博弈论习题集
博弈论习题(1-4)

博弈论作业题第一章4.“囚徒的困境”的内在根源是什么?举出现实中囚徒困境的具体例子。
5.博弈有哪些分类方法?有哪些主要的类型?9.你正在考虑是否投资100万元开设一家饭店。
假设情况是这样的:你决定开,则0.35的概率你将收益300万元(包括投资),而0.65的概率你将全部亏损掉;如果你不开,则你能保住本钱但也不会有利润。
请你(a )用得益矩阵和扩展表示该博弈;(b )如果你是风险中性的,你会怎样选择?(c )如果你是风险规避的,且期望得益的折扣系数为0.9,你的策略选择是什么?(d )如果你是风险偏好的,期望得益折扣系数为1.2,你的选择又是什么?10. 一逃犯从关押他的监狱中逃走,一看守奉命追捕。
如果逃犯逃跑有两条可选择的路线,看守只要追捕方向正确就一定能抓住逃犯。
逃犯逃脱可少坐10年牢,但一旦被抓住则要加刑10年;看守抓住逃犯能得1000元奖金。
请分别用得益矩阵和扩展形表示该博弈,并作简单分析。
第二章4.求出下图中得益矩阵所表示的博弈中的混合策略纳什均衡。
博弈方2T 博弈方1B5.下面的得益矩阵表示两博弈方之间的一个静态博弈。
该博弈有没有纯策略纳什均衡?博弈的结果是什么?博弈方2T 博弈方1 M B6.设古诺模型中有n 家厂商。
q i 为厂商i 的产量,Q=q 1+…+q n 为市场总产量。
P 为市场出清价格,且已知P=P(Q)=a-Q (当Q<a 时,否则P=0)。
假设厂商i 生产q i 产量的总成本为C i =C i (q i )=cq i ,也就是说没有固定成本且各厂商的 边际成本都相同,为常数c (c<a )。
假设各厂商同时选择产量,该模型的纳什均衡是什么?当n 趋于无穷大时博弈分析是否仍然有效?7.两寡头古诺模型,P(Q)=a-Q 等与上题相同,但厂商的边际成本不同,分别为c 1和c 2。
如果0<c i <a/2,问纳什均衡产量各位多少?如果c 1<c 2<a ,但2c 2>a+c 1,则纳什均衡产量又为多少?8.甲、乙两公司分属两个国家,在开发某种新产品方面有下面得益矩阵表示的博弈关系(单位:百万美元)。
博弈论练习题 第一组 参考答案

4
6.一个支付组合是帕累托有效率的,当且仅当没有任何其他的支付组合可以同时 改善所有人的处境。假定A和B两人组成一个社会,可能的支付组合如下:
组合1(200,200),组合2(0,300),组合3(300,0),组合4(100,100), (这里(200,200)表示A的支付为200,B的支付为200)。 (1)假定只有如上可能的四个支付组合。找出下列支付组合中帕累托有效率的
2. 某博弈中甲乙双方各有三个策略,其相应的支付矩阵如下图所示: 问: (1)甲会不会采用策略A,为什么? (2)请剔除上述支付矩阵里的占劣策略。 (3)请找出该博弈的纯策略纳什均衡。
A 甲B
C
D 3,7 4,2 3,7
乙 E
3,5 2,7 4,8
F 1,2 6,4 2,5
1
答案:1)甲不会采用策略A,策略A是甲的劣策略,它是劣于C的。 2)对于甲而言,A是一个劣策略。对于乙而言,F是一个劣策略(做到这一步即
A.S+T>200
B.S<T, T>100
C.S<0,T>100
D.以上都不是
解答,答案为C,写出这个博弈的支付矩阵如下:
乙
合作
斗争
合作 (100, 100) (S, T) 甲 斗争 (T, S) (0, 0)
由于这个博弈是对称的,因此,只需要解其中一个人的最优反应即可(对于 对称博弈而言,两个人的占优策略及其支付是完全相同的),我们不妨解 乙的反应。要使“斗争”为占优策略,意味着无论甲选择什么行动,对于 乙而言,斗争总是好于合作,于是,若甲选择合作,那么乙选择斗争的收 益是T,选择合作的收益是100,因此T>100;若甲选择斗争,乙选择斗争的 收益是0,选择合作的收益是S,要使对于乙而言斗争好于合作,则0>S。因 此要选C。
博弈论习题

博弈论习题一、判断1、纳什均衡即任一博弈方单独改变策略都只能得到更小利益的策略组合。
错,只要任一博弈方单独改变策略不会增加得益,策略组合就是纳什均衡了。
本题说的是严格纳什均衡。
2、若一博弈有两个纯战略纳什均衡则一定还存在一个混合战略纳什均衡。
对的,NE的基本性质之一——奇数性所保证的。
3、博弈中混合策略纳什均衡一定存在,纯战略的不一定存在。
对4、上策均衡一定是帕累托最优的均衡。
错,囚徒困境,(坦白,坦白)是上策均衡但不是帕累托最优。
5、在动态博弈中,因为后行为的博弈方可以先观察到对方行为后再做选择,因此总是有利的。
错,先动优势6、动态博弈本身也是自己的子博弈之一。
错,根据子博弈的定义,整个博弈本身不是自己的子博弈。
7、如果动态博弈的一个策略组合不仅在均衡路径上是纳什均衡,而且在非均衡路径上也是纳什均衡,就是该动态博弈的一个子博弈完美纳什均衡。
对,8逆推归纳法并不能排除所有不可置信的威胁、错,逆推归纳法最基本的特征就是能排除动态博弈中所有不可信行为,包括不可信威胁和不可信承诺。
9、颤抖手均衡与第二章的风险上策均衡都是在有风险和不确定情况下的稳定策略组合,因为她们本质上是一样的。
错,区别很大。
前者是针对很小的犯错误导致的偏离概率的均衡概念,对博弈方的理性假设与完全理性假设基本接近,且本身是纳什均衡。
10、有限次重复博弈的子博弈完美纳什均衡每次重复均采用的都是原博弈的纳什均衡。
错,对于有两个以上纯策略纳什均衡博弈的有限次重复博弈,SPNE在前面某些次重复时采用的可以不是原博弈的NE,例如许多出发策略。
11、有限次重复博弈的子博弈完美纳什均衡的最后一次重复必定是原博弈的一个纳什均衡。
对,因为最后一次重复就是动态博弈对的最后一个阶段,根据SPNE的要求,博弈方在该阶段的选择必须构成纳什均衡。
最后一次博弈就是原博弈本身12、无限次重复博弈的均衡解一定优于原博弈均衡解的得益。
错,对于严格竞争的零和博弈或者不满足合作条件的其他博弈来说,无限次重复博弈并不意味着效率的提高,得益不一定高。
博弈论习题集

PROBLEM SET I OF GAME THEOR Y1. State whether the following gameshave unique pure strategy solutions, and if so whatthey are and how they can be found.2. Draw the normal form gamefor the following gameand identify both thepure-a nd mixed-strategy equilibria. In the mixed-strategy Nash equilibrium determine each firm ' s expected profit level if it enters the market.There are two firms that are con sideri ng en teri ng a new market, and must make their decision without knowing what the other firm has done. Unfortun ately the market is on ly big eno ugh to support one of the two firms. If both firms enter the market, then they will each makea loss of £onlyPlayer 1⑵Player 1⑶Player 1Player 2Player 2one firm enter s the market, th at firm will earn a profit of £ 50m, and the other firm will just break even.3. Con vert the follow ing exte nsive form game into a no rmal form game, and identifythe Nash equilibria and subgame perfect Nash equilibria.Finally, what is the Nash equilibrium if both players maketheir movessimulta neously4. Consider an economy consisting of one government and two people. Let X i be thechoice of the people, where X i € X = {x L, x 叫x H}, and i=1,2, and y the choice of the government, where y € Y={ y L, y M y H}. Thepayoffs to the government-household are given by the values of u i(x 1, X2, y) and u 2(x 1, x 2, y) = u 1 (x 2,x 1, y) . These payoffs are entered in the following table:12government' s policy. Enter the blank with value ranges such that the Nash equilibria are supported.(2) Suppose the government moves first, find Nash Equilibria, the subgame perfect Nashequilibria, and the subgame perfect outcome. Is the outcome efficie nt Why(3) Show whether there exists Nash equilibrium (in pure strategies) forthe one-period economy when households and the government move simultaneously.(4) lf the household choose first, do question (2) again.5. Assume that two players are faced with Rosenthal ' s centipede game.Use Bayes' theorem to calculate the players ' reputation for being co-operative in the follow ing situati ons if they play across.(1) At the beginning of the gameeach player believes that there is a 50/50 chanee thatthe other player is rational or co-operative. It is assumed that a co-operative player always plays across. Furthermore supposethat a rati onal player will play across with a probability of(2) At their second movethe players again moveacross. (Continue to assumethat the probability that a rati onal player plays across rema ins equal .(3) How would the players ' reputation have changed after the first movehad the other player believed that rational players always play across. (Assume all other probabilities rema in the same.)(4) Fin ally, how would the players ' reputati on have cha nged after the first move hadthe other player believed that rati onal players n everplay across. (Aga in assume all other probabilities rema in the same.)6. Assume there are midentical Stackelberg leaders in an industry,indexed j =1,…, m and n identical Stackelberg followers, indexed k=1,…,n. All firms have a constant marginal cost of c and no fixed costs. The market price, Q, is determ ined accord ing to the equatio nP - a - C, where Q is total industry output, and a is a constant. Findthe subgame perfect Nash equilibrium supply for the leaders and the followers.Confirm the duopoly results for both Cour not competiti on and Stackelbergcompetitio n, and the gen eralized Cour not result for n firms derived in Exercise . 7. Assume that there are i =1,…,n identical firms in an industry, each with con sta ntmargi nal costs of c and no fixed costs. If the marketprice, P, is determined by the equation , where Qis totalindustry output and a is a constant, determine the Cournot-Nash equilibrium outputlevel for each firm. Where happe ns as n—、8. Find the separating equilibrium behaviour of the low-cost incumbent in the follow ingtwo-period model. The in cumbe nt has marginal costsequal to either £ 4or £ 2.0nly the incumbent initially knows its exact costs. Theentrant observes the incumbent ' s output decision in thefirst period and only enters the market in the second period if it believes that theincumbent has high marginal costs. If entry does occur, the two firms Cour notcompete, and we assumethat at this stage in the game the incumbent ' s true costs are revealed. Price, P, isdetermined by the following equation 卜「./.::■-』,where Qis the combinedoutput of the two firms. Finally, it is assumed that the firms ' discount factor is equal to .9. In the text we argued that a weak government can exploit the privatesector ' s uncertainty about the government ' s preferences topartially avoid the inflationary bias associated withtime-i neon siste nt mon etary policy .In this exercise we provided a simple model that illustrates this result.Assume that the government, via its monetary policy, can perfectly con trol in flati on. Furthermore the gover nment can be one of two types. Either it is strong or it is weak. A strong government is only concerned about the rate of in flati on, and so n ever in flates the economy. A weak government, however, is concerned about both inflation and unemployment.Specially, its welfare in time-period t is given by the followingequation :where 严and g are the rates of inflation and unemployment intime-period t respectively, and c, d and e are all positive parameters.It is assumed the gover nment does not disco unt future welfare, and so a weak gover nment attempts to maximize the sum of its per-period welfare over all current and future periods. The constraint facing the government is given by the expectations-augmented Phillips curve. This is written as=山-也#where is the expected rate of in flati on in period t determ ined at thebeg inning of that period, and aga in a and b are positive parameters. The private sector formulates its expectati ons rati on ally in accorda nee with Bayes' Theorem. Finally, it is assumed that this policy game lasts foronly two periods.(1) Determine the subgame perfect path of inflation if it is common knowledge thegovernment is weak.(2) Determine the sequential equilibrium path of inflation if there is incomplete informationand the private sector 's prior probabilitythat the government is strong is . (Hint: initially determine the necessary condition for the weak government to be indifferent between inflating and not inflating theeconomy.)。
博弈论习题(1-4)

博弈论习题(1-4)博弈论作业题第⼀章4.“囚徒的困境”的内在根源是什么?举出现实中囚徒困境的具体例⼦。
5.博弈有哪些分类⽅法?有哪些主要的类型?9.你正在考虑是否投资100万元开设⼀家饭店。
假设情况是这样的:你决定开,则0.35的概率你将收益300万元(包括投资),⽽0.65的概率你将全部亏损掉;如果你不开,则你能保住本钱但也不会有利润。
请你(a )⽤得益矩阵和扩展表⽰该博弈;(b )如果你是风险中性的,你会怎样选择?(c )如果你是风险规避的,且期望得益的折扣系数为0.9,你的策略选择是什么?(d )如果你是风险偏好的,期望得益折扣系数为1.2,你的选择⼜是什么?10.⼀逃犯从关押他的监狱中逃⾛,⼀看守奉命追捕。
如果逃犯逃跑有两条可选择的路线,看守只要追捕⽅向正确就⼀定能抓住逃犯。
逃犯逃脱可少坐10年牢,但⼀旦被抓住则要加刑10年;看守抓住逃犯能得1000元奖⾦。
请分别⽤得益矩阵和扩展形表⽰该博弈,并作简单分析。
第⼆章4.求出下图中得益矩阵所表⽰的博弈中的混合策略纳什均衡。
博弈⽅2T 博弈⽅1B5.下⾯的得益矩阵表⽰两博弈⽅之间的⼀个静态博弈。
该博弈有没有纯策略纳什均衡?博弈的结果是什么?博弈⽅2T 博弈⽅1 M B6.设古诺模型中有n 家⼚商。
q i 为⼚商i 的产量,Q=q 1+…+q n 为市场总产量。
P 为市场出清价格,且已知P=P(Q)=a-Q (当Q7.两寡头古诺模型,P(Q)=a-Q 等与上题相同,但⼚商的边际成本不同,分别为c 1和c 2。
如果0a+c 1,则纳什均衡产量⼜为多少?8.甲、⼄两公司分属两个国家,在开发某种新产品⽅⾯有下⾯得益矩阵表⽰的博弈关系(单位:百万美元)。
该博弈的纳什均衡有哪些?如果⼄公司所在国政府想保护本国公司利益,有什么好的⽅法?⼄公司甲公司开发不开发第三章4.如果开⾦矿博弈中第三阶段⼄选择打官司后的结果尚不能确定,即右图中a、b的数值不确定。
试讨论本博弈可能有哪⼏种可能的结果。
“博弈论”习题及参考答案

“博弈论”习题及参考答案《博弈论》习题⼀、单项选择题1.博弈论中,局中⼈从⼀个博弈中得到的结果常被称为()。
A. 效⽤B. ⽀付C. 决策D. 利润2.博弈中通常包括下⾯的内容,除了()。
A.局中⼈B.占优战略均衡C.策略D.⽀付3.在具有占优战略均衡的囚徒困境博弈中()。
A.只有⼀个囚徒会坦⽩B.两个囚徒都没有坦⽩C.两个囚徒都会坦⽩D.任何坦⽩都被法庭否决了4.在多次重复的双头博弈中,每⼀个博弈者努⼒()。
A.使⾏业的总利润达到最⼤B.使另⼀个博弈者的利润最⼩C.使其市场份额最⼤D.使其利润最⼤5.⼀个博弈中,直接决定局中⼈⽀付的因素是()。
A. 策略组合B. 策略C. 信息D. ⾏动6.对博弈中的每⼀个博弈者⽽⾔,⽆论对⼿作何选择,其总是拥有惟⼀最佳⾏为,此时的博弈具有()。
A.囚徒困境式的均衡B.⼀报还⼀报的均衡C.占优策略均衡D.激发战略均衡7.如果另⼀个博弈者在前⼀期合作,博弈者就在现期合作;但如果另⼀个博弈者在前⼀期违约,博弈者在现期也违约的策略称为()。
A.⼀报还⼀报的策略B.激发策略C.双头策略D.主导企业策略8.在囚徒困境的博弈中,合作策略会导致()。
A.博弈双⽅都获胜B.博弈双⽅都失败C.使得先采取⾏动者获胜D.使得后采取⾏动者获胜9.在什么时候,囚徒困境式博弈均衡最可能实现()。
A. 当⼀个垄断竞争⾏业是由⼀个主导企业控制时B.当⼀个寡头⾏业⾯对的是重复博弈时C.当⼀个垄断⾏业被迫重复地与⼀个寡头⾏业博弈时D. 当⼀个寡头⾏业进⾏⼀次博弈时10.⼀个企业采取的⾏为与另⼀个企业在前⼀阶段采取的⾏为⼀致,这种策略是⼀种()。
A.主导策略B.激发策略C.⼀报还⼀报策略D.主导策略11.关于策略式博弈,正确的说法是()。
A. 策略式博弈⽆法刻划动态博弈B. 策略式博弈⽆法表明⾏动顺序C. 策略式博弈更容易求解D. 策略式博弈就是⼀个⽀付矩阵12.下列关于策略的叙述哪个是错误的():A. 策略是局中⼈选择的⼀套⾏动计划;B. 参与博弈的每⼀个局中⼈都有若⼲个策略;C. ⼀个局中⼈在原博弈中的策略和在⼦博弈中的策略是相同的;D. 策略与⾏动是两个不同的概念,策略是⾏动的规则,⽽不是⾏动本⾝。
博弈论习题集

博弈论习题集1.在下表所示的战略式博弈中,找出重复删除劣战略的占优均衡表1.12.(投票博弈)假定有三个参与人(1、2和3)要在三个项目6、B和C)中选中一个。
三人同时投票,不允许弃权,因此,每个参与人的战略空间Si二{A, B, C}。
得票最多的项目被选中,如果没有任何项目得到多数票,项目A被选中。
参与人的支付函数如下:U1(A)=U2(B)=U3(C)=2U1(B)=U2(C)=U3(A)=1U1(C)=U2(A)=U3(B)=0求解以上博弈的所有纯战略纳什均衡。
3.求解以下战略式博弈的所有纳什均衡表1.34.考虑一个工作申请的博弈。
两个学生同时向两家企业申请工作,每家企业只有一个工作岗位。
工作申请规则如下:每个学生只能向其中一家企业申请工作;如果一家企业只有一个学生申请,该学生获得工作;如果一家企业有两个学生申请,则每个学生获得工作的概率为1/2。
现在假定每家企业的工资满足:W1/2<W2<2W1,则问:a.写出以上博弈的战略式描述b.求出以上博弈的所有纳什均衡5.(库诺特博弈)假定有n个库诺特寡头企业,每家企业生产成本函数为cq,市场逆需求函数是P=a-Q,其中P是价格,Q=E qi是总供给,a是大于c的常数。
企业i的战略是选择自身产量qi最大化自己的利润,即其他企业的产量q-i;选择自身产量最大化自己的利润。
求解以上博弈的纳什均衡,以及均衡产量和价格如何随n的变化而变化。
6.(伯川德博弈)假定两个寡头企业之间进行价格竞争,两企业生产的产品是完全替代的,并且两家企业的生产成本函数为cq。
市场逆需求函数是P=a-Q, Q=Z qi是总供给,a是大于c的常数。
求出企业i所面临市场需求以及纳什均衡时的价格。
7.(差异价格竞争)假定两个寡头企业进行价格竞争,但产品并不完全相同,企业i的市场需求q(p ,p ) = a - p + p j(i, j = 1,2),两家企业的生产成本函数为cq,求两个寡头同时选择价格时的纳什均衡。
“博弈论”习题参考附标准答案

《博弈论》习题一、单项选择题1.博弈论中,局中人从一个博弈中得到的结果常被称为()。
A. 效用B. 支付C. 决策D. 利润2.博弈中通常包括下面的内容,除了()。
A.局中人B.占优战略均衡C.策略D.支付3.在具有占优战略均衡的囚徒困境博弈中()。
A.只有一个囚徒会坦白B.两个囚徒都没有坦白C.两个囚徒都会坦白D.任何坦白都被法庭否决了4.在多次重复的双头博弈中,每一个博弈者努力()。
A.使行业的总利润达到最大B.使另一个博弈者的利润最小C.使其市场份额最大D.使其利润最大5.一个博弈中,直接决定局中人支付的因素是()。
A. 策略组合B. 策略C. 信息D. 行动6.对博弈中的每一个博弈者而言,无论对手作何选择,其总是拥有惟一最佳行为,此时的博弈具有()。
A.囚徒困境式的均衡B.一报还一报的均衡C.占优策略均衡D.激发战略均衡7.如果另一个博弈者在前一期合作,博弈者就在现期合作;但如果另一个博弈者在前一期违约,博弈者在现期也违约的策略称为()。
A.一报还一报的策略B.激发策略C.双头策略D.主导企业策略8.在囚徒困境的博弈中,合作策略会导致()。
A.博弈双方都获胜B.博弈双方都失败C.使得先采取行动者获胜D.使得后采取行动者获胜9.在什么时候,囚徒困境式博弈均衡最可能实现()。
A. 当一个垄断竞争行业是由一个主导企业控制时B.当一个寡头行业面对的是重复博弈时C.当一个垄断行业被迫重复地与一个寡头行业博弈时D. 当一个寡头行业进行一次博弈时10.一个企业采取的行为与另一个企业在前一阶段采取的行为一致,这种策略是一种()。
A.主导策略B.激发策略C.一报还一报策略D.主导策略11.关于策略式博弈,正确的说法是()。
A. 策略式博弈无法刻划动态博弈B. 策略式博弈无法表明行动顺序C. 策略式博弈更容易求解D. 策略式博弈就是一个支付矩阵12.下列关于策略的叙述哪个是错误的():A. 策略是局中人选择的一套行动计划;B. 参与博弈的每一个局中人都有若干个策略;C. 一个局中人在原博弈中的策略和在子博弈中的策略是相同的;D. 策略与行动是两个不同的概念,策略是行动的规则,而不是行动本身。
“博弈论”习题及参考答案

《博弈论》习题一、单项选择题1.博弈论中,局中人从一个博弈中得到的结果常被称为( )。
A.效用B.支付C.决策 D.利润2.博弈中通常包括下面的内容,除了( )。
A.局中人 B.占优战略均衡C.策略D.支付3.在具有占优战略均衡的囚徒困境博弈中( )。
A.只有一个囚徒会坦白B.两个囚徒都没有坦白C.两个囚徒都会坦白D.任何坦白都被法庭否决了4.在多次重复的双头博弈中,每一个博弈者努力( )。
A.使行业的总利润达到最大 B.使另一个博弈者的利润最小C.使其市场份额最大D.使其利润最大5.一个博弈中,直接决定局中人支付的因素是( )。
A. 策略组合 B. 策略C. 信息 D. 行动6.对博弈中的每一个博弈者而言,无论对手作何选择,其总是拥有惟一最佳行为,此时的博弈具有()。
A.囚徒困境式的均衡 B.一报还一报的均衡C.占优策略均衡D.激发战略均衡7.如果另一个博弈者在前一期合作,博弈者就在现期合作;但如果另一个博弈者在前一期违约,博弈者在现期也违约的策略称为()。
A.一报还一报的策略 B.激发策略C.双头策略D.主导企业策略8.在囚徒困境的博弈中,合作策略会导致( )。
A.博弈双方都获胜 B.博弈双方都失败C.使得先采取行动者获胜D.使得后采取行动者获胜9.在什么时候,囚徒困境式博弈均衡最可能实现()。
A. 当一个垄断竞争行业是由一个主导企业控制时B.当一个寡头行业面对的是重复博弈时C.当一个垄断行业被迫重复地与一个寡头行业博弈时D.当一个寡头行业进行一次博弈时10.一个企业采取的行为与另一个企业在前一阶段采取的行为一致,这种策略是一种( )。
A.主导策略 B.激发策略C.一报还一报策略D.主导策略11.关于策略式博弈,正确的说法是( )。
A. 策略式博弈无法刻划动态博弈B. 策略式博弈无法表明行动顺序C. 策略式博弈更容易求解D. 策略式博弈就是一个支付矩阵12.下列关于策略的叙述哪个是错误的( ):A. 策略是局中人选择的一套行动计划;B.参与博弈的每一个局中人都有若干个策略;C. 一个局中人在原博弈中的策略和在子博弈中的策略是相同的;D. 策略与行动是两个不同的概念,策略是行动的规则,而不是行动本身。
博弈论习题和参考答案与解析

博弈论?习题一、单项选择题1.博弈论中,局中人从一个博弈中得至口的结果常被称为〔〕. A?效用B.支付C.决策D.利润2.博弈中通常包括下面的内容,除了〔〕.A.局中人B.占优战略均衡C策略D?支付3.在具有占优战略均衡的囚徒困境博弈中〔〕.A.只有一个囚徒会坦白氏两个囚徒都没有坦白C?两个囚徒都会坦白D.任何坦白都被法庭否决了4.在屡次重复的双头博弈中,每一个博弈者努力〔〕.A.使行业的总利润到达最大B?使另一个博弈者的利润最小C?使其市场份额最大D.使其利润最大5.一个博弈中,直接决定局中人支付的因素是〔〕A.策略组合B.策略C信息D.行动6.对博弈中的每一个博弈者而言,无论对手作何选择,其总是拥有惟一最正确行为,此时的博弈具有〔〕0A.囚徒困境式的均衡B.一报还一报的均衡C.占优策略均衡D?激发战略均衡7.如果另一个博弈者在前一期合作,博弈者就在现期合作;但如果另一个博弈者在前一期违约,博弈者在现期也违约的策略称为〔〕.A.一报还一报的策略B.激发策略8.在囚徒困境的博弈中,合作策略会导致〔〕oA博弈双方都获胜B博弈双方都失败C使得先米取行动者获胜D使得后米取行动者获胜9.在什么时候,囚徒困境式博弈均衡最可能实现〔〕oA.当一个垄断竞争行业是由一个主导企业限制时B.当一个寡头行业面对的是重复博弈时C.当一个垄断行业被迫重复地与一个寡头行业博弈时D.当一个寡头行业进行一次博弈时一个企业米取的彳丁为10.与另一个企业在前一阶段采取的行为一致〞这种策略是一种〔〕A.主导策略B.激发策略C.一报还一报策略D.主导策略11-关于策略式博弈,正确的说法是〔〕0A.策略式博弈无法刻划动态博弈B.策略式博弈无法说明行动顺序C.策略式博弈更容易求解D.策略式博弈就是一个支付矩阵12.以下关于策略的表达哪个是错误的〔〕:A.策略是局中人选择的一套行动方案;B.参与博弈的每一个局中人都有假设干个策略;C.一个局中人在原博弈中的策略和在子博弈中的策略是相同的;D.策略与行动是两个不同的概念,策略是行动的规那么,而不是行动本身.13.囚徒困境说明〔〕:A.双方都独立依照自己的利益行事,那么双方不能得到最好的结果;B.如果没有某种约束,局中人也可在〔抵赖,抵赖〕的根底上到达均衡;C.双方都依照自己的利益行事,结果一方赢,一方输;D.每个局中人在做决策时,不需考虑对手的反响14.一个博弈中,直接决定局中人损益的因素是〔〕:A.策略组合B.策略C信息D.行动15.动态博弈参与者在关于博弈过程的信息方面是〔〕A不对称的B对称的C不确定的D无序的16.古诺模型表达了寡头企业的〔〕决策模型A本钱B价格C产量D质量17.伯特兰德模型表达了寡头企业〔〕决策模型.A本钱价格C产量 D 质量18.用囚徒困境来说明两个寡头企业的情况,说明了:〔〕A、每个企业在做决策时,不需考虑竞争对手的反响E. 一个企业制定的价格对其它企业没有影响C、企业为了预防最差的结果,将不能得到更好的结果D、一个企业制定的产量对其它企业的产量没有影响19.子博弈精炼纳什均衡〔〕:A.是一个一般意义上的纳什均衡;B.和纳什均衡没有什么关系;C.要求某一策略组合在每一个子博弈上都构成一个纳什均衡;D.要求某一策略组合在原博弈上都构成一个纳什均衡.20.在一般产品销售市场上,以下哪种原因导致了逆向选择.〔〕A产品质量的不确定性B私人信息C公共信息D产品价格21.完全信息动态博弈参与者的行动是〔〕A无序的B有先后顺序的C不确定的D因环境改变的22.市场交易中普遍存在的讨价还价属于哪种博弈.〔〕A完全信息静态博弈B完全信息动态博弈C不完全信息静态博弈D不完全信息动态博弈23.下面哪种模型是一种动态的寡头市场博弈模型〔〕A古诺模型B伯川德模型C斯塔克尔伯格模型D田忌齐威王赛马24?博弈方根据一组选定的在两种或两种以上可能行为中随机选择的策略为血玄〔、A纯策略B混合策略C激发策略D 一报还一报策略25.影响重复博弈均衡结果的主要因素是〔〕A博弈重复的次数B信息的完备性C支付的大小DA和B26.在动态博弈战略行动中,只有当局中人从实施某一威胁所能获得的总收益()不实施该威胁所获得的总收益时,该威胁才是可信的.A大于B等于C小于D以上都有可能二、判断正误并简要说明理由I,纳什均衡一定是上策均衡,上策均衡一定是纳什均衡.2?在一个博弈中博弈方可以有很多个.3.在一个博弈中只可能存在一个纳什均衡.4.由于零和博弈中博弈方之间关系都是竞争性的、对立的,因此零和博弈就是非合作博弈.5.在一个博弈中如果存在多个纳什均衡那么不存在上策均衡.6.曲于两个罪犯只打算犯罪一次〞所以被捕后才出现了不合作的问题即囚徒困境.但如果他们打算重复合伙屡次,比方说20次,那么对策论预测他们将采取彼此合作的态度,即谁都不招供.7,在博弈中纳什均衡是博弈双方能获得的最好结果.8.在博弈中如果某博弈方改变策略后得益增加那么另一博弈方得益减少.9,纳什均衡即任一博弈方单独改变策略都只能得到更小利益的策略组合.10.囚徒的困境博弈中两个囚徒之所以会处于困境,无法得到较理想的结果,是由于两囚徒都不在乎坐牢时间长短本身,只在乎不能比对方坐牢的时间更长.11.斯塔克博格产量领导者所获得的利润的下限是古诺均衡下它得到的利润.12.在有限次重复博弈中,存在最后一次重复正是破坏重复博弈中局中人利益和行为的相互制约关系〞使重复博弈无法实现更高效率均衡的关键问题.13.子博弈精炼纳什均衡不是一个纳什均衡.14.零和博弈的无限次重复博弈中,可能发生合作,局中人不一定会一直重复原博弈的混合战略纳什均衡.15.原博弈惟一的纳什均衡本身是帕雷托效率意义上最正确战略组合,符合各局中人最大利益:采用原博弈的纯战略纳什均衡本身是各局中人能实现的最好结果,符合所有局中人的利益,因此,不管是重复有限次还是无限次,不会和一次性博弈有区别.16.在动态博弈中,由于后行动的博弈方可以先观察对方行为后再选择行为 ,因此总是有利的.入计算与分析题1、A、B两企业利用广告进行竞争.假设A、B两企业都做广告,在未来销售中,A企业可以获得20万元利润,B企业可获得8万元利润;假设A企业做广告,B企业不做广告,A企业可获得25万元利润,B企业可获得2万元利润;假设A企业不做广告,B企业做广告,A企业可获得10万元利润,B企业可获得12万元利润;假设A、B两企业都不做广告,A企业可获得30万元利润,B企业可获得6万元利润.〔,〕画出A、B两企业的损益矩阵.〔2 〕求纯策略纳什均衡.2、可口可乐与百事可乐〔参与者〕的价格决策:双方都可以保持价格不变或者提升价格〔策略〕;博弈的目标和得失情况表达为利润的多少〔收益〕;利润的大小取决于双方的策略组合〔收益函数〕;博弈有四种策略组合,其结局是:〔1〕双方都不涨价,各得利润10单位;〔2 〕可口可乐不涨价,百事可乐涨价,可口可乐利润100,百事可乐利润-30 ;(3 )可口可乐涨价,百事可乐不涨价,可口可乐利润-20,百事可乐利润30 ;(4 )双方都涨价,可口可乐利润140,百事可乐利润35 ;画出两企业的损益矩阵求纳什均衡.3、假定某博弈的报酬矩阵如下:(1)如果(上,左)是上策均衡,那么,a>?, b>?, g<?, f>?(2 )如果(上,左)是纳什均衡,上述哪几个不等式必须满足4、北方航空公司和新华航空公司分享了从北京到南方冬天度假胜地的市场.如果它们合作,各获得500000元的垄断利润,但不受限制的竞争会使每一方的利润降至60000元.如果一方在价格决策方面选择合作而另一方却选择降低价格,那么合作的厂商获利将为零,竞争厂商将获利900000元.(1)将这一市场用囚徒困境的博弈加以表示.(2 )解释为什么均衡结果可能是两家公司都选择竞争性策略.5、博弈的收益矩阵如下表:⑴如果(上/左)是占优策略均衡/那么a、b、c、d、G、f、g、h之间必然满足哪些关系〔尽量把所有必要的关系式都写出来〕〔2 〕如果〔上,左〕是纳什均衡,那么〔1〕中的关系式哪些必须满足〔3 〕如果〔上,左〕是上策均衡,那么它是否必定是纳什均衡为什么〔4 〕在什么情况下,纯策略纳什均衡不存在6、猪圈里有一头大猪和_头小猪,猪圈的一头有一个饲料槽,另一头装有限制饲料供给的按钮.按一下按钮就会有,0个单位饲料进槽,但谁按谁就要付出2个单位的本钱.谁去按按纽那么谁后到;都去按那么同时到.假设大猪先到,大猪吃到9个单位,小猪吃到一个单位;假设同时到,大猪吃7个单位,小猪吃3个单位;假设小猪先到,大猪吃六个单位,小猪吃4个单位.求〔1〕各种情况组合扣除本钱后的支付矩阵〔2 〕求纳什均衡.7、设啤酒市场上有两家厂商,各自选择是生产高价啤酒还是低价啤酒,相应的利润〔单位:万元〕由以下图的得益矩阵给出:1〕有哪些结果是纳什均衡(2 )两厂商合作的结果是什么8、求出以下博弈的所有纯策略纳什均衡.9、求出下面博弈的纳什均衡(含纯策略和混合10、根据两人博弈的损益绸邛仲I答问题:(1) ◎出两人各自的金部策略.图示均衡点.(2 )求出斯塔克博格rstackelberg )均衡情况下的产量、价格和利润.(3)说明导致上述两种均衡结果差异的原因.13.下面的得益矩阵两博弈方之间的一个静态博弈,该博弈有没有纯策略的纳什均衡,博弈的结果是什么14.两个兄弟分一块冰激凌.哥哥先提出一个分割比例 ,弟弟可以接受或拒绝,接受那么按哥哥的提议分割,假设拒绝就自己提出一个比例.但这时候冰激凌已化得只剩1/2 了,对弟弟提议的比例哥哥也可以接受或拒绝,假设接受那么按弟弟的建议分割,假设拒绝冰激凌会全部化光.由于兄弟之间不应该做损人不利己的是“因此我们假设接受和拒绝利益相同时兄弟俩都会接受.求该博弈的子博弈完美纳什均衡.15?如果学生在测试之前全面复习,考好的概率为90%,如果学生只复习一局部重点,那么有50% 的概率考好.全面复习花费的时间tl = 100小时,重点复习之需要花费t2=20小时.学生的效用函数为:U二W-2巳其中W是测试成绩,有上下两种分数Wh和Wl, e为努力学习的时间.问老师如何才能促使学生全面复习16?在以下监工与工人之间的博弈中,试用划线法分析该博弈有无纯策略纳什均衡;如果没有,那么写出混合策略纳什均衡的结果.监工17 ?求解以下博弈的纳什均衡.博弈方29 18 ?某人正在打一场官司,不请律师肯定会输,请律师后的结果与律师的努力程度有关.假设当律师努力工作〔100小时〕时有50%的概率能赢,律师不努力工作<10小时〕那么只有15%的概率能赢.如果诉讼获胜可得到250万元赔偿,失败那么没有赔偿.由于委托方无法监督律师的工作,因此双方约定根据结果付费,赢官司律师可获赔偿金额的10%,失败那么律师一分钱也得不到.如果律师的效用函数为m 0.05e,其中m是报酬e是努力小时数,且律师有时机本钱5万元.求这个博弈的均衡.四、论述题Is解释"囚犯困境;并举商业案例说明.2、用〃小偷与守卫的博弈"说明〃鼓励〔监管〕悖论"博弈论?习题参考答案>单项选择题r 5 B. B. C.D ' A.11 15. B. C. A.6 10 C. A. A.D. C.16 20 C. B. C.21 26. B. B. C. B. D. A.,判断正误并简要说明理由1. F 上策均衡是比纳什均衡更严格的均衡.所以上策均衡一定是纳什均衡 一定是上策均衡,2. T 博弈类型按局中人数多少分为单人博弈、双人博弈和多人博弈3. IF 博弈双方偏好存在差异的条件下,一个博弈模型中可能存在多个纳什均衡4. T 零和博弈才旨参与博弈各方在严格竞争下,一方收益等于另一方损失与损失之和恒为零,所以双方不存在合作可能性而纳什均衡不 ,如性别战.,博弈各方收益 ,只能有一个5.T上策均衡是通过严格下策消去法〔重复剔除下策〕所得到的占优策略纳什均衡6.IF只要两囚犯只打算合作有限次,其最优策略均为招供.比方最后一次合谋,两小偷被抓住了,由于将来没有合作时机了,最优策略均为招供.回退到倒数第二次,既然已经知道下次不会合作,这次为什么要合作呢.依此类推,对于有限次内的任何一次,两小偷均不可能合作.7.F纳什均衡是上策的集合,指在给定的别人策略情况下,博弈方总是选择利益相对较大的策略,并不保证结果是最好的.团F局中人总是以自己的利益最大化选择自己的策略,并不以对方收益的变化为目标9.T纳什均衡是上策的集合,指在给定的别人策略情况下,没有人会改变自己的策略而减低自己的收益10.F局中人总是以自己的利益最大化选择自己的策略,并不以对方收益的变化为目标11.T虽然斯塔格伯格模型各方利润总和小于古诺模型〞但是领导者的利润比古诺模型时12..T无限次重复博弈没有结束重复确实定时间;而在有限次重复博弈中,存在最后一次重复,并且正是有结束重复确实定时间,使重复博弈无法实现更高效率均衡.13.F子博弈精炼纳什均衡一定是一个纳什均衡.14.F零和博弈的无限次重复博弈中,所有阶段都不可能发生合作,局中人会一直重复原博弈的混合战略纳什均衡.15.T原博弈惟一的纳什均衡本身是帕雷托效率意义上最正确战略组合,因此不管是重复有限次还是无限次,不会和一次性博弈有区别.16.F动态博弈是指各博弈方的选择和行动又先后次序的博弈.动态博弈的信息盯以是不对称的.所以策略分为先发制人和.斯塔克伯格博弈揭示“先发制人〞更有禾L而"后发制人"后行动的博弈方可以先观察对方行为后再选择行为反而处于不利境地.三、计算与分析题Is (1)(2)纯策略纳什均衡为(做广告,做广告),(不做广告,不做广告)得长价-20, 30140,35纳什均衡〔不涨价,不涨价〕,〔涨价,涨价〕.从帕累托均衡角度,为〔涨价,涨价〕3、〔 1〕如果〔上/左〕是上策均衡,那么,a>e b>d, g<c, f>h 〔2 〕如果〔上〕左〕是纳什均衡,a>e b>d,不等式必须满足新华航空北方航空 合作竞争50, 50 90, 00, 90 6, 65、 略纳什均衡为〔按,等〕 7、略8、纯策略纳什均衡〔氏甲〕,〔⑴不存在纯策略纳什均衡合作肓争⑵设甲选择"U"的概率为概率为1-P1乙选择"『的概率为P2,贝V选择" R" 的概率为1-P2对甲而言,最正确策略是按定的概率选〃上"和‘下’,使乙选择“左〃和〃右"的期望值相等即PI*8+ (l-PI) *0-P1*1+ (1-P1) *5解得PI = 5/12即⑸12, 7/12 )按5/12概率选〃上“、7/12概率选〃下"为甲的混合策略Nash均衡对乙而言,最正确策略是按一定的概率选“左“和“右",使乙选择〃上"和‘下’的期望值相等即P2*5+(l-P2)*0- P2*2 + (l-P2)*4即(4/7, 3/7肢4/7概率选‘左’、3/7概率选"右"为乙的混合策略Nash均衡10、略.11、见笔记12、见笔记.13、首先,运用严格下策反复消去法的思想,不难发现在博弈方1的策略中,B是相对于T的严格下策.把博弈方1的B策略消去后又可以发现,博弈方2的策略中C是相对于R的严格下策,从而也可以消去.两个博弈方各消去一个策略后的博弈是如下的两人2X 2博弈,己经不存在任何严格下策.再运用划线或箭头法,很容易发现这个2X2博弈有两个纯策略纳什均衡(M,L )和(1R ) 0由于两个纯策略纳什均衡之间没有帕累托效率意义上的优劣关系,一次性静态博弈的结果不能肯定.由于双方在该博弈中可能采取混合策略,因此实际上该博弈的结果可以是4个纯策略组合中的任何一个.14.假设哥的方案是SI: 1-S1淇中S1是自己的份额,弟的方案是S2: 1-S2, S2是哥的份额,那么可用如下的扩展形表示该博弈:Hi SiC5V2eS? 2)CO O)运用逆推归纳法先分析最后一阶段哥的选择.由于只要接受的利益不少于不接受的利益哥就会接受,因此在这个阶段只要弟的方案满足S2/2 $0,也就是S2$0,哥就会接受,否那么不会接受.由于冰激凌的份额不可能是负数,也就是说由于哥不接受弟的方案冰激凌会全部化掉〞因此任何方案哥都会接受.现在回到前一阶段弟的选择.由于弟知道后一阶段哥的选择方法,因此知道如果不接受前一阶段哥提出的比例,自己可以取S2=0,独享此时还未化掉的1/2块冰激凌;如果选择接受前一阶段哥的提议,那么自己将得到出1,显然只要l-Sn/2 ,即S1W1/2,弟就会接受哥的提议.再回到第一阶段哥的选择.哥清楚后两个阶段双方的选择逻辑和结果 ,因此他在这一阶段选择Sl = 1/2,正是能够被弟接受的自己的最大限度份额,超过这个份额将什么都不能得到,因此SI二1/2是最正确选择.综上,该博弈的子博弈完美纳什均衡是:哥哥开始时就提议按(1/2J/2)分割,弟弟接受.15.此题中老帅的调控于段高分和低分的差距.该博弈的扩•展形如下:只有当Ul» U2时学生才会选择全面复习.根据Ul» U2我们可以算出Wh- WD 400o这就是老师能有效全面复习需要满足的条件.其实在奖学金与成绩挂钩时,Wh- W1也可以理解成不同等奖学金的差额.16泄有纯策略均衡,只有混合策略均衡((0. 25,0.75 ),(0. 5,0. 5 ))17. 可以根据画线法求得有唯一纯策略均衡(上,左)18.参见第15题四、论述题1、解释〃囚犯困境〃,并举商业案例说明.(1)假设条件举例:两囚徒被指控是一宗罪案的同案犯.他们被分别关在不同的牢房无法互通信息.各囚徒都被要求坦白罪行.如果两囚徒都坦白,各将被判入狱5年;如果两人都不坦白,两囚徒可以期望被从轻发落入狱2年;如果一个囚徒坦白而另一个囚徒不坦白,坦白的这个囚徒就只需入狱1年,而不坦白的囚徒将被判入狱10年.(2)囚徒困境的策略矩阵表.每个囚徒都有两种策略:坦白或不坦白.表中的数字分别代表囚徒甲和乙的得益.囚徒乙3〕分析:通过划线法可知:在囚徒困境这个模型中,纳什均衡就是双方都〃坦白〃.给定甲坦白的情况下,乙的最优策略是坦白;给定乙坦白的情况下,甲的最优策略也是坦白.这里双方都坦白不仅是纳什均衡,而且是一个上策均衡,即不管对方如何选择,个人的最优选择是坦白.其结果是双方都坦白.4〕商业案例:寡头垄断厂商经常发现它们自己处于一种囚徒的困境.当寡头厂商选择产量时,如果寡头厂商们联合起来形成卡特尔,选择垄断利润最大化产量,每个厂商都可以得到更多的利润.但卡特尔协定不是一个纳什均衡,由于给尢双方遵守协议的情况下,每个厂商都想增加生产,结果是每个厂商都只得到纳什均衡产量的利润,它远小于卡特尔产量下的利润.2用〞小偷与守卫的博弈〃说明〃鼓励〔监管〕悖论〃.〔1〕假设条件举例:偷窃和预防偷窃是小偷和门卫之间进行博弈的一场游戏.门卫可以不睡觉,或者睡觉.小偷可以采取偷、不偷两种策略.如果小偷知道门卫睡觉, 他的最正确选择就是偷;如果门卫不睡觉,他最好还是不偷.对于门卫,如果他知道小偷想偷,他的最正确选择是不睡觉,如果小偷采取不偷,自己最好去睡觉.〔2 〕小偷与门卫的支付矩阵表〔假定小偷在门卫睡觉时一定偷成功,在门卫不睡觉时偷一定会被抓住〕:。
耶鲁大学公开课博弈论课习题

耶鲁大学公开课:博弈论习题集1(第1-3讲内容)Ben Polak, Econ 159a/MGT522a.由人人影视博弈论制作组Darrencui翻译1.严格劣势策略与弱劣势策略:严格劣势策略的定义是什么?弱劣势策略的定义是什么?请用一个包含两个参与人的博弈矩阵来举例说明,要求其中一个参与人有三个策略且三者之一为严格劣势策略;另一个参与人有三个策略但三者之一为弱劣势策略。
请指出你所举例子中的劣势策略。
2.迭代剔除(弱)劣势策略:请看下面的博弈2(a). 这个博弈中是否存在严格劣势策略和弱劣势策略?如果存在,请指出并说明。
(b). 剔除掉严格劣势策略和弱劣势策略之后,在简化的博弈中是否还有劣势策略呢?如果是,请指出并说明。
最后哪些策略不会被剔除呢?(c). 回顾你第一次剔除劣势策略时哪些策略是劣势策略并给出解释。
把它与第二次剔除的劣势策略作比较。
从中你能得出关于迭代剔除劣势策略的何种结论?3. 霍特林的选址博弈(也称霍特林模型):回顾一下课堂中所讲的选票博弈。
其中有两个参与人,每个参与人都从集合* +中选出自己的立场。
这十个立场均分全部的选票。
选民把选票投给与自己立场最接近的候选人。
如果两个候选人站在同一个立场上,那么持该立场选民的选票平均分给每个候选人。
候选人想要最大化自己的得票率。
举例来说,()。
而() [提示:回答这道题时不必画出整个矩阵](a).课堂中我们指出立场2严格优于立场1,而实际上还有其它的立场也是严格优于立场1的,请找出所有优于立场1的立场并作出解释。
(b).假设现在有三名候选人。
举例来说,()而()。
此时立场2是否严格优于立场1?立场3呢?请作出解释。
另外,假设我们剔除了立场1和10,但是该立场的选票依然存在。
在简化的博弈中,立场2是否严格劣于或弱劣于其它(纯)策略?请作出解释。
4. “到底谁的话语权更重”:由三人组成的评审委员会要决出一场全国艺术大赛的冠军。
经过激烈的讨论之后,有三名选手进入最后的获奖候选人名单,分别是:一名画城市中的羚羊的女画家、一名做铅盒的男工匠、一名做根雕的男雕塑家。
博弈论习题集

博弈论习题集(总4页)--本页仅作为文档封面,使用时请直接删除即可----内页可以根据需求调整合适字体及大小--PROBLEM SET I OF GAME THEORY1. State whether the following games have unique pure strategy solutions, and if so what they are and how they can be found. (1) Player 2Player 1(2) Player 2Player 1(3) Player 2Player 12. Draw the normal form game for the following game and identify both the pure-and mixed-strategy equilibria. In the mixed-strategy Nash equilibrium determine each firm ’s expected profit level if it enters the market.There are two firms that are considering entering a new market, and must make their decision without knowing what the other firm has done. Unfortunately the market is only big enough to support one of the two firms. If both firms enter the market, then they will each make aloss of £ only one firm enter s the market, that firm will earn a profit of £50m, and the other firm will just break even.3. Convert the following extensive form game into a normal form game, and identify the Nash equilibria and subgame perfect Nashequilibria. Finally, what is the Nash equilibrium if both players make their moves simultaneously4. Consider an economy consisting of one government and two people. Let x i be the choice of the people, where x i ∈X = {x L , x M , x H }, and i=1, 2, and y the choice of the government, where y ∈Y= {y L , y M , y H }. The payoffs to the government-household are given by the values of u 1(x 1, x 2, y) and u 2(x 1, x 2, y) = u 1(x 2,x 1, y) . These payoffs are entered in the following table:12the government ’s policy. Enter the blank with value ranges such that the Nash equilibria are supported.(2)Suppose the government moves first, find Nash Equilibria, the subgame perfect Nash equilibria, and the subgame perfect outcome. Is the outcome efficient Why(3)Show whether there exists Nash equilibrium (in pure strategies)for the one-period economy when households and the government move simultaneously.(4)If the household choose first, do question (2) again.5.Assume that two players are faced with Rosenthal’s centipede game.Use Bayes’ theorem to calculate the players’ reputation forbeing co-operative in the following situations if they play across.(1)At the beginning of the game each player believes that there isa 50/50 chance that the other player is rational or co-operative.It is assumed that a co-operative player always plays across.Furthermore suppose that a rational player will play across with a probability of(2)At their second move the players again move across. (Continueto assume that the probability that a rational player plays across remains equal .(3)How would the players’ reputation have changed after the firstmove had the other player believed that rational players always play across. (Assume all other probabilities remain the same.)(4)Finally, how would the players’ reputation have changed afterthe first move had the other player believed that rational players never play across. (Again assume all other probabilities remain the same.)6.Assume there are m identical Stackelberg leaders in an industry,indexed j=1,…, m, and n identical Stackelberg followers, indexed k=1,…, n. All firms have a constant marginal cost of c and nofixed costs. The market price, Q, is determined according to the equation , where Q is total industry output, and ɑ is aconstant. Find the subgame perfect Nash equilibrium supply for the leaders and the followers. Confirm the duopoly results for bothCournot competition and Stackelberg competition, and thegeneralized Cournot result for n firms derived in Exercise .7.Assume that there are i=1,…, n identical firms in an industry,each with constant marginal costs of c and no fixed costs. If the market price, P, is determined by the equation , where Qis total industry output and ɑ is a constant, determine theCournot-Nash equilibrium output level for each firm. Where happens as n8.Find the separating equilibrium behaviour of the low-costincumbent in the following two-period model. The incumbent hasmarginal costs equal to either £4 or £2. Only the incumbentinitially knows its exact costs. The entrant observes theincumbent’s output decision in the first period and only enters the market in the second period if it believes that the incumbent has high marginal costs. If entry does occur, the two firmsCournot compete, and we assume that at this stage in the game the incumbent’s true costs are revealed. Price, P, is determined bythe following equation , where Q is the combined outputof the two firms. Finally, it is assumed that the firms’ discount factor is equal to .9.In the text we argued that a weak government can exploit theprivate sector’s uncertainty about the government’s preferences to partially avoid the inflationary bias associated with time-inconsistent monetary policy. In this exercise we provided asimple model that illustrates this result.Assume that the government, via its monetary policy, can perfectly control inflation. Furthermore the government can be one of two types. Either it is strong or it is weak. A strong government is only concerned about the rate of inflation, and so never inflates the economy. A weak government, however, is concerned about bothinflation and unemployment. Specially, its welfare in time-period tis given by the following equation:,where and are the rates of inflation and unemployment in time-period t respectively, and c, d and e are all positive parameters. It is assumed the government does not discount future welfare, and so a weak government attempts to maximize the sum of its per-periodwelfare over all current and future periods. The constraint facingthe government is given by the expectations-augmented Phillips curve. This is written as,where is the expected rate of inflation in period t determined at the beginning of that period, and again ɑ and b are positive parameters. The private sector formulates its expectations rationally in accordance with Bayes’ Theorem. Finally, it is assumed that this policy game lasts for only two periods.(1)Determine the subgame perfect path of inflation if it is commonknowledge the government is weak.(2)Determine the sequential equilibrium path of inflation if thereis incomplete information and the private sector’s prior probability that the government is strong is . (Hint: initially determine the necessary condition for the weak government to be indifferent between inflating and not inflating the economy.)。
博弈论习题集

博弈论习题集PROBLEM SET I OF GAME THEORY1. State whether the following games have unique pure strategy solutions, and if so what they are and how they can be found.(1) Player 2Player 1(2) Player 2Player 1(3) Player 2Player 12. Draw the normal form game for the following game and identify both the pure-and mixed-strategy equilibria. In the mixed-strategy Nashequilibrium determine each firm ’s expected profit level if it enters the market.There are two firms that are considering entering a new market, and must make their decision without knowing what the other firm has done. Unfortunately the market is only big enough to support one of the two firms. If both firms enter the market, then they will each make a loss of £ onlyone firm enter s the market, th at firm will earn a profit of £50m, and the other firm will just break even.3. Convert the following extensive form game into a normal form game, and identify the Nash equilibria and subgame perfect Nash equilibria. Finally, what is the Nash equilibrium if both players make their moves simultaneously4. Consider an economy consisting of one government and two people. Let x i be the choice of the people, where x i ∈X = {x L , x M , x H }, and i=1, 2, and y the choice of the government, where y ∈Y= {y L , y M , y H }. The payoffs to the government-household are given by the values of u 1(x 1, x 2, y) and u 2(x 1, x 2, y) = u 1(x 2,x 1, y) . These payoffs are entered in the following table:12government ’s policy. Enter the blank with value ranges such that the Nash equilibria are supported.(2)Suppose the government moves first, find Nash Equilibria, the subgame perfect Nash equilibria, and the subgame perfect outcome. Is the outcome efficient Why(3)Show whether there exists Nash equilibrium (in pure strategies) forthe one-period economy when households and the government move simultaneously.(4)If the household choose first, do question (2) again.5.Assume that two players are faced with Rosenthal’s centipede game.Use Bayes’ theorem to calculate the players’ reputation for being co-operative in the following situations if they play across.(1)At the beginning of the game each player believes that there is a 50/50chance that the other player is rational or co-operative. It is assumed that a co-operative player always plays across. Furthermore suppose that a rational player will play across with a probability of (2)At their second move the players again move across. (Continue to assumethat the probability that a rational player plays across remains equal .(3)How would the players’ reputation have changed after the first movehad the other player believed that rational players always play across.(Assume all other probabilities remain the same.)(4)Finally, how would the players’ reputation have changed after thefirst move had the other player believed that rational players never play across. (Again assume all other probabilities remain the same.)6.Assume there are m identical Stackelberg leaders in an industry,indexed j=1,…, m, and n identical Stackelberg followers, indexed k=1,…, n. All firms have a constant marginal cost of c and no fixed costs. The market price, Q, is determined according to the equation , where Q is total industry output, and ɑis a constant. Find the subgame perfect Nash equilibrium supply for the leaders and the followers. Confirm the duopoly results for both Cournot competition and Stackelberg competition, and the generalized Cournot result for n firms derived in Exercise .7.Assume that there are i=1,…, n identical firms in an industry, eachwith constant marginal costs of c and no fixed costs. If the market price, P, is determined by the equation , where Q is total industry output and ɑ is a constant, determine the Cournot-Nash equilibrium output level for each firm. Where happens as n 8.Find the separating equilibrium behaviour of the low-cost incumbentin the following two-period model. The incumbent has marginal costs equal to either £4 or £2. Only the incumbent initially knows its exact costs. The entrant observes the incumbent’s output decision in the first period and only enters the market in the second period if it believes that the incumbent has high marginal costs. If entry does occur, the two firms Cournot compete, and we assume that at this stage in the game the incumbent’s true costs are revealed. Price, P, isdetermined by the following equation , where Q is thecombined output of the two firms. Finally, it is assumed that the firms’ discount factor is equal to .9.In the text we argued that a weak government can exploit the privatesector’s uncertainty about the government’s preferences topartially avoid the inflationary bias associated withtime-inconsistent monetary policy. In this exercise we provided a simple model that illustrates this result.Assume that the government, via its monetary policy, can perfectly control inflation. Furthermore the government can be one of two types. Either it is strong or it is weak. A strong government is only concerned about the rate of inflation, and so never inflates the economy. A weak government, however, is concerned about both inflation and unemployment. Specially, its welfare in time-period t is given by the following equation:,where and are the rates of inflation and unemployment in time-period t respectively, and c, d and e are all positive parameters. It is assumed the government does not discount future welfare, and so a weak government attempts to maximize the sum of its per-period welfare over all current and future periods. The constraint facing the government is given by the expectations-augmented Phillips curve. This is written as,where is the expected rate of inflation in period t determined at the beginning of that period, and again ɑand b are positive parameters. The private sector formulates its expectations rationally in accordance with Bayes’ Theorem. Finally, it is assumed that this policy game lasts foronly two periods.(1)Determine the subgame perfect path of inflation if it is commonknowledge the government is weak.(2)Determine the sequential equilibrium path of inflation if there isincomplete information and the private sector’s prior probability that the government is strong is . (Hint: initially determine the necessary condition for the weak government to be indifferent between inflating and not inflating the economy.)。
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PROBLEM SET I OF GAME THEORY1. State whether the following games have unique pure strategy solutions, and if so what they are and how they can be found.(1) Player 2Player 1(2) Player 2Player 1(3) Player 2Player 12. Draw the normal form game for the following game and identify both the pure-and mixed-strategy equilibria. In the mixed-strategy Nashequilibrium determine each firm ’s expected profit level if it enters the market.There are two firms that are considering entering a new market, and must make their decision without knowing what the other firm has done. Unfortunately the market is only big enough to support one of the two firms. If both firms enter the market, then they will each make a loss of £ onlyone firm enter s the market, th at firm will earn a profit of £50m, and the other firm will just break even.3. Convert the following extensive form game into a normal form game, and identify the Nash equilibria and subgame perfect Nash equilibria. Finally, what is the Nash equilibrium if both players make their moves simultaneously4. Consider an economy consisting of one government and two people. Let x i be the choice of the people, where x i ∈X = {x L , x M , x H }, and i=1, 2, and y the choice of the government, where y ∈Y= {y L , y M , y H }. The payoffs to the government-household are given by the values of u 1(x 1, x 2, y) and u 2(x 1, x 2, y) = u 1(x 2,x 1, y) . These payoffs are entered in the following table:12government ’s policy. Enter the blank with value ranges such that the Nash equilibria are supported.(2)Suppose the government moves first, find Nash Equilibria, the subgame perfect Nash equilibria, and the subgame perfect outcome. Is the outcome efficient Why(3)Show whether there exists Nash equilibrium (in pure strategies) forthe one-period economy when households and the government move simultaneously.(4)If the household choose first, do question (2) again.5.Assume that two players are faced with Rosenthal’s centipede game.Use Bayes’ theorem to calculate the players’ reputation for being co-operative in the following situations if they play across.(1)At the beginning of the game each player believes that there is a 50/50chance that the other player is rational or co-operative. It is assumed that a co-operative player always plays across. Furthermore suppose that a rational player will play across with a probability of (2)At their second move the players again move across. (Continue to assumethat the probability that a rational player plays across remains equal .(3)How would the players’ reputation have changed after the first movehad the other player believed that rational players always play across.(Assume all other probabilities remain the same.)(4)Finally, how would the players’ reputation have changed after thefirst move had the other player believed that rational players never play across. (Again assume all other probabilities remain the same.)6.Assume there are m identical Stackelberg leaders in an industry,indexed j=1,…, m, and n identical Stackelberg followers, indexed k=1,…, n. All firms have a constant marginal cost of c and no fixed costs. The market price, Q, is determined according to the equation , where Q is total industry output, and ɑis a constant. Find the subgame perfect Nash equilibrium supply for the leaders and the followers. Confirm the duopoly results for both Cournot competition and Stackelberg competition, and the generalized Cournot result for n firms derived in Exercise .7.Assume that there are i=1,…, n identical firms in an industry, eachwith constant marginal costs of c and no fixed costs. If the market price, P, is determined by the equation , where Q is total industry output and ɑ is a constant, determine the Cournot-Nash equilibrium output level for each firm. Where happens as n8.Find the separating equilibrium behaviour of the low-cost incumbentin the following two-period model. The incumbent has marginal costs equal to either £4 or £2. Only the incumbent initially knows its exact costs. The entrant observes the incumbent’s output decision in the first period and only enters the market in the second period if it believes that the incumbent has high marginal costs. If entry does occur, the two firms Cournot compete, and we assume that at this stage in the game the incumbent’s true costs are revealed. Price, P, isdetermined by the following equation , where Q is thecombined output of the two firms. Finally, it is assumed that the firms’ discount factor is equal to .9.In the text we argued that a weak government can exploit the privatesector’s uncertainty about the government’s preferences topartially avoid the inflationary bias associated withtime-inconsistent monetary policy. In this exercise we provided a simple model that illustrates this result.Assume that the government, via its monetary policy, can perfectly control inflation. Furthermore the government can be one of two types. Either it is strong or it is weak. A strong government is only concerned about the rate of inflation, and so never inflates the economy. A weak government, however, is concerned about both inflation and unemployment. Specially, its welfare in time-period t is given by the following equation:,where and are the rates of inflation and unemployment in time-period t respectively, and c, d and e are all positive parameters. It is assumed the government does not discount future welfare, and so a weak government attempts to maximize the sum of its per-period welfare over all current and future periods. The constraint facing the government is given by the expectations-augmented Phillips curve. This is written as,where is the expected rate of inflation in period t determined at the beginning of that period, and again ɑand b are positive parameters. The private sector formulates its expectations rationally in accordance with Bayes’ Theorem. Finally, it is assumed that this policy game lasts foronly two periods.(1)Determine the subgame perfect path of inflation if it is commonknowledge the government is weak.(2)Determine the sequential equilibrium path of inflation if there isincomplete information and the private sector’s prior probability that the government is strong is . (Hint: initially determine the necessary condition for the weak government to be indifferent between inflating and not inflating the economy.)。