经济博弈论06

1 序贯博弈和同时博弈的结合 Combining Sequential and Simultaneous Moves 第6章 Chapter 6 Slide 2 序贯博弈和同时博弈的结合 Combining Sequential and Simultaneous Moves 收益（策略形式） Payoff tables (Strategic form) 纳什均衡 Nash equilibrium 纯粹同时博弈 Purely Simultaneous- move games 博弈树（扩展形式） Game Trees (Extensive form) 反转均衡 Rollback equilibrium 纯粹序贯博弈 Purely Sequential- move games 分析技术 Techniques of Analysis 概念 Concepts 博弈类型 Game Type 2 Slide 3 序贯博弈和同时博弈的结合 Combining Sequential and Simultaneous Moves 在现实中，许多策略环境包含了这两种相互作 用的成分。 In reality, many strategic situations contain elements of both types of interaction. 而且，我们还可以使用扩展形式或策略形式分 析任何一种博弈（可以交叉使用）。 Also, we can use either extensive form or strategic form for any type of game. Slide 4 提要 Outline 兼具同时和序贯行动的博弈 Games with both simultaneous and sequential moves 改变博弈中的行动顺序 Changing the order of moves in a game !改变分析方法 Change in the method of analysis *三人博弈 Three-player games 3 Slide 5 兼具同时和序贯行动的博弈 Games with Both Simultaneous and Sequential Moves 典型的一般都是博弈者在一段比较长的时 间内相互作用。 The most obvious examples are those between players over an extended period of time. 这样的博弈是同时利用博弈树和反转，以及收 益表和纳什均衡的工具来分析的。 Such games are analyzed by combing the tools of trees and rollback, and payoff tables and Nash Equilibrium. Slide 6 兼具序贯和同时行动的一个两阶段博弈 A Two-stage Game Combining Sequential and Simultaneous Moves 有两个可能成为电信巨头的企业：C和G。 There’re two would-be telecom giants, CrossTalk and GlobeDialog. 每个企业都需要同时选择是否投资100亿以购置光纤网。 Each can choose whether to invest $10 billion in the purchase of a fiberoptic network, simultaneously. 如果一个企业投资了而另一个没有，投资的企业需要确定其电信 服务的定价。 If one invests and the other does not, then the investor has to make a pricing decision for its telecom service. 如果两个企业都投了，那么他们的定价选择成为一个第二阶段的 同时博弈。 If both invest, then their pricing choices become a second simultaneous-move game. 4 Slide 7 兼具序贯和同时行动的一个两阶段博弈 A Two-stage Game Combining Sequential and Simultaneous Moves , 0Invest 0, 0, 0Don’tCROSS- TALK InvestDon’t GLOBEDIALOG Second stage: GlobalDialog’s pricing decision GLOBAL- DIALOG 14 6 High Low Second stage: GlobalDialog’s pricing decision CROSS- TALK 14 6 High Low -2, -26,-10Low -10, 62, 2HighCROSS- TALK LowHigh GLOBEDIALOG Second stage: pricing game First stage: Investment Game Slide 8 兼具序贯和同时行动的一个两阶段博弈 A Two-stage Game Combining Sequential and Simultaneous Moves -2, -214, 0Invest 0, 140, 0Don’tCROSS- TALK InvestDon’t GLOBEDIALOG Stage one Investment Game (After Substituting Rolled-Back Payoffs from the Equilibrium of the Second Stage) Two Nash Equilibria: A chicken game 5 Slide 9 子博弈 Subgames 一个子博弈是整个博弈的一部分，它自身就构成一个 完备博弈，具有完整的结构：博弈者、策略和收益。 A subgame is a part of a full game, which is also a full-fledged game in its own right, with a fully specified structure of players, strategies, and payoffs. 更一般的，一个子博弈是多行动博弈的一部分，它开 始于原博弈的某一个节点。 More generally, a subgame is the part of a multimove game that begin at a particular node of the original game. „ 一个多行动博弈具有的子博弈数目等于其决策点数目。 A multimove game has as many subgames as it has decision nodes. Slide 10 多阶段博弈的构成：例子 Configurations of Multistage Games: Examples -2, -26,-10Low -10, 62, 2HighCROSS- TALK LowHigh GLOBEDIALOG GLOBAL- DIALOG 0, 14 0, 6 High Low 假设G事先已经投了100亿了 Suppose GlobalDialog has already made the $10 billion investment…… CROSS- TALK Invest Don’t 6 Slide 11 多阶段博弈的构成：例子 Configurations of Multistage Games: Examples 3进攻 阵形 2 1防守 阵形中国女足 进攻 阵形 防守 阵形 德国女足 德国女足 1 2 反应 不反应 中国女足 不变阵 变阵 -1 Simultaneous-move First Stage Followed by Sequential Moves Slide 12 多阶段博弈的构成：例子 Configurations of Multistage Games: Examples 13进攻 阵形 2 1防守 阵形中国女足 进攻 阵形 防守 阵形 德国女足 No pure strategy Nash equilibrium. It turns out our Chinese team should choose the attack lineup with probability 1/3. (Shown in Ch7) 7 Slide 13 改变博弈中的行动顺序 Changing the Order of Moves in a Game 序贯博弈可以变成同时的，如果参与者在做出 自己的选择时，不能观察到对手的行动。 Sequential-move games become simultaneous if the players cannot observe moves made by their rivals before making their own choices. 这样，我们就得去寻找纳什均衡而非反转均衡。 In that case, we would analyze the game by searching for a Nash equilibrium rather than for a rollback equilibrium. Slide 14 改变博弈中的行动顺序 Changing the Order of Moves in a Game 同时博弈也可以变成序贯的，如果某一参与者 能够在做出自己选择前观察到其他人的行动。 A simultaneous-move game could become sequential if one player were able to observe the other’s move before choosing her own. 博弈规则的任何改变，也能改变其结果。 Any changes to the rules of the game can also change its outcome. 8 Slide 15 变同时博弈为序贯博弈 Changing Simultaneous-Move Games into Sequential-Move Games 结果不变 No change in outcome 先行者优势 First-mover advantage 后行者优势 Second-mover advantage 双方都更好 Both Players may be better Slide 16 结果不变 No Change in Outcome 某些博弈，无论是同时的，还是序贯的，也无 论序贯博弈中行动顺序如何，结果都不变。 Certain games have the same outcomes in the equilibrium of both simultaneous and sequential versions and regardless of the order of play in the sequential game. 这一结果通常产生在所有博弈者都具有优势策 略时。This result generally arises only when both or all players have dominant strategies. 9 Slide 17 例：囚徒困境的三个版本 Three Versions of the Prisoners’ Dilemma Game 3 yr, 3 yr25 yr, 1 yrDeny (Cooperate) 1 yr, 25 yr10 yr, 10 yrConfess (Defect) HUSBAND Deny (Cooperate) Confess (Defect) WIFE HUSBAND Confess Confess Confess Deny Deny Deny 10, 10 25, 1 1, 25 3, 3 WIFE WIFE HUSBAND, WIFE WIFE Confess Confess Confess Deny Deny Deny 10, 10 25, 1 1, 25 3, 3 HUSBAND HUSBAND WIFE, HUSBAND (a) Simultaneous play (b) Sequential play: Husband moves first (c) Sequential play: Wife moves first 4？ Slide 18 先行者优势 First-Mover Advantage 先行者优势可能产生在博弈规则从同时行动变 为顺序行动时。 A first-mover advantage may emerge when the rules of a game are changed from simultaneous to sequential play. 至少来说，如果同时博弈具有多重均衡，序贯 博弈可以使先行者从中选择对自己有利的结果。 At a minimum, if the simultaneous- mover version has multiple equilibria, the sequential-move version enables the first mover to choose his preferred outcome. 10 Slide 19 先行者优势：小鸡博弈的例子 First-Mover Advantage: An Example of Chicken -2, -21, -1Straight (Tough) -1, 10, 0Swerve (Chicken) JAMES Straight (Tough) Swerve (Chicken) DEAN JAMES Swerve Swerve Swerve Straight Straight Straight 0, 0 1, -1 -1, 1 -2, -2 DEAN DEAN JAMES, DEAN DEAN Swerve Swerve Swerve Straight Straight Straight 0, 0 1, -1 -1, 1 -2, -2 JAMES JAMES DEAN, JAMES (a) Simultaneous play (b) Sequential play: James moves first (c) Sequential play: Dean moves first Slide 20 后行者优势 Second-Mover Advantage 从同时行动变为序贯行动的博弈，也可 以导致后行者优势。 A second-mover advantage may emerge when simultaneous play is changed into sequential play. 11 Slide 21 后行者优势：运动比赛的例子 Second-Mover Advantage : An Example of Sport 2090CC 8050DLEVERT CCDL NAVRATILOVA EVERT DL DL DL CC CC CC 50, 50 90, 10 80, 20 20, 80 NAVRATILOVA NAVRATILOVA EVERT, NAVRATILOVA NAVRATILOVA DL DL DL CC CC CC 50, 50 20, 80 10, 90 80, 20 EVERT EVERT NAVRATILOVA, EVERT No pure strategy Nash equilibrium. In the mixed strategy Nash equilibrium, Evert gets 62 on average. (a) Simultaneous play (b) Sequential play: EVERT moves first (c) Sequential play: Wife moves first Slide 22 双方都更好 Both Player May Do Better 直观地，一个博弈可能具有先行者或后行者优 势。只不过在同时行动时，这一优势被压制了， 而在给出行动顺序后得到了体现。 Intuitively, A game may have a first- mover or a second-mover advantage, which is suppressed when moves have to be simultaneous but emerges when an order of moves is imposed. 令人惊讶的是，当规则从一种变为另一种时， 双方参与者都有可能变好。 Surprisingly, both player may do better under one set of rules of play than under another. 12 Slide 23 双方都更好：例子 Both Player May Do Better: An Example 2, 24, 1Budget deficit 1, 33, 4Budget balance CONGRESS High interest rates Low interest rates FEDERAL RESERVE Congress has a dominant strategy: Budget deficit. The unique Nash equilibrium is: (Budget deficit, High interest rate) (a) Simultaneous moves Slide 24 双方都更好：例子 Both Player May Do Better: An Example FED Low High Balance Deficit Balance Deficit 4, 3 1, 4 3, 1 2, 2 CONGRESS CONGRESS FED, CONGRESS(b) Sequential moves: Fed moves first 13 Slide 25 双方都更好：例子 Both Player May Do Better: An Example CONGRESS Balance Deficit Low High Low High 3, 4 1, 3 4, 1 2, 2 FED FED CONGRESS, FED(b) Sequential moves: Congress moves first Slide 26 双方都更好：例子 Both Player May Do Better: An Example 更令人奇怪的是，更好的结果来自于当国会先 行时，而它所选择的平衡预算策略是它的劣势 策略。 Even more surprisingly, the better outcome obtain when Congress moves first results from its choosing Balance, which is its dominated strategy. 是否矛盾？ Is this a paradox? 14 Slide 27 双方都更好：例子 Both Player May Do Better: An Example 赤字成为优势策略，从国会的角度讲，就要求它在任意给定的 Fed的行动下，都要比平衡预算来得好。 For deficit to be a dominant strategy, it must be better than Balance from the Congress’s perspective for each given choice of the Fed. 这种比较在同时博弈中是有意义的，因为国会必须： This type of comparison between Deficit and Balance is relevant in the simultaneous-move game because there the Congress must …… „ ……在不知道Fed的选择时决策 make a decision without knowing the Fed’s choice „ ……透彻考虑Fed的行动，或者对之形成信念，以选择自己的最优 反应。也即，选择一个理性化策略。 think through, or formulate a belief about, the Fed’s action, and choose its best response to that. I.e, it must choose a rationalizable strategy. Slide 28 双方都更好：例子 Both Player May Do Better: An Example 这一比较在国会后行动时也是有意义的。它知道Fed的行动后，只需选 择其最优反应，总为赤字。 The comparison is also relevant with sequential moves if the Congress moves second, then it knows what the Fed has already done and merely picks its best response, which is always Deficit. 然而，如果国会先行，它就不能（或无须）将Fed的行动看成既定。 However, if the Congress moves first, it cannot take the Fed’s choice as given. 相反，它必须考虑到Fed对自己先前行动的最优反应，以及由此带来的 收益。然后选择自己喜欢的结果。 Instead, it must recognize the Fed’s best response to each of its own first moves, and payoffs as a result. Then choose its more preferred outcome between them. 因而优势的思想对于先行者来说可能没有意义。 Thus the idea of dominant may cease to be a relevant concept for the first mover. 15 Slide 29 双方都更好：例子 Both Player May Do Better: An Example 在这一博弈中，国会和Fed将会达成，让国会先 行。 In this game, the Congress and the Fed would agree that the Congress should move first. 不过，协议要得以实施，先行的技术条件必须满足， 即：先行的行动必须可以被后行者观察到，并且一旦 行动，不能逆转。 However, to implement the agreement, the technical requirements of a first move – that it be observable to the second mover and not reversible thereafter – must be satisfied. Slide 30 改变分析方法 Change in the Method of Analysis 博弈树是表示序贯博弈的理想方式，收益表是 表示同时博弈的理想方式。 Game trees are the natural way to display sequential-move games, and payoff tables the natural representation of simultaneous-move games. 不过，上述每一方法适当修改都可以用来表示 另一种博弈。 However, each technique can be adapted to the other type of game. 16 Slide 31 用博弈树表示同时博弈 Illustrating Simultaneous-Move Games by Using Trees EVERT DL CC DL CC DL CC 50, 50 80, 20 90, 10 20, 80 NAVRATILOVA NAVRATILOVA EVERT, NAVRATILOVA Information set Slide 32 信息集 Information Set 信息集表明对于当前参与者不完美信息的存在：给定 其可得的信息，她无法辨别该集合中的节点。 An information set indicate the presence of imperfect information for the player who moves there. She cannot distinguish between the nodes in the set, given her available information. 她在单独一个信息集中的策略选择必须对集合中的所 有节点规定相同的行动。 Her strategy choice from within a single information set must specify the same move at all the nodes contained in it. 17 Slide 33 用策略形式表示和分析序贯博弈 Showing and Analyzing Sequential- Move Games in Strategic Form CONGRESS Balance Deficit Low High Low High 3, 4 1, 3 4, 1 2, 2 FED FED CONGRESS, FED How to show this game in normal/strategic form, i.e., by using a payoff table? Slide 34 用策略形式表示和分析序贯博弈 Showing and Analyzing Sequential- Move Games in Strategic Form 2, 24, 14, 12, 2Deficit 1, 33, 41, 33, 4Balance CONGRESS H if B, H if D L if B, L if D H if B, L if D L if B, H if D FED Congress’s strategies Fed’s strategies Two Nash Equilibria: (Balance, LH), (Deficit, HH). But the game produced just one rollback equilibrium – (Balance, LH) when analyzed in its extensive form! 18 Slide 35 用策略形式表示和分析序贯博弈 Showing and Analyzing Sequential- Move Games in Strategic Form 不过，策略组合（赤字，总是高利率）不是该序贯博弈的合理预 测。 Yet, the strategy combination (Deficit, High always) is not a sensible prediction of this sequential game. 在这一策略组合中，Fed从来没有使用过其策略的一部分——“在 平衡时采取高利率”。这一部分仅作为了对国会的一种威胁。 In this strategy combination, Fed never plays one part of his strategy – “ High if Balance”. It’s only a threaten to the Congress. 但这一威胁是不可信的，因为一旦有必要采取它时，Fed总会发 现偏离它是最优的。 But this threaten is incredible since once the need arises to play it, the Fed will always find it optimal to deviate. 国会预计到了这一点，不会对这一威胁当真。 The Congress will foresee this and not take this threaten seriously. Slide 36 用策略形式表示和分析序贯博弈 Showing and Analyzing Sequential- Move Games in Strategic Form 换句话说，该策略组合不合理，是因为Fed在其始于 国会选了平衡后的子博弈中没有出最优反应（也就是 纳什均衡策略）。 In other words, strategy combination (Deficit, High always) is not sensible because the Fed did not play its best response (and also, its Nash equilibrium strategy) in his subgame starting at the node where the Congress has already chosen Balance. 这一策略组合没有构成一个子博弈完美均衡。 This strategy combination does not form a subgame-perfect equilibrium. 19 Slide 37 子博弈完美均衡 Subgame-Perfect Equilibrium 子博弈完美（纳什）均衡（SPE）由来自各参与者的策略组合而 成。在博弈树的每一节点上（无论该节点是否位于博弈的均衡路 径上）都有一个子博弈，而整体策略在子博弈延续的部分，在起 始点上对于相应参与者为最优。 A subgame-perfect (Nash) equilibrium (SPE) is a set of strategies, one for each player, such that at every node of the game tree, whether or not the node lies along the equilibrium path of play, the continuation of the same strategy in the subgame starting at that node is optimal for the player who takes the action there. 简单说，SPE要求参与者的策略在整个博弈的每一个子博弈中都 是纳什均衡策略。 SPE requires players to use strategies that constitute a Nash equilibrium in every subgame of the larger game. Slide 38 子博弈完美均衡和反转均衡 SPE and Rollback Equilibrium 在有限和信息完美的博弈中，除个别不重要的情况， 反转的方法都可以找出唯一的子博弈均衡。 In games with finite trees and perfect information, where players can observe every previous action taken by all other players so that there are no multiple nodes enclosed in one information set, rollback finds the unique (except for trivial and exceptional cases ot ties) subgame-perfect equilibrium of the game. 复杂信息结构和信息集的博弈中，子博弈完美的概念 涵盖更广。 In games with complex information structures and information set, subgame perfectness becomes a rich notion. 20 Slide 39 子博弈完美均衡和反转均衡 SPE and Rollback Equilibrium 因而，反转或SPE是对纳什均衡概念的进一步 检验和补充，以从策略形式的多重均衡中进行 选择。 Thus, rollback, or SPE is a further test, supplementing the requirements of a Nash equilibrium and helping to select from among multiple Nash equilibria of the strategic form. 它是纳什均衡概念的一种精炼。 It is a refinement of the Nash equilibrium concept. Slide 40 Summary 许多博弈都包括了多重成分，包括同时和序贯 行动。 Many games include multiple components, some of which entail simultaneous play and others of which entail sequential play. 在后面的行动中出现的完备博弈成为整个博弈 的子博弈。 Full-fledged games that arise in later stages of play are called subgames of the full game. 21 Slide 41 总结 Summary 改变博弈规则以变更行动顺序可能改变也可能不改变 均衡结果。 Changing the rules of a game to alter the timing of moves may or may not alter the equilibrium outcome of a game. 同时博弈可以用博弈树表示。由于参与者在决策时不 知道他们处于某些节点中的具体哪个，就需要把所有 这些节点归入信息集。 Simultaneous-move games can be illustrated in a game tree by collecting decision nodes in information set when players make decisions without knowing at which specific node they find themselves. Slide 42 总结 Summary 序贯博弈可以用博弈表来表示。 Sequential-move games can be illustrated by using a game table. 用其策略形式来求解一个序贯博弈，可能导致 过多的纳什均衡，减少的办法时使用子博弈完 美均衡（SPE）的概念。 Solving a sequential-move game from its strategic form may lead to many possible Nash equilibria, which can be reduced by using the concept of subgame-perfect equilibrium (SPE).