课程名称︰赛局论 课程性质︰选修 课程教师︰古慧雯 开课学院：社会科学院 开课系所︰经济学系 考试日期（年月日）︰2009.11.13 考试时限（分钟）：100 是否需发放奖励金：是 试题 : 1.(4 points)In the following game, the first elements in the payoff vector is the payoff to A, and the second elements is the payoff to B. Ａ ╱｜╲ 　 a1╱ ｜a2╲ a3 Ｂ　　3,5 Ｂ b1 ╱╲ b2 b3 ╱╲ b4 ╱ ╲ ╱ ╲ 1,2 2,1 2,1 1,2 (a)In the subgame perfect equilibrium, what is A's payoff? (b)In the normal form, B has 4 different pure strategies. Name one. 2.(6 points)Find all the Nash equilibria in the following game. (The first (second) element in the payoff vector is the payoff to the row(column) player.) │ l r ─┼──── u│2,3 1,1 d│1,1 4,2 3.(6 points)Gino and Polly always tell the truth. The state space is Ω={1,2,3,4,5,6,7,8,9}. The players' initial posibility partitions are: Gino's={{1,2,3},{4,5,6},{7,8,9}}, Polly's={{1,2,3,4},{5,6,7,8},9}. The players alternate in announcing how many elements their current possibility set contains and Gino makes the first announcement. (a)What's Polly's possibility partition after Gino's first announcement? (b)What's Gino's possibility partition after Polly's first announcement? (c)What's Polly's possibility partition after Gino's second announcement? 4.(4 points)Consider the game NIM with 8 rows of matches. In the i-th row, there are i matches. To win for sure, what should the first player do in the first round? 5.(6 points)In a duel, each player holds a pistol with one bullet. The distance between them is D when the duel starts, and they start to approach each other as time passes by. Let Pi(d) denote the probability that player i could hit his opponent when he fires from distance d. Pi(D)=0, Pi(0)=1 and Pi'<0. The most important thing for each player is to maximize hos own survival probability. The next most important thing is to minimize his opponent's survival probability. (a)In a Nash equilibrium, could one player fire before his opponent fires? (b)In a Nash equilibrium, could both players fire from distance d where P1(d)+P2(d)>1? 6.(4 points)Consider a two-person zero-sum game. Let M denote the payoff matrix which element is the payoff to the row player. The set off all row player's mixed strategies is denoted by P, and the set of all column player's mixed strategies is T Π(p,q)=p Mq, ~ where p≦P and q≦Q. The maximin v and the row player's security strategy p are define by: ¯ ~ v = max min Π(p,q) = min Π(p,q). ¯ p≦P q≦Q q≦Q ~T Please prove that there exists a column M.j in M such that p M,j=v. ¯ 7.(6 points)Consider the folloing strategic form of a two-person zero-sum game. The shown payoff is the payoff to the row player. │t1 t2 t3 t4 t5 ──┼─────── s1 │ 1 2 3 4 5 s2 │ 9 7 5 3 1 (a)Find the row player's mixed security strategy. (b)Is it possible that the column player only use t1 and t3 in his mixed security strategy? 8.(4 points)On page 355 in the text. it is claimed: Property (K3) is really redundant because it can be deduced from (K2) and (K4). where we have: (K2) KE≦E (K3) KE≦K^2E (K4) PE≦KPE Prove the claim. 注:≦皆代表属於 --

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