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 leads to an inconsistent and cyclic pattern of preferences among voters. In such a scenario, no single option can be said to be the clear preference of the majority, as each option is preferred over another, creating a cycle. This inconsistency prevents determining a majority-approved choice.

Step-by-step solution

Identify the options and voters

To understand the problem, let's assume that there are three options (A, B, and C) and three voters (1, 2, and 3). Each voter has to rank the options according to their preferences.

Assign preferences to voters

Let's assign preferences as follows: - Voter 1: A > B > C - Voter 2: B > C > A - Voter 3: C > A > B The greater than (>) symbol represents preference, so Voter 1 prefers option A to option B and option B to option C, and so on.

Compare preferences

Now let's compare the preferences of the voters by comparing different pairs of options: 1. A vs. B: Voter 1 and Voter 3 prefer A, while Voter 2 prefers B. Therefore, the majority (2 out of 3) prefers A to B. 2. B vs. C: Voter 1 and Voter 2 prefer B, while Voter 3 prefers C. Therefore, the majority (2 out of 3) prefers B to C. 3. A vs. C: Voter 2 and Voter 3 prefer C, while Voter 1 prefers A. Therefore, the majority (2 out of 3) prefers C to A.

Identify the voting cycle

From the comparisons in Step 3, we can observe that the majority prefers A to B, B to C, and C to A. This is a voting cycle, as there is no clear majority-approved choice.

Conclusion

A voting cycle makes it impossible to decide on a majority-approved choice because it leads to an inconsistent and cyclic pattern of preferences. In our example, no single option can be said to be the clear preference of the majority as each option is preferred over another, creating a cycle. Therefore, it is impossible to determine a majority-approved choice.