H L H we L W2 a3 a4 The numbers in each cell of the matrix above, (m1,m2), represents monetary payments that two players receive based on their choices of actions (H, L}. Refer to this table for questions 1-2 below. 1. Assume that the utility of each player equals to the amount of money she gets (that is, the matrix above represent the payoffs in the game). Denote by p the probability that player 1 picks the action H (that is, p summarized Player 1's strategies given that there are only two actions). Similarly, denote by q the probability that player 2 chooses to play H. Which of the following statements is true about the mixed strategy Nash equilibrium (NE) for this game? (a) p has to be such that player 2 is indifferent between choosing H and L (b) 10 has to be such that player 1 is indifferent between choosing H and L (c) Player 1 can never be indifferent between playing H and L in any NE (d) The game has no mixed strategy Nash equilibrium. (e) p and q are both equal to % in the mixed strategy NE 2. Which of the following characterizes the Nash Equilibrium where at least one player uses a strictly mixed strategy? (a) The game has no mixed strategy Nash equilibrium. ( WP%=% (p=%4=% _1 _1 (dlp:Q )p=%q=% 3. Titanium Dioxide, the base chemical used to make white paint, is produced by two base chemical companies. These companies compete a la Cournot. The market inverse demand function is given by P(Q) : 8 Q. Assume that both rms have the total cost functions C(q) = 1 + 29'. Let the star superscript* denote equilibrium quantities /prices/ prots. Which of the following statements is true? 4. Assume the same demand and cost structures as in question 3, but now firm 1 enters the market first and firm 2 follows, as in the Stackelberg model from lecture (both firms are guar- anteed to enter; the only choice is quantities produced). Which of the following statements regarding the equilibrium outcome is FALSE? (a) The first mover produces a greater quantity than the second mover (b) Total market output is Q* = 4.5 (c) The second mover will receive a negative profit (d) The first mover will receive a greater profit than the second mover (e) The difference in profits between the two firms is greater than 2 5. We will modify the game from question 4. Firm 2 now has the possibility of suing Firm 1 for violating environmental prtection laws, after observing the first firm's choices (TiO2 is dirty stuff to make). The lawsuit is costly for both parties, because it requires hiring a bunch of chemistry experts and and lawyers for a while. The lawsuit does not affect quantities or prices in the market. We'll model lawsuit costs as both firms having to pay an extra cost equal to one if there is a lawsuit. That will provide firm 2 with a means to "punish" firm 1 for overproducing. The game has three stages. In the first stage, firm 1 enters and chooses the quantity q1. In stage 2, firm 2 observes firm 1's choices and decides whether to start the lawsuit or not. At the last stage, firm 2 chooses its quantity produced q2 and "the market" determines the price given the quantities produced by both firms. In order to derive the subgame perfect Nash Equilibrium of this game, we proceed using backward induction. What is the object we are solving for in the second-to-last step (stage 2) using backward induction? (a) q2 as a function of q1 and of whether the lawsuit is in place. (b) Whether to create the lawsuit or not as a function of q1 and q2. (c) Whether to create the lawsuit or not as as a function of q1. (d) q1 as a function of q2 and whether the lawsuit is in place. (e) 92 as a function of q1 only. 6. Continue with the game from the last question. Under what conditions will firm 2 decide to engage in a lawsuit? (a) q1 > 2 (b) q1 2 3 (c) 92 > 2 (d) 92 2 3 (e) Firm 2 will never start the lawsuit