A local connection game is given by a set of vertices and graph G where connection is built by adding edges.
If the cost to user a(user at node a) is given by $$C(u)=\alpha n_u + \beta \sum_v(dist(u,v))^2$$
where $n_u$ is the number of edges incident to u that u purchases and $dist(u,v)$ is the shortest distance from u to v. $\alpha$,$\beta$$\ge$$0$
What is the relationship between $\alpha$ and $\beta$ such that
i)the optimum solution is a star
ii)the optimum solution is a complete graph.
iii)the Nash solution is a star
iv)Determine the price of stability in this model.
I have gone through the theory and the game where the cost is given by $C(u)=\alpha n_u +\sum_v dist(u,v)$ ,but this question is pretty confusing. I'd appreciate if anyone could throw some light on this question.Thanks.