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）︰2012.10.12 考试时限（分钟）：60 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 总分22分 答题皆须附说明 未作解释的答案概不计分 1. (2 points) In the following normal form, the 1st (2nd) element in the payoff vector is the payoff to the row (column) player. Find out all the Nash equilibrium. │ t1 t2 ─────── s1│ 1,2 3,-1 s2│ 4,-2 1,5 2. (3 points) Consider Nim. There are 8 matches in the 1st row, 2 matches in the 2nd row, 13 matches in the 3rd row and 5 matches in the 4th row. What should player 1 do in his very first move to win the game. 3.In the following strictly competitive game, the preference of player1 is W>1D>1L. 1 L ╱ ╲R 　 ╱ ╲ 2　　 2 l ╱m|╲ r l ╱m|╲ r ╱ | ╲ ╱ | ╲ W D L W D W (a) (1 point) In the normal form (or strategic form, how many pure strategies does player 2 have? (b) (2 points) What is the value of the game? (c) (2 points) Please find a saddle point in the normal form. 4. A,B,C and D sit in a circle. Each of them wears a hat that is either white or black. A person who could see the hats of other 3 persons, but not his own. E will tell them whether all of them wear black hats or not, and then E will start to count time. Every one minute (long enough for their mental calculation) ,the one who realizes the color of his hat will raise a hand. (a) (1 point) How many different states of the world are in the universe? (b) The true state (ω*) is that A and B wear white hats, and C and D wear black hats. i. (1 point) Before E counts time what is A's possibility set of ω*, P (ω*)? Please define your notation clearly. A ii. (1 point) Will there be any one to raise a hand after the first minute? And who? iii. (2 points) Will there be any one to raise a hand after the second minute ? And who? iv. (2 points) Will there be any one to raise a hand after the third minute? And who? 5. Consider the following axioms: (K0) KΩ = Ω (K1) K(E∩F) = KE∩KF (K2) KE 包含於 E 2 (K3) KE 包含於 K E (K4) PE 包含於 KPE where P is the possibility operator, K is the knowledge operator and E and F are events. (a) (3 points) Please prove that S is a truism then ~S is a truism. Clearly state your reasoning for each step. You can use any axiom except (K3). (b) (2 points) Please prove that (K3) is implied by (K2) and (K4). You can cite result in (a) even if you fail to provide a proof for (a). --

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