Sequential Investments Suppose that two people can invest in a project that would be valuable to both of them if it is completed. Person 1 can only invest in odd period while person 2 can only invest in even periods. There are four periods, with the cost of investment in period t equal to 2t per unit of investment. That is, an investment of 1 costs 2 in period 1, 4 in period 2, 6 in period 3 and 8 in period 4. If there are at least 2 units of investment at the end of period 4 each person receives a payoff of 25 minus the cost of the investments that player has made. If there are 0 or 1 units of investment at the end of period 4 each person just loses the cost of her investment, if any. Suppose that in each period that a player can make an investment, either 0 or 1 can be invested.
1. Write down the game tree for the game.
2. Use backward induction to find the outcome for the game (i.e., what person 1 does in periods 1 and3; and what person 2 does in periods 2 and 4).
3. Suppose now that in each period that a player can make an investment, each player can invest 0, 1, or 2. What is the outcome for the game?
4. (Optional. 10 bonus points) Suppose now that as before, in each period that a player can make an investment, that player can invest 0 or 1, but that there are six periods instead of four periods. Also as before, the cost of making an investment of 1 is 2t, where t is the period the investment is made. Thus, the cost of investing 1 in period 5 is 10 and the cost in period 6 is 12. What is the outcome in this case?
