Assignment 1 ECON 233: Introduction to Game Theory Fall 2021 Total marks: 100 Solve the following questions from your book. You can discuss each problem with your classmates but must submit your own answers. There will be zero tolerance for plagiarism. You can submit the answers either as a scanned pdf (make sure your answers are legible) or a typed word file. Question 1 (5 marks) Represent the following strategic situation as an extensive-form game concerning the owner of a firm (O), the manager of the firm (M), and a potential worker (W). The owner first decides whether to hire the worker, to refuse to hire the worker, or to let the manager make the decision. If the owner lets the manager make the decision, then the manager must choose between hiring the worker or not hiring the worker. If the worker is hired, then he or she chooses between working diligently and shirking. Assume that the worker knows whether he or she was hired by the manager or the owner when he or she makes this decision. If the worker is not hired, then all three players get a payoff of 0. If the worker is hired and shirks, then the owner and manager each get a payoff of −5, whereas the worker gets 1. If the worker is hired by the owner and works diligently, then the owner gets a payoff of 10, the manager gets 0, and the worker gets 0. If the worker is hired by the manager and works diligently, then the owner and manager each gets 5, and the worker gets 1. Question 2 (5 marks) Draw the extensive form for the following game. There is an industry in which two firms compete as follows: First, firm 1 decides whether to set a high price (H) or a low price (L). Without seeing firm 1’s price, firm 2 decides whether to set a high price (H) or a low price (L). If both firms selected the low price, then the game ends with no further interaction. If either or both firms selected the high price, then the court decides whether to prosecute (P) or not (N) for anticompetitive behavior. In this case, the court does not observe which firm selected the high price (or if both firms selected the high price). Invent your own payoffs for each terminal node. Question 3 (5 marks) Consider the following strategic setting. There are three players, numbered 1, 2, and 3. Player 1 has two cards, labeled King and Ace. At the beginning of the game, player 1 deals one of the cards to player 2 and the other card to player 3; that is, player 1 either gives the Ace to player 3 and the King to player 2 (call this the action A) or the King to player 3 and the Ace to player 2 (action K). Player 2 observes the card dealt to him; player 3 does not get to see the card dealt to her. Player 2 then must decide between switching cards with player 3 (S) or not (N). Player 3 observes whether player 2 made the switch, but does not see her card. Finally, player 3 responds to the question “Is your card the Ace?” by saying either “yes” (Y) or “no” (N). If player 3 correctly states whether her card is the Ace, then she obtains a payoff of 1 and the other players get 0; otherwise, players 1 and 2 both get a payoff of 1 and player 3 obtains 0. Represent this game in the extensive form. Question 4 (5 marks) Suppose a manager and a worker interact as follows. The manager decides whether to hire or not hire the worker. If the manager does not hire the worker, then the game ends. When hired, the worker chooses to exert either high effort or low effort. On observing the worker’s effort, the manager chooses to retain or fire the worker. In this game, does “hire” describe a strategy for the manager? Explain and given another example of manager’s strategy. Question 5 (15 marks) Draw the normal-form matrix of each of the following extensive-form games. a) b) c) d) e) Question 6 (5 marks) Consider a version of the Cournot duopoly game, in which two firms (1 and 2) compete in a homogeneous goods market, where the firms produce exactly the same good. The firms simultaneously and independently select quantities to produce. The quantity selected by firm 𝑖 is denoted 𝑞 and must be greater than or equal to zero, for 𝑖 1, 2. The market price is given by 𝑝 𝑞 . Assume that the cost to firm 𝑖 of producing any quantity is 𝑐𝑞 . Further, assume that 2 𝑞 each firm’s payoff is defined as its profit. That is, firm 𝑖’s payoff is 𝑝𝑞 𝑐𝑞 , where 𝑗 denotes firm 𝑖’s opponent in the game. Describe the normal form of this game by expressing the strategy spaces and writing the payoffs as functions of the strategies. Question 7 (10 marks) Consider the following strategic setting involving a cat named Tom, a mouse named Jerry, and a dog named Spike. Tom’s objective is to catch Jerry while avoiding Spike; Jerry wants to tease Tom but avoid getting caught; Spike wants to rest and is unhappy when he is disturbed. In the morning, Tom and Jerry simultaneously decide what activity to engage in. Tom can either nap (N) or hunt (H), where hunting involves moving Spike’s bone. Jerry can either hide (h) or play (p). If nap and hide are chosen, then the game ends. The game also will end immediately if hunt and play are chosen, in which case Tom captures Jerry. On the other hand, if nap and play are chosen, then Jerry observes that Tom is napping and must decide whether to move Spike’s bone (m) or not (n). If he chooses to not move the bone, then the game ends. Finally, in the event that Spike’s bone was moved (either by Tom choosing to hunt or by Jerry moving it later), then Spike learns that his bone was moved but does not observe who moved it; in this contingency, Spike must choose whether to punish Tom (B) or punish Jerry (J). After Spike moves, the game ends. In this game, how many information sets are there for Tom, Jerry, and Spike? List all the strategy profiles in this game? Question 8 (10 marks) Draw an extensive-form representation of following normal form game. Can you think of other extensive forms that correspond to this normal-form game? Question 9 (20 marks) Evaluate the following payoffs for the game given by the normal form pictured here. [Remember, a mixed strategy for player 1 is 𝜎 ∈ ∆ 𝑈, 𝑀, 𝐷 , where 𝜎 𝑈 is the probability that player 1 plays strategy 𝑈, and so forth. For simplicity, we write 𝜎 as 𝜎 𝑈 , 𝜎 𝑀 , 𝜎 𝐷 , and similarly for player 2.] a) 𝑢 𝑈, 𝐶 b) 𝑢 𝑀, 𝐿 c) 𝑢 𝐷, 𝑅 d) 𝑢 𝜎 , 𝐶 for 𝜎 , ,0 e) 𝑢 𝜎 , 𝑅 for 𝜎 , , f) 𝑢 𝜎 , 𝐿 for 𝜎 0,0,1 g) 𝑢 𝜎 , 𝑅 for 𝜎 , ,0 h) 𝑢 𝜎 , 𝜎 for 𝜎 , 0, and 𝜎 , , i) 𝑢 𝜎 , 𝜎 for 𝜎 , 0, and 𝜎 , , j) 𝑢 𝜎 , 𝜎 for 𝜎 0,0,1 and 𝜎 , , Question 10 (10 marks) Evaluate the following payoffs for the game pictured here: a) 𝑢 𝜎 , 𝑂 𝑓𝑜𝑟 𝜎 , , , b) 𝑢 𝜎 , 𝜎 0, , , 𝑓𝑜𝑟 𝜎 ,𝜎 , Question 11 (10 marks) Consider a version of the Cournot duopoly game, where firms 1 and 2 simultaneously and independently select quantities to produce in a market. The quantity selected by firm 𝑖 is denoted 𝑞 and must be greater than or equal to zero, for 𝑖 1, 2. The market price is given by 𝑝 100 2𝑞 2𝑞 . Suppose that each firm produces at a cost of 20 per unit. Further, assume that each firm’s payoff is defined as its profit. Suppose that player 1 has the belief that player 2 is equally likely to select each of the quantities 6, 11, and 13. What is player l’s expected payoff of choosing a quantity of 10?
