game theory

Problem 3: [30 pts + 10 extra] The owner of a firm needs to decide how much to invest to improve its production process. More investment lneans that the workers are more efcient in generating revenue but is costly. The following galne models this investment decision. The timing of the game is as follows: The game begins with the owner, player 1, choosing an investment level (1' 2 0. Two employers, player 2 and 3, who work for the owner observe or and subsequently decide simultaneously on effort levels, 132, 83 Z 0. After the investment and efforts have been decided, revenue for the rm is generated according to the following function: 7r(oa, e2, 63) = (182 + 01133 + 8283. Notice that the higher oz is, the more eicient the workers are at converting effort into revenue. Revenue is shared among the three parties in the following way: ,8 E [0, 1] proportion of the revenue accrues to the owner while the remaining revenue is split among the workers equally so that each worker obtains %(1 ,8} proportion of the revenue. Both investment and effort are costly. The owner incurs a cost of a3 for investment or while each worker 1' incurs a cost of is? by exerting eifort (3g. Therefore, the utility functions of the players of this game are as follows: \"1(0. 62: 63:3.) = 5(062 + 063 + 263) 03: 1 1 use. e2, as) = '5' (082 + ma + as) 1e. 1 1 \"3(1):. 622. 3) = 2 '8 (082 + (163+ 283) - 18%- Part a: {10 guts] Describe the strategies of each of the players in this game. Part I): [10 pts/ Suppose that 13 = 1/2. Solve for the NE} of the simultaneous move in the proper subgame after the manager has already chosen the investment level to be 0:. Part C: {10 pts] Suppose that ,6 = 1/2. Solve for thc SPNE of this game. What is the SPNE} strategy prole? Part (1': [Extra Credit +- 10 pts] Solve for the unique SPNE for ally ,8 > 0. What happens to the manager's prots as ,8 increases? Explain intuitively