Introduction
The health of democracy depends on a crucial technical detail: the electoral procedure. This is not just a clever saying, but the foundation of social choice theory. Changing the method of vote counting leads to different governments, coalitions, and policies. The core of democracy is the preference aggregation mechanism, not marketing. This article, based on mathematical analysis, explains how electoral rules, algorithms, and thresholds shape real power and the strength of our vote.
Electoral Systems: Three Main Families
Electoral systems can be divided into three main families: majoritarian, proportional, and mixed. Majoritarian systems operate on a "winner-takes-all" principle, which fosters stable governments and a two-party political landscape, as described by Duverger's Law. Their drawback is the phenomenon of "wasted votes," cast for losing candidates.
Proportional systems aim for a faithful representation of public support in parliament, using formulas like the d'Hondt method. However, their actual form depends more on electoral thresholds – both legal and natural – than on the algorithms themselves. While ensuring broad representation, they risk political fragmentation. Mixed systems attempt to combine the advantages of both approaches, balancing stability with representativeness.
Vote Power Exceeds Formal Weight
A key concept is vote power, which is the probability that our vote will be decisive. This differs from its formal weight. In a company where the threshold is 51%, a shareholder with 51% of shares has 100% power, while one with 49% has zero. However, the introduction of a third player with a 2% stake dramatically shifts the balance of power, making them a kingmaker in potential coalitions.
This dynamic is utilized in international bodies. Penrose's Square Root Law demonstrates that for the vote power of citizens from small and large states to be equal, the weight of a state's vote should be proportional to the square root of its population. The Jagiellonian Compromise proposal for the Council of the European Union, among others, was based on this principle.
Arrow's Theorem: The Illusion of Ideal Choice
Mathematics proves that no ideal voting system exists. Arrow's Impossibility Theorem shows that no system (for more than two options) can simultaneously satisfy several basic conditions of fairness, such as consistency of outcome or independence from "spoiler candidates." We must always compromise, sacrificing one of the desired characteristics.
This is illustrated by Condorcet's Paradox, where majority preferences can form a cycle (A>B, B>C, but C>A), meaning the outcome depends on the order of voting. Furthermore, the Gibbard–Satterthwaite theorem confirms that in complex systems, strategic voting is not a pathology, but an inevitable consequence of the rules of the game themselves.
Conclusion
Designing electoral systems is an art of balancing two fundamental values: the governability of the state and inclusivity, i.e., broad social representation. The evolution from drawing lots for officials in ancient Athens to modern algorithms is a continuous search for this balance. As Aristotle reminded us, the true art of governance lies in combining different orders to avoid both the tyranny of the majority and the rule of the few.
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