Introduction
Democracy is not just an ideal of freedom, but above all, a precise mathematics of procedures. Social choice theory teaches us that the way we count votes fundamentally shapes the political outcome. Understanding the mechanisms governing electoral systems allows us to move from a naive belief in a "single right method" toward the conscious design of systems that must balance stability and representativeness. In this article, we will examine the mathematical foundations and limitations that define modern states.
Efficiency vs. Fairness: The Axes of Social Choice
In social choice theory, three main axes of contention emerge: consistency (aggregating rankings), strategy (susceptibility to manipulation), and institutions (the influence of process architecture). A key lever here is the size of the electoral district. According to Duverger's law, small districts favor two-party systems, while large ones support pluralism. Choosing a method for converting votes—such as d’Hondt (favoring large players) or Sainte-Laguë (more proportional)—is a decision between government stability and faithfully reflecting the social mosaic. To measure these effects, we use the Gallagher index (disproportionality) and the Laakso–Taagepera index, which determines the effective number of parties.
Strategic Voting: A Barrier to Manipulation
Most systems allow for strategic voting, which is casting a vote contrary to one's preference to achieve a better outcome. The ideal of strategy-proofness is mathematically unattainable under conditions of pluralism. Designers must therefore manage incentives so that tactics do not undermine the legitimacy of power. A threat here is gerrymandering—manipulating district boundaries (using pack and crack techniques)—which can be prevented by independent commissions. Equally important is agenda control; McKelvey’s theorem proves that whoever sets the order of business shapes the final result. Today, these same challenges apply to algorithms in social media, which, by aggregating preferences, fall into the same traps as electoral systems.
Arrow's Theorem: The Mathematical End of Electoral Consistency
Arrow's theorem proves that with more than two options, no system exists that is simultaneously rational, consistent, and free from dictatorship. This makes the "will of the people" a procedural construct rather than a mathematical fact. In complex systems, such as the EU Council, it is crucial to distinguish between voting weight (nominal share) and real decision-making power. The Banzhaf and Shapley–Shubik indices allow us to measure the probability of being a "kingmaker"—both in politics and in joint-stock companies. The answer to these inequalities is the Jagiellonian Compromise, based on Penrose's square root law, which seeks to equalize the voting power of every citizen in supranational structures.
A Fair Electoral System: Three Minimal Institutional Conditions
The mathematics of democracy teaches us humility toward institutions. Although an ideal system does not exist, a fair electoral law must meet three minimal conditions: it must be predictable in its operation, verifiable in its outcome, and communicable to citizens. As John Rawls reminds us, justice is the first virtue of institutions. Everything else—the balance between stability and pluralism—is the domain of politics, practiced within the insurmountable boundaries set by the cold logic of numbers. Democracy is a constant "weighing of values," where every compromise has its quantifiable price.
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