Considers the following undirected network, where each node plays a

coordination game against all of its neighbors. Each node needs to choose

either A or B$.$ As assumed in class, A generates nAa payoffs and B

generates nB b payoff, where nz $,$ z $=$ A$,$ B is the number of the neighbors

who choose z$.$ Assume that a $=1\mathrm{.}5$ and b $=1.$

Answer the following questions.

$($a$)$ Suppose that one node starts playing A when everyone else is play$-$

ing B$,$ which may trigger a cascade of A through the myopic ad$-$

justment process discussed in class. Will everyone end up playing

A in the end of the process? Why?

$($b$)$ What is the minimum number of $$initial adopters$$ of A that con$-$

verts all B$-$nodes to A$-$nodes in the entire network? Find a set of

such initial adopters.

$($c$)$ Can you add one link and pick one initial adopter of A so that all

nodes will adopt A eventually? If so$,$ add a link and find one such

initial adopter.

$($d$)$ Can you cut one link and pick one initial adopter of A so that all

nodes will adopt A eventually? If so$,$ cut a link and find one such

initial adopter.