Little Italy has 100 consumers all of whom buy either one pizza or nothing. Each consumer...

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Little Italy has 100 consumers all of whom buy either one pizza or nothing. Each consumer has a reservation price of $15. That is, each consumer is willing to pay at most $15 for a pizza. There are two pizza stores - A and B - located next to each other. It costs $7 to make a pizza. Let pa and pb denote the price charged by stores A and B respectively. For simplicity, assume Pi € {12, 15} for both i = A, B. Hereafter we ignore the dollar sign. Thus, prices can take just one of the two values - 12 or 15. If p; <pj, all consumers buy pizza from store i. If pi = p; = p then 50 consumers buy from store A while 50 buy from store B. (i) (1 mark) Write down the payoff matrix. State the unique Nash equilibrium of the one-shot game. (ii) (1 mark) Suppose the game is repeated twice. Is there a subgame perfect equilibrium where Pa = Pb = 15 in the first period? Assume that after first period, one-period profits are realised and each store knows whether or not the other store has charged 15 or 12. Explain. (iii) (1.5 marks) Suppose the game is infinitely repeated. Consider the following trigger strategy by player i: Play p; = 15 in the first stage. In the t-th stage, if the outcome of all t - 1 preceding stages has been (15,15) then play p; = 15; otherwise, play p; = 12. Assume a common discount factor of 8 for both stores A and B. Can the two stores sustain Pa = Pb = 15 each period if 8 = ? Explain. (iv) (2.5 marks) Now add p; = 9 to the set of possible prices. That is, suppose Pi € (9, 12, 15) How does the inclusion of this new strategy p; = 9 affect your findings. In particular, • Are there any additional pure-strategy Nash equilibrium in the one-shot game? • Does you answer to part (ii) change with inclusion of this additional strategy? • Does your answer to part (iii) change? [Assume d = Little Italy has 100 consumers all of whom buy either one pizza or nothing. Each consumer has a reservation price of $15. That is, each consumer is willing to pay at most $15 for a pizza. There are two pizza stores - A and B - located next to each other. It costs $7 to make a pizza. Let pa and pb denote the price charged by stores A and B respectively. For simplicity, assume Pi € {12, 15} for both i = A, B. Hereafter we ignore the dollar sign. Thus, prices can take just one of the two values - 12 or 15. If p; <pj, all consumers buy pizza from store i. If pi = p; = p then 50 consumers buy from store A while 50 buy from store B. (i) (1 mark) Write down the payoff matrix. State the unique Nash equilibrium of the one-shot game. (ii) (1 mark) Suppose the game is repeated twice. Is there a subgame perfect equilibrium where Pa = Pb = 15 in the first period? Assume that after first period, one-period profits are realised and each store knows whether or not the other store has charged 15 or 12. Explain. (iii) (1.5 marks) Suppose the game is infinitely repeated. Consider the following trigger strategy by player i: Play p; = 15 in the first stage. In the t-th stage, if the outcome of all t - 1 preceding stages has been (15,15) then play p; = 15; otherwise, play p; = 12. Assume a common discount factor of 8 for both stores A and B. Can the two stores sustain Pa = Pb = 15 each period if 8 = ? Explain. (iv) (2.5 marks) Now add p; = 9 to the set of possible prices. That is, suppose Pi € (9, 12, 15) How does the inclusion of this new strategy p; = 9 affect your findings. In particular, • Are there any additional pure-strategy Nash equilibrium in the one-shot game? • Does you answer to part (ii) change with inclusion of this additional strategy? • Does your answer to part (iii) change? [Assume d =

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