second question. Using the above formalisation, we are able to quantitatively argue that it is sufficient to consider only the vector of swings for the players as the power measure. We apply our methodology to the voting games arising in the decision making processes of the International Monetory Fund (IMF) and the European Union (EU). In the case of IMF, we provide quantitative evidence that the actual winning threshold that is currently used is sub-optimal and instead propose a winning threshold which has a firm analytical backing. On the other hand, in the case of EU, we provide quantitative evidence that the presently used threshold is very close to the optimal.