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博弈论作业（博弈论24讲）数应专业一、1、理性人：指代这一类人，他们只关心自己的利益。
2、如果选择a的结果严格优于b，那么就说a相对于b来说是一个严格优势策略。
结论：不要选择严格略施策略。
3、理性人的理性选择造成了次优的结果4、举例：囚徒困境、宿舍卫生打扫问题、企业打价格战等5、协和谬误收益很重要，“如欲得之，必先知之”6、要学会换位思考，站在别人的立场上看别人会怎么做，在考虑自己受益的同时，要注意别人会怎么选择二、1、打渔问题、全球气候变暖与碳排放问题2、博弈的要素：参与人、策略集合、收益3、如果策略a严格劣于策略b，那么不管他人怎么选择，b总是更好的选择4、军队的入侵与防卫问题5、所有人都从1到100中选个数字，最接近所有人选的数字的均值的2/3者为胜，这个数字是多少呢？作为理性人，每个人都会选择67（100*2/3）以下的数，进一步假设你的对手也是理性的，你会选择45（100*4/9）以下的数……依据哲学观点，如果大家都是理性程度相当的，那么最后数字将为1，然而结果却是9，这说明博弈的复杂性6、共同知识与相互知识的区别三、1、利用迭代剔除法领悟中间选民问题2、迭代剔除法就是严格下策反复消去法，不断地把劣势策略剔除出去，最后只剩下相对优势的策略3、中间选民问题就是，在两党制中，政党表述施政纲领要吸引位于中间位置的选民，他们认为在选举中处于中间标度可以吸引左右两边的选民，并以此获得胜利。
4、中间选民问题理论成立的条件是有两个参与人；政治立场能使选民相信。
5、由此延伸出来的还有加油站选址问题，两家加油站不是在不同的路口选址，而是在不确定哪个位置较佳的时候会选在同一处，这也是“中间选民定理”的凸显6、在迭代剔除法不能运用时，比如说该博弈中博弈方1和2均没有严格下策，可以用二维坐标系画出选择策略之后的收益分布四、1、罚点球：一个经过模型简化的点球模型：罚球者可以选择左路，中路，右路3种路线去踢点球，门将可以选择向左扑救或者向右扑救（门将没有傻站着不动的option）。
耶鲁大学公开课-心理学导论第2课中文课件耶鲁大学开放课程心理学导论第2课基础：这是你的大脑今天我们开始正式上课，心理学导论，我们首先要来探讨的是大脑；我想先提到一种观点，这种观点的提出者是位诺贝尔奖得主,生物学家、费兰西斯.克里克；他将这种观点称为“惊人的假说”，这一惊人的假说大致观点如下：正如他在其著作《惊人的假说》中写道“你、你的喜悦、悲伤、回忆、抱负，你对人格同一性的感知、你的自由意志；事实上这一切都不过是大量神经细胞集与其缔合分子的生理反应而已；正如刘易斯.卡罗尔在《爱丽斯》中所描述的“你不过是一堆神经元罢了”，用“惊人”来形容这个观点是贴切的，这是个古怪又反常的观点：我并不奢望大家在一开始就能接受这个观点，即使到课程结束时能否接受这个观点，依然是你们的自由；相反、你们要现在就能接受，倒是会让我感到很奇怪，我知道多数人接受不了。
实际上、他们持有另外一种观点，大多数人都是二元论者，二元论则是一种截然不同的假说，你能够在有史以来的所有宗教和绝大多数哲学体系中找到这个假说，比如、柏拉图就曾明确提出过；但是最著名最有影响力的二元论拥护者则非哲学家勒奈.笛卡尔莫属，勒奈.笛卡尔明确地提出了一个问题“人类是否仅仅是生理机器、是生理客体而已”他的答复是“不”，他认为动物都是机器，实际上他将动物称之为“野兽机器”；他认为非人类动物全是机器人，但人类是不同的、人类具有二元性；我们和动物一样拥有有形的生理客体，但与动物不同，我们的本质却并非是生理的，我们是拥有生理客体的无形心灵，我们的心灵占据着生理客体，寄居其中与其形成紧密的联系，这便是二元论，因为它主张至少对于人类而言存在两种独立的成分，即有形的生理客体与无形的心理；笛卡尔为二元论提出了两点论据：第一点是基于对人类行为的观察，笛卡尔生活在一个人类社会发展相对成熟的时代，在他生活的年代已经出现了机器人，当然不是我们现代的电动机器人，当时的机器人需要用水力来带动，笛卡尔曾经在法国皇家园林中散步，当时的法国皇家园林被建造得犹如17世纪的迪斯尼乐园一般，园林中有很多人偶造型可以运用水流来控制他们的动作；当你踏上相应的踩踏板，一个剑客便会跳出来向你挥剑，如果你踩到了另一块，一个正在沐浴的美女便会藏到树丛里去；因此笛卡尔说“天呐、这些机器可以对特定的动作做出特定的反应，原来机器也是可以完成某些动作的”；实际上他说道“我们的身体也是这样运作的，如果你轻击某人的膝盖，小腿就会弹出去，或许这就是我们的本质”但是笛卡尔否定了这一观点，因为有些事情人类可以做到，机器永远不可能做到；人类的行为不光只有反射，相反、人类拥有协调能力、创造力以及发起自发性行为的能力；例如我们能够运用语言，当然有时我说出的话会是反射性的，比如有人问我“你好吗？”我会说“很好、你呢？”但有时、我可以选择那些我想要说的话“你好吗”“非常好”我完全可以进行选择;笛卡尔认为机器无法做出这种选择，因此我们不仅仅只是机器；当然、他的第二个论据非常的有名，在这里他用到了怀疑方法，他首先问自己一个问题“我究竟能够确信什么呢”之后他自答到“我相信上帝、但说实话、我不能确定上帝的存在，我相信我生活在一个富有的国度，但我可能是被愚弄了”；他甚至说“我相信我拥有朋友与家人，但或许我只是他们的一颗棋子罢了，或许是有个恶魔在戏弄我，让我产生错觉感受到了一些并不存在的东西”；《黑客帝国》便是这些怀疑的现代版演绎，《黑客帝国》的创意完全基于笛卡尔的哲学，笛卡尔对于恶灵的忧虑，或许你现在所经历的一切都不是真实的，而是某种邪恶生物制造出的幻觉；笛卡尔甚至同样怀疑自己身体的存在，事实上他注意到疯子有时会相信自己有额外的四肢或者相信他们的大小与形状是与实际不同的，笛卡尔问道“我如何让才能确定我不是个疯子，疯子们都认为自己是正常的，所以我觉得自己不是疯子，我又如何能确定我现在不是在做梦呢”；但迪卡尔认为有一点是他无法质疑的，那就是他无法怀疑自我思考的存在，而这又成为了对自己的反驳；因此、笛卡尔运用怀疑方法得出与拥有不可确定的身体不同、拥有心理是肯定的；他用这一论据来支持二元论、来支持身心二元的观点；因此他总结到“我知道我是个实体、是思考主体的本质或本性，而这种思考主体的存在，无需任何空间，也不依赖与任何有形实体；也就是说我的心灵、我的本质完全不同于我的身体”；我之前说过这是个普通观点，现在我要从几个方面来说明这个普遍观点：首先、二元论镶嵌于我们的语言之中，我们对我们所拥有的或是与我们有关的事物有着特定的表述模式，比如“我的胳膊、我的心脏、我的孩子、我的爱车”但我们对自己身体和大脑的表述，却也是如此在我们谈及我们拥有大脑时，似乎大脑是与我们相分的;二元论揭示了直觉上的人格同一性，这也就是在说普通观点告诉我们，一个人、即使经历了身体上的巨变也依旧是原先的那个人，很多虚构的故事非常能说明这个道理；我们完全能够看懂电影中，一个少女睡去一觉醒来却变成了詹妮弗.加纳一个成年人，没人会说“这就是纪录片、我相信这些全部是真的”但同时无论大人、青少年还是孩子、没有人会选择中途离开，说“我完全不知道电影在讲些什么”；相反、我们能够理解电影中的故事情节，我们也能理解很多包含更大转变的故事；比如某人死后又转世投胎到新生儿的身上，你可能有些不同的观点是关于对于转世投胎是否真的存在，在座的各位可能会有很多不同的看法；但我们可以想象一下，我们可以想象有这么个一个人死了，然后出现在另一个身体里；这并非是好莱坞首创上世纪最伟大的短篇小说之一，弗郎茨.卡夫卡所写的名篇是这样开头的“一天早晨，格里高尔.萨姆沙从不安的睡梦中醒来发现自己躺在床上变成了一只巨大的甲虫”这里卡夫卡再带次领我们去想象一觉醒来却变成一只甲虫的场景，而我们是可以想象到的；还有一个古老的例子，在公元前几百年荷马这样描述奥德赛的几个同样的命运，一个女巫将它们变成了猪，实际上这么说并不准确，他并未将它们变成猪，他的做法更加过分，女巫将它们封锁在猪的身体里，他们拥有猪的脑袋、声音、鬃毛和身体，但他们的心里却从未改变，他们被关在猪圈里轻声哭泣，在这里、作者再次邀请我们去想象，我们自己被困在其他动物身体里的场景，如果你能想象如此场景，那是因为你将自己当做了与所依肉体分离的心理实体；我们相信许多人是能够“灵魂附体”的，这是滑稽戏剧的一贯伎俩;同样也出现在经典电影《衰鬼上错身》中史蒂芬.马丁和莉莉.汤普林主演大力推荐，但很多人认为这种事情是真实存在的，对多重人格障碍的一种解释就是在你的身体中存在着多个心灵争夺对身体的控制权；我们将在本学期结束之前探讨多重人格障碍，到时大家会发现，事情要远比这复杂得多，但我想说的并非多重人格障碍究竟是什么，而是我们如何看待它，普通观点告诉我们，你的身体里可以拥有不止一个的心灵，这种普遍观点出现在许多不同的故事之中，也包括恶灵附体；许多信任系统都相信，个体行为，尤其是那些邪恶或非理性行为之所以会出现，是因为某种事物占据了他们的身体。
最近一朋友和我讲，YALE大学把他们上课的内容录了下来，并发布在因特网上，供网络传播浏览。
一个全世界排名第二的大学，竟然把自己课程的内容完全录了下来，并在全世界传播，在大学产业化的中国，有些人会觉得不可思议，你没交学费，怎么可以把我的课程免费的给公众提供呢？可是人家并不是这样认为的，本着人不分贵贱，教育不分你我的原则，耶鲁大学做出了这个匪夷所思的举动，实在是非常钦佩。
好了，不说大话，这个公开课真的非常的好，我有义务分享给我的朋友们首先你得会用电驴，具体怎么用百度去ps:我分享的这个是有字幕的，而且这个字幕是非常考究的，所以出的非常慢，理解一下。
＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝音乐学聆听音乐Listening to Music（教授本人著述）课程简介：本课程培养在对西方音乐理解基础上对音乐的感悟。
它会介绍各种类型的音乐是如何搭配，并教导如何聆听各种类型的音乐，从巴赫，莫扎特，格里高利咏叹调到蓝调关于课程主讲人：Craig Wright在1966年于the Eastman School获得钢琴乐和音乐史双学位，在1972年于哈佛大学获得博士学位。
Craig Wright从1973年开始在耶鲁大学任教，目前是the Henry L. and Lucy G的音乐教授。
在耶鲁大学，Craig Wright的成就包括常年流行的入门课程“聆听音乐”和选择性研讨会“探索大自然的天才”。
每年夏天，他都会带领一些耶鲁大学的社团区法国，德国和意大利采风。
他的六本著述包括：巴黎圣母院中的音乐(1989)、西方文明中的音乐(2005)、聆听音乐(5th edition, 2007)、聆听西方音乐(2007).他目前的工作是在写作“莫扎特：探索大自然的天才”在2004年Craig Wright被芝加哥大学授予人文学名誉博士学位。
课程安排：1. Introduction导言2. Introduction to Instruments and Musical Genres介绍乐器和音乐风格3. Rhythm: Fundamentals节奏：音乐的基础4. Rhythm: Jazz, Pop and Classical节奏：爵士流行和古典5. Melody: Notes, Scales, Nuts and Bolts旋律：音符，音节，基本细节6. Melody: Mozart and Wagner旋律：莫扎特和瓦格纳7. Harmony: Chords and How to Build Them和声：和弦和如何创建主题8. Bass Patterns: Blues and Rock贝斯风格：布鲁斯和摇滚9. Sonata-Allegro Form: Mozart and Beethoven奏鸣曲式：莫扎特和贝多芬10. Sonata-Allegro and Theme and Variations奏鸣曲式和主题以及主题变奏11. Form: Rondo, Sonata-Allegro and Theme and Variations (cont.)曲式：回旋曲式，奏鸣曲式，主题变奏曲12. Guest Conductor: Saybrook Youth Orchestra客席指挥：布鲁克青年交响乐团13. Fugue: Bach, Bizet and Bernstein赋格：巴赫，比才和伯恩斯坦14. Ostinato Form in the Music of Purcell, Pachelbel, Elton John andVitamin C帕赫贝尔，艾尔顿·约翰音乐中的固定音型15. Benedictine Chant and Music in the Sistine Chapel本尼迪克特教团圣歌和的音乐西斯廷教堂16. Baroque Music: The Vocal Music of Johann Sebastian Bach波洛克音乐：巴赫的声乐作品17. Mozart and His Operas莫扎特和他的歌剧18. Piano Music of Mozart and Beethoven莫扎特和贝多芬的钢琴音乐19. Romantic Opera: Verdi's La Traviata, Bocelli, Pavarotti and Domingo浪漫派歌剧：威尔第的《茶花女》，波切利，帕瓦罗蒂和多明戈20. The Colossal Symphony: Beethoven, Berlioz, Mahler and Shostakovich巨人的交响乐：贝多芬，柏辽兹，马勒，肖斯塔科维奇21. Musical Impressionism and Exoticism: Debussy, Ravel and Monet印象派音乐和异国情调：德彪西，拉威尔和莫奈22. Modernism and Mahler现代主义和马勒23. Review of Musical Style音乐风格的回顾下载地址，电驴会用吧？/topics/2832525/＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝经济学金融市场Financial Markets课程简介：金融机构是文明社会的重要支柱。
博弈论_耶鲁公开课__笔记及扩展1.博弈（game theory）构成要素：参与人players：i，j策略集strategy set:Si 策略si s-i 最优战略si*效益（目标）payoff：Ui Ui（s1，。
si。
sn）博弈game：G={S1，S2....Sn；U1，U2....Un}2.博弈论简史理论提出：1944 冯诺依曼（计算机之父、博弈论之父）与摩根斯坦恩合作出版《博弈论与经济行为》提出了博弈概念；提出了零和博弈（Zero-sum game）；引进了合作博弈理论发展：a.奠定非合作博弈基石：1950 Tucker 提出了“囚徒困境”；1950-1951 纳什：引入纳什均衡，将博弈论从零和博弈推进到非零和博弈；定义非合作博弈并证明纳什均衡存在；1994年诺贝尔经济学奖（与selten harsanyi共同拿到）b.1965-1975 泽尔腾Selten 将纳什均衡推广到动态博弈并提出子博弈精炼均衡；发展了倒退归纳分析方法；提出颤抖手均衡c.1967-1968 海萨尼harsanyi将纳什均衡推广到非完全信息博弈并突出贝叶斯均衡3.博弈的分类是否合作合作博弈cooperative ganme 非合作博弈non-operative ganme 一般说博弈指后者后者又分一下四类根据：是否完全信息是否同时进行完全信息静态博弈（囚徒困境prison's dilemma）纳什均衡Nash equilibrium完全信息动态信息（抢劫博弈）子博弈精炼均衡subgame perfect Nash equilibrium非完全信息静态博弈（密封报价拍卖模型）贝叶斯纳什均衡Bayesian Nash equilibrium非完全信息动态博弈（就业市场信号黔驴技穷）精炼贝叶斯纳什均衡perfect Bayesian Nash equilibrium完全信息complete information与完美信息perfect information完全信息指每个参与者都知道其他人的可行策略以及收益（支付函数），如果一个博弈不是完全信息，那么参与者就不可能知道自己的行为对其他博弈者的影响完美信息指参与者对其他参与者行动action的完全知识的状态，并随信息的出现而更新。
课时：2课时教学目标：1. 让学生了解博弈论的基本概念、原理和应用。
2. 培养学生的逻辑思维能力和决策能力。
3. 增强学生对现实生活中的博弈现象的认识。
教学重点：1. 博弈论的基本概念和原理。
2. 博弈论在现实生活中的应用。
教学难点：1. 博弈论模型的构建。
2. 博弈论在实际问题中的应用。
教学过程：第一课时一、导入1. 向学生介绍博弈论的定义和起源。
2. 通过生活中的实例，引导学生关注博弈现象。
二、基本概念1. 介绍博弈论的基本术语，如参与者、策略、收益等。
2. 讲解零和博弈、正和博弈、完全信息博弈和不完全信息博弈等概念。
三、博弈论原理1. 分析纳什均衡、混合策略均衡等核心原理。
2. 通过实例讲解博弈论原理在现实生活中的应用。
四、课堂练习1. 让学生分组讨论，分析一个现实生活中的博弈现象，并尝试运用博弈论原理进行分析。
2. 每组选派代表进行汇报，教师点评。
第二课时一、回顾与总结1. 回顾博弈论的基本概念、原理和应用。
2. 总结学生在课堂练习中的表现。
二、博弈论模型构建1. 介绍博弈论模型的基本要素，如参与者的数量、策略空间、收益矩阵等。
2. 通过实例讲解博弈论模型的构建过程。
三、博弈论在实际问题中的应用1. 分析博弈论在经济学、政治学、心理学等领域的应用。
2. 结合实际案例，讲解博弈论在实际问题中的应用。
四、课堂讨论1. 让学生分组讨论，分析一个与自己专业相关的博弈现象，并尝试运用博弈论原理进行分析。
2. 每组选派代表进行汇报，教师点评。
五、课后作业1. 让学生收集生活中的博弈现象，运用博弈论原理进行分析。
2. 下节课进行汇报。
教学反思：1. 本节课通过理论讲解和实例分析，让学生对博弈论有了初步的认识。
2. 课堂练习和讨论环节，激发了学生的学习兴趣，提高了学生的参与度。
3. 教师在讲解过程中，应注重引导学生思考，培养学生的逻辑思维能力和决策能力。
4. 在今后的教学中，可以结合更多实际案例，让学生更好地理解博弈论的应用。
耶鲁公开课—博弈论笔记第一节、名词解释优势策略（Dominant strategy ）：不论其他局中人采取什么策略，优势策略对一个局中人而言都是最好的策略。
即某些时候它胜于其他策略，且任何时候都不会比其他策略差。
注：1、“优势策略”的优势是指你的这个策略对你的其他策略占有优势，而不是无论对手采用什么策略，都占有优势的策略。
2、采用优势策略得到的最坏的结果不一定比采用另外一个策略得到的最佳的结果略胜一筹。
严格劣势策略(strictly dominated strategy)：被全面的严格优势策略压住的那个策略，也就是说不是严格优势策略以外的策略。
弱劣势策略：原来不是严格劣势策略，但是经过剔除严格劣势策略后，这个策略就成了严格劣势策略。
例：囚徒困境囚徒到底应该选择哪一项策略，才能将自己个人的刑期缩至最短？两名囚徒由于隔绝监禁，并不知道对方选择；而即使他们能交谈，还是未必能够尽信对方不会反口。
就个人的理性选择而言，检举背叛对方所得刑期，总比沉默要来得低。
试设想困境中两名理性囚徒会如何作出选择：若对方沉默、背叛会让我获释，所以会选择背叛。
若对方背叛指控我，我也要指控对方才能得到较低的刑期，所以也是会选择背叛。
二人面对的情况一样，所以二人的理性思考都会得出相同的结论——选择背叛。
背叛是两种策略之中的支配性策略。
因此，这场博弈中唯一可能达到的纳什均衡，就是双方参与者都背叛对方，结果二人同样服刑2年。
例：协和谬误20世纪60年代，英法两国政府联合投资开发大型超音速客机，即协和飞机。
该种飞机机身大、装饰豪华并且速度快，其开发可以说是一场豪赌，单是设计一个新引擎的成本就可能高达数亿元。
难怪政府也会被牵涉进去，竭力要为本国企业提供更大的支持。
项目开展不久，英法两国政府发现：继续投资开发这样的机型，花费会急剧增加，但这样的设计定位能否适应市场还不知道；但是停止研制也是可怕的，因为以前的投资将付诸东流。
随着研制工作的深入，他们更是无法做出停止研制工作的决定。
ECON-159: GAME THEORYLecture 2 - Putting Yourselves into Other People's Shoes [September 10, 2007]Chapter 1.Recap of Previous Lecture: Prisoners' Dilemma and Payoffs [00:00:00]Professor Ben Polak: Okay, so last time we looked at and played this game. You had to choose grades, so you had to choose Alpha and Beta, and this table told us what outcome would arise. In particular, what grade you would get and what grade your pair would get. So, for example, if you had chosen Beta and your pair had chosen Alpha, then you would get a C and your pair would get an A.One of the first things we pointed out, is that this is not quite a game yet. It's missing something. This has outcomes in it, it's an outcome matrix, but it isn't a game, because for a game we need to know payoffs. Then we looked at some possible payoffs, and now it is a game. So this is a game, just to give you some more jargon, this is a normal-form game. And here we've assumed the payoffs are those that arise if players only care about their own grades, which I think was true for a lot of you. It wasn't true for the gentleman who's sitting there now, but it was true for a lot of people.We pointed out, that in this game, Alpha strictly dominates Beta. What do we mean by that? We mean that if these are your payoffs, no matter what your pair does, you attain a higher payoff from choosing Alpha, than you do from choosing Beta. Let's focus on a couple of lessons of the class before I come back to this. One lesson was, do not play a strictly dominated strategy. Everybody remember that lesson? Then much later on, when we looked at some more complicated payoffs and a more complicated game, we looked at a different lesson which was this: put yourself in others' shoes to try and figure out what they're going to do.So in fact, what we learned from that is, it doesn't just matter what your payoffs are -- that's obviously important -- it's also important what other people's payoffs are, because you want to try and figure out what they're going to do and then respond appropriately. So we're going to return to both of these lessons today. Both of these lessons will reoccur today. Now, a lot of today is going to be fairly abstract, so I just want to remind you that Game Theory has some real world relevance.Again, still in the interest of recapping, this particular game is called the Prisoners' Dilemma. It's written there, the Prisoners' Dilemma. Notice, it's Prisoners, plural. And we mentioned some examples last time. Let me just reiterate and mention some more examples which are actually written here, so they'll find their way into your notes. So, for example, if you have a joint project that you're working on, perhaps it's a homework assignment, or perhaps it's a video project like these guys, that can turn into a Prisoners' Dilemma. Why? Because each individual might have an incentive to shirk. Price competition -- two firms competing with one another in prices -- can have a Prisoners' Dilemma aspect about it. Why? Because no matter how the other firm, your competitor, prices you might have an incentive to undercut them. If both firms behave that way, prices will get driven down towards marginal cost and industry profits will suffer.In the first case, if everyone shirks you end up with a bad product. In the second case, if both firms undercut each other, you end up with low prices, that's actually good for consumers but bad for firms. Let me mention a third example. Suppose there's a common resource out there, maybe it's a fish stock or maybe it's the atmosphere. There's a Prisoners' Dilemma aspect to this too. You might have an incentive to over fish. Why? Because if the other countries with this fish stock--let's say the fish stock is the Atlantic--if the other countries are going to fish as normal, you may as well fish as normal too. And if the other countries aren't going to cut down on their fishing, then you want to catch the fish now, because there aren't going to be any there tomorrow.Another example of this would be global warming and carbon emissions. Again, leaving aside the science, about which I'm sure some of you know more than me here, the issue of carbon emissions is a Prisoners' Dilemma. Each of us individually has an incentive to emit carbons as usual. If everyone else is cutting down I don't have too, and if everyone else does cut down I don't have to, I end up using hot water and driving a big car and so on.In each of these cases we end up with a bad outcome, so this is socially important. This is not just some abstract thing going on in a class in Yale. We need to think about solutions to this, right from the start of the class, and we already talked about something. We pointed out, that this is not just a failure of communication. Communication per se will not get you out of a Prisoners' Dilemma. You can talk about it as much as you like, but as long as you're going to go home and still drive your Hummer and have sixteen hot showers a day, we're still going to have high carbon emissions.You can talk about working hard on your joint problem sets, but as long as you go home and you don't work hard, it doesn't help. In fact, if the other person is working hard, or is cutting back on their carbon emissions, you have every bit more incentive to not work hard or to keep high carbon emissions yourself. So we need something more and the kind of things we can see more: we can think about contracts; we can think about treaties between countries; we can think about regulation. All of these things work by changing the payoffs. Not just talking about it, but actually changing the outcomes actually and changing the payoffs, changing the incentives.Another thing we can do, a very important thing, is we can think about changing the game into a game of repeated interaction and seeing how much that helps, and we'll come back and revisit that later in the class. One last thing we can think of doing but we have to be a bit careful here, is we can think about changing the payoffs by education. I think of that as the "Maoist" strategy. Lock people up in classrooms and tell them they should be better people. That may or may not work -- I'm not optimistic -- but at least it's the same idea. We're changing payoffs.So that's enough for recap and I want to move on now. And in particular, we left you hanging at the end last time. We played a game at the very end last time, where each of you chose a number -- all of you chose a number -- and we said the winner was going to be the person who gets closest to two-thirds of the average in the class. Now we've figured that out, we figured out who the winner is, and I know that all of you have been trying to see if you won, is that right? I'm going to leave you in suspense. I am going to tell you today who won. We did figure it out, and we'll get there, but I want to do a little bit of work first. So we're just going to leave it in suspense. That'll stop you walking out early if you want to win the prize.Chapter 2. The Formal Ingredients of a Game [00:06:47]So there's going to be lots of times in this class when we get to play games, we get to have classroom discussions and so on, but there's going to be some times when we have to slow down and do some work, and the next twenty minutes are going to be that. So with apologies for being a bit more boring for twenty minutes, let's do something we'll call formal stuff. In particular, I want to develop and make sure we all understand, what are the ingredients of a game? So in particular, we need to figure out what formally makes something into a game.The formal parts of a game are this. We need players -- and while we're here let's develop some notation. So the standard notation for players, I'm going to use things like little i and little j. So in that numbers game, the game when all of you wrote down a number and handed it in at the end of last time, the players were who? The players were you. You'allwere the players. Useful text and expression meaning you plural. In the numbers game, you'all, were the players.Second ingredient of the game are strategies. (There's a good clue here. If I'm writing you should be writing.) Notation: so I'm going to use little "s i" to be a particular strategy of Player i. So an example in that game might have been choosing the number 13. Everyone understand that? Now I need to distinguish this from the set of possible strategies of Player I, so I'm going to use capital "S i"to be what? To be the set of alternatives.The set ofpossible strategies of Player i. So in that game we played at the end last time, what were the set of strategies? They were the sets 1, 2, 3, all the way up to 100.When distinguishing a particular strategy from the set of possible strategies. While we're here, our third notation for strategy, I'm going to use little "s" without an "i," (no subscripts): little "s" without an "i," to mean a particular play of the game. So what do I mean by that? All of you, at the end last time, wrote down this number and handed them in so we had one number, one strategy choice for each person in the class. So here they are, here's my collected in, sort of strategy choices. Here's the bundle of bits of paper you handed in last time. This is a particular play of the game.I've got each person's name and I've got a number from each person: a strategy from each person. We actually have it on a spreadsheet as well: so here it is written out on a spreadsheet. Each of your names is on this spreadsheet and the number you chose. So that's a particular play of the game and that has a different name. We sometimes call this "a strategy profile." So in the textbook, you'll sometimes see the term a strategy profile or a strategy vector, or a strategy list. It doesn't really matter. What it's saying is one strategy for each player in the game.So in the numbers game this is the spreadsheet -- or an example of this is the spreadsheet. (I need to make it so you can still see that, so I'm going to pull down these boards. And let me clean something.) So you might think we're done right? We've got players. We've got the choices they could make: that's their strategy sets. We've got those individual strategies. And we've got the choices they actually did make: that's the strategy profile. Seems like we've got everything you could possibly want to describe in a game.What are we missing here? Shout it out. "Payoffs." We're missing payoffs. So, to complete the game, we need payoffs. Again, I need notation for payoffs. So in this course, I'll try and use "U" for utile, to be Player i's payoff. So"U i" will depend on Player 1's choice … all the way to Player i's own choice … all the way up to Player N's choices. So Player i's payoff "U i," depends on all the choices in the class, in this case, including her own choice. Of course, a shorter way of writing that would be "U i(s)," it depends on the profile.So in the numbers game what is this? In the numbers game "U i(s)" can be two things. It can be 5 dollars minus your error in pennies, if you won. I guess it could be something if there was a tie, I won't bother writing that now. And it's going to be 0 otherwise. So we've now got all of the ingredients of the game: players, strategies, payoffs. Now we're going to make an assumption today and for the next ten weeks or so; so for almost all the class. We're going to assume that these are known. We're going to assume that everybody knows the possible strategies everyone else could choose and everyone knows everyone else's payoffs. Now that's not a very realistic assumption and we are going to come back and challenge it at the end of semester, but this will be complicated enough to give us a lot of material in the next ten weeks.I need one more piece of notation and then we can get back to having some fun. So one more piece of notation, I'm going to write "s-i" to mean what? It's going to mean a strategy choice for everybody except person "i." It's going to be useful to have that notation around. So this is a choice for all except person "i" or Player i. So, in particular, if you're person 1 and then "s-i" would be "s2, s3, s4" up to "s n" but it wouldn't include "s1." It's useful why? Because sometimes it's useful to think about the payoffs, as coming from "i's" own choice and everyone else's choices. It's just a useful way of thinking about things.Now this is when I want to stop for a second and I know that some of you, from past experience, are somewhat math phobic. You do not have to wave your hands in the air if you're math phobic, but since some of you are, let me just get you all to take a deep breath. This goes for people who are math phobic at home too. So everyone's in a slight panic now. You came here today. You thought everything was going to fine. And now I'm putting math on the board. Take a deep breath. It's not that hard, and in particular, notice that all I'm doing here is writing down notation. There's actually no math going on here at all. I'm just developing notation.I don't want anybody to quit this class because they're worried about math or math notation. So if you are in that category of somebody who might quit it because of that, come and talk to me, come and talk to the TAs. We will get you through it. It's fine to be math phobic. I'm phobic of all sorts of things. Not necessarily math, but all sorts of things. So a serious thing, a lot of people get put off by notation, it looks scarier than it is, there's nothing going on here except for notation at this point.Chapter 3. Weakly Dominant Strategies [00:16:01]So let's have an example to help us fix some ideas. (And again, I'll have to clean the board, so give me a second.) I think an example might help those people who are disturbed by the notation. So here's a game which we're going to discuss briefly. It involves two players and we'll call the Players I and II and Player I has two choices, top and bottom, and Player II has three choices left, center, and right. It's just a very simple abstract example for now. And let's suppose the payoffs are like this. They're not particularly interesting. We're just going to do it for the purpose of illustration. So here are the payoffs: (5, -1), (11, 3), (0, 0), (6, 4), (0, 2), (2, 0).Let's just map the notation we just developed into this game. So first of all, who are the players here? Well there's no secret there, the players are -- let's just write it down why don't we. The players here in this game are Player I and Player II. What about the strategy sets or the strategy alternatives? So here Player I's strategy set, she has two choices top or bottom, represented by the rows, which are hopefully the top row and the bottom row. Player II has three choices, this game is not symmetric, so they have different number of choices, that's fine. Player II has three choices left, center, and right, represented by the left, center, and right column in the matrix.Just to point out in passing, up to now, we've been looking mostly at symmetric games. Notice this game is not symmetric in the payoffs or in the strategies. There's no particular reason why games have to be symmetric. Payoffs: again, this is not rocket science, but let's do it anyway. So just an example of payoffs.So Player I's payoff, if she chooses top and Player II chooses center, we read by looking at the top row and the center column, and Player I's payoff is the first of these payoffs, so it's 11. Player II's payoff, from the same choices, top for Player I, center for Player II, again we go along the top row and the center column, but this time we choose Player II's payoff, which is the second payoff, soit's 3.So again, I'm hoping this is calming down the math phobics in the room. Now how do we think this game is going to be played? It's not a particularly interesting game, but while we're here, why don't we just discuss it for a second. If our mike guys get a little bit ready here. So how do we think this game should be played? Well let's ask somebody at random perhaps. Ale, do you want to ask this guy in the blue shirt here, does Player I have a dominated strategy?Student: No, Player I doesn't have a dominated strategy. For instance, if Player II picks left then Player I wants to pick bottom, but if Player II picks center, Player I wants to pick center.Professor Ben Polak: Good. Excellent.Very good. I should have had you stand up. I forgot that. Never mind. But that was very clear, thank you. Was that loud enough so people could hear it? Did people hear that? People in the back, did you hear it? So even that wasn't loud enough okay, so we we really need to get people--That was very clear, very nice, but we need people to stand up and shout, or these people at the back can't hear. So your name is?Student: Patrick.Professor Ben Polak: What Patrick said was: no, Player I does not have a dominated strategy. Top is better than bottom against left -- sorry, bottom is better than top against left because 6 is bigger than 5, but top is better than bottom against center because 11 is bigger than 0. Everyone see that? So it's not the case that top always beats--it's not the case that top always does better than bottom, or that bottom always does better than top. What about, raise hands this time, what about Player II? Does Player II have a dominated strategy? Everyone's keeping their hands firmly downso as not to get spotted here. Ale, can we try this guy in white? Do you want to stand up and wait until Ale gets there, and really yell it out now.Student: I believe right is a dominated strategy because if Player I chooses top, then Player II will choose center, and if-- I'm getting confused now, it looks better on my paper. But yeah, right is never the best choice.Professor Ben Polak: Okay, good. Let's be a little bit careful here. So your name is?Student: Thomas.Professor Ben Polak: So Thomas said something which was true, but it doesn't quite match with the definition of a dominated strategy. What Thomas said was, right is never a best choice, that's true. But to be a dominated strategy we need something else. We need that there's another strategy of Player II that always does better. That turns out also to be true in this case, but let's just see.So in this particular game, I claim that center dominates right. So let's just see that. If Player I chose top, center yields 3, right yields 0: 3 is bigger than 0. And if Player I chooses bottom, then center yields 2, right yields 0: 2 is bigger than 0 again. So in this game, center strictly dominates right. What you said was true, but I wanted something specifically about domination here. So what we know here, we know that Player II should not choose right. Now, in fact, that's as far as we can get with dominance arguments in this particular game, but nevertheless, let's just stick with it a second.I gave you the definition of strict dominance last time and it's also in the handout. By the way, the handout on the web. But let me write that definition again, using or making use of the notation from the class. So definition so: Player i's strategy "s'i" is strictly dominated by Player i's strategy "s i," and now we can use our notation, if "U I" from choosing "s i," when other people choose "s-i," is strictly bigger than U I(s'i) when other people choose "s-i," and the key part of the definition is, for all "s-i."So to say it in words, Player i's strategy "s'i" is strictly dominated by her strategy "s i," if "s i" always does strictly better -- always yields a higher payoff for Player i -- no matter what the other people do. So this is the same definition we saw last time, just being a little bit more nerdy and putting in some notation. People panicking about that, people look like deer in the headlamp s yet? No, you look all right: all rightish.Let's have a look at another example. People okay, I can move this? All right, so it's a slightly more exciting example now. So imagine the following example, an invader is thinking about invading a country, and there are two ways -- there are two passes if you like -- through which he can lead his army. You are the defender of this country and you have to decide which of these passes or which of these routes into the country, you're going to choose to defend. And the catch is, you can only defend one of these two routes.If you want a real world example of this, think about the third Century B.C., someone can correct me afterwards. I think it's the third Century B.C. when Hannibal is thinking of crossing the Alps. Not Hannibal Lecter: Hannibal the general in the third Century B.C..The one with the elephants. Okay, so the key here is going to be that there are two passes. One of these passes is a hard pass. It goes over the Alps. And the other one is an easy pass. It goes along the coast. If the invader chooses the hard pass he will lose one battalion of his army simply in getting over the mountains, simply in going through the hard pass. If he meets your army, whichever pass he chooses, if he meets your army defending a pass, then he'll lose another battalion.I haven't given you--I've given you roughly the choices, the choice they're going to be for the attacker which pass to choose, and for the defender which pass to defend. But let's put down some payoffs so we can start talking about this. So in this game, the payoffs for this game are going to be as follows. It's a simple two by two game. This is going to be the attacker, this is Hannibal, and this is going to be the defender, (and I've forgotten which general was defending and someone's about to tell me that). And there are two passes you could defend: the easy pass or the hard pass. And there's two you could use to attack through, easy or hard. (Again, easy pass here just means no mountains, we're not talking about something on the New Jersey Turnpike.)So the payoffs here are as follows, and I'll explain them in a second. So his payoff, the attacker's payoff, is how many battalions does he get to bring into your country? He only has two to start with and for you, it's how many battalions of his get destroyed? So just to give an example, if he goes through the hard pass and you defend the hard pass, he loses one of those battalions going over the mountains and the other one because he meets you. So he has none left andyou've managed to destroy two of them. Conversely, if he goes on the hard pass and you defend the easy pass, he's going to lose one of those battalions. He'll have one left. He lost it in the mountains. But that's the only one he's going to lose because you were defending the wrong pass.Everyone understand the payoffs of this game? So now imagine yourself as a Roman general. This is going to be a little bit of a stretch for imagination, but imagination yourself as a Roman general, and let's figure out what you're going to do. You're the defender. What are you going to do? So let's have a show of hands. How many of you think you should defend the easy pass? Raise your hands, let's raise your hands so Jude can see them. Keep them up. Wave them in the air with a bit of motion. Wave them in the air. We should get you flags okay, because these are the Romans defending the easy pass. And how many of you think you're going to defend the hard pass? We have a huge number of people who don't want to be Roman generals here.Let's try it again, no abstentions, right? I'm not going to penalize you for giving the wrong answer. So how many of you think you're going to defend the easy pass? Raise your hands again. And how many think you're going to defend the hard pass? So we have a majority choosing easy pass -- had a large majority. So what's going on here? Is it the case that defending the easy pass dominates defending the hard pass? Is that the case? Is it the case that defending the easy pass dominates defending the hard pass? You can shout out. No, it's not.In fact, we could check that if the attacker attacks through the easy pass, not surprisingly, you do better if you defend the easy pass than the hard pass: 1 versus 0. But if the attacker was to attack through the hard pass, again not surprisingly, you do better if you defend the hard pass than the easy pass. So that's not an unintuitive finding. It isn't the case that defending easy dominates defending hard. You just want to match with the attacker. Nevertheless, almost all of you chose easy. What's going on? Can someone tell me what's going on? Let's get the mikes going a second. So can we catch the guy with the--can we catch this guy with the beard? Just wait for the mike to get there. If you could stand up: stand up and shout. There you go.Student: Because you want to minimize the amount of enemy soldiers that reach Rome or whatever location it is.Professor Ben Polak: You want to minimize the number of soldiers that reach Rome, that's true. On the other hand, we've just argued that you don't have a dominant strategy here; it's not the case that easy dominates hard. What else could be going on? While we've got you up, why don't we get the other guy who's got his hand up there in the middle. Again, stand up and shout in that mike. Point your face towards the mike. Good.Student: It seems as though while you don't have a dominating strategy, it seems like Hannibal is better off attacking through--It seems like he would attack through the easy pass.Professor Ben Polak: Good, why does it seem like that? That's right, we're on the right lines now. Why does it seem like he's going to attack through the easy pass?Student: Well if you're not defending the easy pass, he doesn't lose anyone, and if he attacks through the hard pass he's going to lose at least one battalion.Professor Ben Polak: So let's look at it from--Let's do the exercise--Let's do the second lesson I emphasized at the beginning. Let's put ourselves in Hannibal's shoes, they're probably boots or something. Whatever you do when you're riding an elephant, whatever you wear. Let's put ourselves in Hannibal's shoes and try and figure out what Hannibal's going to do here. So it could be--From Hannibal's point of view he doesn't know which pass you're going to defend, but let's have a look at his payoffs.If you were to defend the easy pass and he goes through the easy pass, he will get into your country with one battalion and that's the same as he would have got if he went through the hard pass. So if you defend the easy pass, from his point of view, it doesn't matter whether he chooses the easy pass and gets one in there or the hard pass, he gets one in there. But if you were to defend the hard pass, if you were to defend the mountains, then if he chooses the easy pass, he gets both battalions in and if he chooses the hard pass, he gets no battalions in. So in this case, easy is better.We have to be a little bit careful. It's not the case that for Hannibal, choosing the easy pass to attack through, strictly dominates choosing the hard pass, but it is the case that there's a weak notion of domination here. It is the case -- to introduce some jargon -- it is the case that the easy pass for the attacker, weakly dominates the hard pass for the attacker. What do I mean by weakly dominate? It means by choosing the easy pass, he does at least as well, and sometimes better, than he would have done had he chosen the hard pass.So here we have a second definition, a new definition for today, and again we can use our jargon. Definition- Player i's strategy, "s'i" is weakly dominated by her strategy "s i" if--now we're going to take advantage of our notation--if Player i's payoff from choosing "s i" against "s-i" is always as big as or equal, to her payoff from choosing "s'i" against "s-i" and this has to be true for all things that anyone else could do. And in addition, Player i's payoff from choosing "s i" against "s-i" is strictly better than her payoff from choosing "s'i" against "s-i," for at least one thing that everyone else could do.Just check, that exactly corresponds to the easy and hard thing we just had before. I'll say it again, Player i's strategy "s'i" is weakly dominated by her strategy "s i" if she always does at least as well by choosing "s i" than choosing "s'i" regardless of what everyone else does, and sometimes she does strictly better. It seems a pretty powerful lesson. Just as we said you should never choose a strictly dominated strategy, you're probably never going to choose a weakly dominated strategy either, but it's a little more subtle.Now that definition, if you're worried about what I've written down here and you want to see it in words, on the handout I've already put on the web that has the summary of the first class, I included this definition in words as well. So compare the definition of words with what's written here in the nerdy notation on the board. Now since we think that Hannibal, the attacker, is not going to play a weakly dominated strategy, we think Hannibal is not going to choose the hard pass. He's going to attack on the easy pass. And given that, what should we defend? We should defend easy which is what most of you chose.So be honest now: was that why most of you chose easy? Yeah, probably was. We're able to read this. So, by putting ourselves in Hannibal's shoes, we could figure out that his hard attack strategy was weakly dominated. He's going to choose easy, so we should defend easy. Having said that, of course, Hannibal went through the mountains which kind of screws up the lesson, but too late now.Chapter 4. Rationality and Common Knowledge [00:35:29]Now then, I promised you we'd get back to the game from last time. So where have we got to so far in this class. We know from last time that you should not choose a dominated strategy, and we also know we probably aren't going to choose a weakly dominated strategy, and we also know that you should put yourself in other people's shoes and figure out that they're not going to play strongly or strictly or weakly dominated strategies. That seems a pretty good way to predict how other people are going to play. So let's take those ideas and go back to the numbers game from last time.Now before I do that, I don't need the people at home to see this, but how many of you were here last time? How many of you were not. I asked the wrong question. How many of you were not here last time? So we handed out again that game. We handed out again the game with the numbers, but just in case, let me just read out the game you played. This was the game you played.。
