博弈论(2)—讲义
博弈论PPT课件
这就是混合策略。
混合策略的纳什均衡定义
如果对于博弈中所有的游戏者i，对于所有的 σi∈Mi，都有ui﹙σ*﹚≥ui﹙σi，σ-i*﹚，则称 σ*就是一个混合策略的纳什均。
如何求混合策略的纳什均衡
猜硬币的博弈中 解：设猜方猜正方的概率为p，猜反方的概率则为1－
无名氏（大众）定理
无名氏定理：在无穷次重复的由n个游戏者参与的 博弈里，如果在每一次重复中博弈的行动集是有限 的，则在满足下列三个条件时，在任何有限次重复 中所观察到的任何行动组合都是某个子博弈完美均 衡的惟一结果：
条件1：贴现因子接近于1； 条件2：在每一次重复中，博弈结束的概率或等于0，或 为非常小的一个正值； 条件3：严格占优于一次性博弈中的最小最大收益组合的 那个收益组合集是n维的。
博弈方
博弈方：独立决策、独立承担博弈结果的个人 或组织
博弈规则面前博弈方之间平等，不因博弈方之 间权利、地位的差异而改变
博弈方数量对博弈结果和分析有影响 根据博弈方数量分单人博弈、两人博弈、多人
博弈等。最常见的是两人博弈，单人博弈是退 化的博弈
策略
策略：博弈中各博弈方的选择内容 策略有定性定量、简单复杂之分 不同博弈方之间不仅可选策略不同，而且可
游戏和经济等决策竞争较量的共同特征：规 则、结果、策略选择，策略和利益相互依存， 策略的关键作用
游戏——下棋、猜大小 经济——寡头产量决策、市场阻入、投标拍卖 政治、军事——美国和伊朗、以色列和巴勒斯 坦、中国和日本等等。
博弈的基本要素
博弈的参加者(Player)——博弈方 各博弈方的策略(Strategies)或行动(Actions) 博弈的次序(Order) 博弈方的收益(Payoffs) （或称支付，或得益）
lecture_2（博弈论讲义GameTheory（MIT））
Last Time:Defined knowledge, common knowledge, meet (of partitions), and reachability.Reminders:• E is common knowledge at ω if ()I K E ω∞∈.• “Reachability Lemma” :'()M ωω∈ if there is a chain of states 01,,...m 'ωωωωω== such that for each k ω there is a player i(k) s.t. ()()1()(i k k i k k h h )ωω+=:• Theorem: Event E is common knowledge at ωiff ()M E ω⊆.How does set of NE change with information structure?Suppose there is a finite number of payoff matrices 1,...,L u u for finite strategy sets 1,...,I S SState space Ω, common prior p, partitions , and a map i H λso that payoff functions in state ω are ()(.)u λω; the strategy spaces are maps from into . i H i SWhen the state space is finite, this is a finite game, and we know that NE is u.h.c. and generically l.h.c. in p. In particular, it will be l.h.c. at strict NE.The “coordinated attack” game8,810,11,100,0A B A B-- 0,010,11,108,8A B A B--a ub uΩ= 0,1,2,….In state 0: payoff functions are given by matrix ; bu In all other states payoff functions are given by . a upartitions of Ω1H : (0), (1,2), (3,4),… (2n-1,2n)... 2H (0,1),(2,3). ..(2n,2n+1)…Prior p : p(0)=2/3, p(k)= for k>0 and 1(1)/3k e e --(0,1)ε∈.Interpretation: coordinated attack/email:Player 1 observes Nature’s choice of payoff matrix, sends a message to player 2.Sending messages isn’t a strategic decision, it’s hard-coded.Suppose state is n=2k >0. Then 1 knows the payoffs, knows 2 knows them. Moreover 2 knows that 1knows that 2 knows, and so on up to strings of length k: . 1(0n I n K n -Î>)But there is no state at which n>0 is c.k. (to see this, use reachability…).When it is c.k. that payoff are given by , (A,A) is a NE. But.. auClaim: the only NE is “play B at every information set.”.Proof: player 1 plays B in state 0 (payoff matrix ) since it strictly dominates A. b uLet , and note that .(0|(0,1))q p =1/2q >Now consider player 2 at information set (0,1).Since player 1 plays B in state 0, and the lowest payoff 2 can get to B in state 1 is 0, player 2’s expected payoff to B at (0,1) is at least 8. qPlaying A gives at most 108(1)q q −+−, and since , playing B is better. 1/2q >Now look at player 1 at 1(1,2)h =. Let q'=p(1|1,2), and note that '1(1)q /2εεεε=>+−.Since 2 plays B in state 1, player 1's payoff to B is at least 8q';1’s payoff to A is at most -10q'+8(1-q) so 1 plays B Now iterate..Conclude that the unique NE is always B- there is no NE in which at some state the outcome is (A,A).But (A,A ) is a strict NE of the payoff matrix . a u And at large n, there is mutual knowledge of the payoffs to high order- 1 knows that 2 knows that …. n/2 times. So “mutual knowledge to large n” has different NE than c.k.Also, consider "expanded games" with state space . 0,1,....,...n Ω=∞For each small positive ε let the distribution p ε be as above: 1(0)2/3,()(1)/3n p p n ee e e -==- for 0 and n <<∞()0p ε∞=.Define distribution by *p *(0)2/3p =,. *()1/3p ∞=As 0ε→, probability mass moves to higher n, andthere is a sense in which is the limit of the *p p εas 0ε→.But if we do say that *p p ε→ we have a failure of lower hemi continuity at a strict NE.So maybe we don’t want to say *p p ε→, and we don’t want to use mutual knowledge to large n as a notion of almost common knowledge.So the questions:• When should we say that one information structure is close to another?• What should we mean by "almost common knowledge"?This last question is related because we would like to say that an information structure where a set of events E is common knowledge is close to another information structure where these events are almost common knowledge.Monderer-Samet: Player i r-believes E at ω if (|())i p E h r ω≥.()r i B E is the set of all ω where player i r- believesE; this is also denoted 1.()ri B ENow do an iterative definition in the style of c.k.: 11()()rr I i i B E B E =Ç (everyone r-believes E) 1(){|(()|())}n r n ri i I B E p B E h r w w -=³ ()()n r n rI i i B E B =ÇEE is common r belief at ω if ()rI B E w ¥ÎAs with c.k., common r-belief can be characterized in terms of public events:• An event is a common r-truism if everyone r -believes it when it occurs.• An event is common r -belief at ω if it is implied by a common r-truism at ω.Now we have one version of "almost ck" : An event is almost ck if it is common r-belief for r near 1.MS show that if two player’s posteriors are common r-belief, they differ by at most 2(1-r): so Aumann's result is robust to almost ck, and holds in the limit.MS also that a strict NE of a game with knownpayoffs is still a NE when payoffs are "almost ck” - a form of lower hemi continuity.More formally:As before consider a family of games with fixed finite action spaces i A for each player i. a set of payoff matrices ,:l I u A R ->a state space W , that is now either finite or countably infinite, a prior p, a map such that :1,,,L l W®payoffs at ω are . ()(,)()w u a u a l w =Payoffs are common r-belief at ω if the event {|()}w l w l = is common r belief at ω.For each λ let λσ be a NE for common- knowledgepayoffs u .lDefine s * by *(())s l w w s =.This assigns each w a NE for the corresponding payoffs.In the email game, one such *s is . **(0)(,),()(,)s B B s n A A n ==0∀>If payoffs are c.k. at each ω, then s* is a NE of overall game G. (discuss)Theorem: Monder-Samet 1989Suppose that for each l , l s is a strict equilibrium for payoffs u λ.Then for any there is 0e >1r < and 1q < such that for all [,1]r r Î and [,1]q q Î,if there is probability q that payoffs are common r- belief, then there is a NE s of G with *(|()())1p s s ωωω=>ε−.Note that the conclusion of the theorem is false in the email game:there is no NE with an appreciable probability of playing A, even though (A,A) is a strict NE of the payoffs in every state but state 0.This is an indirect way of showing that the payoffs are never ACK in the email game.Now many payoff matrices don’t have strictequilibria, and this theorem doesn’t tell us anything about them.But can extend it to show that if for each state ω, *(s )ω is a Nash (but not necessarily strict Nash) equilibrium, then for any there is 0e >1r < and 1q < such that for all [,1]r r Î and [,1]q q Î, if payoffs are common r-belief with probability q, there is an “interim ε equilibria” of G where s * is played with probability 1ε−.Interim ε-equilibria:At each information set, the actions played are within epsilon of maxing expected payoff(((),())|())((',())|())i i i i i i i i E u s s h w E u s s h w w w w e-->=-Note that this implies the earlier result when *s specifies strict equilibria.Outline of proof:At states where some payoff function is common r-belief, specify that players follow s *. The key is that at these states, each player i r-believes that all other players r-believe the payoffs are common r-belief, so each expects the others to play according to s *.*ΩRegardless of play in the other states, playing this way is a best response, where k is a constant that depends on the set of possible payoff functions.4(1)k −rTo define play at states in */ΩΩconsider an artificial game where players are constrained to play s * in - and pick a NE of this game.*ΩThe overall strategy profile is an interim ε-equilibrium that plays like *s with probability q.To see the role of the infinite state space, consider the"truncated email game"player 2 does not respond after receiving n messages, so there are only 2n states.When 2n occurs: 2 knows it occurs.That is, . {}2(0,1),...(22,21,)(2)H n n =−−n n {}1(0),(1,2),...(21,2)H n =−.()2|(21,2)1p n n n ε−=−, so 2n is a "1-ε truism," and thus it is common 1-ε belief when it occurs.So there is an exact equilibrium where players playA in state 2n.More generally: on a finite state space, if the probability of an event is close to 1, then there is high probability that it is common r belief for r near 1.Not true on infinite state spaces…Lipman, “Finite order implications of the common prior assumption.”His point: there basically aren’t any!All of the "bite" of the CPA is in the tails.Set up: parameter Q that people "care about" States s S ∈,:f S →Θ specifies what the payoffs are at state s. Partitions of S, priors .i H i pPlayer i’s first order beliefs at s: the conditional distribution on Q given s.For B ⊆Θ,1()()i s B d =('|(')|())i i p s f s B h s ÎPlayer i’s second order beliefs: beliefs about Q and other players’ first order beliefs.()21()(){'|(('),('))}|()i i j i s B p s f s s B h d d =Îs and so on.The main point can be seen in his exampleTwo possible values of an unknown parameter r .1q q = o 2qStart with a model w/o common prior, relate it to a model with common prior.Starting model has only two states 12{,}S s s =. Each player has the trivial partition- ie no info beyond the prior.1122()()2/3p s p s ==.example: Player 1 owns an asset whose value is 1 at 1θ and 2 at 2θ; ()i i f s θ=.At each state, 1's expected value of the asset 4/3, 2's is 5/3, so it’s common knowledge that there are gains from trade.Lipman shows we can match the players’ beliefs, beliefs about beliefs, etc. to arbitrarily high order in a common prior model.Fix an integer N. construct the Nth model as followsState space'S ={1,...2}N S ´Common prior is that all states equally likely.The value of θ at (s,k) is determined by the s- component.Now we specify the partitions of each player in such a way that the beliefs, beliefs about beliefs, look like the simple model w/o common prior.1's partition: events112{(,1),(,2),(,1)}...s s s 112{(,21),(,2),(,)}s k s k s k -for k up to ; the “left-over” 12N -2s states go into 122{(,21),...(,2)}N N s s -+.At every event but the last one, 1 thinks the probability of is 2/3.1qThe partition for player 2 is similar but reversed: 221{(,21),(,2),(,)}s k s k s k - for k up to . 12N -And at all info sets but one, player 2 thinks the prob. of is 1/3.1qNow we look at beliefs at the state 1(,1)s .We matched the first-order beliefs (beliefs about θ) by construction)Now look at player 1's second-order beliefs.1 thinks there are 3 possible states 1(,1)s , 1(,2)s , 2(,1)s .At 1(,1)s , player 2 knows {1(,1)s ,2(,1)s ,(,}. 22)s At 1(,2)s , 2 knows . 122{(,2),(,3),(,4)}s s s At 2(,1)s , 2 knows {1(,2)s , 2(,1)s ,(,}. 22)sThe support of 1's second-order beliefs at 1(,1)s is the set of 2's beliefs at these info sets.And at each of them 2's beliefs are (1/3 1θ, 2/3 2θ). Same argument works up to N:The point is that the N-state models are "like" the original one in that beliefs at some states are the same as beliefs in the original model to high but finite order.(Beliefs at other states are very different- namely atθ or 2 is sure the states where 1 is sure that state is2θ.)it’s1Conclusion: if we assume that beliefs at a given state are generated by updating from a common prior, this doesn’t pin down their finite order behavior. So the main force of the CPA is on the entire infinite hierarchy of beliefs.Lipman goes on from this to make a point that is correct but potentially misleading: he says that "almost all" priors are close to a common. I think its misleading because here he uses the product topology on the set of hierarchies of beliefs- a.k.a topology of pointwise convergence.And two types that are close in this product topology can have very different behavior in a NE- so in a sense NE is not continuous in this topology.The email game is a counterexample. “Product Belief Convergence”:A sequence of types converges to if thesequence converges pointwise. That is, if for each k,, in t *i t ,,i i k n k *δδ→.Now consider the expanded version of the email game, where we added the state ∞.Let be the hierarchy of beliefs of player 1 when he has sent n messages, and let be the hierarchy atthe point ∞, where it is common knowledge that the payoff matrix is .in t ,*i t a uClaim: the sequence converges pointwise to . in t ,*i t Proof: At , i’s zero-order beliefs assignprobability 1 to , his first-order beliefs assignprobability 1 to ( and j knows it is ) and so onup to level n-1. Hence as n goes to infinity, thehierarchy of beliefs converges pointwise to common knowledge of .in t a u a u a u a uIn other words, if the number of levels of mutual knowledge go to infinity, then beliefs converge to common knowledge in the product topology. But we know that mutual knowledge to high order is not the same as almost common knowledge, and types that are close in the product topology can play very differently in Nash equilibrium.Put differently, the product topology on countably infinite sequences is insensitive to the tail of the sequence, but we know that the tail of the belief hierarchy can matter.Next : B-D JET 93 "Hierarchies of belief and Common Knowledge”.Here the hierarchies of belief are motivated by Harsanyi's idea of modelling incomplete information as imperfect information.Harsanyi introduced the idea of a player's "type" which summarizes the player's beliefs, beliefs about beliefs etc- that is, the infinite belief hierarchy we were working with in Lipman's paper.In Lipman we were taking the state space Ω as given.Harsanyi argued that given any element of the hierarchy of beliefs could be summarized by a single datum called the "type" of the player, so that there was no loss of generality in working with types instead of working explicitly with the hierarchies.I think that the first proof is due to Mertens and Zamir. B-D prove essentially the same result, but they do it in a much clearer and shorter paper.The paper is much more accessible than MZ but it is still a bit technical; also, it involves some hard but important concepts. (Add hindsight disclaimer…)Review of math definitions:A sequence of probability distributions converges weakly to p ifn p n fdp fdp ®òò for every bounded continuous function f. This defines the topology of weak convergence.In the case of distributions on a finite space, this is the same as the usual idea of convergence in norm.A metric space X is complete if every Cauchy sequence in X converges to a point of X.A space X is separable if it has a countable dense subset.A homeomorphism is a map f between two spaces that is 1-1, and onto ( an isomorphism ) and such that f and f-inverse are continuous.The Borel sigma algebra on a topological space S is the sigma-algebra generated by the open sets. (note that this depends on the topology.)Now for Brandenburger-DekelTwo individuals (extension to more is easy)Common underlying space of uncertainty S ( this is called in Lipman)ΘAssume S is a complete separable metric space. (“Polish”)For any metric space, let ()Z D be all probability measures on Borel field of Z, endowed with the topology of weak convergence. ( the “weak topology.”)000111()()()n n n X S X X X X X X --=D =´D =´DSo n X is the space of n-th order beliefs; a point in n X specifies (n-1)st order beliefs and beliefs about the opponent’s (n-1)st order beliefs.A type for player i is a== 0012(,,,...)()n i i i i n t X d d d =¥=δD0T .Now there is the possibility of further iteration: what about i's belief about j's type? Do we need to add more levels of i's beliefs about j, or is i's belief about j's type already pinned down by i's type ?Harsanyi’s insight is that we don't need to iterate further; this is what B-D prove formally.Coherency: a type is coherent if for every n>=2, 21marg n X n n d d --=.So the n and (n-1)st order beliefs agree on the lower orders. We impose this because it’s not clear how to interpret incoherent hierarchies..Let 1T be the set of all coherent typesProposition (Brandenburger-Dekel) : There is a homeomorphism between 1T and . 0()S T D ´.The basis of the proposition is the following Lemma: Suppose n Z are a collection of Polish spaces and let021201...1{(,,...):(...)1, and marg .n n n Z Z n n D Z Z n d d d d d --´´-=ÎD ´"³=Then there is a homeomorphism0:(nn )f D Z ¥=®D ´This is basically the same as Kolmogorov'sextension theorem- the theorem that says that there is a unique product measure on a countable product space that corresponds to specified marginaldistributions and the assumption that each component is independent.To apply the lemma, let 00Z X =, and 1()n n Z X -=D .Then 0...n n Z Z X ´´= and 00n Z S T ¥´=´.If S is complete separable metric than so is .()S DD is the set of coherent types; we have shown it is homeomorphic to the set of beliefs over state and opponent’s type.In words: coherency implies that i's type determines i's belief over j's type.But what about i's belief about j's belief about i's type? This needn’t be determined by i’s type if i thinks that j might not be coherent. So B-D impose “common knowledge of coherency.”Define T T ´ to be the subset of 11T T ´ where coherency is common knowledge.Proposition (Brandenburger-Dekel) : There is a homeomorphism between T and . ()S T D ´Loosely speaking, this says (a) the “universal type space is big enough” and (b) common knowledge of coherency implies that the information structure is common knowledge in an informal sense: each of i’s types can calculate j’s beliefs about i’s first-order beliefs, j’s beliefs about i’s beliefs about j’s beliefs, etc.Caveats:1) In the continuity part of the homeomorphism the argument uses the product topology on types. The drawbacks of the product topology make the homeomorphism part less important, but theisomorphism part of the theorem is independent of the topology on T.2) The space that is identified as“universal” depends on the sigma-algebra used on . Does this matter?(S T D ´)S T ×Loose ideas and conjectures…• There can’t be an isomorphism between a setX and the power set 2X , so something aboutmeasures as opposed to possibilities is being used.• The “right topology” on types looks more like the topology of uniform convergence than the product topology. (this claim isn’t meant to be obvious. the “right topology” hasn’t yet been found, and there may not be one. But Morris’ “Typical Types” suggests that something like this might be true.)•The topology of uniform convergence generates the same Borel sigma-algebra as the product topology, so maybe B-D worked with the right set of types after all.。
博弈论(第二章)讲义
纳什均衡的练习（1）
例1：囚徒困境
囚徒B
坦白
不坦白
坦白 囚徒A
不坦白
-5， -5 -8， 0
0， -8 -1， -1
纳什均衡的练习（2）
例2：智猪博弈
大猪
踩
不踩
小猪
踩 不踩
1.5， 3.5 5， 0.5
- 0.5， 6 0， 0
纳什均衡的练习（3）
例2：猜硬币的博弈
猜硬币者
正
反
正 盖硬币者
反
-1， 1 1， -1
博弈方2
U
L
R
U 博弈方1
D
1， 0 0， 3
1， 2 0， 1
0， 1 2， 0
三、划线法
其中心思想是根据博弈方策略之间的相对优劣关系，导 出博弈分析的“划线法”。
例：下图中的得益矩阵表示两博弈方的一个静态博弈，
试使用划线法进行分析。 博弈方2
左
中
右
上 博弈方1
下
1， 0 0， 4
1， 3 0， 2
二、严格下策反复消去法
（1）如果在一个博弈中，不管其它博弈方的策略如何变 化，一个博弈方的某种策略给他带来的得益，总是 比另一种策略给他带来的得益要小，那么称前一种 策略为相对于后一种策略的一个“严格下策” 。
（2）经“反复消去”博弈方的严格下策以后，每个博弈 方
可选策略都缩小为一个策略。因此，每个博弈方都 选择各自剩下的一个策略所组成的策略组合，是这 个博弈的均衡解 。
0， 1 2， 0
划线法的练习（1） 例2：囚徒困境
坦白 囚徒A
不坦白
囚徒B
坦白
不坦白
-5， -5 -8， 0
博弈论讲义2
第3章 完全信息动态博弈假设博弈的标准表述为。
如果博弈的参与人集合，全部参与人的策略空间和支付函数都是公共信息，则称G 为完全信息（Complete Information ）博弈。
本章和第2章介绍的都是完全信息博弈。
{n n u u u S S S G ,,,;,,,2121L L =}{}n ,,2,1L n S S S ,,,21L n u u u ,,,21L 第2章介绍的是只有一个阶段的博弈，即全部参与人独立地选择策略，然后实现支付，这样博弈就结束了。
在本章介绍的动态博弈中，或者同一个参与人前后有多次决策，或者不同的参与人的决策有先后顺序。
完全信息动态博弈分类为完美信息博弈和不完美信息博弈。
下面我们仅就最简单的2参与人2阶段博弈描述完全和完美信息动态博弈的定义。
定义（完全和完美信息博弈）：假设完全信息博弈的标准表述为{}2121,;,u u S S G =，博弈的行动顺序为：（1）参与人1选择可行策略11S s ∈；（2）参与人2观察到，然后选择可行策略11S s ∈22S s ∈； （3）参与人1和2实现支付()()212211,,,s s u s s u我们称这种博弈是完美信息（Perfect Information ）博弈，因为决策有严格的先后顺序，而且参与人在每次决策之前，对自己和其他参与人以前的决策都有完整的记忆。
下面我们就上述2参与人2阶段博弈介绍求解动态博弈常用的倒推法（Backward Induction ）。
在第（2）阶段，给定参与人1的策略11S s ∈，参与人2的最优策略反应为：()(){}222121*2:,max arg S s s s u s s ∈=在第（1）阶段，理性的参与人1可以准确地预测到参与人2的最优策略反应，于是参与人1的最优策略选择就是：()1*2s s ()[]{}111*211*1:,max arg S s s s s u s ∈= 这样在第（2）阶段，参与人2的最优策略就是：()*1*2*2s s s =这样得到的互为最优策略反应组合()()21*2*1,,S S s s ∈称为博弈G 的Nash 均衡，这种均衡也称为倒推法均衡（Backward Induction Outcome ）。
博弈论讲义完整PPT课件
如果两个企业联合起来形成卡特尔，选择垄断利润最大化的产量，每 个企业都可以得到更多的利润。给定对方遵守协议的情况下，每个企业都 想增加产量，结果是，每个企业都只得到纳什均衡产量的利润，它严格小 于卡特而产量下的利润。
• 请举几个囚徒困境的例子
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第一章 导论-囚徒困境
知识：完全信息博弈和不完全信息博弈。 ❖完全信息：每一个参与人对所有其他参与人的（对手）的特征、
战略空间及支付函数有准确的 知识，否则为不完全信息。
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第一章 导论-基本概念
• 博弈的划分：
行动顺序 信息
完全信息
静态
完全信息静态博弈 纳什均衡
纳什（1950，1951）
不完全信息
不完全信息静态博弈 贝叶斯纳什均衡
0，300 0，300
纳什均衡：进入，默许；不进入，斗争
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第一章 导论
• 人生是永不停歇的博弈过程，博弈意略达到合意的结果。 • 作为博弈者，最佳策略是最大限度地利用游戏规则，最
大化自己的利益； • 作为社会最佳策略，是通过规则使社会整体福利增加。
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第一章 导论-基本概念
一只河蚌正张开壳晒太阳，不料，飞 来了一只鸟，张嘴去啄他的肉，河蚌连忙合 起两张壳，紧紧钳住鸟的嘴巴，鸟说：“今 天不下雨，明天不下雨，就会有死蚌肉。” 河蚌说：“今天不放你，明天不放你，就会 有死鸟。”谁也不肯松口，有一个渔夫看见 了，便过来把他们一起捉走了。
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第一章 导论-囚徒困境
✓“要害”是否在于“利己主义”即“个人理
性”？
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博弈论导论 2
图 2－5 军备竞赛
思考：现实生活中还有哪些情况属于囚徒困境？ 练习：将团队生产问题模型化成囚徒困境；如何理解囚徒困境与“看不见的手”之间 的矛盾？
2.1.5 走出囚徒困境
从社会福利的角度讲，囚徒困境不是帕累托最优的，但这与理性人的假设并不矛盾。
① ②
这实际上是 Betrand 价格竞争模型。 这是 Hardin（1968）发表在 Science 上但是被经济学引用最多的例子。但是，最近有学者提出了“反公地 悲剧”理论。董志强（2007）启发我使用这个简单的收益矩阵而非复杂的数学模型。 白鲨在线 2
2.3.2 性别战
如图 2－12。两个博弈相同的地方在于：（1）存在多重均衡，而且双方各自偏向一个 均衡；（2）任何一个均衡结果都是帕累托最优的。信念扮演了重要的作用。在这个博弈中， 假设男方是一个有名的拳击手，而女方也知道这点，那么（拳击，拳击）应该是一个均衡结 果，而（芭蕾，拳击）不应该出现。
白鲨在线 5
2.3.4 协调博弈
如图 2－14，史密斯公司和琼斯公司独立地决定选择何种智能手机操作系统。若两家公 司选择同样的操作系统，销售会更好。 特征：存在多重均衡，但是一些均衡帕累托优于另一些均衡，这与性别战和斗鸡博弈 都不同。 提示：一定要注意不同博弈模型的结构性特征，而不是过于关注具体数字。 思考：现实生活中有哪些博弈是性别战、斗鸡博弈和协调博弈？
图 2－1 双边优势
图 2－2 单边优势
2.1.2 定义优势策略均衡
并且，我们有 命题：如果一个博弈 N ,{Si }i 1 ,{vi ()}i 1 存在优势策略均衡 s ，那么 s 就是惟一的 优势策略均衡，并且也是惟一的纳什均衡。 证明过程略（可做思考题或作业）。
白鲨在线 1
博弈论讲义2
尽管许多博弈中重复剔除的占优均衡是一个合理 的预测，但并不总是如此，尤其是大概支付是某 些极端值的时候。
参与人B
L
参与人A
R -1000，9
U
8，10
D
7， 6
6， 5
U是A的最优选择，但是，只要有1/1000的概率B选R， A就会选D
14
斗鸡博弈
进 A 独木桥 纳什均衡：A进，B退；A退，B进 对于相当多的博弈，我们无法运用重复剔除劣战略的 方法找出均衡解。
1、Cournot Model of Duopoly
按竞争程度划分的市场类型（就卖方来说）：
A 完全竞争市场 B 寡头竞争市场 C 独家垄断市场
29
市场类型不同，厂商之间行为特征不同，A与C 类型中，厂商的决策都是个体优化决策，而B类 型中寡头垄断竞争的本质就构成博弈，他们都 是理性的决策者，他们的行为既影响自身，又 影响对方。尽管两寡头由于垄断能给他们带来 一些共同的利益，但是他们的根本利益并不是 完全一致的。如果两寡头之间可以签定有约束 力的协议，彼此之间达成合作，形成完全垄断， 此时的博弈是一种合作博弈。然而在大多数情 况下，彼此之间很难达成有约束力的协议，这 样就是非合作博弈。
7
注意：
与占优战略均衡中的占优战略和劣战略不同，
这里的占优战略或劣战略可能只是相对于另一个
特定战略而言。
8
案例1-智猪博弈
小猪 按 大猪 按 5，1 等待 9，-1 等待 4，4 4大于1
0，0
0大于-1
按是小猪的严格 劣战略-剔除 “按”是大猪的占优战略，纳什均衡：大猪按，小猪等待
9
案例2
U 行先生
s * 是一个纳什均衡： 或者用另一种表达方式： 当且仅当 si* 是下述最大化问题的解时，
博弈论(二)—讲义
9.2 完全信息静态博弈9.2.1 博弈的战略式表述Definition A normal (strategic) form game G consists of: (1) a finite set of agent s . {1,2,,}D n = (2) strategy sets .12,,,n S S S (3) payoff functions . 12:(1,2,,)i n u S S S R i n ⨯⨯⨯→=囚徒B囚徒A完全信息静态博弈是一种最简单的博弈，在这种博弈中，战略和行动是一回事。
博弈分析的目的是预测博弈的均衡结果，即给定每个参与人都是理性的，什么是每个参与人的最优战略？什么是所有参与人的最优战略组合？纳什均衡是完全信息静态博弈解的一般概念，也是所有其他类型博弈解的基本要求。
下面，我们先讨论纳什均衡的特殊情况，然后讨论其一般概念。
9.2.2 占优战略（Dominated Strategies ）均衡一般说来，由于每个参与人的效用（支付）是博弈中所有参与人的战略的函数，因此，每个参与人的最优战略选择依赖于所有其他参与人的战略选择。
但是在一些特殊的博弈中，一个参与人的最优战略可能并不依赖于其他参与人的战略选择。
也就是说，不管其他参与人选择什么战略，他的最优战略是唯一的，这样的最优战略被称为“占优战略”。
Definition Strategy s i is strictly dominated for player i if there is some such that i i s S '∈ for al .(,)(,)i i i i i i u s s u s s --'>i i s S --∈Proposition a rational player will not play a strictly dominated strategy.抵赖 is a dominated strategy. A rational player would therefore never 抵赖. This solves the game since every player will 坦白. Notice that I don't have to know anything about the other player . 囚徒困境：个人理性与集体理性之间的矛盾。
博弈论讲义2
三 重复剔除的占优均衡
重复剔除严格劣策略：
思路：首先找到某个参与人的劣策略（假定存 在），把这个劣策略剔除掉，重新构造一个不包 含已剔除策略的新的博弈，然后再剔除这个新的 博弈中的某个参与人的劣策略，一直重复这个过 程，直到只剩下唯一的策略组合为止。 这个唯一剩下的策略组合就是这个博弈的均衡 解，称为“重复剔除的占优均衡”。
独木桥
进
A
退
B
进退 -3，-3 2，0
0，2 0，0
纳什均衡：A进，B退；A退，B进
斗鸡博弈
村子里有两户富户，有两种可能：一家修，另 一家就不修；一家不修，另一家就得修。
冷战期间美苏抢占地盘：一方抢占一块地盘， 另一方就占另一块。
夫妻吵架，一方厉害，另一方就出去躲躲。
注意：在混合策略纳什均衡条件下，也可能两 败俱伤。
注意： 如果所有人都有（严格）占优策略存在，
那么占优策略均衡就是可以预测的唯一 均衡。 占优策略只要求每个参与人是理性的， 而不要求每个参与人知道其他参与人是 理性的（也就是说，不要求理性是共同 知识）。为什么？
二 占优策略均衡
案例-囚徒困境
囚徒A
囚徒 B
坦白
坦白 -8，-8
抵赖
0，-10 -8大于-10
相安无事；第二天，相安无事……；直到第100天 ，突然，每个妻子都把丈夫杀了。为什么会这样？
这是一个推理和行动的过程。如果她的丈夫不忠的话，她就杀 死他；如果没有证据证明她的丈夫不忠的话，她便相信他，不 杀死他。
如果村里只有一个男人是不忠的话，在老太太作了宣布之
后的第一天，这个男人的妻子在老太太宣布之后马上就能知道
两只猪一起去按，然后一起回槽边进食， 由于大猪吃得快可吃下8个单位的食物， 小猪只能吃到2个单位食物。
张维迎《博弈论与信息经济学》讲义-第02章-纳什均衡与一致预期
最优选择
这个博弈只要求一阶理性共识就可以预测均衡 结果: 如果R相信C是理性的,R就知道C不会选择C3, 所以R的最优选择是R1; R R1 如果C相信R是理性的,C就知道R不会选择R2, 所以C的最优选择是C2. 但要C预期R不会选择R3,需要二阶理性共识; 要R不预期C会选择C1,需要三阶理性共识.
– – – – 如果R(b)C 选择C2, 如果R(b)C(b)R会选择R2; 如果R(b)C(b)R(b)C会选择C1; 如果R(b)C(b)R(b)C(b)R会选择R1
Consistently aligned beliefs (CAB)
考虑(R3,C3):对方不会犯预期错误:R选 择R3,如果他认为C会选择C3;C会选择C3, 如果他认为R会选择R3. CAB CAB:每个人对别人行为的预期(信念)是正 确的; Harsanyi doctrine: 如果两个理性的人具有相同 的信息,他们一定会得出相同的推断和相同的 结论; Robert Aumann: rational agents cannot agree to disagree.
重复剔除与理性共识
重复剔除不仅要求每个人是理性的,而且要求每个人 知道其他人是理性的,每个人知道每个人知道每个人 是理性的,如此等等,即理性是"共同知识"(共识) C1 R1 R2 R3 10,4 9, 9 1,98 C2 1, 5 0, 3 0,100 C3 98,4 99,8 100,98 这个博弈只要求 一阶理性共识就 可以预测均衡结 果. 如果把(下-左) 的第一个数字改为 11呢?
纳什均衡与一致预期
张维迎 教授 北京大学光华管理学院
博弈的基本概念(1)
参与人(players):博弈中决策主体的集合:什 么人参与博弈?每个人是什么角色? 行动(actions): 每个人有些什么样行动可以选 择?在什么时候行动? 信息(information):在博弈中的知识;每个人 知道些什么(包括特征,行动等)? 战略(strategies):行动计划;每个人有什么战 略可供选择?战略的完备性;
博弈论讲义2 非合作博弈理论
开发商A
开发 4000，4000 不开发 0，8000
8000，0 0，0
需求小的情况 开发商A
开发商B 开发 不开发
开发 -3000，-3000 1000，0
不开发 0，1000
0，0
博弈的战略式表述
一 、博弈的基本概念及战略表述
博弈论的基本概念包括： ✓ 参与人：博弈论中选择行动以最大化自己效用的决策主体； ✓ 行动：参与人的决策变量 ✓ 战略：参与人选择行动的规则 ✓ 信息：参与人在博弈中的知识，特别是有关其他参与人的特征和
一 、博弈的基本概念及战略表述
需求大的情况
开发商B 开发 不开发
开发商A
开发 4000，4000 不开发 0，8000
8000，0 0，0
需求小的情况 Βιβλιοθήκη 发商A开发商B 开发 不开发
开发 -3000，-3000 1000，0
不开发 0，1000
0，0
博弈的战略式表述
不完全信息动态博弈-精练贝叶斯纳什均衡 泽尔腾（1965）
一 、博弈的基本概念及战略表述
博弈的战略式表述：
战略式表述给出： 1、博弈的参与人集合：i , (1,2, , n); 2、每个参与人的战略空间：Si，i 1,2, , n; 3、每个参与人的支付函数：ui (s1, , si , , sn ),i 1,2, , n)
用G S1， ，Sn;u1, , un代表战略式表述博弈。
✓ 完美信息：指一个参与人对其他参与人（包括“自然”）的 行动选择有准确了解的情况，即每一个信息集只包含一个值。
✓ 完全信息：指自然不首先行动或自然的行动的初始行动所有 参与人观察到的情况。
✓ 共同知识：指“所有参与人知道所有参与人知道所有参与人 知道….”的知识。
第二讲-新制度经济学与博弈论
莱茵哈德·泽尔滕 阿尔钦
博弈论概述
演进博弈论的发展
➢ 演进博弈论的推广和发展，应当归功于史密斯和普瑞斯， 他们通过对动物冲突逻辑的研究，提出了演进博弈论中的 基本概念——进化稳定策略（ESS） 。
➢博弈论冲击着新古典经济学的思维方式，也正引 导着人们彻底的重新认识经济世界和人们的经济 行为。
新制度经济学与博弈论的关系
新古典经济学与博弈论的比较比ຫໍສະໝຸດ 项目新古典经济学博弈论
研究对象 研究假设
价格制度 完全竞争/完全信息
主体之间行动决策以及在相 互影响下个人如何决策
非对称信息
个人决策依据
实践性
既无需考虑自己对别人选 择的影响，也无须考虑别 人选择对自己决策的影响
讲义结构
• 学习目的和要求 • 博弈论概述 • 新制度经济学与博弈论的关系 • 博弈论在制度分析中的应用
第二节 新制度经济学与博弈论的关系
博弈论与新古典经济学
➢博弈论对新古典经济学的冲击
➢自从博弈论被引入经济学以来，现代经济学的许 多领域都发生了巨大变化，诸多按新古典经济学 思想写作出版的教材纷纷被作者利用博弈理论来 重新演绎。
➢怀疑完全理性假设 ➢用传统博弈理论预测结果去揭示现实经济问题，
有可能起误导作用 ➢当参与者1对1谈判博弈分配金额时，对社会公平
性这一概念的理解方式在很大程度上影响着分配 结果。 ➢人们心灵深处存在着某种公平性的观念，这实际 上是人们在历史上无数次博弈中形成的规则，即 影响人们行为的习惯、文化、习俗或制度，它们 指导着人们在各种战略性状况中应该如何行动。
《博弈论与信息经济学讲义》第2章 纳什均衡与一致预期
• 但要C预期R不会选择R3，需要二阶理性共识； 要R不预期C会选择C1，需要三阶理性共识。
R排除C选择C1
R believes C believes R believes C is rational
(C1,C2) R1 C2
• 例如：R理性化选择R1：
– 如果R(b)C 选择C2， – 如果R（b）C（b）R会选择R2； – 如果R（b）C（b）R（b）C会选择C1； – 如果R（b）C（b）R（b）C（b）R会选择R1
Consistently aligned beliefs
(CAB)
• 考虑（R3，C3）：对方不会犯预期错误：R选 择R3，如果他认为C会选择C3；C会选择C3， 如果他认为R会选择R3。
• CAB：每个人对别人行为的预期（信念）是正 确的；
• Harsanyi doctrine: 如果两个理性的人具有相同 的信息，他们一定会得出相同的推断和相同的 结论;
• Robert Aumann: rational agents cannot agree to disagree.
纳什均衡与一致预期
• 如果这样的解存在，我们说该博弈是“重复剔除占优 可解的”(iterated dominance solvable).
理性共识
（common knowledge of rationality)
• (1)Zero-order CKR: 每个人都是理性的，但不知 道其他人是否是理性的；
• (2)first-order CKR: 每个人是理性的，并且知道 其他每个人也都是理性的，但并不知道其他人 是否知道自己是理性的；
所有权配置与等级结构
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9.2 完全信息静态博弈9.2.1 博弈的战略式表述Definition A normal (strategic) form game G consists of: (1) a finite set of agent s {1,2,,}D n = . (2) strategy sets 12,,,n S S S .(3) payoff functions 12:(1,2,,)i n u S S S R i n ⨯⨯⨯→= .囚徒B囚徒A完全信息静态博弈是一种最简单的博弈，在这种博弈中，战略和行动是一回事。
博弈分析的目的是预测博弈的均衡结果，即给定每个参与人都是理性的，什么是每个参与人的最优战略？什么是所有参与人的最优战略组合？纳什均衡是完全信息静态博弈解的一般概念，也是所有其他类型博弈解的基本要求。
下面，我们先讨论纳什均衡的特殊情况，然后讨论其一般概念。
9.2.2 占优战略（Dominated Strategies ）均衡一般说来，由于每个参与人的效用（支付）是博弈中所有参与人的战略的函数，因此，每个参与人的最优战略选择依赖于所有其他参与人的战略选择。
但是在一些特殊的博弈中，一个参与人的最优战略可能并不依赖于其他参与人的战略选择。
也就是说，不管其他参与人选择什么战略，他的最优战略是唯一的，这样的最优战略被称为“占优战略”。
Definition Strategy s i is strictly dominated for player i if there is some i i s S '∈ such that (,)(,)i i i i i i u s s u s s --'> for al i i s S --∈.Proposition a rational player will not play a strictly dominated strategy.抵赖 is a dominated strategy. A rational player would therefore never 抵赖. This solves the game since every player will 坦白. Notice that I don't have to know anything about the other player . 囚徒困境：个人理性与集体理性之间的矛盾。
This result highlights the value of commitment in the Prisoner's dilemma – commitment consists of credibly playing strategy 抵赖.囚徒困境的广泛应用：军备竞赛、卡特尔、公共品的供给。
9.2.3 Iterated Deletion of Dominated Strategies (重复剔除劣战略)智猪博弈（boxed pigs ）小猪大猪按 等待此博弈没有占优战略均衡。
因为尽管“等待”是小猪的占优战略，但是大猪没有占优战略。
大猪的最优战略依赖于小猪的占略: ---。
大猪会正确地预测到小猪会选择“等待”；给定此预测，大猪的最优选择只能是“按”。
这样，（按，等待）就是唯一的均衡。
重复剔除的占优均衡：先剔除某个参与人的劣策略，重新构造新的博弈，再剔除，---。
应用：大股东监督经理，小股东搭便车；大企业研发，小企业模仿。
9.2.4 Nash equilibrium性别战博弈（battle of the sexes ）:女男足球赛 演唱会在上面的博弈中，两个参与者都没有占优策略，每个参与者的最优策略都依赖于另一个参与人的战略。
所以，没有重复剔除的占优均衡。
Definition A strategy profile s * is a pure strategy Nash equilibrium of G if and only if***(,)(,)i i i i i i u s s u s s --≥ for all players i and all i i s S ∈求解Nash 均衡的方法。
A Nash equilibrium captures the idea of equilibrium : Both players know what strategy the other player is going to choose, and no player has an incentive to deviate from equilibrium play because her strategy is a best response to her belief about the other player's strategy.对纳什均衡的理解：设想所有参与者在博弈之前达成一个（没有约束力的）协议，规定每个参与人选择一个特定的战略。
那么，给定其他参与人都遵守此协议，是否有人不愿意遵守此协议？如果没有参与人有积极性单方面背离此协议，我们说这个协议是可以自动实施的（self-enforcing ），这个协议就构成一个纳什均衡。
否则，它就不是一个纳什均衡。
问题：纳什均衡与重复剔除（严格）劣战略均衡之间的关系。
9.2.5 Cournot Competition （古诺竞争）This game has an infinite strategy space .Two firms choose output levels q i ，cost function c i (q i ) = cq i .market demand determines a price 1212()()p f q q q q αβ=+=-+：the products of both firms are perfect substitutes, i.e. they are homogenous products. D = {1; 2} S 1 = S 2 = R +u 1 (q 1, q 2) = q 1 f (q 1 + q 2) -c 1 (q 1) u 2 (q 1, q 2) = q 2 f (q 1 + q 2) - c 2 (q 2)the 'best-response' function BR (q j ) of each firm i to the quantity choice q j of the other firm: 由11121[()]q q q cq παβ=-+-，得FOC: 21211()0 22q c q q q c q ααβββ--+--=⇒=-；又120 cq q αβ-≥⇒≤。
因 , ()220, otherwise j j i j q c c if q BR q ααββ⎧---≤⎪=⎨⎪⎩q 1q 2BR 1(q 2)BR 2(q 1)(q 1*,q 2* )()(2)c αβ-()c αβ-The best-response function is decreasing in my belief of the other firm's action.Using our new result it is easy to see that the unique Nash equilibrium of the Cournot game is the intersection of the two BR functions.Because of symmetry we know that q 1 = q 2 = q*.Hence we obtain **2c q q αβ-=-, This gives us the solution 2()*3c q αβ-=.问题：将寡头竞争的古诺均衡与垄断企业的最优产量和利润进行比较。
9.2.6 Bertrand Competition （伯特兰竞争）Firms compete in a homogenous product market but they set prices . Consumers buy from the lowest cost firm. demand curve q = D (p )Therefore, each firm faces demand12() (,)()2 = 0i i j i i i j i j D p if p p D p p D p if p p if p p ⎧<⎪=⎨⎪>⎩We also assume that D (c ) > 0, i.e. firms can sell a positive quantity if they price at marginal cost.Lemma The Bertrand game has the unique NE **12(,)p p = (c; c ).Proof: First we must show that (c,c) is a NE . It is easy to see that each firm makes zero profits. Deviating to a price below c would cause losses to the deviating firm. If any firm sets a higher price it does not sell any output and also makes zero profits. Therefore, there is no incentive to deviate.To show uniqueness we must show that any other strategy profile (p 1; p 2) is not a NE . It's easiest to distinguish lots of cases.Case I: p 1 < c or p 2 < c. In this case one (or both players) makes negative losses. This player should set a price above his rival's price and cut his losses by not selling any output.Case II: c < p 1 < p 2 or c < p 2 < p 1. In this case the firm with the higher price makes zero profits. It could profitably deviate by setting a price equal to the rival's price and thus capture at least half of his market, and make strictly positive profits.Case III: c = p 1 < p 2 or c = p 2 < p 1. Now the lower price firm can charge a price slightly above marginal cos t (but still below the price of the rival) and make strictly positive profits.Case IV: c < p 1 = p 2. Firm 1 could profitably deviate by setting a price 12p p c ε=->. The firm's profits before and after the deviation are:22()()2B D p p πε=- 22()()A D p p c πεε=---Note that the demand function is decreasing, so 22()()D p D p ε->. We can therefore deduce:222()()()2A B D p p c D p πππεε∆=->--- This expression (the gain from deviating) is strictly positive for sufficiently small ε.Therefore, (p 1; p 2) cannot be a NE.9.2.7 Mixed Strategies （混合战略）猜谜游戏 matching pennies game:儿童BH(正面) T(反面) 儿童AH(正面)T(反面)每一个参与者都想猜透对方的战略，而又不能让对方猜透自己的战略。