There are N professors in a department that is spread over two campus locations: U (uptown) and M (main). Simultaneously and independently, each professor chooses between having an office in U or in M. The value to each professor for each location only depends on the number of other professors who decide to occupy the same location. Specifically, if k professors choose M, each of those professors obtain a payoff of 5kk^2+ 50; if lambda professors choose U, each of them obtains a payoff of 48lambda.

(a) Suppose N= 10. Assume the professors play pure strategies. In equilibrium how many professors are in each location? Justify your answer.

(b) For general N, and assuming the professors play pure strategies, write down all the conditions for a pair (k,lambda) to be an equilibrium outcome, where k is the number of professors choosing M and lambda is the number of professors choosing U.