博弈论-game-theory-两人轮流进行游戏
博弈论作业——精选推荐
博弈论作业1.海盗分金中如果假设需要同意的人超过半数提议才能通过，则理性结局又会是什么？如果200个人参加（只要半数即可通过）又将如何？500人呢？解：半数即可通过——倒推分析结果如下（1000，0）（999,0,1）（999,0,1,0）（998,0,1,0，1）下一步的分钱方案中，只需要把上一步得钱非0的强盗的得钱数改为0，而其它强盗则反之。
进而有非0和0的间隔分布，除了提出方案的强盗外，其它得钱非0的强盗得到1块钱。
因此有200个强盗分1000个金币的情形为 ( X, 0, 1, …, 0 )可得X=901因此有500个强盗分1000个金币的情形为 ( X, 0, 1, …, 0 )可得X=751超过半数才可通过——倒推结果如下（0,1000）（999,1,0）（997,0,2,1）（997,0,1,0，2）因此，5个强盗分1000个金币的情形为（997,0,1,0，2）2.在无限期的鲁宾斯坦模型中，假设分割只能是0.01的整数倍，即X只能为0，0.01；0.02；…….0.99或1，求δ=0.5和δ非常接近于1时的子博弈完美均衡(假设两个人的折现因子相同)两个人要分一块冰淇淋，甲将分得冰淇淋的x份额（x ≥ 0），乙将得到1-x的份额（1-x ≥ 0）。
两人进行轮流出价。
首先，甲提出一个划分方法(x，1-x)，乙可以接受或拒绝这个提议，如果他接受了，则博弈结束，他们按照这种划分去切割冰淇淋；如果乙拒绝这个提议，那么他会提出一个划分方法(y，1-y)，甲可以接受或者拒绝，博弈过程将这个方式持续进行下去，直到他们达成一个协议。
每当协议的达成拖延时，他们的得益会有一个折扣（贴现），两人的贴现因子由iδ (0<iδ<1)表示。
这种折扣代表了讨价还价的成本。
其它条件相同，对参与者而言，达成一个协议所需的时间越长，冰淇淋就会越小。
两人贴现相同，如果假定1δ＝2δ＝δ的话，上述讨价还价博弈的唯一的均衡结果将会是（1 / (1 + δ), δ / (1 + δ)）。
博弈论
博弈论是一种处理竞争与合作问题的数学决策方法；研究竞争中参加者为争取最大利益应当如何做出决策的数学方法；根据信息分析及能力判断，研究多决策主体之间行为相互作用及其相互平衡，以使收益或效用最大化的一种对策理论；研究决策主体的行为发生直接相互作用时候的决策以及这种决策的均衡问题。
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，达到取胜的目的。
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著作。
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。
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。
近代对于博弈论的研究，开始于策墨洛（Zermelo），波雷尔（Borel）及冯·诺伊曼（von Neumann）。
1928年，冯·诺依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·诺依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。
1950～1951年，约翰·福布斯·纳什（John Forbes Nash Jr）利用不动点定理证明了均衡点的存在，为博弈论的一般化奠定了坚实的基础。
纳什的开创性论文《n人博弈的均衡点》（1950），《非合作博弈》（1951）等等，给出了纳什均衡的概念和均衡存在定理。
此外，塞尔顿、哈桑尼的研究也对博弈论发展起到推动作用。
今天博弈论已发展成一门较完善的学科。
博弈的分类根据不同的基准也有所不同。
一般认为，博弈主要可以分为合作博弈和非合作博弈。
它们的区别在于相互发生作用的当事人之间有没有一个具有约束力的协议，如果有，就是合作博弈，如果没有，就是非合作博弈。
从行为的时间序列性，博弈论进一步分为两类：静态博弈是指在博弈中，参与人同时选择或虽非同时选择但后行动者并不知道先行动者采取了什么具体行动；动态博弈是指在博弈中，参与人的行动有先后顺序，且后行动者能够观察到先行动者所选择的行动。
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".。
「算法笔记」博弈论入门
「算法笔记」博弈论⼊门⼀、公平组合游戏 ICG1. 公平组合游戏的定义若⼀个游戏满⾜：1. 游戏有两个⼈参与，⼆者轮流做出决策。
2. 在游戏进程的任意时刻，可以执⾏的合法⾏动与轮到哪名玩家⽆关。
3. 不能⾏动的玩家判负。
则称该游戏为⼀个公平组合游戏。
2. ⼀些说明我们把游戏过程中⾯临的状态称为局⾯，整局游戏第⼀个⾏动的为先⼿，第⼆个⾏动的为后⼿。
我们讨论的博弈问题⼀般只考虑理想情况，即两⼈均⽆失误，都采取最优策略⾏动时游戏的结果。
定义必胜态为先⼿必胜的状态，必败态为先⼿必败的状态。
注意，在⼀般确定操作状态的组合游戏中，只会存在这两种状态，如果先⼿和后⼿都⾜够聪明，不会出现介于必胜态和必败态之间的状态。
⼀个重要的性质：⼀个状态是必败态当且仅当它的所有后继都是必胜态。
⼀个状态是必胜态当且仅当它⾄少有⼀个后继是必败态。
特别地，没有后继状态的状态是必败态（因为⽆法操作则负）。
⼆、Nim 博弈\(\text{Nim}\) 游戏是⼀个公平组合游戏。
⼤概是这样的：现在有 \(n\) 堆⽯⼦，第 \(i\) 堆有 \(a_i\) 个。
两⼈轮流操作，每⼈每次可以从任选⼀堆中取⾛任意多个⽯⼦，但是不能不取。
取⾛最后⼀个⽯⼦的⼈获胜（即⽆法再取的⼈就输了）。
结论：\(\text{Nim}\) 博弈先⼿必胜，当且仅当 \(a_1\oplus a_2\oplus \cdots \oplus a_n\neq 0\)。
证明：为了证明这个结论，我们需要证明：1. 所有⽯⼦都被取⾛是⼀个必败局⾯。
2. 对于任意⼀个局⾯，若 \(a_1\oplus a_2\oplus \cdots \oplus a_n\neq 0\)，⼀定能得到⼀个 \(a_1\oplus a_2\oplus \cdots \oplusa_n=0\) 的局⾯。
3. 对于任意⼀个局⾯，若 \(a_1\oplus a_2\oplus \cdots \oplus a_n=0\)，⼀定不能得到⼀个 \(a_1\oplus a_2\oplus \cdots \oplusa_n=0\) 的局⾯。
范里安-微观经济学现代观点(第7版)-28博弈论(含习题解答)
Chapter 28: Game TheoryIntermediate Microeconomics:A Modern Approach (7th Edition)Hal R. Varian(University of California at Berkeley)第28章:博弈论（含习题解答）含习题解答）中级微观经济学：现代方法（第7版）范里安著（加州大学伯克利）曹乾译（东南大学caoqianseu@）简短说明：翻译此书的原因是教学的需要，当然也因为对现行中文翻译版教材的不满。
市场中的翻译版翻译生硬错误百出。
此次翻译的错误是微不足道的，但仍欢迎指出。
仅供教学和学习参考。
28博弈理论我们在上一章阐述的寡头理论，是企业间策略性互动的经典经济理论解释。
但这只是冰山一角。
经济行为人（agents）的策略性互动有多种方式，经济学家借助博弈理论（game theory）这个工具已研究了很多种策略性互动的行为。
博弈理论关注的是策略性互动的一般分析。
人们可使用博弈理论研究室内游戏（parlor games）、政治协商和经济行为（一）。
在本章，我们将简要分析这一迷人的学科，目的是让你感受一下它是如何运行的，以及让你初步知道如何使用博弈理论分析寡头市场中的经济行为。
28.1博弈的收益矩阵策略性互动可能涉及很多选手和很多策略，但是我们仅限于分析两个选手之间的博弈，而且限于分析策略的数量有限的情形。
这样做的好处是可以用收益矩阵（payoff matrix）描述博弈。
最好举例进行分析。
假设两人玩一种简单的游戏。
选手A在纸上写出“上”或“下”。
与此同时，选手B独立地写出“左”或“右”。
在两人写好后，经过分析，将他们的收益标记于表28.1中。
若A 选上且B选左，我们看矩阵的左上角的小方格。
在该小方格中，A的收益是第一个数，B 的收益是第二个数。
类似地，如果A选下B选右，则A得到收益为1，B得到的收益为0.表28.1：一个博弈的收益矩阵选手A有两个策略：上或下。
博弈论板子
博弈知识汇总有一种很有意思的游戏，就是有物体若干堆，可以是火柴棍或是围棋子等等均可。
两个人轮流从堆中取物体若干，规定最后取光物体者取胜。
这是我国民间很古老的一个游戏，别看这游戏极其简单，却蕴含着深刻的数学原理。
下面我们来分析一下要如何才能够取胜。
（一）巴什博奕（Bash Game）：只有一堆n个物品，两个人轮流从这堆物品中取物，规定每次至少取一个，最多取m个。
最后取光者得胜。
显然，如果n=m+1，那么由于一次最多只能取m个，所以，无论先取者拿走多少个，后取者都能够一次拿走剩余的物品，后者取胜。
因此我们发现了如何取胜的法则：如果n=（m+1）r+s，（r为任意自然数，s≤m),那么先取者要拿走s个物品，如果后取者拿走k（≤m)个，那么先取者再拿走m+1-k个，结果剩下（m+1）（r-1）个，以后保持这样的取法，那么先取者肯定获胜。
总之，要保持给对手留下（m+1）的倍数，就能最后获胜。
这个游戏还可以有一种变相的玩法：两个人轮流报数，每次至少报一个，最多报十个，谁能报到100者胜。
（二）威佐夫博奕（Wythoff Game）：有两堆各若干个物品，两个人轮流从某一堆或同时从两堆中取同样多的物品，规定每次至少取一个，多者不限，最后取光者得胜。
这种情况下是颇为复杂的。
我们用（ak，bk）（ak ≤ bk ,k=0，1，2，…,n)表示两堆物品的数量并称其为局势，如果甲面对（0，0），那么甲已经输了，这种局势我们称为奇异局势。
前几个奇异局势是：（0，0）、（1，2）、（3，5）、（4，7）、（6，10）、（8，13）、（9，15）、（11，18）、（12，20）。
可以看出,a0=b0=0,ak是未在前面出现过的最小自然数,而 bk= ak + k，奇异局势有如下三条性质：1。
任何自然数都包含在一个且仅有一个奇异局势中。
由于ak 是未在前面出现过的最小自然数，所以有ak > ak-1 ，而 bk= ak + k > ak-1 + k-1 = bk-1 > ak-1 。
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。
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.
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
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。
继续扩展
……
我们需要理论！
公平组合博弈
Impartial Combinatorial Game
它是两人参与的游戏 游戏局面的状态集合是有限的 对于同一个局面，两个游戏者的可操作集合完全相同 游戏者轮流进行游戏 当无法进行操作时，游戏结束，此时不能进行操作的
F(x) = F(x1, x2, … , xn ) = F1(x1)*{x1} *{x2} *... *{xn}
U {x1} *F2(x2) *{x3} *... *{xn}
U…
U {x1} *{x2} *... *{xn-1} *Fn(xn)
也就是说，每一次每一个游戏者只能选择其中一个子游戏
进行一步操作。
姚金宇
先玩一个游戏☺
游戏0：有10枚石子组成的石碓，两人轮流进行游戏。 每次游戏者可以取走石碓中不超过3枚的石子。谁取走 最后一颗石子为胜。
你会选先取还是后取？ 为什么？
抽象
游戏1：有n枚石子组成的石碓，两人轮流进行游戏。 每次游戏者取走石碓中至少1枚，至多k枚的石子。谁 取走最后一颗石子为胜。
一般性的结论？
n = a(k+1)时，先手必败。
扩展
游戏2：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为胜。
怎么办？
再扩展
游戏3：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枚石子。谁取走最后一 颗石子为胜。
Problems from POJ
2960 1704 2975 2068
Thanks for your attention!
Sprague-Grundy 定理
定理2（Sprague-Grundy 定理）：设gi(x)为Gi的SG函 数(1≤i≤n)，则G=G1+G2+…+Gn的SG函数g(x1, x2, …, xn)=g1(x1)⊕g2(x2)⊕…⊕gn(xn)（其中⊕为抑或运算）
如何证明？
游戏2的解决
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 ……
则它是一个N局面 对于一个局面，无论如何操作都必然变成一个N局面，
则它是一个P局面
Sprague-Grundy函数
定义：在非负整数集上定义局面x的SG函数g(x)如下：
其中F(x)表示x的后继局面集合
定理1：对于任意的局面x，若g(x)=0则x是P局面，否 则x是N局面.
再看游戏1
以k=3为例 x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 …… g(x) 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 ……