gametheory3-1
博弈论(第一、二章)
游戏2：摘柿子
甲 跑
摇 跑
乙
摇 跑
甲
摇
乙 跑
摇 跑
甲
不跑 （2,2）
（0,0）
（0,1） （2,0）
（0,3） （4,0）
游戏3：免费彩票博弈
每个人可以免费购买任意数量彩票，随机 抽取1张彩票中奖，奖金总额为1000万元/n，n 为彩票数量。
博弈论：研究理性人行为选择的理论
博弈论作用：帮助个人、组织等决策主 体深刻理解策略并明智的选择行动。
第二章 完全信息静态博弈
� 基本分析思路和方法 � 纳什均衡 � 混合策略 � 纳什均衡的选择
第一节 基本分析思路和方法
行动或策略（acቤተ መጻሕፍቲ ባይዱion or strategy）
si：局中人i的一个特定策略 Si：局中人i的策略集（strategy set）或策略空间 （strategy space），可以是离散的或连续的。
纳什的基本贡献是证明了非合作博弈均衡解 及其存在性，建立了作为博弈论基础的“纳 什均衡”概念；海萨尼则把不完全信息纳入 到博弈论方法体系中；泽尔腾的贡献在于将 博弈论由静态向动态的扩展，建立了“子博 弈精练纳什均衡”的概念。
1996莫里斯（James A.Mirrlees）和维克瑞（William Vickrey）
游戏1：军事游戏-进攻和防守
博弈结果表
守方
B 攻方 a -1 b -1 c +1 +1 +1 -1 +1 -1 -1
C -1 +1 +1
游戏1：军事游戏-进攻和防守
博弈结果表
守方
B 攻方 a -1 b -1 c +1 +1 +1 -1 +1 -1 -1
博弈论答案（Gametheoryanswer）
博弈论答案（Game theory answer）Game theory, exercises, reference answers (second assignments)First, the multiple-choice question1.B,2.C,3.A,4.A,5.B,6.ABCD7.C 8.B 9.CTwo, judge and explain the reason1.F best balance is an equilibrium more rigorous than the Nash equilibrium2.T best balance is an equilibrium more rigorous than the Nash equilibrium3.T game types are divided into single game, double game and multiplayer game according to the number of players in the gameUnder the condition that both sides of the 4.F game have different preferences, there may be 2 Nash equilibria in a game model, such as the sex war5.T zero sum game refers to the participation of all parties in the game, under strict competition, one side of revenue is equal to the other party's loss, the sum of gains and losses of the game is always zero, so there is no possibility of cooperation between the two sides6.T is strictly dominated equilibrium through the worstelimination method (excluding repeat decision) the dominant strategy, there is only one Nash equilibrium7.F Nash equilibrium is a collection of best policies, which means that in the case of a given strategy, the game side always chooses a relatively large strategy, and does not guarantee the outcome to be the best.In the 8.F game, people always choose their own strategies to maximize their interests and not aim at the change of the other's earnings9.T Nash equilibrium is a collection of best policies, which means that when given someone else's strategy, no one changes his strategy to reduce his earningsIn the 10.F game, people always choose their own strategies to maximize their interests and not aim at the change of the other's earningsIn the 11.F game, people always choose their own strategies to maximize their interests and not aim at the change of the other's earnings12.T although Berg Stagg model profit is less than the sum of the Cournot model, but the profit model of high Bigunuo leaderThree, calculation and analysis questions1, (1) draw A, B two enterprise profit and loss matrix.B enterpriseAdvertise without advertisingA enterprises advertise 20, 825, 2No advertising 10, 1230, 6(2) pure strategy Nash equilibrium.(advertising, advertising)2, draw two enterprise profit and loss matrix, seek Nash equilibrium.(1) draw the profit and loss matrix of A and B two enterprisesPepsi ColaOriginal price increaseCoca-Cola's original price is 10, 10100, -30Price increases -20, 30140, 35(2) seeking Nash equilibrium.Two: (the original price, the original price), (prices, prices)3, suppose the payoff matrix of a game is as follows:Methyl ethylLeft and rightOn a, B, C, DNext, e, F, G, H(1) if (on, left) is the best balance, then, a>, b>, g<, f>?Answer: a>e, b>d, f>h, g<c(2) what inequalities must be satisfied if (upper, left) is the Nash equilibrium?Answer: a>e, b>d4, answer: (1) this market is represented by the game of prisoner's dilemma.Northern AirlinesCooperative competitionXinhua Airlines cooperation 500000500000090000Competition 900000, 06000060000(2) explain why the equilibrium result may be that both companies choose competitive strategies.Answer: if Xinhua chooses "competition", then the north will choose "60000>0"; if Xinhua chooses "cooperation", the north will still choose "900000>500000".If the North chooses "competition", Xinhua will choose "60000>0"; if the North chooses "cooperation", Xinhua will still choose "900000>0".Because the competition is the dominant strategy of both sides, the equilibrium result is that both companies choose competitive strategy.5. The payoff matrix of the game is shown as follows:BLeft and rightA, a, B, C, DNext, e, F, G, H(1) if the (top, left) is the dominant policy equilibrium, what relation must be satisfied between a, B, C, D, e, F, G, and H?Answer: starting from the definition of dominant strategy equilibrium:For the one, the strategy "g" (a) is better than "C" (E);For B., the policy "left" (B, f) is superior to the policy"right" (D, H).So the conclusions are: a>e, b>d, f>h, c>g(2) if the (upper, left) is Nash equilibrium, what relation must be satisfied in (1)?Answer: Nash equilibrium only needs to meet: a>e, b>d,(3) if the (top, left) is the best balance, then is it necessarily a Nash equilibrium? Why?Answer: the equilibrium of dominant strategy must be Nash equilibrium, because the equilibrium condition of dominant strategy contains the condition of Nash equilibrium.(4) under what circumstances does the pure strategy Nash equilibrium exist?A: when each of these strategies does not satisfy the Nash equilibrium, the pure strategic Nash equilibrium does not exist.7, seek the Nash equilibrium.PigPress waitBig pigs press 5, 14, 4Wait 9, -1 0, 0The Nash equilibrium is: big pig, press, pig, etc., namely (press, etc.)6,BLow priceA low price of 10080050, 50High priced -20, -30 900600(1) what are the results of Nash equilibrium?Answer: (low price, low price), (high price, high price)(2) what is the result of the cooperation between the two firms?Answer: (high price, high price)8. The pure Nash equilibrium of the following games is obtained by using the reaction function method and the marking method.Participants 1 participants 2A, B, C, DingA, 2,3, 3,2, 3,4, 0,3B, 4,4, 5,2, 0,1, 1,2C, 3,1, 4,1, 1,4, 10,2D, 3,1, 4,1, -1,2, 10,1Participant 1's response function:R1 (2) =B, if 2 chooses a=B, if 2 chooses B.=A, if 2, choose C=C or D, if 2, choose DingParticipant 2's response function:R2 (1) = C, if 2, select A= a, if 2, select B= C, if 2, select C= C, if 2, select DFor the common set, the pure strategy Nash equilibrium is (B, a) and (A, c)9, the following game Nash equilibrium (including pure strategyand mixed strategy).Methyl ethylL RU 5,0 0,8D 2,6 4,5Solution: (1) pure strategy Nash equilibrium: we can see from the scratch method that there is no pure strategy Nash equilibrium in the matrix game.(2) mixed strategy Nash equilibriumThe probability of setting a "U" is P1, and the probability of "D" is 1-P1B. the probability of selecting "L" is P2, and the probability of "R" is 1-P2For a, the best policy is to choose "U" and "D" by a certain probability, so that the second choice of "L" and "R" is equal to the expected valueThat is, P1*0+ (1-P1), *6=, P1*8+ (1-P1), *5Xie P1=1/9That is, (1/9,8/9) Nash policy is chosen according to 1/9probability, U and 8/9 probability, and D is chosen as a mixed strategyFor B, the best strategy is to choose "L" and "R" by a certain probability, so that the second is equal to the expected value of "U" and "D"That is, P2*5+ (1-P2), *0=, P2*2+ (1-P2), *4Xie P2=4/7That is, (4/7,3/7) according to the probability of 4/7, "L", "3/7", "R" is chosen as "B", the mixed strategy Nash equilibrium10, answer the question according to the profit and loss matrix of two player game:Methyl ethylLeft and rightGo to 2,3 0,0Lower 0,0 4,2(1) write out all the strategies of the two men.Answer: all strategies: (upper, left), (upper, right), (lower, left), (lower, right)(2) find all the pure strategy Nash equilibrium of the game.A: by the scratch method, we can see that the matrix game is purely strategic and the Nash equilibrium is(upper, left) and (lower, right) two(3) the mixed strategy Nash equilibrium of the game is obtained.Solution: the probability of setting a "up" is P1, and the probability of selecting "down" is 1-P1B. the probability of "left" is P2, and the probability of "right" is 1-P2For a, the best strategy is to choose "upper" and "lower" according to a certain probability, so that the left and right of the second are equal to the expected valueThat is, P1*3+ (1-P1), *0=, P1*0+ (1-P1), *2Xie P1=2/5That is, (2/5,3/5) a mixed strategy Nash equilibrium based on the "2/5 probability", "upper", "3/5" probability, and "next"For b.,The best strategy is to choose "left" and "right" according to a certain probability, so that the candidate's "upper" and "lower" expectations are equalThat is, P2*2+ (1-P2), *0=, P2*0+ (1-P2), *4Xie P2=2/3That is, (2/3,1/3) Nash policy is chosen by the 2/3 probability "left" and "1/3", and the "right" is b11, an oligopoly market has two manufacturers, the total cost is 20 times the output of their own, the market demand letterThe number is Q=200-P.Answer: (1) if two manufacturers decide the output at the same time, how much is the output?(2) if the two firms reach an agreement to monopolize the market and arrange production together, what about their respective profits?(3) use the case to explain the prisoner's dilemma.Answer: (1) by the known conditions Q=200-P, P=200-QTC1=20q1, TC2=20q2, q1+q2=QThe profit functions obtained by 1,2 manufacturers are:K1=Pq1-TC1= (200- (q1+q2)) q1-20q1=180q1-q12-q1q2K2=Pq2-TC2= (200- (q1+q2)) q2-20q2=180q2-q22-q1q2The dK/dq1=0's 1 response function is 180-2Q1-Q2=0,The dK/dq2=0's 2 response function is 180-Q1-2Q2=0,The joint solution can be obtained by q1=q2=60K1=K2=3600(2) by the known condition Q=200-P, P=200-QTC=TC1+TC2=20q1+20q2 =20QThe total profit function of the 1,2 manufacturer is:K=PQ-TC= (200-Q) Q-20Q=180Q-Q2Order dK/dQ=0, Q=90, q1=q2=45K=PQ-TC= (200-Q) Q-20Q=180Q-Q2=8100K1=K2=4050(3) q1=45, q2=60 and q1=60, q2=45, respectively, into the profit function of 1,2 manufacturersThe profits of the 1,2 manufacturers are:K1 (q1=45, q2=60) =Pq1-TC1= (200- (q1+q2))q1-20q1=180q1-q12-q1q2=3375K1 (q1=60, q2=45) =Pq1-TC1= (200- (q1+q2))q1-20q1=180q1-q12-q1q2=4500K2 (q1=45, q2=60) =Pq2-TC2= (200- (q1+q2))q2-20q2=180q2-q22-q1q2=4500K1 (q1=60, q2=45) =Pq1-TC1= (200- (q1+q2))q1-20q1=180q1-q12-q1q2=3375Vendor 2Cooperation (q2=45), non cooperation (q2=60);Vendor 1 Cooperation (q1=45) 4050405033754500Non cooperative (q1=60) 4500337536003600According to the marking method, the best way for the manufacturer is 1.2 (non cooperation, non cooperation), that is, (36003600)The profits of both sides were lower than (cooperation, cooperation). (40504050) obviously it belonged to the prisoner's dilemma"13, consider the following (market deterrence) a dynamic game: first of all, the potential in a market entrants to choose whether or not to enter, and then on the market for enterprise (incumbent) is selected to compete with the new enterprise. The incumbent may have two types of gentle type (left) and cruel type (right), answer the following questions..Left: gentle right: cruel type(1) find the corresponding Nash equilibrium for two types of incumbent, and the sub game perfect Nash equilibrium(1) the Nash equilibrium of the gentle type of incumbent is (access, acquiescence)The Nash of the cruel type is balanced (not entering, entering, struggling)(2) when the existing enterprise is tender, at least how many times will the new enterprise be willing to enter?Four. Discussion questions1, explain the prisoner's Dilemma and explain the business case.(1) assumptions for example: two prisoners were accused of a crime is an accomplice. They were kept in separate cells, unable to communicate information. Prisoners are required to confess crimes. If two prisoners confess, each shall be sent to prison for 5 years; if two men do not confess, two prisoners may expect to be sent from prison to prison for 2 years; if a prisoner confesses, another prisoner does not confess,Frankly, the prisoner will only go to prison for 1 years, and the prisoner without confession will be sentenced to 10 yearsin prison.(2) the strategy matrix of prisoners' dilemma. Each prisoner has two strategies: to confess or not to confess. The numbers in the table represent the benefits of prisoner a and B.Prisoner BConfessPrisoner frank, -5, -5, -1, -10Don't confess, -10, -1, -2, -2(3) analysis: through the marking method, we can see that in the model of prisoner's dilemma, Nash equilibrium is that both sides confess". Given a frank case, the best strategy for B. is to confess; the optimal policy given by B. is also frank. And here both sides confess, not only is the Nash equilibrium, but also is a best balance, that is, regardless of how the other side of the choice, the individual's best choice is to confess. As a result, both sides confess.(4) business cases: oligopoly firms often find themselves ina prisoner's dilemma. When the oligarchic manufacturer chooses the output, every manufacturer can gain more profits if the oligopoly firms combine to form cartels and choose monopoly profits to maximize the output. But the cartel agreement is not a Nash equilibrium, because given both comply with the agreement, each firm to increase production, the result is that each vendor has only been Nash equilibrium yield profits, itis far less than the yield of profit under the cartel.2. Explain and discuss the Nash equilibrium of Cournot duopoly model. Why is balance a prisoner's dilemma?See class notesOr calculation questions eleventh3, use the game of thief and guard to explain the paradox of encouragement (regulation)".(1) assume the conditions for example: stealing and preventing theft is a game between thieves and guards. The guard can sleep or sleep. Thieves can take two tactics: stealing and stealing. If the thief knows that the guard is sleeping, his best bet is to steal. If the guard doesn't sleep, he'd better not steal. For the doorman, if he knows the thief wants to steal, his best choice is not to sleep, and if the thief take it without stealing, he'd better go to sleep.(2) the payment matrix of the thief and the doorman (assuming that the thief must have succeeded in stealing when the guard sleeps, and that the thief will be caught when the guard does not sleep.):GuardGo to bed without sleepThieves steal 1, -1 -2, 0Do not steal 0, 20, 0(3) analysis: through the marking method, we can see that there is no Nash equilibrium in this game. The thieves do not steal, do not sleep, neither gains nor loss; the guard did not sleep, the thief, because the job is not to reward, the thief was sentenced to 2 unit failure loss; guard sleeping, thieves do not steal, the sleeping happily get 2 utility unit, the thief did not return no loss of sleep; the guard, the thief, the guard was punished because of dereliction of duty and his failure in 1 units, 1 units of utility thieves to steal success.(4) "incentive (regulatory) paradox" shows: in reality, we can see that when the doorman without sleep, stealing a crackdown of the convergence of molecules; time, molecular theft began to make waves, the thief can not tolerate when too rampant, the guard had to begin again. The more the thief, so the guard will not sleep more, steal the thief less, not sleeping guard will be less; in turn, the more don't sleep, steal the thief less, do not sleep the less, the more the thief stole. If you steal group selection is out in force, so the guard all don't sleep, but the once all don't sleep, the best choice not to steal all the thief, the thief stole all the guard once chose not to, all the best choose to sleep.(5) conclusion: increasing penalties for thieves can not prevent theft in the long run (but only to make the guard lazy); Aggravating Punishment, dereliction of duty is just to reduce the probability of theft. This game of gatekeeper and thief reveals that the unexpected relationship between policyobjectives and policy outcomes is often called the paradox of motivation".。
第三节博弈论（GameTheory）
第三节博弈论(Game Theory)在国际关系的研究过程中，我们时常会运用到博弈论这样一个工具。
博弈论在英语中称之为“Game Theory”。
很多人会认为这是一种所谓的游戏理论，其实不然，我们不能把Games 与Fun 同论，而应该将博弈论称之为是一种“Strategic interaction”（策略性互动）。
“博弈”一词现如今在我们的生活中出现的已经很频繁，我们经常会听说各种类型的国家间博弈（如：中美博弈），“博弈论”已经深刻的影响了世界局势和地区局势的发展。
在iChange创设的危机联动体系中，博弈论将得到充分利用，代表也将有机会运用博弈论的知识来解决iChange 核心学术委员会设计的危机。
在这一节中，我将对博弈论进行一个初步的介绍与讨论，代表们可以从这一节中了解到博弈论的相关历史以及一些经典案例的剖析。
（请注意：博弈论的应用范围非常广泛，涵盖数学、经济学、生物学、计算机科学、国际关系、政治学及军事战略等多种学科，对博弈论案例的一些深入分析有时需要运用到高等数学知识，在本节中我们不会涉及较多的数学概念，仅会通过一些基本的数学分析和逻辑推理来方便理解将要讨论的经典博弈案例。
）3.1 从“叙利亚局势”到“零和博弈”在先前关于现实主义理论的讨论中，我们对国家间博弈已经有了初步的了解，那就是国家是有目的的行为体，他们总为了实现自己利益的最大化而选择对自己最有利的战略，其次，政治结果不仅仅只取决于一个国家的战略选择还取决于其他国家的战略选择，多种选择的互相作用，或者策略性互动会产生不同的结果。
因此，国家行为体在选择战略前会预判他国的战略。
在这样的条件下，让我们用一个简单的模型分析一下发生在2013年叙利亚局势1：叙利亚危机从2011年发展至今已经将进入第四个年头。
叙利亚危机从叙利亚政府军屠杀平民和儿童再到使用化学武器而骤然升级，以2013年8月底美国欲对叙利亚动武达到最为紧张的状态，同年9月中旬，叙利亚阿萨德政府以愿意向国际社会交出化学武器并同意立即加入《禁止化学武器公约》的态度而使得局势趋向缓和。
GameTheory：博弈论
EE693H Fall2007Game TheoryTR,12:00pm–1:15pm,Holmes389Course InformationGame theory provides the most natural framework to study the strategic interactions between self-interested decision makers.Due to the emergence of distributed complex systems made up of many autonomous agents (such as the Internet),there has been a resurgence of interest in game theory within the engineering and the computer science communities.This course will introduce the students to the fundamentals of noncoopera-tive game theory as well as the computational tools provided by noncooperative game theory.Emphasis will be on the engineering applications such as control,communications,transportation systems,and resource allocation problems.The course is intended for mathematically inclined students with some background on probability theory.Instructor:G¨u rdal Arslan,Holmes440,Phone:956–3432,E-mail:*****************Office Hours:OpenRecommended Texts:Dynamic Noncooperative Game Theory by Bas.ar and OlsderGame Theory by Fudenberg and Tirole,Webpage:/∼gurdal/EE693H.htmSite of announcements,handouts,homeworks,etc.Grading:Homework30%;Mid-term35%;Project35%.Policies:No credit will be given to late homeworks.Exams must be taken at the announced times.(Tentative)Topics•Introduction(1Lecture)–Examples and various solution concepts•Zero-Sum Finite Games in Normal Form(2Lecture)–Security strategies–Lower and upper values–Saddle-point equilibrium–Mixed strategies–Minmax theorem–Computation of saddle-point equilibria by graphical solution and LP approaches–Dominated strategies–Iterative elimination of dominated strategies•Normal Form Games(6Lecture)–Pure and mixed strategies–Dominated strategies and solution by iterated dominance–Nash equilibrium–Pure equilibrium,Strict equilibrium–Examples of pure equilibrium(Cournot’s model of oligopoly,CDMA uplink power control)–Existence of mixed equilibria infinite normal games(Best response correspondence,Kakutani’s fixed point theorem)–Existence of pure equilibrium in infinite games with continuous payoffs(Quasi-concavity of player payoffs in its own decisions)–Sufficient conditions for the uniqueness of pure equilibrium in infinite games with continuous payoffs(Diagonal strict concavity condition)–Existence of mixed equilibrium in infinite games with continuous payoffs–Discontinuous games–Computation of Nash equilibria infinite normal-form games(algebraic approach,optimization approach)–Correlated equilibrium,coarse correlated equilibrium,correlated equilibrium with information partitions•Well-known Classes of Non-Zero-Sum Games(7Lecture)–Generalized ordinal potential games and existence of pure equilibria–Finite improvement property–Characterization of potential games–Weighted potential games–Congestion games–Inefficiency of Nash equilibria in congestion games,Tolls minimizing the total congestion,Braess’paradox–Price of anarchy and price of stability in congestion games–Infinite potential games–Efficiency loss in resource allocation games–(Weakly)acyclic games–Consensus problem–Supermodular games•Learning in games(8Lecture)–Cournot’s adjustment process–Fictitious play,Asymptotic behavior,Convergence of beliefs in certain classes of games,Shapley’s example,Lack of payoffconsistency,–Stochasticfictitious play,Payoffconsistency,Perturbed equilibria,Convergence of intended be-havior via stochastic approximation theory–Computation,memory,and observation requirements offictitious play–Regret based dynamics,Utility basedfictitious play–Finite memory variants offictitious play,Adaptive play,Elements of Markov processes,Perturbed Markov processes,Stochastic stability•Repeated Games•Auctions;Mechanism design;Incentive design•Games with incomplete/imperfect information;•Extensive form games•Dynamic games;Markov games。
博弈论 Game theory (全)
博弈论 Game Theory博弈论亦名“对策论”、“赛局理论”，属应用数学的一个分支, 目前在生物学，经济学，国际关系，计算机科学, 政治学，军事战略和其他很多学科都有广泛的应用。
在《博弈圣经》中写到：博弈论是二人在平等的对局中各自利用对方的策略变换自己的对抗策略，达到取胜的意义。
主要研究公式化了的激励结构间的相互作用。
是研究具有斗争或竞争性质现象的数学理论和方法。
也是运筹学的一个重要学科。
博弈论考虑游戏中的个体的预测行为和实际行为，并研究它们的优化策略。
表面上不同的相互作用可能表现出相似的激励结构（incentive structure），所以他们是同一个游戏的特例。
其中一个有名有趣的应用例子是囚徒困境（Prisoner's dilemma）。
具有竞争或对抗性质的行为称为博弈行为。
在这类行为中，参加斗争或竞争的各方各自具有不同的目标或利益。
为了达到各自的目标和利益，各方必须考虑对手的各种可能的行动方案，并力图选取对自己最为有利或最为合理的方案。
比如日常生活中的下棋，打牌等。
博弈论就是研究博弈行为中斗争各方是否存在着最合理的行为方案，以及如何找到这个合理的行为方案的数学理论和方法。
生物学家使用博弈理论来理解和预测演化（论）的某些结果。
例如，约翰·史密斯（John Maynard Smith）和乔治·普莱斯（George R. Price）在1973年发表于《自然》杂志上的论文中提出的“evolutionarily stable strategy”的这个概念就是使用了博弈理论。
其余可参见演化博弈理论（evolutionary game theory）和行为生态学（behavioral ecology）。
博弈论也应用于数学的其他分支，如概率，统计和线性规划等。
历史博弈论思想古已有之，我国古代的《孙子兵法》就不仅是一部军事著作，而且算是最早的一部博弈论专著。
博弈论最初主要研究象棋、桥牌、赌博中的胜负问题，人们对博弈局势的把握只停留在经验上，没有向理论化发展。
英语第一章阅读 game theory 原文及翻译
The essence of a game is the interdependence of player strategies. There are two distinct types of strategic interdependence: sequential and simultaneous. In the former the players move in sequence, each aware of the others’ previous actions. In the latter the players act at the same time, each ignorant of the others’ actions.
Game theory was pioneered by Princeton mathematician john von Neumann. In the early years the emphasis was on games of pure conflict (zero-sum games). Other games were considered in a cooperative form. That is, the participants were supposed to choose and implement their actions jointly. Recent research has focused on games that are neither zero sum nor purely cooperative. In these games the players choose their actions separately, but their links to others involve elements of both competition and cooperation.
博弈论-game-theory-两人轮流进行游戏
当k∞时 x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 …… g(x) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 …… 这有啥用
游戏的联合
定义：对于n个给定的公平组合博弈G1, G2, …, Gn，定义他 们集的合联 ；合对为于G一=个G1局+G面2+x…i属+G于n.X对i，于设游F戏i(xGi)i表Байду номын сангаас示设xXi的i为后它继的局局面面集 合对。于G那的么一G个的局局面面x集=合{x1X,x=2,X…1*,xXn2}*，…它*X的n（后其继中局*为面笛集卡合儿积）；
gn(x1,x2,…,xn) = g(x1)⊕g(x2)⊕…⊕g(xn)
= x1⊕x2⊕…⊕xn
经典Nim游戏
图的游戏
3
0
2 0
1
3 ⊕0 ⊕0=3
0 0
1 0
1
Anti-Nim
有n堆石子，每堆ai个，两个人轮流游戏，每次游戏者 取走某一石碓中至少1枚，至多k枚的石子。谁取走最 后一颗石子算谁输。
一方算输 无论游戏如何进行，总可以在有限步之内结束。(the
Ending Condition)
N局面，P局面
N局面——先手必胜局面
winning for the Next player
P局面——后手必胜局面
winning for the Previous player
定义：
每一个最终局面都是P局面 对于一个局面，若至少有一种操作使它变成一个P局面，
还扩展
游戏4：游戏有n堆石子，第i堆有ai枚，两人轮流进行 游戏，每次游戏者可以从任意一堆取走任意多枚石子， 也可以将任意的一堆石子任意的分成两堆。谁取走最 后一颗石子为胜。
博弈论又被称为对策论（GameTheory）博弈论和经济行为
博弈论又被称为对策论（GameTheory)博弈论和经济行为本文话题：博弈论和经济行为一帆风顺协同作用博弈论策略博弈论又被称为对策论（Game Theory)目录博弈论的概念博弈论的发展博弈论的基本概念基本概念博弈论的意义纳什博弈论的原理与应用囚徒困境博弈企业博弈老三论小释老三论小释博弈论的概念博弈论又被称为对策论（GameTheory)，它是现代数学的1个新分支，也是运筹学的1个重要组成内容。
按照2005年因对博弈论的贡献而获得诺贝尔经济学奖的RobertAumann教授的说法，博弈论就是研究互动决策的理论。
所谓互动决策，即各行动方（即局中人[player]）的决策是相互影响的，每个人在决策之际必须将他人的决策纳入自己的决策考虑之中，当然也需要把别人对于自己的考虑也要纳入考虑之中……在如此迭代考虑情形进行决策，选择最有利于自己的战略(strategy)。
博弈论的应用领域十分广泛，在经济学、政治科学（国内的以及国际的）、军事战略问题、进化生物学以及当代的计算机科学等领域都已成为重要的研究和分析工具。
此外，它还与会计学、统计学、数学基础、社会心理学以及诸如认识论与伦理学等哲学分支有重要联系。
按照Aumann所撰写的《新帕尔格雷夫经济学大辞典》“博弈论”辞条的看法，标准的博弈论分析出发点是理性的，而不是心理的或社会的角度。
不过，近20年来结合心理学和行为科学、实验经济学的研究成就而对博弈论进行一定改造的行为博弈论(behavoiralgame theory )也日益兴起。
博弈论的发展博弈论思想古已有之，我国古代的《孙子兵法》就不仅是一部军事著作，而且算是最早的一部博弈论专著。
博弈论最初主要研究象棋、桥牌、赌博中的胜负问题，人们对博弈局势的把握只停留在经验上,没有向理论化发展，正式发展成一门学科则是在20世纪初。
1928年冯·诺意曼证明了博弈论的基本原理，从而宣告了博弈论的正式诞生。
1944年，冯·诺意曼和摩根斯坦共著的划时代巨著《博弈论与经济行为》将二人博弈推广到n 人博弈结构并将博弈论系统的应用于经济领域，从而奠定了这一学科的基础和理论体系。
博弈论基础课程教学大纲
博弈论基础课程教学大纲课程名称：博弈论基础英文名称：Game Theory课程编号：X4080251学时数：32其中实验（实训）学时数：0课外学时数：0学分数：2适用专业：金融学一、课程的性质和任务本课程是经济类专业选修课程之一。
本课程的任务是使学生从应用角度出发，在理论和实践上掌握博弈论的基本概念和基本方法，使学生具有应用博弈论的方法分析实际问题的初步能力。
二、课程教学内容的基本要求、重点和难点1.博弈的基本理论基本要求：理解策略形式的博弈，掌握博弈三要素和博弈的基本分类，理解囚徒困境、“抓钱博弈”。
重点和难点：博弈要素、囚徒困境2.同时决策博弈基本要求：掌握纳什均衡的定义，理解优势策略均衡，理解纳什均衡的应用。
重点和难点：纳什均衡3.混合策略纳什均衡基本要求：理解混合策略与期望支付，了解反响函数法，掌握纳什定理和奇数定理，了解多重纳什均衡及其甄别。
重点和难点：纳什定理4.序贯决策博弈基本要求：掌握序贯决策博弈与博弈树，理解策略与行动，了解序贯博弈的纳什均衡, 了解倒推法。
重点和难点：序贯决策博弈与博弈树5.同时博弈与序贯博弈基本要求：掌握正规型表示与展开型表示，理解同时决策与序贯决策的混合博弈，了解树型博弈的子博弈，了解子博弈精炼纳什均衡重点和难点：同时博弈与序贯博弈的正规型表示与展开型表示6.重复博弈和策略性行动基本要求：理解囚徒困境的有限次重复，理解囚徒困境的无限次重复，掌握重复次数不确定的情形，.掌握策略性行动的分类。
重点和难点：囚徒困境的有限次、无限次重复7.零和博弈基本要求：掌握零和博弈与非零和博弈，了解最小最大方法、直线交叉法，理解零和博弈的线性规划解法，了解霍特林模型。
重点和难点：零和博弈与非零和博弈，零和博弈的线性规划解法三、教学方式及学时分配四、课程其它教学环节的要求本课程以教师讲课为主，并适当安排课堂讨论，以学生课后实践为辅，同时鼓励学生参与经济实践与经济讨论的活动如举行经济辩论、撰写小论文等。
game theory3 博弈论 英文
Strict & Weak
Best Response
Mixed Strategy NE
Summary
Example:
1 2 L R
T
B
2,2
2,2
1,1
2,3
Nonstrict NE
Strict NE
If we eliminate T which is weakly dominated by B, and then eliminate L which is dominated by R we lose nonstrict NE {T,L}
Summary
• Definition: player i’s action ai weakly dominates her action bi if ui(ai,a−i)≥ui(bi,a−i) for every list a−i of the other players’ actions, where ui is a payoff function that represents player i’s preferences
a*i is in Bi(a*−i) for every player i This is why “method of circles”, i.e. looking for best responses leads to NE
18 / 43
Review
Strict & Weak
Best Response
• Elimination method is sometimes imprecise, NE (Circle Method, Best responses) is stronger.
运筹学-第六讲对策论
【定义 】 在对策G={S1，S2…，Sn；h1，h2…hn}中，假如由各个对策方旳各 选用一种策略构成旳某个策略组合(S1*，S2*…，Sn*)中，任一对策方i 旳策略 Si*，都是对其他策略方策略旳组合 (S1*，…，S*i-1，S*i+1…，Sn*)旳最佳策略， 即h i(S1*, … , S*i-1, Si*, S*i+1,…Sn*)≥hi(S1*, …, S*i-1, Sij, S*i+1 , …, Sn*)对任意 Sij∈Si 都成立，则称(S1*，…，Sn*)为G旳一种纯策略意义下旳“纳什均 衡”(Nash Equilibrium)．
(2,0)
(4,0)
反应函数法
对策论 game theory
【例4】 考虑上述模型旳另一种情况即各厂商所选择旳是价格而不是产量，假 设产量与价格旳函数关系为：
q1 ( p2 ) a1 b1 p1 d1 p2
q2 ( p1 ) a2 b2 p2 d 2 p1
其他条件不变，边际成本为C1、C2，试求解其纳什均衡。
P2
R2 ( p1 )
1 2b2
(a2
b2 c2
d 2 p1 )
p1*
p2*
1 2b1 1 2b2
(a1b1c1ຫໍສະໝຸດ d1p* 2
)
(a2 b2c2 d 2 p1* )
P1*
d1 4b1b2 d1d 2
(a2
b2c2 )
2b2 4b1b2 d1d 2
(a1
b1c1 )
P2*
d2 4b1b2 d1d 2
Nash对对策论旳贡献有： (i) 合作对策中旳讨价还价模型，称为Nash讨价还价解； (ii) 非合作对策旳均衡分析。
博弈论 Game Theory
• •
信息是博弈论中重要的内容。 完全博弈是指在博弈过程中，每一位博弈 者对其他博弈者的特征、策略空间及收益函数 有准确的信息。严格地讲，完全信息博弈是指 博弈者的策略空间及策略组合下的支付，是博 弈中所有博弈者的“公共知识”(Commom Knowledge)的博弈。 • 完美信息是指博弈者完全清楚到他决策时 为止时, 所有其他博弈者的所有决策信息，或者 说，了解博弈已进行过程的所有信息。
• 2 . 猜硬币游戏
猜方 正面 盖 正面 方 反面 -1，1 1，-1 反面 1，-1 -1，1
• 3. “田忌赛马” • “田忌赛马”是我国古代一个非常有名的故 事，讲的是发生在齐威王与大将田忌之间的赛 马的故事。田忌在谋士孙膑的帮助下，运用谋 略帮助田忌以弱胜强战胜了齐威王。这个故事 讲的其实是一个很典型的博弈问题。
田 上 中 下 上中下 上下中 齐 中上下 威 中下上 王 下上中 下中上 3，-3 1，-1 1，-1 -1，1 1，-1 1，-1 上 下 中 1，-1 3，-3 -1，1 1，-1 1，-1 1，-1 中 上 下 1，-1 1，-1 3，-3 1，-1 1，-1 -1，1
忌 中 下 上 1，-1 1，-1 1，-1 3，-3 -1，1 1，-1 下 上 中 -1，1 1，-1 1，-1 1，-1 3，-3 1，-1 下 中 上 1，-1 -1，1 1，-1 1，-1 1，-1 3，-3
•
动态博弈是指在博弈中，博弈者的行动有 先后顺序（Sequential-Move），且后行动者能 够观察到先行动者所选择的行动或策略,因此, 动态博弈又叫做序贯博弈。
•
2.如果按照博弈者对其他博弈者所掌握的 信息的完全与完备程度进行分类，博弈可以划 分为完全信息博弈（Game with Complete Information）与不完全信息的博弈（Game with Incomplete Information），以及完美信息的博弈 （Game with Perfect Information）与不完美信息 的博弈（Game with Imperfect Information),确定 的博弈(Game of Certainty)与不确定的博弈 （Game of Uncertainty),对称信息的博弈（Game of Symmetric Information）与非对称信息的博弈 （Game of Asymmetric Information）等。
GAME THEORY
1.2.2 重复剔除的占优均衡
Definition：In a normal-form game, if for each player i , si" is i‟s dominant strategy, than we call the strategies profile (s1″, …, sn" ) the „dominantstrategy equilibrium‟.
1.2.1占优战略均衡
定义2：一个博弈G,若对博弈方i及所用s-i都有 ui (si*,s-i) > ui (si ‟,s-i)，则称si*是si ‟的严格上 策， si ‟是si*的严格下策。 定义3：若在博弈G中对每个博弈方i都存在策 略si*是其它所有策略的严格上策,则称策略组 合s*=(s1*,s2*, … ,sn*)是G的上策均衡。
In the normal-form representation of a game ,each
player simultaneously chooses a strategy, and the combination of strategies chosen by the players determines a payoff for each player. Definition: The normal-form representation of an-n-player game specifies the players‟ strategy spaces S1 , … , Sn and their payoff functions u1 ,…, un. We denote this game
《科技英语》课后习题答案
MainContent:UNIT1MATHEMATICSI.TextOrganizationParts ParagraphsMainIdeasPartOne Paras.1-3 Gametheorycanbedefinedasthescienceofstrategywhichstudiesbothpureconflicts(zero-sumgames)andconflictsincooperativeforms.PartTwo Paras.4-11 Therearetwodistincttypesofstrategicinterdep endence:sequential-movegameandsimultaneous-movegame.PartThre e Paras.12-19Thetypicalexamplesofgametheoryaregivenasthebasicprinciplessuchasprisoners’dilemma,mixingmoves,strategicmoves,bargaining,concealingandrevealinginformation.PartFour Para.20 Theresearchofgametheoryhassucceededinillustratingstrategiesinsituationsofconflictandcooperationanditwillfocusonthedesignofsuccessfulstrategyinfuture.nguagePointsThegamesitstudiesrangefromchesstochildrearingandfromtennistotak eovers.(Para.1)Paraphrase:Thegamesit(gametheory)studiesextendsfromchesstochild bringing-upandfromtennistohandovers.range:v.tovarybetweenlimits,extend,runinalinee.g.(1)Thepricerangesfrom$30to$80.(2)Theboundaryrangesfromnorth tosouth.takeover:n.theactoraninstanceofassumingcontrolormanagementoforr esponsibilityforsth.接收、接管e.g.TheeconomyofHongkonggoeswellafteritstakeover. GametheorywaspioneeredbyPrincetonmathematicianJohnvonNeumann.(P ara.2)pioneer:v.tobeapioneer;tooriginate(courseofactionetc.,followedl aterbyothers)e.g.Thenewtreatmentforcancerwaspioneeredbytheexpertsofstatehosp ital.pioneer:n.originalinvestigatorofsubjectorexplorerorsettler;init iatorofenterprisee.g.Theyounggenerationwasgreatlymotivatedbythepioneers’exploit s.Thatis,theparticipantsweresupposedtochooseandimplementtheiracti onsjointly.(Para.2)Paraphrase:Thatis,theplayerswereexpectedtoselectandcarryoutthei ractionstogether. …hemustanticipateandovercomeresistancetohisplans.(Para.3) anticipate:v.1)toexpectorrealizebeforehand;toforeseee.g.Theexpertsareanticipatingthenegativeeffectsofairpollution. anticipate:v.2)todealwithorusebeforepropertime预支e.g.Tedwasnotusedtosavingmonthlyandhewouldalwaysanticipatehisin come. Theessenceofagameistheinterdependenceofplayerstrategies.(Para.4 )Paraphrase:Thekeyprincipalofagameisthatplayerstrategiesaredepen dentoneachother.essence:n.1)thequalitywhichmakesathingwhatitis;theinnernatureor mostimportantqualityofathinge.g.Thetwothingsarethesameinoutwardformbutdifferentinessence. essence:n.2)extractobtainedfromasubstancebytakingoutasmuchofthe massaspossiblekessence;essenceofpeppermint(椒薄荷、椒薄荷油) interdependence:n.thequalityorfactofdependingoneachotherinter-为前缀，意为betweeneachother,类似的词还有interchange、intermarry、international、interview等。
博弈论-Game Theory
4.Understand the application of game theory in economics, management, computer science and other disciplines through group cooperation and topical discussion.
5.Apply the theory in the course to solve a variety of problems at an appropriate level of difficulty.
6.Demonstrate skills in communicating mathematics.
Pre-requisites
Advanced Algebra, Mathematical Analysis, Probability Theory, Operations Research
Intended Learning Outcomes
On successful completion of this course, students will be able to:
Syllabus ofMathematics and Applied Mathematicsat Haide College
Introduction to Game Theory
Description
Game theory is a theory that studies the decision-maker's strategic choice and related equilibrium problems under the conditions of competition or conflict factors and mutual dependence of interests. It has extensive and profound applications in economics, management, political science, computer science, military strategy and many other disciplines. This course mainly introduces the mathematical models, equilibrium theory and applications of game theory. The content includes: the basic concept of game theory, static and dynamic game theory (complete information static game, incomplete information static game, complete information dynamic game, incomplete information dynamic game), cooperative game theory, etc. Through the study of this course, students can understand the background, establishment and analysis of various game models, master the basic theories and methods of game theory, think and explore with the ideological methods of game theory, and improve the ability to analyze practical problems by establishing corresponding mathematical models and solving them.
GameTheory.ppt
Firm B
Firm A
Option C
Option C πac,πbc
Option D πac,πbd
Option D πad,πbc
πad,πbd
8
New toy
• Cost to firm is $5 • Two firms will sell • Can sell the toy at a high ($20/toy) or low
Don’t
5, -5
-10,-10
27
Nash Equilibrium
• No player has an incentive to deviate from equilibrium play because their strategy is best response to her belief about the other player’s strategy
– If one exists, will play the dominant strategy… – But, none may exist
• To solve the game, eliminate dominated strategies
• Key assumption – player is acting in own self interest – doing the same thing you are doing
9
P1 Q1 Profits1 P2 Q2 Profits2
$10 30
$150 $10 30
$150
$20 15
$225 $20 25
$225
$10 60
$300 20 0 $0
20 0
Game theory博弈论
that is:
* * i, xi Si : f i ( xi* , x ) f ( x , x i i i i )
To prove the existence of a Nash Equilibrium, we need using the Kakutani fixed point theorem, John Nash have proved it in 1951.
The end,thank you!
General and applied uses
As a method of applied mathematics, game theory has been used to study a wide variety of human and animal behaviors. It was developed in economics to understand a large collection of economic behaviors. Game theory in the social sciences has been applied to political, sociological, and psychological behaviors as well.
let xi be a strategy profile of player i and
x i be a strategy profile of player i , When each player i {1,, n},
xi resulting in strategy profile x ( x1 xn),
The games studied in game theory are welldefined mathematical objects. To be fully defined, a game must specify the following elements: the players of the game the information the actions available to each player at each decision point the payoffs for each outcome
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13 完全且完美信息动态博弈3 完全且完美信息动态博弈3.1 动态博弈的表示法和特点3.2 可信性和纳什均衡的问题3.3 子博弈和子博弈完美纳什均衡3.4 几个经典动态博弈模型3.5 有同时选择的动态博弈模型3.6 动态博弈分析的问题和扩展讨论动态博弈引例：抢吃饼（1）阶段和扩展型表示动态博弈中一个博弈方的一次选择行为称为一个“阶段”。
•动态博弈中可能存在几个博弈方同时选择的情况，这些博弈方的同时选择构成一个阶段；•一个动态博弈至少有两个阶段；•动态博弈也称为“多阶段博弈”或“序列博弈”。
z 扩展型表示（博弈树）(30, 50)(-10,0)进入不进入在位者进入者默许斗争(0,100)例：市场阻击博弈几个要素：参与人行动顺序行动空间信息集得益外生事件（自然）状态概率分布给出每个策略的动态描述¾节点决策节点末端节点¾枝（棱）¾信息集：引例1A开发不开发(4, 4)(8,0)B开发不开发(0, 8)(0,0)开发不开发B引例21坦白抵赖(-3, -3)(0,-5)2坦白抵赖(-5, 0)(-1,-1)坦白抵赖2决策结的集合•每个决策结是同一个参与人的决策结•该参与人知道进入该集合的某个决策结，但不知道自己究竟处于哪个决策结每个参与人分别有几个信息集？例：房地产开发博弈房地产开发商A、B正考虑是否要在某地投资开发一座商住楼，他们面临的选择是开发还是不开发，如果决定开发，就必须投入资金1亿元；如果决定不开发，就不需要投资。
假定房地产市场可能出现两种状态（市场行情）：市场需求大与市场需求小，两种状态出现的概率分别为1/2。
楼盘售价与市场需求大小密切相关：如果市场上同时有两座出售，需求大时，每座楼售价为1.4亿元，需求小时，每座楼售价为0.7亿元；如果市场上只有一座楼出售，需求大时，售价为1.8亿元，需求小时，每座楼售价为1.1亿元。
房地产开发博弈的扩展型表示需求大需求小NA开发不开发(4, 4)(8,0)B开发不开发(-3, -3)(1,0)开发不开发B N(0, 8)(0,0)B开发不开发(0, 1)(0,0)开发不开发B需求大需求小（单位：千万元）博弈过程？•A 不知道市场行情•B 知道市场行情•A 先决策，B 后决策需求大需求小NA开发不开发房地产开发博弈2(4, 4)(8,0)B开发不开发(-3, -3)(1,0)开发不开发B N(0, 8)(0,0)B开发不开发(0, 1)(0,0)开发不开发B•A 不知道市场行情•B 不知道市场行情需求大需求小•A 先决策，B 后决策博弈过需求大需求小NA开发不开发房地产开发博弈3(4, 4)(8,0)B开发不开发(-3, -3)(1,0)开发不开发B N(0, 8)(0,0)B开发不开发(0, 1)(0,0)开发不开发B需求大需求小博弈过•A 不知道市场行情•B 知道市场行情•A 和B 同时决策需求大需求小NA开发不开发房地产开发博弈4(4, 4)(8,0)B开发不开发(-3, -3)(1,0)开发不开发B N(0, 8)(0,0)B开发不开发(0, 1)(0,0)开发不开发B需求大需求小博弈过•A 不知道市场行情•B 不知道市场行情•A 和B 同时决策一个信息集可能包含多个决策结，也可能只包含一个决策结信息集包含了一个参与人有关其它参与人之前行动的知识。
只包含一个决策结的信息集称为单结信息集。
如果博弈树的所有信息集都是单结的，该博弈称为完美信息博弈。
自然的信息集总是假定为单结的，自然在参与人决策之后行动等价于自然在参与人之前行动但参与人不能观测到自然的行动。
不同的博弈树可以代表相同的博弈，但是有一个基本规则：一个参与人在决策之前知道的事情，必须出现在该参与人决策结之前。
需求大需求小NA开发不开发练习1：房地产开发博弈3(4, 4)(8,0)B开发不开发(-3, -3)(1,0)开发不开发B N(0, 8)(0,0)B开发不开发(0, 1)(0,0)开发不开发B需求大需求小另外的表述？练习2：房地产开发博弈在房地产博弈中，试就以下两种情况画出博弈的扩展式表述的博弈树。
（1）开发商A不知道市场行情，而开发商B知道市场行情，但开发商A先决策，开发商B在观测到A的行动后决策;（2）开发商A、B都知道市场行情，但开发商A和B同时决策。
练习3：将下面的动态博弈用博弈树表述出来（1）参与人1最先行动，从L 1和R 1中选择行动a 1（2）参与人2观测到a 1，然后从L 2和R 2中选择行动a 2（3）参与人3只能观测到（a 1，a 2）是否为（R 1，R 2），然后从L 3和R 3中选择行动a 3L 2R 221L 1R 13L 3R 3L 3R 3323L 3R 3L 3R 33L 2R 2（2）动态博弈的基本特点z 策略：关于行动的一个完整计划。
它指明参与人在可能会遇到的每种情况下行动的选择。
策略是行动的规则。
策略规定参与人在什么时候选择什么行动。
策略必须是完备的。
静态博弈中策略和行动是相同的。
动态博弈的决策内容是选择策略。
ABB(-3,-3)(1,0)(0,1)(0,0)开发不开发开发不开发开发不开发如房地产开发博弈A首先行动，两个可选择的行动，策略空间为:(开发，不开发）；有四个策略：9开发策略:不论A开发不开发，我开发:{开发，开发}；9追随策略:A开发我开发，A不开发我不开发:{开发，不开发};9对抗策略:A开发我不开发，A不开发我开发:{不开发，开发}；9不开发策略:不论A开发不开发我不开发:{不开发，不开发}。
B有两个信息集z 结果：各个博弈方的策略组合各个博弈方的策略组合形成的一条联结各个阶段的路径各个博弈方的得益ABB(-3,-3)(1,0)(0,1)(0,0)开发不开发开发不开发开发不开发如房地产开发博弈18房地产开发博弈：B{开发，开发}{开发，不开发}{不开发，开发}{不开发，不开发}A开发-3，-3-3 ，-31，01，0不开发0，10，00，10，0ABB(-3,-3)(1,0)(0,1)(0,0)开发不开发开发不开发开发不开发扩展式矩阵式3.1 动态博弈的表示法和特点练习4：将下列博弈的扩展型表示转换为矩阵型12(3,2)(1,1)(0,2)(4,5)U DL R L R 2参与人2参与人1{L ，L}{L ，R}{R ，L}{R ，R}U 3，2 3 ，21，11，1D0，24，50，24，5矩阵式z动态博弈的非对称性：动态博弈中各博弈方的地位是不对称的。
动态博弈中先行动有利？后行动有利？一般来说，由于后行动的博弈方有更多的信息帮助自己选择行动，可减少决策的盲目性，针对性地选择，因此处于较有利的地位。
但后行动一定较有利并不总是成立。
213 完全且完美信息动态博弈3 完全且完美信息动态博弈3.1 动态博弈的表示法和特点3.2 可信性和纳什均衡的问题3.3 子博弈和子博弈完美纳什均衡3.4 几个经典动态博弈模型3.5 有同时选择的动态博弈模型3.6 动态博弈分析的问题和扩展讨论22 3.2 可信性和纳什均衡问题借不借分不分(2,0)乙甲(3,3)乙打不打(-1, 0)(0,6)甲分不分乙不借2，02，0(借,打)3，3-1，0(借,不打)3，30，6（1）纳什均衡的问题两个纳什均衡：1（乙:不借；甲:如果乙借,不分）2 （乙:第一阶段借，如果第二阶段甲不分，第三阶段打；甲:如果乙第一阶段借,第二阶段分）×23房地产开发博弈：B{开发，开发}{开发，不开发}{不开发，开发}{不开发，不开发}A 开发-3，-3-3 ，-31，01，0不开发0，10，00，10，0ABB(-3,-3)(1,0)(0,1)(0,0)开发不开发开发不开发开发不开发扩展式矩阵式3.2 可信性和纳什均衡问题纯策略纳什均衡：1.（开发，{不开发，开发}）2.（开发，{不开发，不开发}）3.（不开发，{开发，开发}）不可置信（2）相机选择和策略中的可信性问题动态博弈中各博弈方的策略是他们自己设定，但是这些策略没有强制力，在博弈过程中，博弈方完全可以根据自己的利益而改变原来的计划。
——“相机选择”由于相机选择的存在使得在博弈方的策略中，所设定的各个阶段、各种情况下会采取行动的“可信性”有了疑问。
可信性：动态博弈中先行动的博弈方是否该相信后行动的博弈方会采取对自己有利或不利的行动。
许诺：后行动方将来会采取对先行动方有利的行动。
威胁：后行动方将来会采取对先行动方不利的行动。
(3, 3)(0,6)借不借分不分(2,0)例：开金矿博弈甲在开采一价值6万元的金矿时缺2万元资金，而乙正好有2万元资金。
甲希望乙把2万元资金借给自己，并许诺在采到金子后与乙对半分。
乙是否应该将钱借给甲呢？许诺是“可信”的吗？乙甲假定乙威胁在甲违约时将采用法律武器“打官司”来保护自己的利益。
假设打官司的结果是使乙收回本钱2万元，甲失去全部采金收入。
(3, 3)(0,6)借不借分不分(2,0)乙甲借不借分不分(2,0)乙甲(3,3)乙打不打(2, 0)(0,6)威胁是“可信”的吗？假定乙威胁在甲违约时将采用法律武器“打官司”来保护自己的利益。
假设打官司的结果是甲失去全部采金收入，但乙也要承受1万元的损失。
借不借分不分(2,0)乙甲(3,3)乙打不打(-1, 0)(0,6)威胁是“可信”的吗？283 完全且完美信息动态博弈3 完全且完美信息动态博弈3.1 动态博弈的表示法和特点3.2 可信性和纳什均衡的问题3.3 子博弈和子博弈完美纳什均衡3.4 几个经典动态博弈模型3.5 有同时选择的动态博弈模型3.6 动态博弈分析的问题和扩展讨论（1）子博弈由一个动态博弈（T）第一阶段以外的某阶段开始的后续博弈阶段构成的，有初始信息集和进行博弈所需要的全部信息，能够自成一个博弈的原博弈的一部分，称为原动态博弈的一个子博弈（S）。
（1）S的博弈树由T中的一个决策结点（不包括T 的初始结点）和它后面所有的后续结点组成；（2）博弈S不能分割博弈T的信息集；（3）S的末端结点处的得益向量是T的得益向量中适用于S的部分。
例1：房地产开发博弈ABB(-3,-3)(1,0)(0,1)(0,0)开发不开发开发不开发开发不开发x'x 子博弈I子博弈II存在子博弈吗？甲乙乙(-3,-3)(0,-5)(-5,0)(-1,-1)坦白抵赖坦白抵赖坦白抵赖例2：囚徒困境x'x 存在子博弈吗？2133233例3：博弈3博弈方2的两个决策结能开始子博弈吗？2145367例4：博弈4存在几个子博弈？89101112131415借不借分不分(2,0)乙甲(3,3)乙打不打(-1, 0)(0,6)例5：开金矿博弈子博弈I子博弈II（2）子博弈完美纳什均衡在一个完美信息的动态博弈中，如果各博弈方的策略构成的一个策略组合，在整个动态博弈及它的所有子博弈中都构成纳什均衡，那么这个策略组合称为该动态博弈的一个“子博弈完美纳什均衡”。