Game theory question

In what follows, I will give you some statements and ask you to prove or disprove them. If you think the statement is true, then you must (1) provide full strategies of all the players and {2) show that these strategies form an equilibrium. If you think the arguments are false, then you must provide the proof. Your analysis must be well structured and rigorous: If you are using proof by contradiction (or indirect derivation), then you must be explicit in all your assumptions and conclusions. If you are using direct approach and case analysis, you must state all your cases, assumptions and conclusions carefully. I will provide an answer to the rst question just to give you an idea how you can answer the remaining questions. (110 Points total) Suppose that the following stage game is played twice and players don't discount the future (i.e. 6 = 1): A B C D 5.5 0, 10 -6,0 -1,0 B 10, 0 2, 2 -5,5 0, 3 0, -6 5, -5 -7, -7 0, -8 D 0, -1 3,0 -8,0 1, 1 (a) (10 Points) Find all NE (pure only) of this stage game. (b) (50 Points) Prove/Disprove: There is an SPNE of this game in which both players play A in the first stage. (c) (40 Points) Prove/Disprove: There is a NE of the twice-repeated game, where both players play C in the first stage, that is NOT SPNE. (d) (10 Points) Now suppose this stage game is played infinitely many times and players discount the future with 6 6 (0, 1). Solve for the values of & for which the following strategy profile is an SPNE (show all your work): "Both players play A in the first stage. Both players continue playing A as long as these two actions were played in all previous stages. Otherwise, they play D forever." HINT: 1 + 8 + 82 + 83 + ...= 1-8