Condorcet paradox

The Condorcet paradox is a fundamental discovery in social choice theory. It shows that majority rule can be inherently self-contradictory. This means it is logically impossible for any voting system to guarantee a winner that a majority of voters support. Even if every single voter has a rational and consistent list of preferences, the group as a whole can fall into a loop. These loops are often called Condorcet cycles or cyclic ties. In a cycle, every possible choice is rejected by the group in favor of another alternative. This happens because more than half of the voters prefer the next option instead.

To understand the mechanism, imagine three candidates: A, B, and C. Suppose there are three voters with specific ranked preferences. Voter 1 prefers A, then B, then C. Voter 2 prefers B, then C, then A. Voter 3 prefers C, then A, then B. If we compare candidates in pairs, a pattern emerges. A majority of voters prefer A to B. A majority also prefers B to C. However, a majority also prefers C to A. This creates a circle where A > B > C > A. No matter which candidate is selected, a majority would have preferred someone else.

This phenomenon is a special case of Arrow's impossibility theorem. That theorem shows that any social decision-making process is either self-contradictory, a dictatorship, or must use information about the strength of voter preferences. Systems that try to minimize these cycles are called Condorcet methods. These methods aim to find a winner who can win a one-on-one election against every other candidate. When no such winner exists, mathematicians look for the Smith set. The Smith set is the smallest group of candidates where every member can beat anyone outside the group in a head-to-head vote.

History shows that this idea is much older than the Marquis de Condorcet. A Catalan philosopher and theologian named Ramon Llull first discovered the paradox in the 13th century. He was investigating how the church was governed during his time. However, his specific work on this topic was lost for centuries. It was only rediscovered in the 21st century. Later, in the late 18th century, the mathematician and philosopher Marquis de Condorcet rediscovered the paradox. His work helped identify key results regarding how ranked voting systems can suffer from spoiler effects.

Researchers use mathematical models to estimate how often these cycles occur. One model is the "impartial culture" model. This model assumes voter preferences are distributed uniformly, which is considered a worst-case scenario. In this model, the asymptotic probability of a cycle is approximately 8.77%. Other models, like the Impartial Anonymous Culture (IAC) or Uniform Culture (UC), show a lower probability of 6.25%. More realistic models, such as the spatial model, suggest cycles become much rarer as groups grow. For example, in a spatial model, the likelihood drops from 5% with 100 voters to only 0.06% with 10,000 voters.

Real-world examples of the paradox are rare but do exist. In 2021, the Minneapolis City Council Ward 2 election featured a narrow circular tie. Candidates Cam Gordon, Yusra Arab, and Robin Wonsley created a cycle where Arab was preferred over Gordon, Gordon over Wonsley, and Wonsley over Arab. Another instance occurred in a 2022 Oakland, California, school election. Candidates Manigo, Hutchinson, and Resnick formed a cycle with very small margins. In the 2014 Victorian state election in Australia, a similar cycle occurred between the Greens, Liberal, and Labor candidates.

Understanding these cycles is important for the study of political science and mathematics. The paradox can cause voting mechanisms to violate the axiom of independence of irrelevant alternatives. This means the winner of an election might be influenced by whether or not a losing candidate is present. By studying these mathematical loops, researchers can better understand the limits of majoritarianism. They work to build systems that handle complex human preferences as fairly as possible.