1Homework # 2 - ISyE 6230 – Economic Decision Analysis II – Spring 2010 Due Thursday Feb 11, 2010 Grading: Up to 12.5 points per problem for completing each problem; the remaining 50 points will come from grading one randomly selected problem for correctness. 1. Consider a simultaneous-move auction in which two players simultaneously choose bids, which must be in nonnegative integer multiples of one cent. The higher bidder wins a dollar bill. If the bids are equal, neither player receives the dollar. Each player must pay his own bid, whether or not he wins the dollar. (The loser pays too.) Each player's utility is simply his net winnings; that is, the players are risk neutral. Construct a symmetric mixed-strategy equilibrium in which every bid less than 100 cents, e.g. bi א {0, 1, 2,…, 99}, has a positive probability. (Hint: each player’s payoff consists of two main parts; the total money he earns (if he wins) and the money he lose (if the other wins). 2. Consider the following game: Player 2 O P M 4, 3 5, 4.5 N 7.5, 4 0,0 (i.) What is/are the Nash Equilibriums (in pure strategies)? (ii.) Now think about this game in a Stackelberg setting. What is the Equilibrium if Player 1 is the leader? What is the Equilibrium if Player 2 is the leader? 3. Two firms, A and B supply all public transportation to a rapidly developing country. Current demand is characterized by P = 101 – 10Q, where P is the price of the product, and Q is the quantity (Q= QA+ QB). Assume costs are CA = 10 + QA and CB = 10 + QB for firm A and B, respectively. (i.) (Monopoly) If B is tied up in patent lawsuits and cannot supply the market, what is the best quantity for A to supply? (ii.) Specify the price equilibrium price of (i). (iii.) (Cournot duopoly) Now assume B has successfully fended off the lawsuit and will enter the market. B is not certain what quantity to offer, but has decided to focus on a given quantity (Q(B)) to supply. What is the best response function for A to B’s quantity? (iv.) What is the market price of (iii)? (v.) What is the total quantity on the market? (vi.) (Bertrand duopoly) B decides that the decision variable should be price, rather than quantity. If so, what is the best response price function to the price A charges? Player 12 4. Two firms compete in a market. Firm 1 is an incumbent (already operating). Firm 2 is potential entrant. Consider the following sequence of events: - First, firm 2 decides whether or not to enter. - Then, firm 1 chooses its quantity of production, q1; assume that firm 1 can only choose q1 = 10 or q1 = 5. - Finally, if firm 2 had decided to enter, it chooses its quantity of production q2; assume that firm 2 can only choose q2 = 5 or q2 = 2. Production costs are zero for both firms, and the market price is set as a function of total quantity as follows: p(q1, q2) = 15 − q1 − q2. Firms act to maximize profit. (i.) Assume all decisions are perfectly observable to all firms. Find all subgame perfect Nash equilibria of the game (pure and mixed). (ii.) Now assume that if firm 2 enters, it does not observe the production decision of firm 1 at the second stage; however, firm 1 observes the entry decision of firm 2 at the first stage. Find all subgame perfect Nash equilibria of the game (pure and
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