課程名稱︰賽局與訊息 課程性質︰選修 課程教師︰蔡崇聖 開課學院：社會科學院 開課系所︰經濟學系 考試日期（年月日）︰2024/4/10 考試時限（分鐘）：130 試題 : 1. [20%] Consider the game with the following extensive form: 3,0 X／ 2 ／ ● ／︰＼ C／ ︰ Y＼ 1 ／ ︰ 7,5 ● ︰ ／ ＼ ︰ 4,6 A／ D＼ ︰ X／ 1 ／ ＼︰／ ● ● ＼ 2 ＼ B＼ Y＼ ＼ 2,1 6,4 a. [3%] Draw the normal form of this game b. [3%] Find the set of rationalizable strategies. c. [6%] Find all the Nash equilibria. d. [8%] Find all the subgame perfect Nash equilibria (SPNE). Is there any Nash equilibrium that is not a SPNE? If so, explain the reason for that. 2. [20%] Consider the following normal-form game (x > 0): 2 L M N ┌───┬───┬───┐ U │ 5,3 │ 4,2 │ 2,1 │ ├───┼───┼───┤ 1 C │ x,3 │ 3,1 │ 0,6 │ ├───┼───┼───┤ D │ 7,x │ 0,5 │ 1,4 │ └───┴───┴───┘ The value of x is such that "C" is a dominated strategy. a. [3%] Find the range of x. b. [3%] After imposing the condition you found in a., find the set of rationalizable strategies. c. [14%] Find all the Nash equilibria, including the pure- and mixed-strategy ones. Denote the probability that player 1 plays U by p, and the probability that player 2 plays L by q. In each Nash equilibrium, describe the best response functions and draw them on the (p,q) coordinates. Indicate the Nash equilibrium on the figure. 3. [20%] Consider a game between a government (player 1) and a criminal (player 2). The government makes an effort to stop the crime, x ∈ [0,1], which is also the probability that the criminal is caught. The criminal chooses the level of crime, y≧0. If the criminal is caught (with probability x), there is a penalty with a rate f > 0 imposed on the crime level, resulting in a payment of fy paid by the criminal to the government. If the criminal is at large (with probability 1-x), the he/she gains y, which causes a social cost -y to the government. Therefore, the government's payoff is: u_1 = xfy - (1 - x)y - k_1‧x^2 where k_1‧x^2 is the effort cost incurred by the government, and the criminal's payoff is: u_2 = -xfy + (1 - x)y - k_2‧y^2 where k_2‧y^2 is the cost of committing y level of crime. They make the decisions simultaneously. a. [7%] Find the best response function for each player. Draw a figure to show both of them. b. [5%] What is the Nash equilibrium (x*,y*)? c. [8%] Determine whether each of the following statements is true or false by showing the changes of the best response functions and the equilibrium. Explain your finding. (i) Raising the penalty rate f can effectively reduce the crime level. (ii) The more difficult for the government to stop a crime (a higher k_1), the higher the crime level is. 4. [20%] Consider the following alternating-offer bargaining game with three periods. Two players, 1 and 2 want to divide $100. The game proceeds as follows: Period 1: Player 1 first makes a proposal, (x,100 - x), which means that player 1 gets x and player 2 gets 100 - x. If player 2 accepts the proposal, then each player obtains what is proposed. If player 2 rejects it, they continue to the next period. Period 2: Player 2 makes another proposal, (y,100 - y). If player 1 accepts it, then each player obtains what is proposed. If player 1 rejects it, they continue to next period. Period 3: Player 1 makes the final proposal, (z,100 - z). If player 2 accepts it, then each player obtains what is proposed. If player 2 rejects it, the game ends and they split $100 equally, which means that they each obtain $50. There is a discount factor δ_i ∈ (0,1) for player i between any two periods. a. [3%] Draw the extensive form of this game. b. [13%] Fully characterize the SPNE of this game. In which period will the proposal pass? How much does each player obtain in the equilibrium? c. [4%] Discuss the effect of the level of "patience" on the players' equilibrium payoffs. More specifically, when δ_1 increases (given δ_2), who will get more and who will get less? When δ_2 increases (given δ_1) , who will get more and who will get less? Explain your finding. 5. [20%] Consider the lawsuit game taught in class with some modification. A plaintiff (player 1, she) first decides whether or not to bring a lawsuit against a defendant (player 2, he). If no lawsuit is filed, then they both obtain 0. If a lawsuit is filed, then it costs c to the plaintiff. She then makes a settlement offer of s > 0, and the defendant decides whether to accept it or to reject it. If the defendant accepts the offer, then he needs to pay s to the plaintiff. If the defendant rejects the offer, the plaintiff then decides either to give up or to go to trial. If the plaintiff gives up, then no money transfer is made (although the plaintiff still needs to pay c). If the case goes to trial, the plaintiff receives a compensation from the defendant with an amount x if she wins, while she receives nothing if she loses. It is believed that the plaintiff will win the case with probability γ and lose with probability 1 - γ. Moreover, during the trial, the plaintiff incurs an attorney fee k_p, and the defendant incurs an attorney fee k_d. However, after the outcome of the trial is determined, the loser needs to pay the attorney fees for both players. That is, if the plaintiff wins the lawsuit, the defendant needs to pay the fees k_p + k_d and the plaintiff does not pay any fee; while if the defendant wins, the plaintiff needs to pay the fees k_p + k_d while the defendant does not pay any fee. a. [3%] Draw the extensive form of this game. b. [11%] Fully characterize the SPNE of the game. In particular, describe the conditions for the equilibrium where a positive amount of settlement is made and accepted, and those for the equilibrium where no lawsuit is filed. c. [6%] Determine each of the following statements is true or false. Explain your finding. (i) The higher probability that the plaintiff wins the case (γ is higher ), the more likely that the lawsuit is filed and a settlement is made in the equilibrium. (ii) The attorney fee for the plaintiff k_p does not affect the amount of settlement (if it takes place in the equilibrium). --

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