A group of twenty students living on the third and fourth floors of a college dorm like to play online games. When a new game appears on campus, each of these students needs to decide whether to join, by registering, creating a player account, and taking a few other steps necessary in order to start playing.
When a student evaluates whether to join a new online game, she bases her decision on how many of her friends in this group are involved in the game as well. (Not all pairs of people in this twenty-person group are friends, and it is more important if your friends are playing than if many people in the group overall are playing.)
To make the story concrete, let’s suppose that each game goes through the following “life cycle” within this group of students:
(a) The game has some initial players in the group, who have discovered it and are already involved in it.
(b) Each other student outside this set of initial players is willing to join the game if at least half of her friends in the group are playing it.
(c) Rule (b) is applied repeatedly over time, as in our model from Chapter 19 for the diffusion of a new behavior through a social network.
Suppose that in this group of twenty students, ten live on the third floor of the dorm and ten live on the fourth floor. Suppose that each student in this group has two friends on their own floor, and one friend on the other floor. Now, a new game appears, and five students all living on the fourth floor each begin playing it.
If the other students use the preceding rule to evaluate whether to join the game, will this new game eventually be adopted by all twenty students in the group? There are three possible answers to this question: yes, no, or there is not information in the setup of the question to be able to tell. Say which answer you think is correct, and explain.
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