3. Simultaneous and Sequential Moves. (40 points) Monica and Nancy are working together on a joint project. If Monica puts in effort m and Nancy puts in effort n then the joint profits of the project are 4m + In +mn. As equal partners they each receive fifty per cent of the joint profits. The (private) cost to Monica of putting in effort m is m2, and the (private) cost to Nancy of putting in effort n is n2. Hence Monica is choosing m to maximize her payoff 2m + 2n + mn/2 -m2 (holding n fixed) and Nancy is choosing n to maximize her payoff 2m + 2n + mn/2-n2 (holding m fixed). (a) (10 points) Given the rate of change of Monica's payoff as she increases m is given by 2+ - n - 2m and the rate of change of Nancy's payoff as she increases n is given by m 2+ 5 - 2n, calculate the Nash equilibrium amounts of effort put in by the two partners if they choose their effort levels simultaneously. (b) (10 points) Calculate the backward induction solution if Monica commits to her choice of effort first and on observing this decision, Nancy then commits to her own effort. Now suppose Monica and Nancy need to decide which (if either) of them will commit to an effort level first. To do this, each of them simultaneously writes on a separate slip of paper whether or not she will commit first. If they both write "yes" or they both write "no", they choose effort levels simultaneously, as in part (a). If Monica writes "yes" and Nancy writes "no", then Monica commits to her move first, as in part (b). If Monica writes "no" and Nancy writes "yes", then Nancy commits first. (c) (20 points) Find the (pure-strategy) SPE of this game. [WARNING: This question requires you to write down the subgame perfect equilibriumstrategy of each player. Recall a strategy for Monica (respectively, Nancy) is a complete contingent plan and so must specify what Monica (respectively, Nancy) will do at every point in the extensive game where she can be called upon to make a move. In particular, it is not sufficient to simply write down what Monica or Nancy will do in the SPE play of the game.]