[試題] 110-1 黃景沂 賽局論 期末考
課程名稱︰賽局論
課程性質︰選修
課程教師︰黃景沂
開課學院：社會科學院
開課系所︰經濟學系
考試日期（年月日）︰2022.1.10
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 Chinese.
1. Two firms (i = 1, 2) simultaneously plan to open a store in a new market. Each firm can choose either to open (O) or not to open (N). In addition, each firm is either strong (S) or weak (W) with equal probability, and the realizations for each firm are independent. Furthermore, the type of each firm is known only to that firm’s CEO.
When a firm chooses not to enter the market, its payoff is zero. When a firm enters as a monopoly, its payoff is 12 if it is strong, and 4 if it is weak. When both firms enter to compete, there are three possible cases: each firm’s payoff is 4 if both are strong; each firm’s payoff is -4 if both are weak; the strong firm gets 8 and the weak firm gets -8 if one is strong and the other one is weak.
(a) (5%) How many pure strategies does Firm 1 have in the normal form of this game?
(b) (10%) How to represent the normal form of the game in a matrix?
(c) (10%) Please fund out all pure-strategy Bayesian Nash equilibria of this game. What is the probability of duopoly in equilibrium?
2. Imagine an auction for a single item with n bidders whose valuations θi are drawn independently uniformly from the interval [0, 1]. Assume that the players are risk averse and that the utility to player of type θi from winning the item at price p is √θi p for p θi. (We will assume that no bidder ever bids or pays more than his valuation so that θi p 0.)
(a) (5%) Write down a bidder’s expected payoff function for the first- and second-price sealed- bid auctions when bidder i’s submitted bid is represented as bi for i = 1, 2, · · · n.
(b) (10%) Show that in the first-price sealed-bid auction each bidder of type θi choosing the bidding strategy bi = s(θi) = 2n 2θi forms a symmetric Bayesian Nash equilibrium.
2n 1
(c) (5%) Show that in a second-price sealed-bid auction it is a (weakly) dominant strategy for
each bidder to bid his valuation θi.
(d) (5%) What is the seller’s expected revenue from the first-price sealed-bid auction?
3. Dutch auctions and first-price sealed-bid auctions are strategically equivalent under the (IPV) independent private value setting.
(a) (10%) Please explain the reason.
(b) (5%) When bidders’ payoffs have a common value component, does this equivalence result still hold? Why or why not?
4. Consider the game shown in the follow game tree. Nature chooses player 1’s and 2’s types, which affect their payoffs. The probabilities of moving to nodes x1 and x2 are 13 and 23, respectively.
（樹狀圖見下）
https://i.imgur.com/laEVCVf.jpg
(a) (10%) 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) (10%) What are the perfect Bayesian Nash equilibria in this game?
(e) (5%) What are the sequential equilibria in this game?
https://i.imgur.com/43k2JoL.jpg
(a) (10%) 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) (10%) What are the perfect Bayesian Nash equilibria in this game?
(e) (5%) What are the sequential equilibria in this game?
參考解答：
https://i.imgur.com/6RdHNGV.jpg
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