清华大学经济博弈论期末考试05

经济博弈论（2005年秋季学期）

期末测验题

(2005/12/29)

注意：请将所有题目的答案写在答题册上，写在本试题页上一律无效（需要的图表请重画）。

1、(20 points) The following is an interpretation of the rivalry between the United States and the Soviet Union for geopolitical influence in the 1970s and 1980s. Each side has the choice of two strategies: Aggressive and Restrained. The Soviet Union wants to achieve world domination, so being Aggressive is its dominant strategy. The United States wants to prevent the Soviet Union from achieving world domination; it will match Soviet aggressiveness with aggressiveness, and restraint with restraint. Specifically, the payoff table is:

Soviet Union

Restrained Aggressive

United States

Restrained 4, 3 1, 4

Aggressive 3, 1 2, 2

For each player, 4 is best and 1 is worst.

(a) Consider this game when the two countries move simultaneously. Find the Nash

equilibrium.(5 points)

(b) Next consider three different and alternative ways in which the game could be played with

sequential moves: (i) The United States moves first and the Soviet Union moves second. (ii) The Soviet Union moves first and the United States moves second. (iii) The Soviet Union moves first and the United States moves second, but the Soviet Union has a further move in which it can change its first move. For each case, draw the game tree and find the subgame-perfect equilibrium. (3 points each for (i) and (ii); 5 points for (iii)

(c) What are the key strategic matters (commitment, credibility, and so on) for the two

countries?( 4 points)

(a) Soviet Union has a dominant strategy of Aggressive and the Unique Nash equilibrium is (Aggressive, Aggressive), with payoffs (2, 2).

(b) See the attached figure.

(c) Commitment for the SU; or a promise for the SU “ R if R”. US have nothing to do. He cannot commit to move first (then the SU will choose his dominant strategy of Aggressive, the outcome remains to be the status quo); or not be necessary to threat or promise (since if the SU moves first, he must choose to Restrain, and the US need only to follow its best choice.)

2、(20 points) Consider the following game. A neutral referee runs the game. There are two players, Row and Column. The referee gives two cards to each: 2 and 7 to Row and 4 and 8 to Column. This is common knowledge. Then, playing simultaneously and independently, each player is asked to hand over to the referee either his high card or his low card. The referee hands out payoffs – which come from a central kitty, not from the players – that are measured in dollars and depend on the cards that he collects. If Row chooses his Low card 2, then Row gets $2; if he chooses his

High card 7, then Column gets $7. If Column chooses his Low card 4, then Column gets $4; if he

chooses his High card 8, then Row gets $8.

(a) Draw out the payoff table. (4 points)

(b) What is the Nash equilibrium? Is it a prisoners’ dilemma? (4 points)

Now suppose that the game has the following stages. The referee hands out cards as before; who

gets what cards is common knowledge. Then at stage I, each player, out of his own pocket, can

hand over a sum of money, which the referee is to hold in an escrow account. This amount can be

zero but not negative. The rules for the treatment of these sums are as follows and are also

common knowledge. If Column chooses his High card, then the referee hands over to Column the

sum put by Row in the escrow account; if Column chooses Low, then Row’s sum reverts back to

Row. The disposition of the sum deposited by Column depends similarly on what Row does. At

stage II, each player hands over one of his cards, High or Low, to the referee. Then the referee

hands out payoffs from the central kitty, according to the payoff table in section (a), and disposes

of the sums (if any) in the escrow account, according to the rules just given.

(c) Find the rollback (subgame-perfect) equilibrium of this two-stage game. Does it resolve the

prisoners’ dilemma? What is the role of the escrow account? (12 points)

(a)

COLUMN

Low High ROW

Low 2,

4 10,

High 0, 11 8, 7

(b) (Low, Low). It is a prisoners’ dilemma.

(c) The second stage game can be shown as the following table (2 points):

COLUMN

Low High ROW

x

Low 2,

4 10-x,

8-x+y,

7+x-y

11-y

High y,

(x is Row’s sum deposited in the escrow account, y is Column’s.)

If x<4, Column’s dominant strategy is Low; If y<2, Row’s dominant strategy is Low. Thus (Low,

Low) will be the final second-stage equilibrium, with payoffs (2, 4).

Similarly, if x>4, and y<2, the equilibrium would be (Low, High), with payoffs (10-x, x). If such a equilibrium is really wanted by both. Row will decrease x as much as possible, so x=4+ε. (ε>0

and can be as small as possible) The payoffs thus are 6-ε and 4+ε respectively.

If x<4 and y>2, the equilibrium would be (High, Low), with optimal payoffs (2+ε, 9-ε), where

y=2+ε.

If x>4 and y>2, the equilibrium would be (high, high), with payoffs (6, 9), where x=4+ε, y=2+ε.

(4 points)

So the payoff table for the first stage is (2 points),

COLUMN

<2 >2

ROW

4

<4 2,

2+ε, 9-ε

>4 6-ε, 4+ε6, 9

Each player has a dominant strategy of “>4” and “>2”. The only Nash equilibrium is (“>4”, “>2”), with payoffs (6, 9) (or numbers very close to 6 and 9). (2 points)

Yes, it solves the prisoners’ dilemma. The escrow account is used to reward one who plays High.

(2 points)

3、(20 points) Ali and Baba are bargaining to split a total that starts out at $100. Ali makes the first offer, stating how the $100 will be divided between them. If Baba accepts this offer, the game is over. If Baba rejects it, a dollar is withdrawn from the total, and so it is now only $99. Then Baba gets the second turn to make an offer of a division. The turns alternate in this way, a dollar being removed from the total after each rejection. Ali’s BATNA is $2.25 and Baba’s BATNA is $3.50. What is the roll back equilibrium outcome of the game? (Please write down your process, otherwise you can only get half of the points.)

Since the sum of two parties’ BATNA is 2.25+3.5=5.75, is bigger than 5 and less than 6. The bargaining game will end at round 95, where v=6.In round 96, the bargaining will breakdown and each takes his BATNA.

In round 95, v=6. A makes the offer and will give B his BATNA, $3.5. A keeps $2.5.

In round 94, v=7. B makes the offer and will give A $2.5. B keeps $4.5.

In round 93, v=8. A makes the offer and will give B $4.5=3.5+(95-93)/2. A keeps 3.5=2.5+(95-93)/2

……

In round 1, v=100. A makes the offer and will give B 3.5+(95-1)/2=50.5. A keeps 2.5+(95-1)/2=49.5.

As a whole, the agreement will reach in round 1. A gets $49.5 and B gets $50.5.

4、（15分）魏国的统帅司马懿举大军兵临蜀国城下，兵力空虚的蜀国危在旦夕。蜀国的军师诸葛亮想出奇计：打开所有城门以迷惑敌人。果然，多疑的司马懿担心有伏兵，放弃了进攻，从而蜀国得以保全。这就是有名的“空城计”的故事。下面试图对其中的策略博弈进行分析。

考虑有两种类型的蜀国军队——强大或软弱。只有蜀国军队自己知道自己属于哪种类型就；魏国军队并不知道，但有一个事前的概率估计，蜀军强大的概率为p。蜀军有两种选择：打开城门和紧闭城门。一旦魏军选择了进攻，对于软弱的蜀军来说，都会被击败，但紧闭城门比打开城门损失更小，设此时打开和紧闭成本的损失

．．各为A和B，A>B>0。而一旦魏军选择了进攻，对于强大的蜀军来说，都能击败魏军，但打开城门比紧闭城门更有利于蜀

军，设此时打开和紧闭城门对蜀军的收益

．．各为C和D， C>D>0。魏军任何时候的收益都是蜀军收益的相反数（二者收益之和总为零）。如果魏军选择了不进攻，则任何情况下双方都得零。

（1）画出该博弈的博弈树。（3分）

如图所示。

（2）用图形和文字共同说明两种类型的分离均衡（强大的蜀军选择打开城门，弱小的蜀军选择紧闭城门；或者相反）均不存在。并直观地解释之。（4分）

如图。对于第一种分离均衡，魏军会在看见城门打开时断定蜀军是强大的，从而放弃进攻，而在看见城门紧闭时断定蜀军是软弱的，从而进攻。这样，软弱的蜀军肯定会偏离到打开城门。不仅如此，可以分析强大的蜀军也会偏离到紧闭城门。因此该均衡不成立。

类似分析第二种分离均衡不成立（图形略）。

原因是分离均衡均使得软弱的蜀军被进攻，而强大的蜀军不被进攻，这都不是蜀军想要得到的。这类似于零和博弈的“让对手猜”的策略。

（3）用图形和文字共同说明当p

如图。假设这样的混同均衡存在。则在给定的条件p-A。因此，该混同均衡不成立。

更容易存在，因为满足条件的p值的范围缩小了。

（4）求出半分离均衡“强大的蜀军选择打开城门，弱小的蜀军选择打开和紧闭城门的混合策略”，并说明这个均衡成立的条件。（4分）

如图。求得：x=Cp/[A(1-p)]，y=B/A。要使x<1，必须Cp

验证这个均衡成立，得到另一个条件：D/C

5、（15分）一家银行有两类储蓄——“短期储蓄”和“长期储蓄”。短期储蓄存期为1年，长期储蓄存期为两年。年利率（不含本金）为r (r>0)；为简便起见，不计算复利，因此短期和长期储蓄的利率分别为r和2r。在所有全部N个储户中，有比例为t（00)。（15分）（1）如果银行的利润为零，说明储蓄和贷款利率满足：t(1+r)+(1-t)(1+2r)/(1+R)=1。写出表示r和R大小关系的不等式。并解释之。（2分）

银行零利润应该使得它在第二年末的支付等于其净收入，而净收入是第一年末支付短期储蓄者后余额所产生的本息。即：N (1-t) (1+2r)=N[1-t(1+r)](1+R)。变换得上式。r和R的关系满足：2r

（2）考虑在第一年末。此时，所有短期储蓄者都要取走全部本息。长期储蓄者实际上也面临两种选择：提前支取本息，或者继续储蓄到第二年结束。如果提前支取，在储蓄余额大于零时，银行仍然按照利率r支付；一旦储蓄余额小于零，银行就会因为“挤兑”而破产，假设此时选择在第一年末支取的全部储蓄者平分全部的储蓄额。

用字母x表示长期

．．储蓄者中提前支取者的比例。这一比例为多大时，第一年末的储蓄余额将正好变成零？记这一比例为 x。写出当x> x时，提前支取者能够获得的本息额？当所有人都提前支取时，这一金额是多大？（3分）

x=[1-t(1+r)]/[(1-t)(1+r)]。

这一本息额设为a, 满足1=ta+x(1-t)a，则a=1/[t+x(1-t)]。

所有人都提前支取时，这一金额为1。

（3）刚才讨论的是长期储蓄者中提前支取者的收益，下面讨论继续储蓄到第二年的人的收益。他们只能获得第一年末支付提前支取者本息后剩下的储蓄额所产生的本息。

写出他们的收益与x的关系式。（2分）

设这一金额为b。当x> x时，显然等于零。

当x< x时，可以求得b={1-[t+x(1-t)](1+r)}(1+R)/[(1-x)(1-t)]，也可以将第（1）问求得的R 代入，得到b={1-[t+x(1-t)](1+r)}(1+2r)/ {(1-x)[1-t(1+r)]}。注意到，当x=0时，b=1+2r。

（4）利用适当的图形，求出所有长期

．．储蓄者参与的集体博弈的全部（稳定的）均衡解。

有几个均衡解？长期储蓄者的收益各为多少？哪一个（些）均衡对长期储蓄者更有利？（6分）

如图。这是一个保证博弈。有两个均衡：所有长期储蓄者都不提前支取，各自获得1+2r；所有长期储蓄者都提前支取，各自获得1（这是一个“挤兑”均衡）。显然，前者对提前支取者更有利。

（5）现在政府决定对对提前支取的长期储蓄者征税。这将如何影响集体博弈取得不同均衡的可能性？是否有利于防止“挤兑”？（2分）

这会使得图形中提前支取的收益曲线下降。使得在更大范围的x初始值下结果都趋向于好的均衡。这有利于防止“挤兑”。

6、（10分）一个清华学生经常从城铁“大钟寺”站上车，而在“五道口”站下车返校。城铁从大钟寺站出发，途中依次经过的车站是“知春路”、“五道口”、“上地”、“回龙观”、“霍营”（终点站）。如果他（她）下错了站，不能及时返回学校，损失均为-1。而如果正确地从五道口站下车，则所得为1。

（1）画出只包含这一个学生的博弈树。求出反转均衡，这个同学的收益是多少？（1分）如图（注意哪些分支打勾不能画错）。收益为1。

（2）某一天，这个同学仍然按照通常的路线返校。但不幸的是，他喝醉了。从列车驶出大钟寺门站开始，每当车门打开，他（她）都不记得已经过了多少站了。当然，如果到了终点站，列车员会要求他必须下车。画出此时的博弈树。（提示：利用信息集(information set)的概念。）（2分）

如图。

（3）在这一新的博弈中，该学生都有哪些纯策略？所有可能的混合策略是哪些？（2分）该同学有两个纯策略：下车和继续（不下车）。所有的混合策略可表述为：以p的概率选择继续，以1-p的概率选择下车。0

（4）该学生从他的每一个纯策略中所得各为多少？该学生从它的混合策略中所得为多少？（3分）

下车纯策略带来收益-1（因为肯定会在知春路下车）；继续纯策略带来收益-1（肯定会在终点站下车）。

混合策略的收益表述为：-1(1-p)+p(1-p)-p2(1-p)-p3(1-p)-p4= -1+2p-2p2。

（5）求出该学生的最优（纯或者混合）策略。他的最优策略是混合策略还是纯策略？并直观地解释为什么出这一类．策略。（2分）

最优的策略是混合策略：p*=1/2，收益为-1/2。因为任何纯策略都只能使之得到-1（坐错站），而一个混合的策略至少可能“坐对站”而改进其收益。