Question

Why does a voting cycle make it impossible to decide on a majority-approved choice?

Short answer

A voting cycle makes it impossible to decide on a majority-approved choice because it demonstrates that there is no clear majority for any single option due to the continuous shift in voters' preferences among the available options. This occurs due to the transitive preferences (A > B > C > A) in a preference-based voting system with at least three options. In such a situation, a "Condorcet winner" does not exist, making it impossible to identify a majority-approved choice in the voting process.

Step-by-step solution

Understanding the Voting Cycle

A voting cycle occurs when there is no clear majority for any of the choices in a preference-based voting system with at least three options. In this system, voters are asked to rank their preferences for each option, and the options are compared in head-to-head match-ups. The voting cycle is a result of transitive preferences (A > B > C > A), which means that majority approval can continuously shift from one choice to another.

Example Voting Scenario

Let's consider an example where there are three choices - A, B, and C. Suppose that we conduct a preference-based vote and receive the following voter preferences: - 10 voters prefer A > B > C - 10 voters prefer B > C > A - 10 voters prefer C > A > B

Comparing Choices Head-to-Head

Now, let's compare the choices head-to-head for each possible match-up: 1. A vs B: A gets 10 votes (+10 for A > B > C preference), B gets 20 votes (as both B > C > A and C > A > B voters prefer B over A). So, B wins over A. 2. B vs C: B gets 10 votes (+10 for A > B > C preference), C gets 20 votes (as both B > C > A and C > A > B voters prefer C over B). So, C wins over B. 3. C vs A: C gets 10 votes (+10 for B > C > A preference), A gets 20 votes (as both A > B > C and C > A > B voters prefer A over C). So, A wins over C.

Identifying the Voting Cycle

From the head-to-head comparisons, we observe the following cycle: A > B (with B winning over A), B > C (with C winning over B), and C > A (with A winning over C). This represents a voting cycle and demonstrates that there is no clear majority-approved choice.

Conclusion

A voting cycle makes it impossible to decide on a majority-approved choice because the cycle demonstrates that the voters' preferences continuously shift from one option to another, preventing a clear majority for any single option. In this case, a "Condorcet winner," which is a candidate preferred by a majority of voters over all other candidates in all pairwise contests, does not exist, making it impossible to identify a majority-approved choice in the voting process.