Final answer:
Anastasia, Emma, and Greta's preferences create a voting cycle where there is no clear winner among mountain biking, canoeing, and the beach, making a unanimous decision hard to attain.
Step-by-step explanation:
The issue presented centers on a voting cycle within a group decision-making scenario. Anastasia, Emma, and Greta each have a different set of preferences for their weekend getaway activities, resulting in a situation where no single activity is the definitive group favorite. They prefer mountain biking over canoeing by a 2-1 vote, canoeing over the beach by the same margin, but then the beach trumps mountain biking in a head-to-head contest, also by a 2-1 vote.
This outcome illustrates a non-transitive voting cycle where the collective preference isn't transitive — i.e., if A is preferred over B, and B is preferred over C, it does not mean A is necessarily preferred over C. The impossibility of a transitive group preference makes reaching a unanimous decision very challenging. For instance, even though mountain biking is preferred to canoeing, and canoeing is preferred to the beach, the group paradoxically prefers the beach over mountain biking.