The payoffs for coordination are a = 11 for both nodes in an edge choosing behavior A and b = 10 for both nodes in an edge choosing behavior B. Initially, nodes u, r, s, and t have chosen product A and nodes z, v, x, and w have chosen product B. Note that under this initialization, no one will change their behavior in the diffusion process, i.e., the network is in equilibrium.

Suppose we want to spread the adoption of behavior A in the network. In this question, we explore several ways in which one could change the network and diffusion process to encourage the spread of A.

a) The first strategy we will try is direct persuasion. Suppose that we can change one person's product choice from B to A by persuading them that A is better than B, perhaps through targeted advertising. Once they have been directly persuaded, such a node will never go back to product B. Is there a node such that if they were persuaded to switch their choice from B to A, then everyone in the entire network will adopt product A?

If you think the answer is yes, identify such a node and show how the cascade would happen. If you think the answer is no, explain why.

b) Instead of persuading someone, we could just try to make product A more appealing, which would have the effect of increasing the value of a to be greater than 11 in the coordination payoffs. What is the minimum value of a that will result in everyone in the network adopting product A? (Recall that following the definitions in class and the textbook, if a node is indifferent between A and B in terms of payoffs, then that node will choose A.) Explain your answer and show how the cascade would happen.

c) Rather than directly persuading someone or changing the product, we could instead modify the network structure to encourage the diffusion of product A. What is the smallest number of edges that you could add to the network that will result in everyone adopting product A? Explain your answer by identifying the added edges and showing the cascade, along with reasoning for why you cannot use a smaller number of edges.

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