1. Suppose there are two firms with constant marginal cost Player 2 MC =3 and the market demand is P = 63 - 5Q. C D (a) Calculate the market price and profits for each firm in each of the following settings: C 3 3 -1 4 . Cartel Player 1 . Cournot duopoly . Bertrand duopoly (assume firm can set any D 4 -1 0 0 price, not just full dollar amounts) (b) Using part a), construct a 3 x 3 payoff matrix where the firms are choosing prices. The actions available to each of two players are to charge the price from Suppose that players simultaneously choose their strat- the three settings above. If there is a tie, they split egy before the repeated game, and can't change it once the demand. If one has a lower price, the firm with the repeated game has started. Players can choose one the lower price gets all of the demand. of the three following strategies as defined in class (c) What are the Nash equilibria of this 3 x 3 game? . Always Defect . Grim-Trigger 2. Two gas stations are competing on one corner. Each firm . Tit-For-Tat is trying to pick a strategy. The market conditions are such that they face the following payoffs: (a) Draw the 3x3 matrix game with the payoffs for each strategy pair calculated over the 10 periods. (Each row and each column represent one of the Station 2 three strategies) a3 (b) What are the pure strategy Nash equilibria of game C D E in part (a)? (c) Now suppose now that player 1 make a mistake in A 2 5 0 3 5 4 period 2. That is if his strategy is supposed to play C then he accidentally plays D, and if his strategy is supposed to play D then he accidentally plays C. Station 1 Draw the 3x3 matrix game with the payoffs for each strategy pair calculated over the 10 periods. B 4 1 1 2 3 0 (d) What are the pure strategy Nash equilibria of game in part (c)? (e) Now consider the new strategy called Win-Stay, Lose Shift (WSLS): (a) Does either stations have any dominant or domi- . Start with C nated strategies? . If the other person plays C, then I repeat MY last action (b) If the stations choose their actions simultaneously, . If the other person plays D, then I switch MY what are the possible Nash equilibria, and what last action would be the payoffs? What sequence of actions is played if both players (c) If station 1 moves first, what is the Nash equilib play WSLS against each other, and player 1 makes rium? Did they increase their payoff over part (b)? a mistake in period 2. What are the payoffs for each player? (d) If station 2 moves first, what is the Nash equilib (f) Is this better than when both players play TFT rium? Did they increase their payoff over part (b)? against each other, and player 1 makes a mistake in period 2? 3. Suppose two players play the following prisoner's dilemma for 10 periods (periods 1 through 10)
