14.12 Problem Set 1Dennis V. Perepelitsa23 September 2006Muhamet Yildiz, instructorProblem 1(a) These are equivalent. In both cases, u1(T L) > u1(BL) = u1(BR) > u1(T R), andu2(BL) = u2(T R) > u2(T L) = u2(BR).(b) These are equivalent. Player One’s payoff in the second game is his twice payoff in the firs tgame, minus one in all cases. Player Two’s payoff in the second game is the first game minus onein all cases.(c) This is not the same decision process. In the first game, u1(BL) > u1(T L), but in thesecond game, u1(T L) > u1(BL).Problem 2(a) In strategic form, we want to list the set of strategies and payoffs for each player. For PlayerOne, the strategies are LlX, LlY, LrA, LrB, and R. T he payoffs for this are, respectively,u1(LlX) = 1, u1(LlY ) = 0, u1(LrA) = 0, u1(LrB) = 1, u1(R) = 0u1(R) = 0 above, since Player Two will immediately choose l if Player One chooses R. Now weconsider Player Two’s strategies and payoffs. The strategies are Ll, Lr, Rl, Rrx, and Rry. Thefirst two of these will imm ed iately be followed by a Player On e choice of X and B, respectively.Thus, the payoffs for Player Two for these five strategies areu2(Ll) = 0, u2(Lr) = 1, u2(Rl) = 4, u2(Rrx) = 0, u2(Rry) = 1Thus, written in strategic form, our game isG = {{LX, LY, LA, LB, R}, {Ll, Lr, Rl, Rrx, Rry}; {1, 0, 0, 1, 0}, {0, 1, 4, 0, 1}}(b) We begin to iteratively eliminate all weakly dominated strategies. Let’s start with the rightbranch. Player Two would rather opt out at l, to get a payoff of 4, rather than the payoff of 0 or 1that rx or ry provide.Consider the second decision Player Once has to make, in the lower-left corner of the tree. IfPlayer Two has chosen l, Player One will choose X and give hims elf a p ayoff of 1. If Player Twohas chosen r, Player One will choose B and give himself a payoff of 1.Player Two looks at the left branch of the updated extensive form and sees that if he picks l,Player One will then pick X and he will receive a payoff of 0. If he picks r, Player One will thenpick B and give himself a payoff of 1. So at this juncture, Player Two will p ick r.Player One now looks at the updated extensive game, from the top nod e. If he picks L, heknows that Player Two will then pick r and he will pick B, with a payoff of 1 for Player One. Ifhe picks R, he knows that Player Two will then pick l, with a payoff of 0 for Player One. Thedominating strategy is therefore (L, r, B), with a payoff of (1, 1).1Problem 3Problem 1 solution goes here.Problem 4Problem 1 solution goes here.Problem 5Problem 1 solution goes