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）︰1/8/2010
考試時限（分鐘）：110分鐘
是否需發放獎勵金：是
試題 :
總分47分。答題皆須附說明，未做解釋的答案概不計分。
1.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 4,1
d│0,5 3,2
(a)(2 points) Find all the Nash equilibrium.
(b)Consider the game in (a) as a stage game.
i.(3 points) If this stage game is repeated twice, what is the row player's
strategy in a subgame perfect equilibrium?
ii.Now consider the game in (a) to be repeated infinitele many times.
A.(6 points) Suppose the row player and column player use strategies
depicted by the following automata, what is the payoff to the row player at
the 2nd stage? And what is the average payoff(in the long-run) to the row
player?
┌───┬────┐
↓ │r │r
┌─┐ l ┌─┐ l ┌─┐
───→│u │─→│d │─→│u │
└─┘ └─┘ └─┘
r│↑ ↑ │l
└┘ └────┘
┌┐ ┌┐
u│↓ d│↓
┌─┐ ┌─┐
───→│l │─→│r │
└─┘ └─┘
↑ │u
└────┘ _ _
B.(3 points) What is the minimax point (m1,m2) in the stage game?
C.(4 points) According to the folk theorem, please specify all the possible
average payoffs to the row player in a Nash equilibrium with pure
strategies.
2.Reconsider the story of dodo whose day is a fraction τ of a year. During the
daylight hours, dodos search for food. Due to the limited food supply, only N
dodos survive the day. In the evening, the surviving dodoes fight for nesting
sites in random pairs. The fighting determines thier reporduction rate in the
following manner:
│ g1 g2
─┼────────
g1│u+1,u+1 u+3,u-1
g2│u-1,u+3 u,u
For instance, when a dodo with gene g1 fights with a dodo with g2, next morning
, the g1-typed dodo will have τ(u+3) baby dodoes and the g2-typed dodo will
have τ(u-1) baby dodoes. The other fighting results could be similarly read
from the table. Let p(t) denote the proportion of g2-typed dodoes at time t
when dodoes just finish searching for food and haven't met their rivals for
the fight.
(a)(3 points) Given current p(t), how many babies does a g2-typed mother
expect next morning? _
(b)(3 points) It is known that on average a mother expects to have τf(p(t))
babies next morning, where _
f(p(t))=u+(1-p)(1+p)
What is p(t+τ)?
(c)(8 points) The replica equation is:
p'=-p(1-p)(2+p)
Give all the rest point(s) and asymptotic attractor(s).
3.Consider the Nash bargaining solution F(X,d) (X and d being the set of
feasible payoff pairs and the disagreement point, respectively) that satisfy
the following axioms:
˙Group rationality and individual rationality.
˙The affine transformation of a player's utility will not cause any change to
the real barganing result.
˙Independence of irrelevant alternatives.
˙Symmetry.
(a)(4 points) What is F(X',d') if X'={(x1,x2):x1+2x2≦11} and d'=(1,2)?
(b)(6 points) Consider affine transformations of two players' utilities
τ(x1,x2)=(τ(x1),τ(x2)) where
τ1(x1)=rx1+s,
τ2(x2)=tx2+u.
How could you choose r,s,t and u to make τ(X')={(x1,x2):x1+x2≦1} and
τ(d')=(0,0).
4.(5 points) Consider C≦R^2 and C is not a convex set. f:R^2→R^2 is a
one-to-one affine function. Is it possible that f(C) is a convex set? Give an
example if your answer is positive and give a proof if your answer is negative.
(≦表示屬於)
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