Chapter 18: Problem 12

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

Short Answer

Expert verified

A voting cycle makes it impossible to decide on a majority-approved choice because voter preferences form a circular arrangement in which no single option defeats every other option by a majority vote. The cycle prevents a stable outcome and demonstrates the limitations of traditional majority-rule voting, highlighting the need for alternative voting methods or decision rules in such scenarios.

Step by step solution

Understand a Voting Cycle

A voting cycle is a phenomenon that occurs in ranked voting, where three or more options are involved, and voter preferences form a circular pattern. As a result, no clear majority-approved choice emerges because the circularity prevents a stable outcome. In a voting cycle, each choice defeats another by a majority vote, but none can claim a majority over every other choice.

Present an Example of a Voting Cycle

Suppose we have a group of voters with the following preferences: - 40% prefer Option A to Option B, and Option B to Option C - 35% prefer Option B to Option C, and Option C to Option A - 25% prefer Option C to Option A, and Option A to Option B The preferences form a circular arrangement as follows: - Option A is preferred to Option B by 40% of the voters - Option B is preferred to Option C by 75% of the voters (40% + 35%) - Option C is preferred to Option A by 60% of the voters (35% + 25%)

Analyze the Voting Outcomes

Among the three options, none of them emerge as the majority-approved choice because: 1. There is no option that defeats every other option by a majority vote. For instance, although Option B defeats Option C by a significant 75% majority, it still loses to Option A by 40%. 2. The preferences form a circular pattern. Option A defeats Option B, Option B defeats Option C, but Option C leads back to defeating Option A, creating a cycle. This cycle means that no stable outcome can be reached where a single choice is accepted by the majority.

Conclusion

A voting cycle makes it impossible to decide on a majority-approved choice because the circular pattern of voter preferences prevents any single choice from securing a majority over every other choice. In this case, traditional-majority rule voting breaks down, and alternative voting methods or decision rules may be necessary to reach a collective decision.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ranked Voting

In ranked voting systems, voters arrange candidates or options in order of preference. This method can more accurately reflect the voters' wishes compared to simple plurality or 'first-past-the-post' systems, where voters choose only one candidate, and the one with the most votes wins.

However, ranked voting can lead to complex situations, especially when there are three or more choices. If the preferences among options form a circular pattern, as in a voting cycle, no clear winner emerges based on majority preference. This can make it challenging to determine a single option that fairly represents the choice of the majority. To address these complexities, various forms of ranked voting systems, such as instant-runoff voting, are sometimes used to ensure that a winner can be determined even in the presence of a voting cycle.

Majority Vote

The majority vote principle is straightforward: the option that receives more than half of the votes wins. It's a fundamental concept in democratic decision-making because it ensures that the decision reflects the preference of more than half of the participants.

However, the example from the exercise illustrates that a voting cycle can subvert this principle. With circular preferences, no single option commands a consistent majority against all others, signifying a paradox within the majority rule system. Consequently, this paradox leads to the consideration of additional decision-making tactics to achieve a result that aligns with the collective will of the voters, potentially involving more rounds of voting or different voting mechanisms.

Collective Decision-Making

Collective decision-making refers to the process by which a group arrives at a decision that has collective legitimacy. Ideally, it should reflect the group's preferences, which is why a majority-approved choice is typically sought.

In the context of a voting cycle, conventional majority rule fails, and the collective decision-making process encounters a deadlock. To navigate this, groups might use alternative methods, such as ranking with instant runoffs, using points systems, or seeking consensus through discussion and compromise. Each method has its merits and is chosen based on the specific need to ensure a fair representation of the group's overall views. Thus, collective decision-making remains a dynamic field that continually adapts to handle situations where traditional methods, like the majority vote, fall short of providing a clear outcome.