The Mathematics of Democracy: From Choice Theory to Practice

This article provides an in-depth analysis of the intersection of mathematics and political science, focusing on social choice theory. The author explains why an ideal electoral system is mathematically impossible, citing the fundamental theorems of Arrow, May, and Condorcet. The text provides a detailed discussion of systemic manipulation mechanisms, such as strategic voting and gerrymandering, and tools for measuring electoral fairness, including the Gallagher index. The reader will also learn about advanced models of institutional engineering, including Penrose's square root law and the Polish proposal for reforming the EU voting system—the Jagiellonian Compromise. This is a compendium of knowledge on how algorithms and vote counting rules actually impact the division of power and state stability.