課程名稱︰賽局論
課程性質︰大學部選修
課程教師︰古慧雯
開課系所︰經濟系
考試時間︰2006.05.01 星期一 下午1:20~3:10
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
總分41分
1.Consider the following zero-sum game. The payoff matrix only shows the
payoffs to the row player.
│ t1 t2 t3 (<==1 2 3 為下標)
─┼───────
s1│ 1 2 3
│
s2│ 6 5 4
│
s3│ 3 2 1
│
(a) (2分) What's the row player's security level?
(b) (2分) What's the column player's security level?
(c) (2分) What's the saddle point?
2.Consider 3 of Gale's Roulette wheels. Each wheel has 3 numbers on it, and
after spinning one wheel, it could stop at each number with equal chance.
Numbers on the 1st wheel are: 2, 4 and 9; on the 2nd wheel: 1, 6 and 8; and
on the 3rd wheel: 3, 5 and 7. Player I begins by choosing a wheel and
spinning it. While player I's wheel is still spinning, player II chooses one
of the remaining wheels and spin it. The player whose wheel stops on the
larger number wins. Each player wants to maximinize his probability to win.
(a) (4分) Draw a game tree for this game without chance moves.
(b) (2分) In a strategic form, how many different pure strategies does player
II have?
(c) (3分) In a subgame perfect equilibrium, what is player II's strategy?
(d) (2分) What's the value of this game?
3.考慮Ω={1,2,3,4,5,6,7,8}, A之possibility partition為{{1,2,3},{4,5},{6,7,8}},
B的為{{1,2},{3,4,5},{6,7},8}. 兩人皆誠實, 也都知道彼此的possibility partition
為何. 由A先開始, 兩人輪流報出其目前possibility set中元素的個數.
(a) (2分) A第一次報完之後, B之possibility partition為何?
(b) (2分) B第一次報完之後, A之possibility partition為何?
(c) (2分) 兩人各報三次以後, A在哪些情況下會知道true state為何?
4.(8分) 考慮Nash談判問題(X,d),X為談判空間,d為談判破裂時兩人的效用組合.Nash談判
解滿足以下4個公設：(A1)對個人效用做affine transformation不會影響談判的實質結
果; (A2)Pareto效率性與個人理性(individual rationality); (A3)independence of
irrelevant alternatives; (A4)對稱.
Raiffa-Kalsi-smorodinsky(RKS)定義的談判解G(X,d)如下. 令Ui表i之效用,
定義：
M1 ≡ max{U1: (U1,U2)為X的元素, U2≧d2}
M2 ≡ max{U2: (U1,U2)為X的元素, U1≧d1}
由此定義 M≧ (M1,M2). G(X,d) 即線段Md 與X邊界(boundary)之交點,如下圖所示
U2
│
M2│.............M
│ .
│ .
│ .
│ .
│.d .
┼──────────U1
M1
(X因為是個斜橢圓,在bbs上畫不太出來,只能大致上說一下他的位置)
(X以直線Md方向為長軸方向,中心大致上在M與d之間,但有些靠近d,
然後X這個橢圓剛好分別切在線段MM2與線段MM1)
請問RKS之解滿足哪幾個Nash的公設?
5.Let's play Nim. There are 4 rows of matches on the table. In the i-th row,
there are i matches. You move first. Show me you best to win.
(a) (2分) Before you start, is this game balanced or unbalanced?
(b) (2分) What will you do in your first move?
(c) (2分) In my first move, I'll take 1 match from the first row. What will
you do in the next round.
6.K is the knowledge operator. P is the possibility operator. and PE=~K~E.
Consider the following axioms.
(P0) P(空集合) = 空集合
(P1) P(E∪F) = PE∪PF
(P2) E 包含於 PE
(P3) PPE 包含於 PE
(P4) PKE 包含於 KE
Please use these 5 axioms to prove the following. State clearly which axiom
you need for the proof.
(a) (兩分) If S is a truism, then PS = S.
(b) (兩分) If S is a truism, then ~S is also a truism. (You could quoto (a)
even you fail to prove it.)
Game13.ctx
※ 編輯: greensunlife 來自: 140.112.250.104 (07/01 18:00)