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︰經濟系
考試日期（年月日）︰103/6/20
考試時限（分鐘）：110分鐘
試題 :
賽局論期末考 (6/20/2014) 古慧雯
總分35分。答題皆須附說明，未做解釋的答案概不計分。
1. Reconsider the fitness game exemplified by the story of dodos. There are
two types of genes, s_1 and s_2. The fertility rate of a mother depends on
how she interacts with the opponent mother, and is given as follows:
│ s_1 s_2
─────────
s_1│ 0,0 1,0
s_2│ 0,1 2,2
(a) (2 points) Please find all Nash equilibrium of this fitness game.
(b) (2 points) Please find all evolutionarily stable strategies.
(c) (3 points) Let p_i denote the fraction of the population hosting gene s_i.
It is proved that the replicator equations are:
_
(p_i)' = p_i(f_i(p) – f(p)), i = 1, 2,
_
where f_i(p) is called "fitness" of type i, and f(p) is the average
fitness. Please give the replicator equation of p1 using data in the above
matrix.(No points are given for the following two questions, if you do not
answer this question correctly.)
(d) (2 points) What are rest points (p1, p2) of the replicator equations?
(e) (2 points) Find all asymptotic attractor(s) of the replicator dynamics.
2. The following game will be repeated twice:
│ slow speed
────────────
slow │ 4,4 1,5
speed│ 5,1 -2,-2
The repeated game payoffs are just the sum of the stage-game payoffs.
Consider a strategy s that tells you to choose slow at the 1st stage and to
use slow with probability p at the 2nd stage, unless the two players have
failed to use the same actions in the 1st stage. If such a coordination
failure has occurred in the past, s tells a player to use whatever action
that person didn't play at the 1st stage.
(a) (3 points) Which value of p will make (s, s) a Nash equilibrium?
(Credit is granted only when clear arguments are made.)
(b) (2 points) What does s ask a player to do if (speed, speed) is played in
the 1st stage? Is (s, s) a subgame perfect equilibrium
(with p you calculated above)?
(c) (2 points) Give a different subgame perfect equilibrium other than (s, s).
3. The following stage game will be repeated infinitely many times, and two
players care about the long-run average payoffs.
│ t_1 t_2
──────────
s_1│ 0,2 0,-3
s_2│ -1,-1 3,0
Instead of using strategies prescribed by finite automata, both players
consider to use mixed strategies.
(a) (2 points) The row player considers to coordinate their moves, and she
warns the column player that if he does not follow the plan, she will
punish him in the future. What is the row player's most sever punishing
strategy?
(b) (4 points) How should the Folk Theorem presented in the textbook be
modified in this situation? Draw a graph with two players' payoffs on two
axes and illustrate the result.
4. (5 points) Reconsider the finite version of the duel problem in the
textbook. Two players are armed with pistols loaded with just one bullet,
and they walk towards each other. The initial distance between them is D.
Each player is allowed to fire at either distance D, or at distance 0.
(Note they could fire at the same time.) The following table shows the
probability that player i will hit his target when he fires at distance d,
p_i(d), i = 1, 2.
d = │ 0 D
──────────
p_1(d)│ 1/2 1/3
p_2(d)│ 1/3 1/4
For each player, the primary goal is to maximize his own survival
probability, and the secondary goal is to minimize his opponent's survival
probability. In a subgame perfect equilibrium, what will player 1 do at
distance D?
5. Alice confiscates Bob's and Carol's wallets and uses an English auction to
sell their combined contents back to whoever bids higher. Bob's wallet
contains b dollars, and Carol's wallet contains c dollars, but they know
only how much was in their own wallet. We shall solve step by step a
symmetric equilibrium in which a person whose wallet contained x dollars
plans to bid up to B(x), where B(x) is a strictly increasing function of x.
(a) (2 points) If Carol quits first when the price is raised to $10, what
condition about b and B(.) can you infer (besides (B(b) > 10)?
(b) (4 points) Let p denote the price that Bob plans to quit, i.e. p = B(b).
From your answer above, write a condition that helps solve B(.) and solve
it.