Two investors have each deposited $100 million in a bank. The bank has investedthese deposits in a long-term project. If the bank is forced to liquidate the investment beforethe project matures, a total of $160 million can be recovered. On the other hand, if the bankallows 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: date1, before the bank's investment matures, and date 2, after its maturity. If both investors makewithdrawals at date 1 then each receives $80 and the game ends. If only one investor makes awithdrawal at date 1 then that investor receives $100, the other receives $60, and the gameends. If neither investor makes a withdrawal at date 1 then the project matures and theinvestors make withdrawal decisions at date 2.
If both investors make withdrawals at date 2then each receives $150. If only one investor makes a withdrawal at date 2 then that investorreceives $200 and the other receives $100. Finally, if neither investor makes a withdrawal atdate 2 then the bank returns $150 to each investor. At each date the decision whether or not tomake a withdrawal is made simultaneously by both investors. Each investor is selfish andgreedy, 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 subgames are there?
(c) Find the pure-strategy subgame-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 subgame perfect?