Game Theory Homework Problem Set 2 Dr Ilja Neustadt 1 Homework Problem Set 2 to be solved for submission by the 17th of October Question 1. (Modied Driving Conventions Game)1 Consider the following modied coordination (driving conventions) game from the lecture. In this game, each player has a third strategy: to zigzag on the road. Suppose that if a player chooses zigzag, the chances of an accident are the same whether the other player drives on the left, drives on the right, or zigzags as well. Let that payo be 0, so that it lies between 1, the payo when a collision occurs for sure, and 1, the payo when a collision does not occur. Find all Nash equilibria (in pure strategies). Colin drive left drive left Rowena drive right zigzag drive right zigzag 1, 1 1, 1 0, 0 1, 1 1, 1 0, 0 0, 0 0, 0 0, 0 Question 2. (Market Entry Game2 ) Two companies are deciding, at what point to enter a market. The market lasts for four periods and companies decide simultaneously, whether to enter in period 1, 2, 3, or 4, or not to enter at all. Thus, the strategy set of a company is {1, 2, 3, 4, do not enter}. The market is growing over time, which is reected in growing prot from being in the market. Assume that the prot received by a monopolist in period t (where a monopoly means that only one company has entered) is 10t 15, whereas each duopolist (so both have entered) would earn 4t 15. A company earns zero prot for any period that it is not in the market. For example, if company 1 entered in period 2 and company 2 entered in period 3, then company 1 earns zero prot in period 1; 5 (= 10 2 15) in period 2; 3 (= 4 3 15) in period 3; and 1 (= 4 4 15) in period 4, for a total payo of 3. Company 2 earns zero prot in periods 1 and 2, -3 in period 3, and 1 in period 4, for a total payo of -2. (a) Derive the payo matrix. (b) Find the strategies that survive the iterative elimination of strictly dominated strategies. 1 2 Source: Harrington, Ch. 4. Source: Harrington, Ch. 4 Game Theory Homework Problem Set 2 Dr Ilja Neustadt 2 (c) Derive best replies of a company for each strategy of the other company. (d) Consider the remaining game matrix and nd all Nash equilibria (in pure strategies). Question 3. (Bertrand duopoly with asymmetric costs3 ) Two rms in a market with identical products compete in prices simultaneously. As in the lecture, the market demand is given by Q = D1 (p1 , p2 ) + D2 (p1 , p2 ) = 100 p, where Q denotes the total quantity traded in the market, D(pi ) the demand faced by rm i = 1, 2, pi the price set by rm i, and p = min p1 , p2 , the market price. Assume that rm 2 has a cost of 15, while rm 1 has a cost of 10. Make the (admittedly arbitrary) assumption that if both shops set the same price, then all shoppers buy from rm 1. Firm 1's prot is { (p1 10)(100 p1 ) if p1 p2 0 if p2 p1 , whereas rm 2's payo function is { 0 if p1 p2 (p2 15)(100 p2 ) if p2 p1 Find all Nash equilibria in this market game. 3 Source: Harrington, Ch. 6