Two investors have each deposited $100 million in a bank. The bank has invested these deposits in a long-term project. If the bank is forced to liquidate the investment before the project matures, a total of $160 million can be recovered. On the other hand, if the bank allows the investment to reach maturity, the project will pay out a total of $300 million.
There are two dates at which the investors can make withdrawals from the bank: date 1, before the bank’s investment matures, and date 2, after its maturity. If both investors make withdrawals at date 1 then each receives $80 and the game ends. If only one investor makes a withdrawal at date 1 then that investor receives $100, the other receives $60, and the game ends. If neither investor makes a withdrawal at date 1 then the project matures and the investors make withdrawal decisions at date 2. If both investors make withdrawals at date 2 then each receives $150. If only one investor makes a withdrawal at date 2 then that investor receives $200 and the other receives $100. Finally, if neither investor makes a withdrawal at date 2 then the bank returns $150 to each investor. At each date the decision whether or not to make a withdrawal is made simultaneously by both investors. Each investor is selfish and greedy, that is, cares only about his own wealth and prefers more money to less.
(a) Represent this game in extensive form.
(b) How many proper sub games are there?
(c) Find the pure-strategy sub game-perfect equilibria.
(d) Convert the original extensive-form game (of part a) into a strategic-form game.
(e) Find all the pure-strategy Nash equilibria of the game of part (d).
(f) Are all the pure-strategy Nash equilibria sub game perfect?