課程名稱︰賽局論 課程性質︰選修 課程教師︰古慧雯 開課學院：社會科學院 開課系所︰經濟學系 考試日期（年月日）︰2012.10.12 考試時限（分鐘）：60 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 總分22分 答題皆須附說明 未作解釋的答案概不計分 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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