Final answer:
The statement is true; majority rule can fail to produce a single preferred outcome when there are more than two options, due to a phenomenon known as the Condorcet paradox, which occurs in voting situations with multiple preferences.
Step-by-step explanation:
The statement is true. Majority rule can indeed fail to produce a single preferred outcome in situations where more than two options are available. This phenomenon is due to the possibility of a cyclical preference or Condorcet paradox, where collective preferences can be non-transitive, even if individual preferences are transitive.
For example, consider a scenario where three friends, Anastasia, Emma, and Greta, are deciding what to do on a weekend getaway. They have three options: mountain biking, canoeing, and going to the beach. Anastasia prefers mountain biking to canoeing to the beach, Emma prefers the beach to mountain biking to canoeing, and Greta prefers canoeing to the beach to mountain biking. Each of them ranks these activities in order of preference and then they vote. The group may find it difficult to come to a consensus on what activity to do because there is no clear majority favoring one option over the others.
When evaluating the preferences, mountain biking might be preferred over canoeing, canoeing might be preferred over the beach, but the beach might be preferred over mountain biking, making it impossible to determine a clear majority preference. This is an illustration of how majority rule may not yield a single outcome that satisfies everyone involved when multiple choices are present.