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）︰109.11.09 考試時限（分鐘）：180 試題 : 註：部分數學符號與式子以LaTeX語法表示。 1. Two roommates each need to choose to clean their apartment, and each can ch- oose an amount of time t_i \in [0, 10] to clean. If their choice are t_i and t_j, then player i's payoff is (10 - t_j)t_i - t_i^2. (This payoff function im- plies that the more one roommate cleans, the less valuable is cleaning for the other roommate.) Consider only pure strategies for the following questions. (a) (10%) What is the best response correspondence of each player i? (b) (10%) Which choices survive one round of IESDS? (c) (10%) Which choices survive IESDS? (d) (5%) Which choices survive the first round elimination of rationalizability 2. Consider the following Bertrand game. There are two firms producing an iden- tical product. Each firm posts a price for the product simultaneously. Suppose that there is a minimal unit of $0.1 for the prices. Therefore, p_1, p_2 \in { 0, 0.1, 0.2, ...}. Assume that demand is given by q = 100 - p. The cost functi- on is c_1(q_1) = 10q_1 for firm 1 and c_2(q_2) = 12q_2 for firm 2. All buyers buy from the firm whose price is the lowest, p = \min {p_1, p_2}. When p_1 = p_2, the market is splits equally between the two firms. Assume that both firms only use pure strategies. (a) (10%) What is firm 1's best response correspondence BR_1(p_2)? (b) (5%) What is firm 2's best response correspondence BR_2(p_1)? (c) (10%) What are the Nash equilibria in this game? 3. Two players are playing the rock-paper-scissors game. We use R (rock), P (p- aper), and S (scissors) to denote the three possible actions. Their payoffs can be described by the following matrix. https://imgur.com/QBrTwxO (a) (5%) Please find out all pure-strategy Nash equilibria of this game. (b) (10%) Please find out all mixed-strategy Nash equilibria of this game. 4. Consider the following game of cards. Player 1 and 2 put a dollar each in a pot, and player 1 pulls a card out of a deck of kings and aces, with an equal probability of getting a king (K) or an ace (A). Player 1 observes his card and then decides whether not to play the game (N), and forfeit his dollar to player 2, or proceed with the game (Y). If player 1 proceeds with the game, then with- out knowing which card player 1 drew player 2 can fold (F) and forfeit his dol- lar to player 1 or call (C), in which case each player must add another dollar to the pot. After this, if player 1 has a king then player 2 wins the pot, whi- le if player 1 has an ace then player 1 wins the pot. As we mentioned in the class, we can express this game as the following game t- ree. https://imgur.com/undefined (a) (10%) Please present this dynamic game in a game matrix. (b) (10%) What are the Nash equilibria of this game (including both pure and m- ixed strategies)? (c) (5%) What are the subgame perfect Nash equilibria of this game (including both pure and mixed strategies)? --

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