Evaluating Influence in Weighted Voting Systems

The dynamics of decision-making are often influenced by the distribution of power, particularly in systems where votes have different weights. In this Apply It, we will delve into the concept of weighted voting, exploring how it differs from equal voting and how it can significantly impact outcomes in various settings. Through a scenario involving a company’s board of directors, you will apply the Banzhaf power index and the Shapley-Shubik power index to understand and evaluate the influence of each voter in a weighted voting system. This exercise aims to provide practical insights into these mathematical concepts and their real-world applications.

Consider a board of directors at an international environmental organization, GreenFuture, where strategic decisions are made through weighted voting. The board consists of five diverse members: Amina (voting weight [latex]4[/latex]), Raj (voting weight [latex]3[/latex]), Carlos (voting weight [latex]3[/latex]), Yuna (voting weight [latex]2[/latex]), and Kwame (voting weight [latex]1[/latex]). A minimum of [latex]7[/latex] votes is required to pass any decision. Currently, the board is deciding on a significant investment in a global renewable energy project. Your task is to assess the power each member holds in this crucial decision using the principles of weighted voting.