Question

1. Consider the coupon game. But suppose that instead of decisions being made simultaneously, they are made sequentially, with Firm 1 choosing first, and its choice observed by Firm 2 before Firm 2 makes its choice.

a. Draw a game tree representing this game.

b. Use backward induction to find the solution. (Remember that your solution should include both firms’ strategies, and that Firm 2’s strategy should be complete!)

2. Two duopolists produce a homogeneous product, and each has a constant average cost of 2. The firms compete by choosing the quantities ( q1 and q2 ) they produce. Market demand is given by p = 50 - 4Q,

where Q = q1 + q2 . For this question, assume that each firm is restricted to producing quantities of 3, 4 and 6 only (no other quantities are possible).

a. Suppose the firms make their choices simultaneously. Draw a payoff matrix representing this game, and find all Nash equilibria. Also find the payoffs resulting from each Nash equilibrium.

b. Suppose the two firms choose their prices sequentially, with Firm 2 observing Firm 1’s price choice before making its own choice. Draw a game tree representing this game, and use backward induction to find the solution. Also find the payoffs resulting from this solution.

Example (a simple type of duopoly): Two competing firms independently choose whether to print discount coupons for their products. Printing coupons is costly, but captures market share from the rival firnm Players: 1,2 (Firm 1 and Firm 2). Strategies: for both players, C (coupon) and N (no coupon Pavoffs: We assume these are just the firms' profits, If both print coupons, $150,000 each. If neither prints coupons, $200,000 each. If one prints coupons, $250,000 for that firm and $100,000 for the other. Each firm's decision is made without knowing the rival firm's decision. When there are only two players, and the number of strategies is small for both, it is easy to represent the game using a payoffmatrix. Firm 2 15, 15 25, 10 Firm 10, 25 20, 20 (payoffs expressed in 10,000s) Notes: One player's strategy determines a row; the other's determines a column. So, the strategy profile determines an cell of the matrix - Each cell of the matrix has two numbers. By convention, the first number is the payoff of the player who chose the row

Answer 1

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