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

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.

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 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.