经济博弈论13

1 进化博弈 Evolutionary Games 第13章 Chapter 13 Slide 2 进化博弈 Evolutionary Games 目前为止我们学过了具有多种不同特征的博弈： We have so far studied games with many different features: „ 同时和序贯博弈 Simultaneous and sequential moves „ 零和与非零和博弈 Zero-sum and non-zero-sum payoffs „ 操纵未来博弈规则的策略性行动 Strategic moves to manipulate rules of games to come „ 一次性和重复博弈 One-shot and repeated play „ 许多人同时进行的集体博弈 Games of collective action in which a large number of people play simultaneously 2 Slide 3 进化博弈 Evolutionary Games 所有这些博弈中的参与者都是理性的：每个参 与者…… All the players in all these games are rational: each player…… „ ……具有内在一致的价值体系 has an internally consistent value system „ ……能够计算其策略选择的后果 can calculate the consequences of her strategic choices „ ……作出最符合其利益的选择 makes choice that best favors her interests Slide 4 进化博弈 Evolutionary Games 对理性可能的替代方法可以从生物学的进化和进化动 力学中找到，在那里…… One possible alternative to rationality can be found in the biological theory of evolution and evolutionary dynamics, where…… „ ……好的策略可以得到更多的奖励 good strategies will be rewarded with higher payoffs „ ……参与者可以观察或模仿成功者并试验新的策略 players can observe or imitate success and experiment with new strategies „ ……随着参与者在参加博弈中获得经验，好的策略将会得到 更经常的使用，坏的策略得到更少的使用。 good strategies will be used more often and bad strategies less often, as players gain experience playing the game. 3 Slide 5 内容提要 Outline *框架 The framework （*重点或难点） *囚徒困境 Prisoners’ dilemma 小鸡 Chicken 保证博弈 The assurance game *不同物种间作用 Interactions across species 鹰鸽博弈 The hawk-dove game *种群中有三种现型 Three Phenotypes in the Population 一般理论 Some General Theory 群体博弈（略） Playing the field 合作与利他的进化 Slide 6 框架 The Framework 生物学中的进化过程提供了社会科学家使用的 博弈论的平行物。 The process of evolution in biology offers a parallel to the theory of games used by social scientists. 这一理论建立在三个基本原则上： This theory rests on three fundamentals: „ 异质性 Heterogeneity „ 适应性 Fitness „ 选择 Selection 4 Slide 7 框架 The Framework 动物行为的相当一部分是由物种决定的；一个或多个 基因的联合体（基因型）支配着某一特定的行为模式 （称为行为表现型）。 A significant part of animal behavior is generically determined; a complex of one or more genes (genotype) governs a particular pattern of behavior, called a behavior phenotype. 例子 Examples „ 鸟翅膀的空气动力学特征 Aerodynamic characteristics of a bird’s wings „ 好斗或者合作的行为 Aggressive or cooperative behavior „ 筑巢的位置 Locations of nesting sites Slide 8 框架 The Framework 自然界基因库的多样性保证了种群当中表现型 的异质性。 Natural diversity of the gene pool ensures a heterogeneity of phenotypes in the population. 某些行为比其他行为更适合于当前环境，一种 表现型的成功可以用适应性来定量测量。 Some behaviors are better suited than others to the prevailing conditions, and the success of a phenotype is given a quantitative measure called its fitness. 5 Slide 9 框架 The Framework 更加适应的表现型在下一代中就会比更不适应 的表现型数量更多。 The fitter phenotypes then become relatively more numerous in the next generation than the less fit phenotypes. 这一选择过程就是一个改变基因型和表现型的 比例并最终导致稳定状态的动态过程。 This process of selection is the dynamic that changes the mix of genotypes and phenotypes and perhaps leads eventually to a stable state. Slide 10 框架 The Framework 不时的，偶然因素导致新的基因变异。 From time to time, chance produces new genetic mutations. 许多这样的突变产生的行为（表现型）都不适应环境，逐 渐消失。 Many of these mutations produce behavior (that is, phenotypes) that are ill suited to the environment, and they die out. 偶尔地，变异导致的新的表现型更加适应环境。这样的变 异基因就能够成功地侵入种群，即扩展成为群体的重要部 分。 But occasionally a mutation leads to a new phenotype that is fitter. Then such a mutant gene can successfully invade a population – that is, spread to become a significant proportion of the population. 6 Slide 11 框架 The Framework 如果一个群体不能被任何变异侵入， 生物学家就将这一群 体的构成及其当前的表现型称为进化稳定的。 Biologists call a configuration of a population and its current phenotypes evolutionary stable if the population cannot be invaded successfully by any mutant. 这是一个静态的检验；通常应用的是一个更加动态的： 一个构成是进化稳定的，如果它是从群体的任意给定的表 现型构成开始的、动态选择过程的极限结果。 This is a static test; but often a more dynamic criterion is applied: a configuration is evolutionary stable if it is the limiting outcome of the dynamic of selection, starting from any arbitrary mixture of phenotypes in the population. Slide 12 框架 The Framework 一种表现型的适应性取决于个别生物体与环境的关系。 The fitness of a phenotype depends on the relationship of the individual organism to its environment. 它还取决于存在于环境中的不同表现型的比例。 It also depends on the whole complex of the proportions of different phenotypes that exist in the environment. 对于我们的目的来说，这一物种内部表现型之间的相 互作用是最有意思的部分。 For our purpose, this interaction between phenotypes within a species is the most interesting part of the story. 7 Slide 13 框架 The Framework 进化的生物过程和博弈论是非常平行的。 The biological process of evolution finds a ready parallel in game theory. „ 表现型与策略 Phenotype vs. Strategy „ 适应性与收益 Fitness vs. Payoffs Slide 14 框架 The Framework 因为种群是表现型的混合，从种群中选出的不同配对 就将不同的策略组合带入他们的相互作用。 Because the population is a mix of phenotypes, different pairs selected from it will bring to their interactions different combinations of strategies. 某一表现型的适应性的实际定量测度标准是它在与种 群中其他表现型的所有相互作用中得到的平均收益。 The actual quantitative measure of the fitness of a phenotype is the average payoff that it gets in all its interactions with others in the population. 8 Slide 15 框架 The Framework 进化博弈理论看起来像是通往博弈论的新途径 的一个现成框架，它放松了理性行为的假设。 The theory of evolutionary games seems a ready-made framework for a new approach to game theory, relaxing the assumption of rational behavior. 策略遗传的思想可以在生物学之外其他的理论 应用中得到更广泛的阐释。 The idea of inheritance of strategies can be interpreted more broadly in applications of the theory other than biology. Slide 16 框架 The Framework 在社会经济博弈中，策略“优胜劣汰”的原因有 别于生物学中严格的遗传机制： The reasons that the fitter strategies proliferate and the less fit ones die out in socioeconomic games differs from the strict genetic mechanism of biology: „ 观察和模仿 Observations and Imitations „ 有目的的思考和对以往经验方法的修改 Purposive thinking and revision of previous rules of thumb „ 有意识的实验 Conscious experimentation 9 Slide 17 框架 The Framework 为什么参与者要出这样的策略？ Why a player plays such a strategy? „ 理性选择 Rational choices „ 遗传 Genetics „ 社会化、文化背景、教育 Socialization, cultural background, educations „ 依据过去经历的经验方法 A rule of thumb based on past experience 社会会不会最后变成所有的政治家都只关心重新当选， 所有企业都只关心利润？ Will society end up with a situation in which all politicians are concerned with reelection, and all firms with profit? Slide 18 框架 The Framework 生物博弈的进化稳定构成可以有两种。 Evolutionary stable configurations of biological games can be of two kinds. 单态：单独一种表现型被比其他表现型更适应，种群变 为仅由它构成。 Monomorphism: A single phenotype proves fitter than any others and the population comes to consist of it alone. 在这种情况下，这个唯一主导的策略被称为进化稳定策略。 In this case, the unique prevailing strategy is called an evolutionary stable strategy (ESS). 多态：两个或更多表现型同样适应（并比其他没有出现的更 适应）；因此他们可能以某种比例共存。 Polymorphism: Two or more phenotypes are equally fit (and fitter than some others not played); so they may be able to coexist in certain proportions. 10 Slide 19 框架 The Framework 组成进化博弈的完整设定是： The whole set-up which constitutes an evolutionary game is: „ 种群 The Population „ 其可能的表现型的集合 Its conceivable collection of phenotypes „ 表现型相互作用的收益矩阵 The payoffs matrix in the interactions of the phenotypes „ 与其适应性相关的、表现型在种群中比例的进化规则 The rule for the evolution of population proportions of the phenotypes in relation to their fitness 种群的进化稳定的构成可以称为进化博弈的一个均衡。 An evolutionary stable configuration of the population can be called an equilibrium of the evolutionary game. Slide 20 囚徒困境 Prisoners’ Dilemma 假定种群由两种表现型组成：合作者和背叛者。 Suppose a population is made up of two phenotypes: cooperators, defectors. 种群中的每一个体（合作者或者背叛者）被随 机地选择与另一个随机选择的对手竞争。 Each individual (either a cooperator or a defectors) in the population is chosen at random to compete against another random rival. 11 Slide 21 囚徒困境 Prisoners’ Dilemma 324, 324216, 36026 (Cooperate) 360, 216288, 28820 (Defect) ROW 26 (Cooperate) 20 (Defect) COLUMN Slide 22 囚徒困境 Prisoners’ Dilemma 用x表示种群中合作者的比例。 Let x be the proportion of cooperators in the population. 则一个典型的合作者的预期收益为， Therefore a typical cooperator’s expected payoff is, 324x+216(1-x) 一个典型的背叛者的预期收益为， A typical defector’s expected payoff is, 360x+288(1-x) 显然有，It is immediately apparent that, 360x+288(1-x)>324x+216(1-x), for all x between 0 and 1. 12 Slide 23 囚徒困境 Prisoners’ Dilemma 因而背叛者有更高的预期收益，比合作者更适应。 Therefore a defector has a higher expected payoff and is fitter than a cooperator. 这会导致背叛者比例的逐“代”上升（x下降），直到整 个种群都由背叛者组成。 This will lead to an increase in the proportion of defectors (a decrease in x)from one “generation” of players to the next, until the whole population consists of defectors. Slide 24 囚徒困境 Prisoners’ Dilemma 如果整个种群都由背叛者组成呢？ What if the population initially consists of all defectors? 那么这种情况下不会有变异（试验性）的合作者可以 生存和繁殖以改变种群。 Then in this case no mutant (experimental) cooperator will survive and multiply to take over the population. 换句话说，背叛者的种群不能被变异的合作者成功侵 入。 In other words, the defector population cannot be invaded successfully by mutant cooperators. 13 Slide 25 囚徒困境 Prisoners’ Dilemma 我们的分析表明背叛者比合作者有更高的适应性，一 个完全由背叛者组成的种群不能被变异的合作者侵入。 Our analysis shows that both that defectors have higher fitness than cooperators and that an all-defector population cannot be invaded by mutant cooperators. 因而种群的进化稳定构成是单态的，由单一的策略或 表现型“背叛”组成。 Thus the evolutionary stable configuration of the population is monomorphic, consisting of the single strategy or phenotype Defect. Slide 26 囚徒困境 Prisoners’ Dilemma 我们就把“背叛”称为这一进行困境博弈种群的 进化稳定策略。 We therefore call Defect the evolutionary stable strategy for this population engaged in this dilemma game. 如果博弈有一个严格的优势策略，那么该策略 也将是ESS。 If a game has a strictly dominant strategy, that strategy will also be the ESS. 14 Slide 27 重复囚徒困境 The Repeated Prisoner’s Dilemma 648, 648504, 648T (Tit-for tat) 648,504576, 576A (Always defect) ROW T (Tit-for tat) A (Always defect) COLUMNTwice-Repeated Prisoners’ dilemma 576=288*2, 648=324*2=360+288 504=216+288 Slide 28 重复囚徒困境 The Repeated Prisoner’s Dilemma A只是弱优势的。容易看到A也是ESS。 A is only weakly dominant. And it is easy to see that A is an ESS. T是不是另一个ESS呢？ Is T another ESS? „ 注意到：(T, T)是该博弈的理性博弈理论分析的纳 什均衡。 Notice that: (T, T) is a Nash equilibrium in the rational game theoretic analysis of this game. 15 Slide 29 重复囚徒困境 The Repeated Prisoner’s Dilemma 如果种群一开始全是T，有少数几个变异者进入，那么变异者在大多数时 间内会遇到占统治地位的T型，在与T型的对决中，会和T型本身做得一 样好。 If the population is initially all T and a few mutants entered, then the mutants would meet the predominant T types most of the time and would do as well as T does against another T. 但是，偶尔的，一个A变异者将会遇到另一个A变异者，在这一对决中， 她会比T遇到A时做得更好。 But occasionally an A mutant would meet another A mutant, and in this match she does better than would a T against A. 因此，变异者会比占统治地位表现型中的成员有略高的适应性。 Thus the mutants have just slightly higher fitness than that of a member of the predominant phenotype. 这一优势导致种群中变异者的比例增加（虽然较慢）。因而全T种群可以 被A变异者成功入侵；T不是ESS。 This advantage leads to an increase, albeit a slow one, in the proportion of mutants in the population.Therefore an all-T population can be invaded successfully by A mutants; T is not an ESS. Slide 30 重复囚徒困境 The Repeated Prisoner’s Dilemma 我们的推理依赖于对ESS的二重检验。 Our reasoning relies on two tests for an ESS. 首先我们看当，当遇到占统治地位的类型时，变异者是否比占统 治地位的类型做得更好。 First we see if the mutant does better or worse than the predominant phenotype when each is matched against the predominant type. 如果这一主标准给出一个清楚的，问题就解决了。 If this primary criterion give a clear answer, that settles the matter. 如果该主标准给出平分，我们就使用一个“加时赛”，或次标准： 在遇到变异者时，变异者是否比占统治地位的类型做得更好？ But if the primary criterion gives a tie, then we use a tie- breaking, or secondary, criterion: does the mutant fare better or worse than a predominant phenotype when each is matched against a mutant? 16 Slide 31 重复囚徒困境 The Repeated Prisoner’s Dilemma 972, 972792, 936T (Tit-for tat) 936, 792864, 864A (Always defect) ROW T (Tit-for tat) A (Always defect) COLUMNThrice-Repeated Prisoners’ dilemma 864=288*3, 972=324*3 792=216+288*2 936=360+288*2 Slide 32 重复囚徒困境 The Repeated Prisoner’s Dilemma 两种类型的相对适应性依赖于种群构成：每种类型在 它已经在种群中占统治地位时都更适应。 The relative fitness of the two types depend on the composition of the population: each type is fitter when it already predominates in the population. 因而当种群全是A时，T不能成功侵入，反之亦然。 Therefore T cannot invade successfully when the population is all A, and vice versa. 现在有两个可能的种群的进化稳定构成：全A或全T。 Now there are two possible evolutionary stable configurations of the population: all-A or all-T. 17 Slide 33 重复囚徒困境 The Repeated Prisoner’s Dilemma x*=2/3 10 864 792 936 972 Fitness Proportion x of T types in Population A type T type Slide 34 重复囚徒困境 The Repeated Prisoner’s Dilemma 如果开始时种群中恰好是x=2/3？ What if the initial population has exactly x=2/3? 一旦任何一种类型的变异者出现，该构成就不能维持。 Such a configuration can sustain only until a mutant of either type surfaces. 它可以被看成是一个不稳定的均衡；但从严格的生物 过程的逻辑来看，它根本不是一个均衡。 It can be regarded as an unstable equilibrium;but in the strict logic of the biological process, it is not an equilibrium at all. „ 不过，注意到双方出(2/3T, 1/3A)构成了该博弈的理性博弈 版本的混合策略纳什均衡。 Notice however (2/3T, 1/3A) for both players forms a mixed-strategy Nash equilibrium in the rational- player version of this game. 18 Slide 35 重复囚徒困境 The Repeated Prisoner’s Dilemma 324n, 324n288n-72, 288n+72 T (Tit-for tat) 288n+72, 288n-72 288n, 288nA (Always defect) ROW T (Tit-for tat) A (Always defect) COLUMNn-fold-Repeated Dilemma (n>2) Slide 36 重复囚徒困境 The Repeated Prisoner’s Dilemma n=3, x*=2/3 n=4, x*=2/4=1/2 …… n=10, x*=2/10=0.2, …… 但博弈重复次数越多，合作就可以从越大范围的初始 条件中产生。 Cooperation emerges from a larger range of the initial conditions when the game is repeated more times. 19 Slide 37 比较进化和理性参与者模型 Comparing the Evolutionary and Rational-player Models 一个ESS必然是有相同收益结构、由有意识的 理性参与者进行的博弈的纳什均衡解。 An ESS must be a Nash equilibrium of the game played by consciously rational players with the same payoff structure. 因而，进化方法提供了对理性方法的隐含的支 持。 Thus the evolutionary approach provides a backdoor justification for the rational approach. Slide 38 比较进化和理性参与者模型 Comparing the Evolutionary and Rational-player Models 虽然ESS必然是对应的理性参与者博弈的纳什 均衡，反过来却未必。 Although an ESS must be a Nash equilibrium of the corresponding rational-player game, the converse is not true. 因而稳定的生物学概念可以帮助我们从理性博 弈的多重纳什均衡中进行选择。 Thus the biological concept of stability can help us select from a multiplicity of Nash equilibria of a rationally played game. 20 Slide 39 小鸡 Chicken -2, -21, -1Macho (Always straight) -1, 10, 0Wimp (Always swerve) A Macho (Always straight) Wimp (Always swerve) B Slide 40 小鸡 Chicken 10 1 -2 -1 Fitness Proportion x of Machos in Population 0 Macho Wimp 1/2 50-50的混合是稳定的多态ESS。 The 50-50 mix will be the stable polymorphic ESS. 21 Slide 41 小鸡 Chicken 这一理性参与者的小鸡博弈有三个纳什均衡：两个纯 策略和一个混合策略。 The rational-player Chicken game has three Nash equilibria: two in pure strategies and one in mixed strategies. 均衡混合策略中的混合比例恰好是进化博弈中的种群 比例。 The mixture proportions in the equilibrium mixed strategy are exactly the same as the population proportions in the evolutionary game. Slide 42 保证博弈 The Assurance Game 2, 20, 0L (Local Latte) 0, 01, 1S (Starbucks) MEN L (Local Latte) S (Starbucks) WOMAN 22 Slide 43 保证博弈 The Assurance Game 10 2 1 Fitness Proportion x of Machos in Population L type S type 仅有两种极端的单态种群构成是可能的进化稳态。 Only the two extreme monomorphic configurations of the population are possible evolutionary stable states. 2/3 Slide 44 物种之间的相互作用 Interaction Across Species 假定“男人”和“女人”仍然对在星巴克（S）或地方小店 （L）相遇感兴趣——无法相遇每人收益都是0——但 现在每一类型偏爱的咖啡店不同：L带给女人的收益为 2，男人为1，S恰好相反。 Assume that “men” and “women” are still interested in meeting at either Starbucks (S) or Local Latte(L) – no meeting yields each a payoff of 0 – but now each type prefers a different cafe: L gives a payoff of 2 to women and 1 to men, and S the other way around. 这些偏好将两种类型区分开来。在生物学语言中，他 们属于不同物种。 These preferences distinguish the two types. In the language of biology, they belong to different species. 23 Slide 45 物种之间的相互作用 Interaction Across Species 假定在每一种群内部，所有成员都一致同意他们对S， L和不能相遇的评价（收益）。 Suppose within each population, all the members agree among themselves about the valuation (payoffs) of S, L and no meeting. 但一些成员为强硬者，另一些为折衷者。 But some members are hard-liners and others are compromisers. 一个强硬者永远选择他（她）自己所属物种偏爱的咖 啡店。一个折衷者认识到对方物种想法相反，为了和 睦，去了对方的地方。 A hard-liner will always go to his or her species’ preferred cafe. A compromiser recognizes that the other species wants the opposite and goes to that location, to get along. Slide 46 物种之间的相互作用 Interaction Across Species 0, 01, 2Compromiser 2, 10, 0Hard-linerMEN CompromiserHard-liner WOMAN 24 Slide 47 物种之间的相互作用 Interaction Across Species 用x表示男人中强硬者的比例，y表示女人中该比例。 Let x be the proportion of hard-liners among the men and y that among the women. 某一强硬男人的预期收益（适应性）为： A particular hard-liner man’s expected payoff (fitness) is, y*0+(1-y)*2=2(1-y) 一个折衷男人的预期收益（适应性）为， A compromising man’s expected payoff (fitness) is, y*1+(1-y)*0=y 则强硬男人更适应和增长得更快（x上升），当： Therefore the hard-liner man is fitter and reproduce faster (x is increase) when, 2(1-y)>y, or y<2/3 以此类推 And so on…… Slide 48 物种之间的相互作用 Interaction Across Species 2/3 2/30 1Proportion y of hard-liners among women Proportion x of hard-liners among women x→ y↑ x← y↑ x← y↓ x → y↓ 1 ESS ESS 25 Slide 49 鹰鸽博弈 The Hawk-Dove Game 该博弈不是由所指的两种鸟，而是由同一物种的两个 个体进行的，“鹰”和“鸽”只是他们策略的名字。 The game is played not by birds of these two species, but by two animals of the same species, and Hawk and Dove are merely the names for their strategies. 鹰策略是侵略性的，试图抢夺价值为V的所有资源。 The hawk strategy is aggressive and fights to try to get the whole resource of value V. 鸽策略是愿意分享，但如遇战斗则退缩。 The Dove strategy is to offer to share but to shirk from a fight. Slide 50 鹰鸽博弈 The Hawk-Dove Game V/2, V/20, VDove V, 0(V-C)/2, (V-C)/2HawkA DoveHawk B 26 Slide 51 理性策略选择与均衡 Rational Strategic Choice and Equilibrium 如果V>C，那么该博弈为囚徒困境，每个参与者的优 势策略都是“鹰”，但（鸽，鸽）为双方都更好的结果。 If V>C, then the game is a prisoners’ dilemma in which Hawk is the dominant strategy for each, but (Dove, Dove) is the jointly better outcome. 如果VC（囚徒困境），鹰策略是唯一的ESS。 If V>C (prisoners’ dilemma), the Hawk strategy is the only ESS. 如果V