A marketing firm is interested in learning about the dynamics of viral video dissemination on the web. In order to do this they show three videos to students at a university and ask them to fill a survey related to the videos. The videos are the epic splits of van Dame (a), the baby (b), and Chuck Norris (c). The survey contains six questions as follows. For each pair of videos, say a and b, they ask the students "is a at least as good as b." In order to make this operational, they provide the students with a picture in which the three videos are depicted as circles with their description next to it. The students are asked to draw arrows between the videos whenever the answer to a question is affirmative. That is, if a student finds video a is at least as good as video b, the student is asked to draw an arrow from the circle with label a to the circle with label b. (No question of the type, "is a at least as good as a" was asked; assume that the answer to each of these trivial questions is affirmative.) In what follows we write a->b when the student draw an arrow from a to b, and so on. The following are the answers to the survey of five different students. Which student has complete preferences?

your question is incomplete, as you have not provided the answer to the five different student. Please let me assume this to be your complete question;

A marketing firm is interested in learning about the dynamics of viral video dissemination on the web. In order to do this they show three videos to students at a university and ask them to fill a survey related to the videos. The videos are the epic splits of van Dame (a), the baby (b), and Chuck Norris (c). The survey contains six questions as follows. For each pair of videos, say a and b, they ask the students "is a at least as good as b." In order to make this operational, they provide the students with a picture in which the three videos are depicted as circles with their description next to it. The students are asked to draw arrows between the videos whenever the answer to a question is affirmative. That is, if a student finds video a is at least as good as video b, the student is asked to draw an arrow from the circle with label a to the circle with label b. (No question of the type, "is a at least as good as a" was asked; assume that the answer to each of these trivial questions is affirmative.) In what follows we write a->b when the student draw an arrow from a to b, and so on. The following are the answers to the survey of five different students. Which student has complete preferences?

1. a-> b, b->a, c->b.

2. a-> b, b->a, c->a.

3. a-> b, b->c, c->a.

4. c->a, c->b.

5. c->a, a->b.

ANSWER: 3. a->b, b->c, c->a

Explanation: The student with a complete preference is the student that has compared each movie with another one at a time. The third student has a complete preference because, the preference shows a complete comparison between each movie with another, among the three movies.

Though the result of the comparison of the third student is contradicting, because if he said movie a is better than b, and movie b is better than c, it is now contradicting to say that movie c is more better than a. The first student gave a clear answer, because the answers are not contradicting, movie c is better than a, and movie a is better than b, that means that movie b is the least and c is the best. But the result is not complete because the student has not showed a comparison of movie c and b.

Because the question is to identify the student that has showed a complete preference. Only the third student has shown a complete preference.