1. (Dynamic Game with Perfect Information) Consider the following three player game of perfect information depicted below Player 1 Player 3 h " R Player 2 a b Player 3 Player 3 2 -1 0 S h 12 b. 1 3 5 0 4 Identify all pure strategy Nash Equilibria for the game. Verify which pure strategies are also subgame perfect equilibria 2. (Finitely Repeated Game). Consider the following simultaneous move game as depicted below is played twice. The players observe the actions chosen in the first stage of the game prior to the second stage. What are the pure strategy subgame perfect Nash equilibria of this game? Player 2 B1 B2 B3 Player 1 A1 10, 10 2,12 0,13 A2 12,2 5,5 0,0 A3 13,0 0,0 1,1 3. (Infinitely Repeated Game). Two banks are competing for loans from clients. They compete through charging different interest rates (e.g. 4%, 6%, and 8%). The amount of loans (in million pesos) that each bank secures and the combination of interest rates are shown in the payoff matrix below Bank 2 4% 6% 8% Bank 1 4% 0,0 2,1 4, -1 6% 1,2 4,4 7,1 8% -1,4 1,7 5,5 a. What is the pure strategy Nash equilibrium if this game is played once? b. Can cooperation (i.e. both banks charging 8% and earning 5 million pesos each) be sustained as a subgame perfect nash equilibria (SGPNE) through the following trigger strategy? or not At t=1: Cooperate by charging 8% In period t>1: Charge 8% if both banks charged 8% in the previous period, charge 6% forever if one or both banks in all previous periods did charge 8% c. Consider now the following strategy (Temporary punishment in the Trigger Strategy): period, At t=1: Cooperate by charging 8% In period t>1: Charge 8% as long as both banks charge 8% in the previous period. If one of the banks did not charge 8% in the previous charge 6% for three periods then return to 8% afterwards. 4. (Bayesian Nash Equilibrium) Consider the following strategic situation. Two opposed armies are poised to seize an island. Each army's general can choose either "attack" or "not attack." In addition, each army is either "strong" or "weak" with equal probability (the draws for each army are independent), and an army's type is known only to its general. Payoffs are as follows: The island is worth M if captured. An army can capture the island either by attacking when its opponent does not or by attacking when its rival does if it is strong and its rival is weak. If two armies of equal strength both attack, neither captures the island. An army also has a "cost" of fighting, which is s if it is strong and w if it is weak, where s < w: There is no cost of attacking if its rival does not. Identify all pure strategy Bayesian Nash equilibria of this game.