经济博弈论05

1 纯策略同时博弈II：连续策略 和III：讨论与证据 Simultaneous-Move Games with Pure Strategies II: Continuous Strategies and III: Discussion and Evidence 第5章 Chapter 5 Slide 2 离散与连续策略 Discrete and Continuous Strategies 在一个离散策略的博弈中，每个博弈者只有少数几个 定义明确的招术（离散的行动集合）。 In a discrete strategy game, each player has a small number of well-defined plays (i.e., a discrete set of actions) from which to choose. „ 足球里的罚点球 Soccer penalty kicks „ 囚徒困境 Prisoners’ dilemma 这样的博弈可以用博弈来，至少在参与者人数 及其行为的个数不多时是如此。 Such games are amenable to analysis with the use of a game table, at least for situations with a reasonable number of players and available actions. 2 Slide 3 离散与连续策略 Discrete and Continuous Strategies 在一个连续策略的博弈中，参与者从一个大的可能性 范围，也即本质上无限的集合中做出选择。 In a continuous strategy game, players choose from a wide range of possibilities, a virtually infinite set of choices. „ 制造商选择产品价格 Manufacturers choose product prices „ 慈善家选择慈善捐款的数量 Philanthropists choose charitable contribution amounts „ 建筑承包商选择工程的投标价格 Contractors choose project bid levels 在这些情形下，博弈表实际上就不能作为分析工具了。 In these cases, game tables become virtually useless as analytical tools. Slide 4 作为连续变量的纯策略 Pure Strategies That Are Continuous Variables 问：如何求解连续纯策略的同时博弈？ Q: How can we solve a simultaneous- move game with continuous pure strategies? 答：最优反应分析。 A: Best response analysis. 问：哪一种最优反应分析？ Q: What kind of best response analysis? 答：最优反应规则（或曲线） A: Best response rules (or curves). 3 Slide 5 价格竞争 Price Competition 在一个名为雅皮天堂的小镇上，有两个餐馆，泽维尔 的塔帕斯吧和伊冯的比斯托。 In a small town, Yuppie Heaven, there are two restaurants, Xavier’s Tapas Bar and Yvonne’s Bistro. 他们需要为其各自预备的菜单制定价格。 They have to set the prices of their respective set menus. „ 价格＝策略 Prices=strategies 每个餐馆的目标都是利润最大化。 Each restaurant’s goal is to maximize profit. „ 利润＝收益 Profits=payoffs Slide 6 价格竞争 Price Competition 假设每个餐馆服务一个顾客的成本都是8美元。 Suppose the cost of serving each customer is $8 for each restaurateur. 假设当泽维尔的价格为Px，伊冯的价格为Py 时，他们各自的顾客数（分别用Qx 和Qy 表示）由下列等式给出： Suppose when Xavier’s price is Px and Yvonne’s price is Py , the number of their respective customers, respectively Qx and Qy, are given by the equations: Qx = 44 - 2Px + Py , Qy = 44 - 2Py + PX 4 Slide 7 价格竞争 Price Competition 泽维尔的利润（用Bx表示）为： Xavier’s profit, called it Bx , is given by: Bx = (Px - 8) Qx = (PX – 8) (44 – 2Px + Py ) 问：写出伊冯的利润（ By）的表达式。 Q: How about Yvonne’s profit By ? Slide 8 价格竞争 Price Competition 我们已经得出了对这个价格竞争博弈的完整描 述。Until now we already have had a complete specification of the price competition game. 为了求解博弈，我们需要知道：对于伊冯的每 一个可能的价格（ Py），泽维尔的最优反应价格（ Px），即最优反应规则，是什么？ We start to solve it by asking: For each possible level of Yvonne’s price (Py), what would be the best response price (Px) – Best response rules － of Xavier? 5 Slide 9 价格竞争 Price Competition 也即，给定对手价格Py，泽维尔设定一个自己的价格Px以最大化其利润。 That is, given the opponent’s price Py , Xavier sets his price Px to maximize his own profit. 这可以通过使用一阶条件求得： This can be done by using the first-order condition: dBx/dPx =0 (60 + Py) - 4Px = 0 Px = 15 + 0.25Py Slide 10 价格竞争 Price Competition 因此，泽维尔的最优反应规则就是： Thus the rule for Xavier’s best response is: Px = 15 + 0.25Py 类似地，伊冯的最优反应规则是： Similarly Yvonne’s best-response rule is: Py = 15 + 0.25Px 这两个最优反应关系可以图示为最优反应曲线。 These two best-response relations can be shown graphically as best-response curves. 6 Slide 11 价格竞争 Price Competition 10 20 30 10 20 30 Xavier’s price Px Yvonne’s price Py Xavier’s best response Px = 15 + 0.25Py Yvonne’s best response Py = 15 + 0.25Px 纳什均衡 Nash Equilibrium Slide 12 价格竞争 Price Competition 两条最优反应曲线的交点就是该定价博弈的纳什均衡 点。 The point of intersection of the two best- response curve is the Nash equilibrium of the pricing game. „ 该点表示一组价格（每个企业对应其中一个），互为对方的 最优反应。 That point represents the set of prices, one for each firm, that are best responses to each other. 每个餐馆定价策略的具体取值为： The specific values for each restaurant’s pricing strategy in equilibrium are: Px = Py = 20 7 Slide 13 价格竞争 Price Competition 每个餐馆为其菜单收20美元，从每个顾客身上 赚得20-12=8美元，顾客数量为44- 2*20+20=24百，因此每月总利润为 12*2400=28,800。 Each restaurant charges $20 for its menu and makes a profit of $(20-8)=12 on each of the (44-2*20+20)=24 (hundred) customers, for a total profit of $12*2400=28,800 per month. 如果他们联合提高价格呢？ What if they jointly raise their prices? Slide 14 价格竞争 Price Competition 当某一个餐馆将其价格提高1美元…… When one restaurant raises its price by $1 …… 它的一些顾客就转头去了另一家餐馆…… Some of its customers switch to the other restaurant …… 由此它的对手通过提价就可以最大化利润…… And its rival can then best profit from them by raising its price …… 提价幅度为0.25美元。 By $0.25. 这样，一个餐馆提价就帮助了另一个餐馆提高利润。 Thus a restaurant that raise its price is helping to increase the other’s profit. 8 Slide 15 价格竞争 Price Competition 在纳什均衡中，每个餐馆独立地选择其价格，它只考 虑自己的利润，而不关心象这里出现的由它转移给其 他人的好处。 In Nash equilibrium, where each restaurant chooses its price independently and out of concern for its own profit, it doe not take into account this benefit that it conveys to the other. 他们能否合作一致提高价格，从而提高双方的利润呢？ Could they get together and cooperatively agree to raise their prices, thereby raising both profits? Slide 16 价格竞争 Price Competition 可以！假设两个餐馆都把价格定在24美元。则在每个 顾客身上赚得16美元，每月顾客人数各为 44- 2*24+24=2,000，总利润各为32,000美元。 Yes! Suppose the two restaurant charged $24 each. Then each would make a profit of $16 on each of the 2,000 customers [2000=(44- 2*24+24) hundred] that it would serve each month, for a total profit of $32,000. 定价博弈恰如囚徒困境博弈。 The pricing game is exactly like the prisoners’ dilemma game. 9 Slide 17 价格竞争 Price Competition 哪个价格使得两个餐馆的总利益最大？ What price is jointly best for the two restaurants? 需要找到一组价格(Px , Py) ，使得两个餐馆加总的总利润 Bx+By最大： We need to find a pair of prices (Px , Py) such that the aggregated total profit of two restaurants, Bx+By , is maximized: ∂(Bx+By)/∂Px = 0, ∂(Bx+By)/∂Py = 0 (60 + Py) - 4Px + (Py – 8) = 0, (60 + Px) - 4Py + (Px – 8) = 0 Px = Py = 26 最后各自利润为32,400美元。 The resulting profit for each restaurant is $32,400 per month. Slide 18 价格竞争 Price Competition 10 20 30 10 20 30 Xavier’s price Px Yvonne’s price Py Xavier’s best response Px = 15 + 0.25Py Yvonne’s best response Py = 15 + 0.25Px Nash Equilibrium Profit=$28,000 each 共同最优 Joint Best Profit=$32,400 each 10 Slide 19 价格竞争 Price Competition 在经济学的行话中，这样一些提高价格到共同 最优水平的合谋就是卡特尔。 In the jargon of economics, such collusion to raise prices to the jointly optimal level is called a cartel. 高价格损害了消费者，政府的管制部门常常会 试图阻止卡特尔的组成，并使企业相互竞争。 The high prices hurt consumers, and regulatory agencies of government often try to prevent the formation of cartels and to make firms compete with one another. Slide 20 价格竞争 Price Competition 合谋不总是导致高价。 Collusion need not always lead to higher prices. 如果两个企业的产品是相互替代的，合谋导致高价格。 If two firms are selling products that are substitutes for each other, collusion leads to higher prices. 如果两个企业的产品是相互互补的，合谋导致低价格。 If two firms are selling products that are complements for each other, collusion leads to lower prices. „ 例子：硬件和软件。E.g., hardware and software „ 允许他们合作对消费者也是有利的。 Allowing them to cooperate would be beneficial to the consumers as well. 11 Slide 21 政治竞选广告 Political Campaign Advertising 在由两个政党或候选人（L和R）角逐的选举中，每个 人都试图通过广告把选票从对手那边拉过来。 In an election contested by two parties or candidates, L and R, each is trying to win votes away from the other by advertising. 假设某一政党的选票份额等于它在双方所进行的竞选 广告中所占的份额。 Suppose the vote share of a party equals its share of the total campaign advertising that is done. 筹款为这些广告支付费用是有成本的。假设所有成本 与直接的竞选支出成比例。 Raising money to pay for these ads includes a cost. Suppose all these costs are proportional to the direct campaign expenditures. Slide 22 政治竞选广告 Political Campaign Advertising 因而，当L在广告上花了x百万美元，R花了y百万美元，L就会获 得x/(x+y)份额的选票，成本为x。 Thus, when L spends $x million on advertising and R spends $y million, L will get a share x/(x+y) of the votes, with a cost of x. 假设L党的收益为， Suppose that party L’s payoff is, x/(x+y)-x 用微积分方法，政党L的最优反应函数为： Using calculus, party L’s best-response function is x = y1/2-y. 类似的，R党的最优反应函数为， Similarly, party R’s best-response function is y= x1/2-x. 12 Slide 23 政治竞选广告 Political Campaign Advertising 0 1 0.25 0.25 1 Party L’s best response Party R’s best response Nash Equilibrium Party L’s ad, x ($ millions) Party R’s ad, y ($ millions) Slide 24 政治竞选广告 Political Campaign Advertising 随着对手广告的增加，作为每个政党最优反应的广告 数量先是增加，而后下降。 Each party’s best-response advertising increases for a while as the other’s ad increases and then decreases. 此处又是一个囚徒困境：如果双方都以相同的比例削 减他们的广告，他们的选票份额完全不受影响，但双 方都节省了支出。 We have another prisoners’ dilemma here: if both parties cut back on their ads in equal proportions, their vote shares would be entirely unaffected, but both would save on their expenditures. 13 Slide 25 政治竞选广告 Political Campaign Advertising 问：如果双方地位不对称又如何？ Q: what if the parties are not symmetrically situated ? „ 一个政党（R）可以以更低的成本做广告。 One party (say, R) may be able to advertise at a lower cost. „ R的广告花费可能比L更有效；例如，L的选票份额 为x/(x+2y)，R的为2y/(x+2y)。 R’s advertising dollars may be more effective than L’s; for example, L’s vote share may be x/(x+2y), while R’s is 2y/(x+2y) Slide 26 关于纳什均衡的经验证据 Empirical Evidence Concerning Nash Equilibrium 关于纳什均衡的经验证据来自： Empirical evidence concerning Nash Equilibrium came from： „ 对现实中实际进行的博弈的观察 observations on games actually played in the world, „ 在实验室或者教室中人为构造的用以检验理 论的博弈 games deliberately constructed for testing the theory in the laboratory or the classroom. 14 Slide 27 实验室或教室试验 Laboratory and Classroom Experiments 在相对简单的、只动一次的、只有唯一纳什均衡的博弈中，理论 结果“具有相当大的推断力……在与不同的同伴重复若干次博弈 以后。” In relatively simple single-move games with a unique Nash equilibrium, that outcome “has considerable drawing power … after some repetitions with different partners.” 但在更加复杂、反复多次、或者由于多重纳什均衡的存在而要求 进行协调时，或者寻找纳什均衡的计算相当复杂时，理论有时成 功有时失败。 But in more complex or repeated situations or when coordination is required because there are multiple Nash equilibria or when the calculations required for finding a Nash equilibrium are more complex, the theory’ success is more mixed. Slide 28 在多重博弈中选择 Choosing Among Multiple Equilibrium 当存在多重均衡时，参与者往往不能协调，除 非他们有某些共同的文化背景（并且这一事实 为所有参与者共知）以便寻找到焦点。 When there are multiple equilibria, players generally fail to coordinate unless they have some common cultural background (and this fact is common knowledge among them) that is needed for locating focal points. 15 Slide 29 在多重博弈中选择 Choosing Among Multiple Equilibrium 一个学生分到了波士顿，另一个分到了旧金山。 One student was assigned Boston, and the other was assigned San Francisco. 每个人都得到了关于美国其他九个城市的名单：亚特兰大、芝加 哥、达拉斯、丹佛、休斯顿、洛杉矶、纽约、费城和西雅图。他 们被要求选择其中一些城市。两人同时独立作选。 Each was given a list of nine other U.S. cities – Atlanta, Chicago, Dallas, Denver, Houston, Los Angeles, New York, Philadelphia, and Seattle – and asked to choose a subset of these cities. The two chose simultaneously and independently. 如果两人的选择恰好将9个城市完整分开，没有重叠，两人都得 到奖励。否则，均一无所获。 If their choices divided up the nine cities completely and without any overlap between them, both got a prize. Otherwise, neither got anything. Slide 30 在多重博弈中选择 Choosing Among Multiple Equilibrium 这个博弈有多达512种的纯策略纳什均衡。 This game has numerous (512) Nash equilibria in pure strategies 不过当两个博弈者都是美国人或长期居住美国，有80％的时候他 们会选择按地理位置来划分；分到波士顿的学生选择所有位于密 西西比以东的城市，分到旧金山的学生选择密西西比以西的城市。 But when both players were Americans or long-time U.S. residents, more than 80% of the time they chose the division geographically; the student assigned Boston chose all the cities east of Mississippi, and the student assigned San Francisco chose those west of the Mississippi. 这样的协调在两个学生或其一不是美国居民时就很难出现。 Such coordination was much less likely when one or both students were non-U.S. residents. 16 Slide 31 在实验中显示内在的利他主义和公共精神 Revelation of Innate Altruism or Public- Spiritedness in Experiments 在某些实验环境下，参与者的行为常常与实验者预测 的纳什均衡不符。人们看起来会误入友善和公平的“歧 途”。 The behavior of players in some experimental situations does not often conform to the experimenter’s predicted Nash equilibrium. People seem to “err” on the side of niceness or fairness. „ 在囚徒困境博弈中，可以观察到“太多”的合作。 In prisoners’ dilemma games, “too much” cooperation is observed. „ 在谈判博弈中，人们会向对手做出“太多”让步。 In bargaining games, people concede “too much” to opponents. Slide 32 在实验中显示内在的利他主义和公共精神 Revelation of Innate Altruism or Public- Spiritedness in Experiments 存在重复的可能性，或者另外发展的与其他博弈者的 关系可能影响人们在博弈中的选择。 The possibility of repetition or a separate ongoing relationship with the other player may affect their choices in this game. 参与者的价值体系可能已经将某些在广阔社会背景下 被证明有用的、友善与公正的社会规范内在化，因而 在实验博弈中延续了这些行为。 The players’ value systems may have internalized some social norms of niceness and fairness that have proved useful in the larger social context and that therefore carry over to their behavior in the experimental game. 17 Slide 33 在实验中显示内在的利他主义和公共精神 Revelation of Innate Altruism or Public- Spiritedness in Experiments 这些观察并不能说明纳什均衡概念本身有何缺陷。 These observations do no show any deficiency of the Nash equilibrium concept itself. 然而，它们警告我们，不要在对人们的收益做出过于 天真或错误的假设下使用这一概念。 However, they do warn us against using the concept under naive or mistaken assumptions about people’s payoffs. 例如，假设人们总是被追求金钱的私欲所驱动，就可 能是错误的。 It might be a mistake, for example, to assume that players are always driven by the selfish pursuit of money. Slide 34 从经历中学习 Learning From Experience 一个在教室或实验室中广为使用的博弈，要求 每个参与者选择一个0到100的数字。 One game, often used in classroom or laboratories, asks each participant to choose a number between 0 and 100. 选择数字最接近平均数的某个特定比例（例如 一半）者得胜。 The person whose choice is closest to a specified fraction – say half – of the average is the winner. 18 Slide 35 从经历中学习 Learning From Experience 这一博弈的纳什均衡是每个人都选0。 The Nash equilibrium of this game is for everyone to choose 0. 然而，当一个群体第一次实际地进行这一博弈时，获 胜者往往是选择了一个略小于25的数字的人。 However, when a group actually plays this game for the first time, the winner is typically a player who has chosen a number just a little less than 25. 获胜的选择在后续回合中迅速下降。到第3轮时，胜者 的选择通常低到2或3。 The winning choices falls rapidly in successive plays. By the third round, the winner’s choice was usually as low as 2 or 3. Slide 36 从经历中学习 Learning From Experience 如果你有充分的理由相信其他人不会选择纳什均衡策略，那么你 的最优选择也不是你的纳什均衡策略。 If you have good reason to believe that other players will not play their Nash Equilibrium strategies, then your best choice is not your Nash equilibrium strategy. 不过，实验实际上还是证明了理论。因为它的过程显示出人们通 过积累经验进行学习，使得其策略接近纳什均衡。 However, the experiment indeed produce a vindication of the theory since the process shows that people gather experience and learn to play strategies close to Nash equilibrium. 可以发现，人们从观察别人进行博弈中进行学习，比他们自己进 行博弈时学得更快。 It can be found that people learn somewhat faster by observing others play a game than while they play it themselves. 19 Slide 37 真实世界的证据 Real-World Evidence 解释力 Explanatory Power „ 理论能否解释某些普遍观察到的现象？ Can the theory explain some observed phenomena in general terms? „ 例如，价格竞争，古巴导弹危机，拍卖，投票和谈判。 E.g., price competition, Cuban missile crisis, auctions, voting, and bargaining. 统计检验 Statistical Testing „ 对于理论的含义利用数据进行统计检验 Test statistically some implications of the theory against data „ 例如，关于定价、生产能力以及创新方面的商业决策 E.g., Business decisions on pricing, capacity, innovation. Slide 38 我们能够相信纳什均衡吗？ Can we trust Nash Equilibrium? 当面对的博弈是由来自足够稳定总体的参与者，在相 对不变的规则和条件下，频繁地进行，那么就有足够 的理由使用纳什均衡的概念。 When the game in question is played frequently by players from a reasonably stable population and under relatively unchanging rules and conditions, you should have considerable confidence in using the Nash equilibrium concept. 即使博弈是新的或者只进行一次，博弈者都是新手， 分析的第一步仍然是寻找纳什均衡。 Even when the game is new or is played just once and the players are inexperienced, your first step in the analysis should be to look for a Nash equilibrium. 20 Slide 39 对于纳什均衡概念的批判性讨论 Critical Discussion of the Nash Equilibrium Concept 纳什均衡对于风险的处理 The treatment of risk in Nash equilibrium 多重纳什均衡 Multiplicity of Nash equilibrium 纳什均衡对于理性的要求 Requirements of Rationality for Nash equilibrium Slide 40 纳什均衡对于风险的处理 The Treatment of Risk in Nash Equilibrium 一些批评认为，纳什均衡概念没有对风险足够 重视。 Some critics argue that the Nash equilibrium concept does not pay due attention to risk. 在一些博弈中，人们发现不是纳什均衡策略的 策略更加安全，因而可能去选择那些策略。 In some games, people might find strategies different from their Nash equilibrium strategies to be safer and may therefore choose those strategies. 21 Slide 41 存在可疑纳什均衡的一个博弈 A Game with a Questionable Nash Equilibrium 2, 22, 32, 0C 3, 22, 21, 3B 0, 23, 12, 2A ROW CBA COLUMN (A, A) is the unique Nash Equilibrium, yielding the payoffs (2, 2) However, playing C gives both players the same payoffs (2 for each), and always! Slide 42 灾难性的纳什均衡 Disastrous Nash Equilibrium -1000, 9.910, 10逃走 Down 8, 9.99, 10留下 Up 小妹 （章子怡） A 追踪 Right 放弃 Left 刘捕头（刘德华） B 刘捕头的优势策略是放弃。 唯一的纳什均衡是（逃走，放 弃），收益为 (10,10) 给定小妹逃走，如果刘捕头哪怕有一点点可能 会追踪，情况会如何呢？ 22 Slide 43 灾难性的纳什均衡 Disastrous Nash Equilibrium 如果小妹对刘捕头的收益或者理性哪怕有一点怀疑， 那么她留下比出纳什均衡策略逃走都要安全很多。 If A has any doubts about either B’s payoffs or B’s rationality, then it is a lot safer for A to play Up than to play her Nash equilibrium strategy of Down. 然而，真正的问题不是纳什均衡的概念本身不合理， 而是这个例子以一种过于简化的方式使用了它。 However, the real problem is not that the Nash equilibrium concept is inappropriate but that the example choose to use it in an inappropriately simplistic way. Slide 44 灾难性的纳什均衡 Disastrous Nash Equilibrium 假设小妹认为存在概率p，使得刘捕头来自放弃和追 踪的收益恰好与上例所示相反。 Suppose A thinks there is a probability p that B’s payoffs from Left and Right are the reverse of those shown in the example above. 则刘捕头的优势策略以(1-p) 的概率为放弃，以p的 概率为追踪。 With probability, B’s dominant strategy is Left and, with probability p , it is Right. 则无论小妹选择留下还是逃走，她都以(1-p) 概率遇 上刘捕头放弃，以p的概率遇上刘捕头追踪。 Then no matter what A chooses (Up or Down), with probability he will meet B playing Left, and with probability p, B playing Right. 23 Slide 45 灾难性的纳什均衡 Disastrous Nash Equilibrium 则小妹留下和逃走的统计或概率加权的平均收益各为： Thus A’s statistical or probability-weighted average payoff from playing Up and Down are respectively, 9(1-p) + 8p, 10(1-p) –1000p 小妹留下更好，如果： Therefore it is better for A to choose Up if 9(1-p) + 8p > 10(1-p) – 1000p, or p > 1/1009. 小妹留下为最优，即使刘捕头的收益仅有很小的可能 性与例子中的相反。 It is optimal for A to play Up, even if there is only a very slight chance that B’s payoffs are opposite of our example. Slide 46 多重纳什均衡 Multiplicity of Nash Equilibrium 多重纳什均衡并不自动地要求我们抛弃纳什均衡的概 念。 The multiplicity of Nash equilibrium does not automatically require us to abandon the Nash equilibrium concept. 它只说明仅有这一概念是不够的，它必须为其他的考 虑所补充，以选择一个有理由被认为比其他更好的均 衡。 It merely shows that the concept is not enough by itself and must be supplemented by some other consideration that selects the one equilibrium with better claims to be the outcome than the rest. 24 Slide 47 多重纳什均衡 Multiplicity of Nash Equilibrium 在一个存在多重均衡的协调博弈中，参与者可能能选 择一个焦点…… In a coordination game with multiple equilibria, the players may be able to select one as a focal point …… „ 如果他们具有某些共同的社会、文化或历史背景。 if they have some common social, cultural, or historical knowledge „ 或者，如果博弈有某些有意识的或偶然的特征使得他们的预 期可以收敛。 or if the game has some deliberate or accidental features that enable their expectations to converge. Slide 48 多重纳什均衡 Multiplicity of Nash Equilibrium 我们将会发现序贯博弈也可以有多重纳什均衡。 We will see sequential-move games can have multiple Nash equilibria. 在那里，我们将引入对可信性的要求，以使我 们能够选择一个特定的均衡。 There, we introduce the requirement of credibility that enables us to select a particular equilibrium. 这一特定的均衡最后发现就是反转均衡。 It turns out that this one is in fact the rollback equilibrium. 25 Slide 49 多重纳什均衡 Multiplicity of Nash Equilibrium 在更加复杂的博弈中，例如信息不对称或其他 复杂性时，我们会引入其他的限制（称为精 炼），以识别和排除从某个角度看来不合理的 纳什均衡。 In more complex games with information asymmetries or additional complications, other restrictions called refinements have been developed to identify and rule out Nash