《博弈论讲义》英文版 Problem Set 4: Due Tuesday, November

Eco514 Game Theory Problem Set 4: Due Tuesday, November 9 1. Machines Extend Proposition 151. 1(the Perfect Folk Theorem with discounting)to arbitrary mixtures of payoff profiles of the original game G=(N, (Ai, lilieN Allow for both rational and real weights on the set of profiles u(a): aE A]; note that the statement of the result will involve an approximation of the payoff profile Construct a machine that implements the strategies in your proof Hint: You may wish to refer to the proof of Proposition 146.2 2. Repeated Prisoner's Dilemma with a Twist Consider the following variant of the usual repeated Prisoner's Dilemma game(Figure 134.1 in OR). There are four players in the population. At even times t, Player 1 plays Prisoner's Dilemma with Player 2 and Player 3 plays it with Player 4; at odd times t, 1 plays with 3 and 2 plays with 4. There is perfect information: that is, at the end of each stage, every player can observe the actions chosen by the others. Assume for simplicity that the four players use the same discount factor 8. Determine the set of discount factors for which cooperation at each t is a subgame- perfect equilibrium outcome. Next, consider the usual version of this repeated game(two players, I and 2, who share a common discount factor 8, play with each other repeatedly and determine the set of ds for which cooperation at each t is a subgame-perfect equilibrium outcome. Compare the two sets 3. War of attrition Two small grocery stores on the same block are feeling the effects of a large supermarket that was recently constructed a half-mile away. As long as both remain in business, each will lose $1000 per month. On the first day of every month, when the monthly rent for the stores is due, each grocer who is still in business must independently decide whether to stay in business for another month or quit. If one grocer quits, then the grocer who remains

Eco514—Game Theory Problem Set 4: Due Tuesday, November 9 1. Machines Extend Proposition 151.1 (the Perfect Folk Theorem with discounting) to arbitrary mixtures of payoff profiles of the original game G = (N,(Ai , ui)i∈N ). Allow for both rational and real weights on the set of profiles {u(a) : a ∈ A}; note that the statement of the result will involve an approximation of the payoff profile. Construct a machine that implements the strategies in your proof. [Hint: You may wish to refer to the proof of Proposition 146.2] 2. Repeated Prisoner’s Dilemma with a Twist Consider the following variant of the usual repeated Prisoner’s Dilemma game (Figure 134.1 in OR). There are four players in the population. At even times t, Player 1 plays Prisoner’s Dilemma with Player 2 and Player 3 plays it with Player 4; at odd times t, 1 plays with 3 and 2 plays with 4. There is perfect information: that is, at the end of each stage, every player can observe the actions chosen by the others. Assume for simplicity that the four players use the same discount factor δ. Determine the set of discount factors for which cooperation at each t is a subgame￾perfect equilibrium outcome. Next, consider the usual version of this repeated game (two players, 1 and 2, who share a common discount factor δ, play with each other repeatedly) and determine the set of δ’s for which cooperation at each t is a subgame-perfect equilibrium outcome. Compare the two sets. 3. War of Attrition Two small grocery stores on the same block are feeling the effects of a large supermarket that was recently constructed a half-mile away. As long as both remain in business, each will lose $1000 per month. On the first day of every month, when the monthly rent for the stores is due, each grocer who is still in business must independently decide whether to stay in business for another month or quit. If one grocer quits, then the grocer who remains 1

will make $500 per month profit thereafter. Assume that, once a gr his or her lease will be taken by some other merchant (not a grocer), so he or she will not be able to reopen a grocery store in this block, even if the other grocer also quits. Each grocer wants to maximize the expected discounted average value of his or her monthly profits, using a discount factor per month of 8=99 a. Find an equilibrium of this situation in which both grocers randomize between staying and quitting every month until at least one grocer quits. Is this the only equilibrium of this b. Suppose now that grocer 1 has a slightly larger store than grocer 2. As long as both ores remain in business, grocer 1 loses $1, 200 per month, and grocer 2 loses $900 per month If grocer 1 had the only grocery store on the block, she would earn $700 profit per month If grocer 2 had the only grocery store on the block, he would earn $400 per month. Find an equilibrium of this situation in which both grocers randomize between staying and quitting every month, until somebody actually quits. In this equilibrium, which grocer is more likely to quit first? 4. From Or:148.1.152.1.153.2

will make $500 per month profit thereafter. Assume that, once a grocer quits, his or her lease will be taken by some other merchant (not a grocer), so he or she will not be able to reopen a grocery store in this block, even if the other grocer also quits. Each grocer wants to maximize the expected discounted average value of his or her monthly profits, using a discount factor per month of δ = .99. a. Find an equilibrium of this situation in which both grocers randomize between staying and quitting every month until at least one grocer quits. Is this the only equilibrium of this game? b. Suppose now that grocer 1 has a slightly larger store than grocer 2. As long as both stores remain in business, grocer 1 loses $1,200 per month, and grocer 2 loses $900 per month. If grocer 1 had the only grocery store on the block, she would earn $700 profit per month. If grocer 2 had the only grocery store on the block, he would earn $400 per month. Find an equilibrium of this situation in which both grocers randomize between staying and quitting every month, until somebody actually quits. In this equilibrium, which grocer is more likely to quit first? 4. From OR: 148.1, 152.1, 153.2 2