1.Mixed Nash Equilibria (3pt) (a) Find a pure action equilibrium of the following game, and a mixed action where player 1 mixes between T and B and player 2 between L and C. P1.2 L C R T 0, 2 3, 1 2, 1 Pl.1 M 2, 1 1,4 1, 2 B 1, -1 2,0 3,3 (b) Find all (pure and mixed) Equilibria of the following game. (HINT: first, perform IDSDA)Pl. 1 ((9 Find a mixed equilibrium of the following game in which player 3 chooses action X. Find one where he chooses action Y. You will have to check that those actions are optimal against the mixed actions of player 1 and 2. Blue Blue L R L R T303012 T021301 RecHan-2,21 RedB_ Purple: X Purple: Y Figure 1: A three player strategic game, in which Purple chooses X or Y. 2: Dynamic Games (3pt) For the following game: (1) write the game in Normal form (1.5 points); (2) count how many subgames there are; (.5pt) (3) find one Subgame Per- fect Equilibrium. (HINT: First, find the equilibria of the subgames at the bottom of the game tree. Then solve for the optimal choice of player 1 be- tween X and Y). Y N 2 1 . . . . . . ................ . . . . . ................... . . . .....................' u d u d u u 3, 2 1, 2 0, 1 1, 3 0, 2 2, -1 1, 1 3, -2 3. Dynamic Games and Empty threats For the following game, find a Subgame Perfect Equilibrium (1.5 pts); also, find a Nash Equilibrium which is not subgame Perfect and involves an empty threat(1.5 pts). Remember to express the complete strategy of each player when you declare that a strategy profile is an Equilibrium (whether ot is perfect )Y N P r A B 2, 4 2, 1 u d u d 3, 2 0. 3 1, 1 2, 0 4:Stackelberg Model with 3 firms Consider the following variation of the Stackelberg model that we saw in class. There are 3 firms in this market: firm 1 sets its production q1 first; firm 2 observe the production set by firm 1, then sets its own production, 92; finally, firm 3 observes the production set by both firms 1 and 2, and sets its production q3. The total quantity produced in the market is Q = 91 + 92 + 93. The (inverse) demand function is4:Stackelberg Model with 3 firms Consider the following variation of the Stackelberg model that we saw in class. There are 3 firms in this market: firm 1 sets its production q1 first; firm 2 observe the production set by firm 1, then sets its own production, 92; finally, firm 3 observes the production set by both firms 1 and 2, and sets its production q3. The total quantity produced in the market is Q = q1 + 92 + 93. The (inverse) demand function is P(Q) = 18 - 2Q = 18 - 291 - 292 - 293. The cost function of each firm is, for i = 1, 2, 3, Ci(qi) = 2qi. (a) (1pt) What is the best-reply of firm 3 if the other two firms produce q1 and q2, respectively?(b) (2 pts) Find the quantity produced by each rm in the unique Subgame Perfect Equilibrium of the game. (HINT: This is a game with perfect Information, which you can solve by Backward Induction.) 5: Repeated Games (3pts) Consider repeating the following Stage Game twice. Carefully write down (with a sentence) a grim trigger strategy that either player uses. (2pt) Then, argue that the grim trigger strategies constitute an equilibrium of the repeated game. (1 pt)