課程名稱︰賽局論 課程性質︰選修 課程教師︰黃景沂 開課學院：經濟學系 開課系所︰社會科學院 考試日期（年月日）︰2021.11.29 考試時限（分鐘）： 試題 : You have to provide explanations for all of your answers. You can answer in either English or tradi- tional Chinese. 1. Consider the following normal-form game and assume that rationality is common knowledge. （表格見下圖） https://i.imgur.com/LDNpVWJ.jpg When answering parts (a)–(c), assume that player only use pure strategies. (a) (10%) Which strategy profiles are rationalizable? (b) (5%) Which strategy profiles survive iterated elimination of strictly dominated strategies? (c) (5%) Find out all strategy profiles which are Pareto optimal in this game. (d) (10%) If we also consider mixed-strategies, which strategy profiles survive iterated elimina- tion of strictly dominated strategies? 2. Three firms are considering entering a new market. The revenue for each firm that enters is 240 , n where n is the number of firms that enter. The cost of entering is 105 for each firm. The payoff of an entering firm is its revenue minus its cost. The payoff is zero if a firm does not enter. (a) (15%) Find all the pure-strategy Nash equilibria. (b) (10%) Find the symmetric mix-strategy Nash equilibrium in which all three players enter with the same probability. 3. (15%) Please prove the following proposition: Consider a strategy profile s = (s 1, s 2, . . . , s n) in a complete information static game. If s is a strictly dominant strategy equilibrium, then s is the unique Nash equilibrium. 4. Three oligopolists operate in a market with inverse demand given by P (Q) = 100 Q, where Q = q1 + q2 + q3, and qi is the quantity produced by firm i. Each firm has a constant marginal cost of production $10, and no fixed cost. The firms choose their quantities dynamically as follows: Firm 1, which is the industry leader, chooses q1 0. Firms 2 and 3 observe q1 and then simultaneously choose q2 and q3, respectively. (a) (5%) How many proper subgames does this dynamic game have? Explain briefly. (b) (5%) Is it a game of perfect information or imperfect information? Explain briefly. (c) (10%) What is the subgame-perfect Nash equilibrium of this game? Show that it is unique. (d) (10%) Find a Nash equilibrium that is not a subgame-perfect equilibrium. 參考解答： https://i.imgur.com/wqseDj7.jpg https://i.imgur.com/iC01nmi.jpg https://i.imgur.com/MJqqjVl.jpg ----- Sent from JPTT on my iPhone --

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