What is the difference between the voting paradox and the Arrow impossibility theorem?

The Condorcet Paradox, also recognized within the realm of social choice theory, surfaces when collective preferences among three or more options become cyclical and incoherent—even when individual voter preferences are orderly and transitive. To illustrate, in a group of voters where each has a clear preference ranking of A over B, B over C, and C over A, the group decision becomes inconsistent. This paradox underlines a fundamental challenge in designing social decision-making processes: the majority rule does not always yield a rational outcome for society as a whole. This realization helps us understand why constructing a fair and logical voting system is more complex than it might initially seem.

Social choice theory explores the methods and implications of aggregating individual preferences, judgments, and welfare to reach a collective decision or social welfare. It intersects economics, political science, and philosophy, examining how we can reconcile individual opinions to reach a decision that represents the group. Social choice theory involves devising rules and systems that reflect the society's overall wants and needs from the diverse individual preferences. It confronts numerous challenges, including the Condorcet Paradox and Arrow’s Theorem, which highlight the difficulties in creating a flawless system that translates individual preferences into a coherent group choice.

Rank-order electoral systems, also known as preference voting systems, enable voters to list candidates or options in order of preference. These systems aim to provide a more nuanced method for voters to express their choices. They are used to derive a winner based on the rankings, with various methods like instant-runoff voting, single transferable vote, and Borda count. Despite their attempt to accurately reflect voter preferences, Kenneth Arrow's theorem suggests the impossibility of creating a perfect rank-order voting system that adheres to certain fairness criteria.

The criterion of Non-Dictatorship is pivotal to Arrow's Impossibility Theorem. It mandates that no single voter possesses the authority to always determine the group's preference, regardless of the other voters' wishes. It is a critical component of a democratic voting system, ensuring that the collective decision reflects the group's opinion rather than being imposed by a single individual. This principle is a bulwark against autocratic rule within a voting system, reinforcing the concept that the final outcome must be the product of fair and equal consideration of all members' preferences.

In the context of social choice and voting systems, Pareto efficiency is a scenario where no individual's situation can be improved without worsening another's. Arrow's theorem incorporates this concept, suggesting that a collective decision-making rule should satisfy Pareto efficiency, in the sense that if every individual prefers one option over another, the group's ranking should reflect the same preference. It's a principle of unanimity; if everyone agrees that option A is better than option B, the social choice should naturally be A over B. This criterion emphasizes the minimal standard for a group's decision to be considered rational and socially beneficial.

The principle of Independence of Irrelevant Alternatives (IIA) states that the social preference between two options should not be affected by the introduction or removal of a third, unrelated option. In simpler terms, if choice A is preferred over choice B, introducing a choice C should not change this preference, unless C becomes the most preferred option. This criterion, outlined as part of Arrow’s Impossibility Theorem, insists on consistent preferences in the presence of additional choices. It asserts that the integrity of voters' preferences between any two options should not be distorted by the addition of non-preferred alternatives. Arrow's theorem shows that no rank-order voting system can guarantee this independence, alongside meeting other fairness criteria.