Condorcet paradox

Sometimes, picking a winner is hard.

A man named Condorcet found a strange rule. He saw that voting can make a circle.

Imagine three choices. People like choice A more than B. Then they like B more than C. But then they like C more than A!

This means no one truly wins. Every choice can be beaten by another. It is like a loop that never ends.

This can happen in real elections. It can make it very tough to pick a leader.

It is a puzzle for everyone to solve.

Sometimes, picking a winner is very hard. A thinker named the Marquis de Condorcet found a strange rule. He showed that voting can create a loop. This is called the Condorcet paradox.

Imagine there are three choices: A, B, and C. Most people like A better than B. Then, most people like B better than C. But then, most people like C better than A! This creates a circle. No single choice can win against all others. Every choice is beaten by something else.

This can happen in real life. In 2021, a city election in Minneapolis had this problem. Three candidates were in a tie. Voters liked one more than the next, but it went in a circle. This makes it tough to pick a leader. It shows that majority rule is not always simple. Even when people have clear likes, the final result can be a puzzle.

Have you ever tried to pick a movie with friends and ended up in a loop? You might pick one, but then someone else suggests a better one, and then a third person suggests a different one. This strange loop is a real idea in math called the Condorcet paradox. It happens when a group of people cannot agree on a single winner. Even if every person has a clear list of favorites, the group as a whole might end up in a circle. In this circle, every choice is beaten by another choice by a majority vote. This means that majority rule can sometimes lead to a logical contradiction.

To understand how this works, imagine three candidates named A, B, and C. Suppose there are three voters who each rank these candidates in a different order. Voter 1 likes A best, then B, then C. Voter 2 likes B best, then C, then A. Voter 3 likes C best, then A, then B. If we look at them in pairs, we see a problem. A majority prefers A over B. A majority prefers B over C. But a majority also prefers C over A. This creates a cycle where no one can truly win.

People have been studying this puzzle for a very long time. A philosopher named Ramon Llull first discovered this idea in the 13th century. He was looking at how the church was run, but his work was lost for hundreds of years. It was not found again until the 21st century. Later, in the late 1700s, a mathematician named the Marquis de Condorcet found it again. He used it to show that voting systems can be very tricky. His work helped lead to other big ideas, like Arrow's impossibility theorem.

This paradox does happen in real elections, even if it is rare. In 2021, a city council election in Minneapolis, Minnesota, had this exact problem. Three candidates—Cam Gordon, Yusra Arab, and Robin Wonsley—were stuck in a circular tie. Voters preferred Arab over Gordon, Gordon over Wonsley, and Wonsley over Arab. Another example happened in 2022 in Oakland, California. In a school election, three candidates named Manigo, Hutchinson, and Resnick created a cycle too. In those cases, the margins between the voters were very small.

Math experts try to guess how often these cycles happen using different models. One way is called the "impartial culture" model, which assumes voters pick favorites randomly. In that model, the chance of a cycle is about 8.77 percent. However, real life is usually more organized than that. Most studies show that in large groups, these loops happen much less often. For example, one study of real elections found the chance was only about 0.7 percent. Understanding these loops helps us build better ways to make fair decisions for everyone.

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.