推 tigerflyer :數學系的好像比較難耶 11/22 10:56
推 TINTINH :已收精華區:) 11/22 11:56
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）︰2009.11.13
考試時限（分鐘）：100
是否需發放獎勵金：是
試題 :
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.
註:≦皆代表屬於
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