課程名稱︰賽局論 課程性質︰經濟系選修 課程教師︰黃景沂 開課學院：社科院 開課系所︰經濟系 考試日期（年月日）︰110.01.11 考試時限（分鐘）：120 試題 : 註：以下部分數學符號與式子以LaTeX語法表示。 1. Consider the Cournot duopoly model in which two firms, 1 and 2, simultaneou- sly choose the quantities they supply, q_1 and q_2. The price each will face is determined by the market demand function p(q_1, q_2) = a - b(q_1 + q_2). Both firms have zero fixed cost. Each firm has a probability 1/2 of having a margin- al unit cost of c_L and a probability 1/2 of having a marginal unit cost c_H. The distribution of the marginal unit cost is independent across the two firms. These probabilities are common knowledge, but the true type is revealed only to each firm individually. (a) (5%) When you represent this static incomplete information in the normal f- orm, what is the type-dependent payoff function for Frim 1? (b) (10%) What are the players' pure strategies in the Bayesian Nash equilibri- um. (c) (5%) What is the ex ante expected price in the Bayesian Nash equilibrium? 2. Imagine an auction for a single item with n bidders whose valuations \theta_ i are drawn independently uniformly from the interval [0,1]. Assume that the p- layers are risk averse and that the utility to player of type \theta_i from wi- nning the item at price p is \sqrt{\theta_i - p} for p \leq \theta_i. (We will assume that no bidder ever bids or pays more than his valuation.) (a) (5%) Write down a bidder;s expected payoff function for the first- and sec- ond-price sealed-bid auctions when bidder i's submitted bid is represented as as b_i for i = 1,...,n. (b) (10%) Show that in the first-price sealed-bid auction each bidder of type \theta_i choosing the bidding strategy b_i = s(\theta_i) = (2n-2)\theta_i/(2n-1) forms a symmetric Bayesian Nash equilibrium. (c) (5%) Show that in a second-price sealed-bid auction it is a (weakly) domin- ant strategy for each bidder to bid his valuation \theta_i. (d) (5%) What is the seller's expected revenue from the first-price sealed-bid auction? 3. Please answer the following questions on "winner's curse" in auctions? (a) (5%) Please explain what is "winner's curse"? (b) (5%) When you participate in an auction, how could you avoid winner's curse ? 4. Consider a nation with n citizens. Suppose that the value of spending x \in R amount of the government budget on national defense is (x - \theta_i)^2 for i = 1,...,n where \theta_i \geq 0 is citizen i's private information. Each cit- izen's payoff is (x - \theta_i)^2 - t_i, where t_i \in R is his/her tax payment . We allow t_i to be negative when citizen receives a subsidy from the governm- ent. Suppose the government cannot observe \theta = (\theta_1,...,\theta_n) and wants to design a tax scheme to achieve the Pareto optimal outcome. (a) (5%) What is the first-best decision rule on the amount of national defense x^*(\theta)? (b) (10%) Please design a mechanism to implement x^*(\theta) in dominant strat- egies. (c) (5%) Verify that the mechanism you designed in part (b) actually implement x^*(\theta) in dominant strategies. 5. COnsider the game shown in the following game tree. Nature chooses player 1's and 2's types, which affect their payoffs. The probabilities of moving to node x_1 and x_2 are 1/4 and 3/4, respectively. https://imgur.com/undefined (a) (5%) How to represent the normal form of the game in a matrix? (b) (5%) What are the Bayesian Nash equilibria in this game? (c) (5%) What are the subgame-perfect Nash equilibria in this game? (d) (5%) What are the perfect Bayesian Nash equilibria in this game? (e) (5%) What are the sequential equilibria in this game? --

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