Game theory

Players 1 and 2 play the following sequential-move game with perfect information. Player 1 rst picks a number 31 E {1,2,3,4, 5,6} and tells it to player 2. Player 2 then picks a number 32 E {1, 2, 3,4, 5, 6} in response to 51. Player 1 has to pay (31 52)2 to player 2, so payoffs are given by v1(sl,sg) = \"(31 ~ 82)2 and 92(51, 32) = (51 5'2)? (a) [3 points] Draw the extensive-form of the game. How many subgames does the game have? How many pure strategies does player 2 have in the extensive-form game? (b) [2 points] What is player 2's optimal response 33(31) to every 31? (c) [4 points] Find all the pure-strategy subgame perfect equilibria of the game. What payoffs does each player get in these equilibria? (d) [1 point] Does the game have a pure-strategy Nash equilibrium that leads to different payoffs than those you found in part (0)? Explain your