課程名稱︰賽局論 課程性質︰選修 課程教師︰古慧雯 開課學院：社科院 開課系所︰經濟系 考試日期（年月日）︰2008/4/18 考試時限（分鐘）：110 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 總分42分。答題須附說明，未做解釋的答案概不計分。 1. Consider the following game １ A ╱╲ B Regarding the outcome, player 1's preference ordering ╱ ╲ is: Ｗ ＞1 Ｄ ＞1 Ｌ, and player 2's preference Ｄ　　　２ ordering is: Ｌ ＞2 Ｄ ＞2 Ｗ a ╱╲ b ╱ ╲ Ｗ Ｌ (a) (2 points) What is the value of the game? (b) (4 points) Please write out the strategic form of the game, and find the saddle point of the strategic form. (c) Now, turn to consider the following zero-sum game. The payoff in the following strategic form is the payoff to the row player. Each player only considers to use a pure strategy. | a | b ---------- A | 1 | 0 B | 0 | 1 i. (2 points) What is row player's security level? ii. (2 points) You could verify that different from the game in part (b), there is no saddle point in this zero-sum game. But at the beginning of this course, didn't we always talk about the existence of value in a strictly competitive game? So this zero-sum game must violate some assumption that is used to prove the existence theorem. Which assumption does it violate? 2. (6 points) Consider the game Nim. There are 4 rows of matches on the table. The first row has 10 matches, the second row has 15 matches, the third row has 2 matches, and the fourth row has 3 matches. Two players take turns to take matches. When it is his turn, a player has to select a row and take n matches from that row, n>0. The player that has no more match to take loses. Which player has a winning strategy? And how should this player play in his first round? 3. (6 points) Consider the game Parchessi. Different from the rule in the text book, a player will roll a fair dice to decide his move. If the number of dice turns out to be smaller than, or equal to, 4, the player could move his counter two squares or pass. Otherwise, the player could move his counter one square or pass. In the following, the left-most board shows the route of each player. We also depict two possible positions A and B when the X player is about to roll the dice. Please calculate his winning probability in each position. ┌─┬─┐　　Ａ ┌─┬─┐　Ｂ ┌─┬─┐ │Ｘ╪╗│ │ │Ｘ│ │Ｘ│ │ 　 ├─┼╫┤ ├─┼─┤ ├─┼─┤ 　 │←╪╣│ │ │Ｏ│ │ │Ｏ│ 　　├─┼╫┤ ├─┼─┤ ├─┼─┤ 　　│Ｏ╪╝│ │ │ │ │ │ │ 　　└─┴─┘ └─┴─┘ └─┴─┘ 4. A, B and C sit in a circle. Each of them wears a hat that is either white or black. A person could see the hats of the other 2 persons, but not his own. They are told that at least one of them wears a white hat. D will start to count time, and every one minute (long enough for their mental calculation), the one who realizes the color of his hat will raise a hand. (a) (2 points) 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 wears a black hat. i. (2 points) Before D counts time, what is A's possibility set of ω*, Pa(ω*)? ii. (2 points) After the first minute, who will raise hand? iii. (2 points) After the second minute, who will raise hand? iv. (2 points) After the third minute, who will raise hand? 5. (10 points) Consider a finite universe Ω. Let ω denote an element of Ω. Let K and P denote the knowledge operator and the possibility operator, respectively. P(ω) is the possibility set. Consider the following axioms: (p0) Pψ=ψ (p1) P(E∪F)=PE∪PF (p2) E≦PE (註: ≦為集合符號) (p3) (P^2)E≦PE (p4) PKE≦KE and the additional result: P(ω) = P{ω}, all ω屬於Ω Consider an event E ≦ Ω, E≠ψ and E is NOT a truism. Please compare the number of elements in E, KE and PE. (If you use additional result or theorem, please state it clearly.) --

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