Condorcet Voting Paradox: Videos & Practice Problems

The Condorcet Voting Paradox illustrates how majority voting can lead to inconsistent outcomes when there are multiple options. In this scenario, three groups have distinct preferences among three choices: A, B, and C. The first group prefers A over B over C, the second group prefers B over C over A, and the third group prefers C over A over B.

When comparing A and B, the first group votes for A, the second group votes for B, and the third group, whose first choice is C, also votes for A since C is not an option. Consequently, A wins this round. Next, when B faces C, the first group opts for B as their second choice, the second group votes for B as their first choice, and the third group votes for C. Here, B wins. Finally, in the matchup between C and A, the first group votes for A, while the second group, unable to vote for their first choice B, votes for C, and the third group votes for C as well. Thus, C wins this round.

This sequence of votes reveals a paradox: A beats B, B beats C, but C beats A. This situation contradicts the transitive property familiar from algebra, where if A equals B and B equals C, then A should equal C. However, in this voting scenario, the order of the votes significantly influences the outcomes, demonstrating that the structure of the voting agenda can manipulate results. Understanding this paradox is crucial for analyzing how voting systems can be designed or influenced to achieve desired outcomes.

Understanding how the order of the voting agenda influences outcomes is crucial for both policymakers and policyholders. The strategic manipulation of the voting sequence can significantly affect which policy is ultimately adopted. For instance, if the goal is to ensure that policy A wins, the first step is to eliminate policy C, as C has previously been shown to defeat A in a direct comparison. By arranging a vote between policies B and C first, where B is preferred over C, B can be established as the winner, effectively removing C from contention.

Next, a vote between A and B can be conducted. Historical data indicates that A prevails in this matchup, leading to A being the final policy adopted. This illustrates how the initial choice of agenda can lead to a desired outcome through strategic elimination of competitors.

Conversely, if the objective is to have policy C win, the process must begin with eliminating policy B, as B has been shown to defeat C. Thus, a vote between A and B is initiated, where A wins, removing B from the equation. The final vote between A and C can then take place, resulting in C's victory. This demonstrates the importance of the voting sequence in determining policy outcomes.

Moreover, this analysis reveals a critical insight: majority voting alone does not guarantee that the outcomes reflect the preferences of society. The potential for manipulation in the voting process highlights the complexities of democratic decision-making and the need for careful consideration of how voting agendas are structured.