gametheory2博弈论英文精品PPT课件
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.。
吉本斯博弈论2课件
■ 与x0相邻的节点是x0的后 续节 （successors ）. x0的后续节点 是x1, x2
■ 对任何两个相邻的节点来说, 与 根相连接的路径更长的那个节点 是另一个节点的后续节.
■ 例3: x7 是x3的后续节点, 因为它
们相邻, 而且x7到 x0的路径比x3
到x0的路径更长
x4
x0 x1
x5
-1, 1
1 , -1
TT 1 , -1 -1, 1
Game theory-Chapter 2
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Nash equilibrium
■完全信息动态博弈中的纳什均衡集（the set of Nash equilibrium）就是它的标准式的纳什均衡 集合.
Game theory-Chapter 2
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弈可能的终点
x4
■ 例4: x4, x5, x6, x7, x8 都是终点 节
x0 x1
x5 x7
x2 x3
x6 x8
Game theory-Chapter 2
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Game tree
■ 除终点节以外的任何节 点都代表了某个参与人.
■ 对于终点节以外的任意 后节续点节来的说边, 连缘接代它表和了它这的Player 2 个节点所代表的参与人 H 可能采取的行动
A
F
1, 2
A
F
2, 1
0, 0
2, 1
0, 0
Accommodate is the Nash equilibrium in this subgame.
Game theory-Chapter 2
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Find subgame perfect Nash equilibria backward induction
博弈论完整版PPT课件
2-阶理性： C相信R相信C是理性的，C会将R4从R的战略空间中剔除, 所以 C不会选择C1；
3-阶理性： R相信C相信R相信C是理性的， R会将C1从C的战略空间中剔 除, R不会选择R1；
基本假设：完全竞争，完美信息
个人决策是在给定一个价格参数和收入的条 件下最大化自己的效用，个人的效用与其他人 无涉，所有其他人的行为都被总结在“价格”参数 之中
一般均衡理论是整个经济学的理论基石 和道义基础，市场机制是完美的，帕累托 最优成立，平等与效率可以兼顾。
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然而在以下情况,上述结论不成立：
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理性共识
0-阶理性共识：每个人都是理性的，但不知道其 他人是否是理性的；
1-阶理性共识：每个人都是理性的，并且知道其 他人也是理性的，但不知道其他人是否知道自己 是理性的；
2-阶理性共识：每个人都是理性的，并且知道其
他人也是理性的，同时知道其他人也知道自己是
理性的；但不知道其他人是否知道自己知道他们
如果你预期我会选择X，我就真的会选择X。
如果参与人事前达成一个协议，在不存在外部强 制的情况下，每个人都有积极性遵守这个协议，这 个协议就是纳什均衡。
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应用1——古诺的双寡头垄断模型(1938)
假定：
只有两个厂商 面对相同的线形需求曲线，P(Q)=a－Q， Q=q1+q2 两厂商同时做决策； 假定成本函数为C(qi)＝ciqi
劣策略：如果一个博弈中，某个参与人有占优策略，那么
该参与人的其他可选择策略就被称为“劣策略”。
波恩大学博弈论 讲义 GameT-2
Game Theory Lecture 2A reminderLet G be a finite, two player game of perfect information without chance moves. Theorem (Zermelo, 1913):Either player 1 can force an outcome in T or player 2 can force an outcome in T’A reminder Zermelo’s proof uses Backwards InductionA reminderA game G is strictly Competitive if for any twoterminal nodes a,bab b 2a1An application of Zermelo’s theorem toStrictly Competitive GamesLet a 1,a 2,….a n be the terminal nodes of a strictly competitive game (with no chance moves and with perfectinformation) and let:a n 1a n-1 1 …. 1a 2 1a 1(i.e. a n 2a n-1 2 …. 2a 2 2a 1).?Then there exists k ,n k 1 s.t. player 1 can forcean outcome in a n , a n-1……a kAnd player 2can force an outcome in a k , a k-1……a 1?a n 1a n-1 1.. 1 a k 1 .. 1a 2 1a 1G(s,t)=a kPlayer 1has a strategy swhich forces an outcomebetter or equal to a k ( 1)Player 2has a strategy twhich forces an outcomebetter or equal to a k ( 2)Let w j= a n, a n-1……,a j wn+1=an , an-1…aj…, a2, a1w1w2w jwn+1wnPlayer 1can force an outcome in W 1 = a n , a n-1…,a 1 ,and cannot force an outcome in w n+1= .Let w j = a n , a n-1……,a j w n+1= w 1, w 2, ….w n ,w n+1can force cannot forcecan force ??Let k be the maximal integer s.t. player 1can force an outcome in W kProof :w 1, w 2, … wk , w k+1...,w n+1Player 1 can force Player 1 cannot forceLet k be the maximal integer s.t. player 1can force an outcome in W ka n , a n-1…a k+1 ,a k …, a 2, a 1 w 1w k+1w k Player 2can force an outcome in T -w k+1by Zermelo’s theorem!!!!!a n 1a n-1 1.. 1 a k 1 .. 1a 2 1a 1G(s,t)=a k Player 1has a strategy s which forces an outcome better or equal to a k ( 1)Player 2has a strategy t which forces an outcome better or equal to a k ( 2)Now consider the implications of this result for thestrategic form game s ta kplayer 1’s strategy s guarantees at least a kplayer 2’s strategy t guarantees him at least a k------+++++i.e.at most a k for player 1stak------+++++The point (s,t)is a Saddle pointstak------+++++Given that player 2plays t,Player 1hasno better strategythan sstrategy s is player 1’sbest responseto player 2’s strategy tSimilarly, strategy t is player 2’sbest responseA pair of strategies (s,t)such that eachis a best response to the other isa Nash EquilibriumJohn F. Nash Jr.This definition holds for any game,not only for strict competitive ones12 221WW LWWrlRML121 2Example3R LRrL lbackwards Induction(Zermelo)r( l , r ) ( R, , )2 221WW LWWrlRML1223RLRrLl rAll thosestrategy pairs areNash equilibriaBut there are otherNash equilibria …….( l , r ) ( L,( l , r ) ( L, , )2 221WW LWWrlRML1223RLRrLl rThe strategies obtained bybackwards inductionAre Sub-Game Perfect equilibriain each sub-game they prescribe2 221WW LWWrlRML1223RLRrLl rWhereas, thenon Sub-Game PerfectNash equilibriumprescribes a non equilibrium( l , r ) ( L, , )A Sub-Game Perfect equilibriaprescribes a Nash equilibriumin each sub-gameR. SeltenAwarding the Nobel Prize in Economics -1994Chance MovesNature (player 0),chooses randomly, with known probabilities, among some actions.+ + + = 11/61111111/6123456information setN.S .S .N.S .S .N.S .S .N.S .S .N.S .S .N.S .S .Payoffs:W (when the other dies, or when the other chosenot shoot in his turn)D (when not shooting)L (when dead)1/61111111/6123456N.S .S .N.S .S .N.S .S .N.S .S .N.S .S .N.S .S .Payoffs:W (when the other dies, or when the other didnot shoot in his turn)D (when not shooting)L (when dead)W D L1/61111111/6123456N.S .S .N.S .S .N.S .S .N.S .S .N.S .S .N.S .S .DDDDDDL222221/61111111/6123456N.S .S .N.S .S .N.S .S .N.S .S .N.S .S .N.S .S .DDDDDDL22222N.S .S .DLN.S .S .DN.S .S .DN.S .S .DN.S .S .D。
博弈论最全完整-讲解PPT课件
王则柯、李杰编著，《博弈论教程》，中国人民大学 出版社，2004年版。
艾里克.拉斯缪森（Eric Rasmusen）著，《博弈与信 息：博弈论概论》，北京大学出版社，2003年版。
因内思·马可-斯达德勒,J.大卫·佩雷斯-卡斯特里罗著， 《信息经济学引论：激励与合约》,上海财经大学出版 社，2004年版。
常和博弈也是利益对抗程度最高的博弈。 非常和（变和）博弈蕴含双赢或多赢。
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导论
四、主要参考文献
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张维迎著，《博弈论与信息经济学》，上海三联书店、 上海人民出版社，1996年版。
Roger B. Myerson著：Game Theory（原文版、译文 版），中国经济出版社，2001年版。
是关于动态博弈进行过程之中面临决策 或者行动的参与人对于博弈进行迄今的 历史是否清楚的一种刻划。
如果在博弈进行过程中的每一时刻，面 临决策或者行动的参与人，对于博弈进 行到这个时刻为止所有参与人曾经采取 的决策或者行动完全清楚，则称为完美 信息博弈；否则位不完美信息。
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零和博弈与非零和博弈
了解自己行动的限制和约束，然后以精心策划的方式 选择自己的行为，按照自己的标准做到最好。 • 博弈论对理性的行为又从新的角度赋予其新的含义— —与其他同样具有理性的决策者进行相互作用。 • 博弈论是关于相互作用情况下的理性行为的科学。
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4
如何在博弈中获胜？
…… 真的能在博弈中（总是）获 胜吗？
对手和你一样聪明！ 许多博弈相当复杂，博弈论并不
施锡铨编著，《博弈论》上海财大出版社，2000年版。
谢识予编著，《经济博弈论》，复旦大学出版社， 2002年版。
谢识予主编，《经济博弈论习题指南》，复旦大学出 版社，2003年版。
game theory lecture2
• we cab use the common knowledge of rationality to • build an iterative process that takes this reasoning to the limit. • After employing this reasoning one time, we can eliminate all the strategies that are never a best response, resulting in a possibly “smaller” reduced game that includes only strategies that can be a best response in the original game. • Then we can employ this reasoning again and again, in a similar way that we did for IESDS, in order to eliminate outcomes that should not be played by players who share a common knowledge of rationality
A cournot Duopoly Game
• Two identical firms, players 1 and 2, produce some good. • Let the demand be given by p(q) = 100 − q, where q = q1+ q2. • Assume that there are no fixed costs of production, the firms have linear cost for producing quantity qi is given by ci(qi) =10qi for i ∈ {1, 2}. • What is the production plans of the firms?
game2PPT课件
大户与散户的博弈模型
大户|散户 分析并进入 跟随大户进入
分析并进入
0.7p-c， 0.3p-c
跟随大户进入 0.7p， 0.3pc
0.7p-c ， 0.3p
0，0
低价位情况下庄家之间的博弈
庄家甲（在位者）| 庄家乙（进入者）
默许
斗争
进入 3，2 0，-2
不进入 5，0 5，0
高价位情况下庄家之间的博弈
泽尔腾（1975） Krep和swilson（1982） Fudenberg和Tirole（1991）
不同均衡概念之间的关系
纳什均衡 子博弈完美均衡
精练Bayes均衡 序贯均衡 颤抖手均衡
第一部分 完全信息静态博弈
第一章 策略型博弈与Nash均衡
1. 博弈的正则型
• prisoner’s dilemma：简单博弈的老祖宗。 最早由Melvin Dresher 和 Al Tucker 于1953 年在兰德公司（Rand Corporation，博弈论 许多早期工作的丰产地）进行分析。
L
M
U
4，3
5，1
M
2，1
8，4
D
3，0
9，6
R 6，2 3，6 2，8
1/2
L
U
4，3
M
2，1
D
3，0
R 6，2 3，6 2，8
1/2
L
U
4，3
R 6，2
• 合理，符合逻辑的过程，得到累次 严优的解为：
1/2
L
U
4，3
• 累次严优的局限性
累次严优的应用
• 囚徒困境
甲/乙
坦白
抗拒
坦白
0,0
GAME THEORY MODELS(博弈模型) 尼科尔森中级微观ppt
• Each player has the ability to choose among a set of possible actions • The specific identity of the players is irrelevant
• This means that A will also choose to play music loudly • The A:L,B:L strategy choice obeys the criterion for a Nash equilibrium
• because L is a dominant strategy for B, it is the best choice no matter what A does • if A knows that B will follow his best strategy, then L is the best choice for A
The Prisoners’ Dilemma
• The most famous two-person game with an undesirable Nash equilibrium outcome
• games in which the strategies chosen by A and B are alternate levels of a single continuous variable • games where players use mixed strategies
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Existence of Nash Equilibria
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博弈论game theory ppt课件
按产品性质：
纯粹寡头垄断
P=f（Q1+Q2）
差别寡头垄断
P1=f（Q1，Q2） P2=f（Q2，Q1）
按决策变量 ：
联合定产模型（Cournot） 联合定价模型（Bertrand）
Cournot模型的假定：同时决策;决策变量是产量;对手 的反应方式保持不变;产品相同,线性需求曲线,MC=0。
ppt课件
智猪博弈（剔除博弈）
大猪 小猪 按钮 不按
按钮 4，8 －4，20 不按 10，6 0，0
ppt课件
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2.1.3划线法
A 坦 白不 坦 白 B
坦
白 -8，-8 0，-10
不 坦 白 -10，0 -1，-1
猜硬币者
正 盖硬币者
面反 面
正
面 -1，1 1，-1
反
面 1，-1 -1，1
注A：并非所有的博弈均有稳定的解。如右图所示抛硬币博弈
R2 q1
q1
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2.3.4公地的悲剧 （1968年，哈丁） 外部性往往是产权界定不清的结果 一个乡村，村民在公地上放牛。两种放牧机制： （1）让私人拥有这块土地；私人决定放牧规模 （2）让村民共同拥有这块地免费放牧没有限制 结论：公共牧地一定是过度放牧。 例子：土地承包责任制，永佃权
ui qiV Q qiC
A坦 B
白不 坦 白
坦
白 -8，-8
0，-10
不 坦 白 -10，0
-1，-1
(2)扩展型——博弈树 由棱和节点构成
outcome
B
A
root
ppt课件
5
§1.2一些典型博弈
1.2.1 Tucker的囚徒困境
B
博弈论概述.ppt
Glossary
Payoff支付
A payoff is a number, also called utility, that reflects the desirability of an outcome to a player, for whatever reason. When the outcome is random, payoffs are usually weighted with their probabilities. The expected payoff incorporates the play’s attitude towards risk.
Glossary
Mixed strategy混合战略 A mixed strategy is an active randomization,
with given probabilities, that determines the player’s decision. As a special case, a mixed strategy can be the deterministic choice of one of the given pure strategies.
know it, and know that they all know it, and so on. The structure of the game is often assumed to be common knowledge among the players.
如果一个事实被所有的参与人知道，并且每个参与 人都知道所有的人都知道，并且每个参与人都知道每 个参与人都知道所有的人都知道，如此等等，以致无 穷，那么，这个事实就是共同知识。
博弈论完整课件[浙江大学]GAME_Cha(2)
![博弈论完整课件[浙江大学]GAME_Cha(2)](https://img.taocdn.com/s3/m/1115d0800b1c59eef9c7b41e.png)
一、有限次重复博弈 （以two-stage repeated games为例）
（一）阶段博弈只有唯一NE
考虑曾经给出的囚徒困境的标准式（回忆并 画出支付矩阵）。
假设两个参与者要把这样一个同时行动博弈重 复两次，并且在第二次博弈开始之前可观测第一 次的结果。再假设整个过程博弈的收益等于两阶 段各自收益的简单相加（即不考虑贴现因素）， 我们称这一重复进行的博弈为两阶段囚徒困境。
一方的机会主义行为将触发其他参与人策略中 的惩罚机制发生。
触发策略(Trigger strategies):我们把这种包含 着奖励和惩罚机制的策略称为触发策略。正是 由于害怕“触发”其他参与人的惩罚机制，所以 不敢利用机会使自己在该阶段利益最大化，从 而使该阶段的“合作”出现。从这个意义上看， 触发策略是“温柔的”。
perfect outcome:the Nash equilibrium of
played in every stage.
☺注
☺注：在阶段博弈G为完全且完美信息动态博弈 时类似的结论同样成立。设G属于第三章所定义 的完全且完美信息动态博弈，如果G有唯一的逆 推归纳解，则G(T)有唯一的子博弈完美NE：其 中每一阶段的结果都是G的逆推归纳解。类似的， 设G为第三章所定义的有同时选择的两阶段动态 博弈，如果G有唯一的子博弈NE，则G(T)也有唯 一的子博弈完美NE：G的子博弈完美NE重复进 行T次。
Repeated game，顾名思义，就是同样结构的 博弈重复多次，其中每一次博弈称为阶段博 弈Stage game（也称为原博弈）。 Repeated Game的基本特征有3项：
1、阶段博弈之间没有“物质上”的联系（no Physical links），也就是说，前一阶段博弈 不改变后一阶段博弈的结构（对比之下，序惯 博弈涉及到物质上的联系）；
博弈论课件 2
Game Theory1Spring201511Lecture 2Dominance and Best Response22Readings•Watson: Strategy_ An introduction to game theory–Ch 6,1rd ed p.43-55;–Ch6, 3rd ed p.49-66.32Outline•Dominance.•Pareto Efficiency.•Best Response.•Undominated Strategy Sets.425262Consider a mixed strategy of player 1: σ1=(½, ½ , 0).72•A pure strategy s i of player i is dominated if there is a strategy (pure or mixed)σi ∈∆S i such that u i (σi , s -i )>u i (s i , s -i ),for all strategy profiles s -i ∈∆S -i of the other players.•An important component of the definition of dominance is the “strict” inequality.–That is, in mathematical terms, we have the expression u i (σi , s −i ) > u i (s i , s −i ) rather than u i (σi , s −i ) ≥u i (s i , s −i ).•Note: A rational player will never play a dominated strategy.8Game 1The “Grade Game”Without showing your neighbor what you are doing, write down on a form either the letter αor the letter β. Think of this as a ‘grade bid’. We will randomly pair your form with one other form. Neither you nor your pair will ever know with whom you were paired.10Game 1The “Grade Game”☐Here is how grades may be assigned for this course.☐If you put αand your pair putsβ, then you will getgrade A, and your pair grade C.☐if both you and your pair putα, then you both will get grade B-.☐if you put βand your pair putsα, then you will getgrade C, and your pair grade A.☐if both you and your pair putβ, then you will both get grade B+.11Definition☐Definition 1. My strategy αstrictly dominates my strategy βif my payoff from αis strictly higher than that from βregardless of others' choices.☐Definition 2. My strategy αweakly dominates my strategy βif my payoff from αis as high as that from βregardless of others' choices, and is strictly higher for at least one such choice.14Lesson 1. You should never play a strictly dominated strategy.15Lesson 2. Rational play by rational players can lead to bad outcomes.16Prisoners’ Dilemma217Prisoners’ Dilemma•The prisoners’ dilemma illustrates one of the major tensions in strategic settings:–the clash between individual and group interests.218Prisoners’ Dilemmas☐Examples☐Joint project: incentive to shirk☐Price competition: incentive to undercut price☐Common resource: incentive to “overfish” orpollute19Prisoners’ Dilemmas☐Remedies☐Contracts☐Treaties change payoffs☐Regulations☐Repeated play☐Education→ change payoffs22Pareto Efficiency•s is more efficient than s’ if all of the players prefer the outcome of s to the outcome of s’ and if the preference is strict for at least one player.–In mathematical terms, s is more efficient than s’ if u i(s) ≥u i(s’) for each player i and if the inequality is strict for atleast one player.• A strategy profile s is called Pareto efficient if there is no other strategy profile that is more efficient; that is, there is no other strategy profile s’such that u i(s’) ≥u i(s) for every player i and u j(s’) > u j(s) for some player j.•For prisoner's dilemma,CC is more efficient than DD. CC,CD and DC are Pareto efficient.•Suppose player i has a belief θ-i∈ΔS-i about the strategies played by the other players.•Player i's strategy s i∈S i is a best response to a beliefθif u i(s i,θ-i)≥u i(s i', θ-i)for all s i'∈S i.•Define BR i(-i)as the set of strategies that are best responses to the belief-i.•R is the BR when p <4/11.M is the BR when p >4/11.pu 24/11Dominance and Best Response Compared •Let B i be the set of strategies for player i that can berationalized as best responses to some beliefs.–B i={s i|there exists aθ-i∈∆S-i such that s i∈BR i(θ-i)}–For the previous example,B2={M, R}.•Let UD i be the set of undominated strategies for i.–UD i={s i∈S i|there is noσi∈∆S i that dominates s i}–For the previous example,UD2={M, R}•Result: In any finite two-player game,B i=UD i.Procedure for Calculating B i=UD i• 1. Look for strategies that are best responses to the simplest beliefs—those beliefs that put all probability on just one of the other player’s strategies. These best responses are obviously in the set B i so they are also in UD i.• 2. Look for strategies that are dominated by other pure strategies; these dominated strategies are not in UD i and thus they are also not in B i.• 3. Test each remaining strategy to see if it is dominated by a mixed strategy. This final step is the most difficult, but if you are lucky, then you will rarely have to perform it.。
博弈论的理论与方法优质课件
这种厂商的策略选择行为，在博弈论中 称为“从最小收益中选择最大收益 （Maximize the Minimun Payoffs）”, 其数学表达式形式为：
min a1j=a11=50 j
min a2j=a21=80 j
max min aij=a21=80 ij
同样，对于寡头垄断厂商B来说，如果 它也是一个在决策中非常谨慎的风险回避者， 也会在自己所选择的价格策略可能产生的最 糟糕的结果中，选择相对而言能产生较好结 果的价格策略，即：
如果A采用A2，B仍采用B1价格策略时，A所能获得的最小收益为80（TRA=a21=80）。
3
的概率采用策略A ，那么，根据上述厂商A采 在上述例子中，设厂商A的最优混合策略为：以概率ρa采用策略A1，以概率（1-ρa）采用策略A2，则在厂商B同时相应采用策略B1，
或者策略B2时，厂商A的预期收益为：
传统Micro研究效用（函数）最大化，生产（函 数）最大化，主要涉及人与物（商品、生产要 素）的关系，较少涉及人与人的关系。
当经济研究涉及人与人（企业与企业）的关系时 ，例如厂商的价格战，博弈论就成了一个有用 的分析工具。
博弈论的发展
① 博弈论产生于30-50年代
A、1944年，冯·诺依曼、摩根斯坦恩合作发表 《博弈论与经济行为》，将博弈论引入关于 经济不确定性分析（预期效用概念），是博 弈论正式诞生的标志；
ij
ji
这一博弈论模型的分析结论表明，厂商A和 厂商B都一致地选择了它们各自的价格策略 的组合a21（或者b21），结果产生了一个稳 定的博弈解或者均衡解。
因为，此时a21=80，既不是厂商A的最大收 益（或者厂商B的最大损失），也不是厂商A的 最小收益（或者厂商B的最小损失）。在博弈论 中，这一博弈的均衡解被称为“纳什均衡” （ Nash Eguilibrium ） 或 被 称 为 “ 鞍 点 ” （Saddle Point）。所谓“鞍点”，就是博弈所 具有的确定的解。存在“鞍点”的博弈，也被称 为 严 格 确 定 的 博 弈 （ Strictly Determined Game）。相应地，求解“鞍点”的方法在博弈 论 模 型 中 被 称 为 “ 极 小 — 极 大 定 理 ” （ Min— Max Theorem）。
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Review
Elimination
Nash Equilibrium
Summary
Games
Course topics:
• Games of complete and perfect information • Static Games (Nash Equilibrium) • Dynamic Games (Backward Induction)
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Review
Elimination
Nash Equilibrium
Summary
Games
• Consider the following game:
• Two players • Each player chooses between two actions: A and B • Payoff for all outcomes is in the table below:
1
2
A
B
your ID keyword
A 50,50 0,200
B 200,0 80,80
1
2
A
B
A 50,50 0,200
B 200,0 80,80
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Review
Elimination
Nash Equilibrium
Summary
Games
• Game Participation:
• you can win up to 200 CZK • send SMS with your action • phone numbers + IDs will be strictly protected • participant will be matched randomly in pairs, one pair will be
chosen to get the payoff
1
2
A
B
A 50,50 0,200
B 200,0 80,80
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Review
Elimination
Nash Equilibrium
Hale Waihona Puke SummaryGames• To participate:
• SMS 773 038 630: xaaa02 game1 0 xaaa02 game1 1
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Review
Elimination
Nash Equilibrium
Summary
Games
• Game theory is about economic models
• Economic models help us understand behavior of agents, they do not tell us what their optimal action is
• Each game represents some economic situation (Prisoner’s dilemma = Duopoly)
• By solving the game (finding equilibrium) we find plausible outcome of a given situation
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Review
Elimination
Nash Equilibrium
Summary
Games
• Strategic game consists of
• set of players • for each player set of actions • for each player set of preferences over the set of action
• Games of complete but imperfect information • Dynamic Games (Subgame perfect NE)
• Games of incomplete information • Static Games (Auctions) • Dynamic Games (Signaling)
profiles • preferences represented by payoff function • static games: players simultaneously chose actions
normal form game representation (table)
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• Osborne, M. J. – An Introduction to Game Theory Gibbons, R. – A Primer in Game Theory Suggested articles
• Important information on webpage
• Grading: Midterm 30%, Final 60%, Homework 10%, Experiments up to 5%
Introduction to Game Theory
Lecture 2
Disclaimer: this presentation is only a supporting material and is not sufficient to master the topics covered during the lecture. Study of relevant books is strongly recommended.
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Review
Elimination
Nash Equilibrium
Summary
Games
…previously: “what are models?”… …today: “how to solve them?”
• Elimination of strictly dominated strategies • Nash Equilibrium • Experiment
Review
Elimination
Nash Equilibrium
Summary
Games
• Contact: home.cerge-ei.cz/kalovcova/teaching.html
• Office hours: Wed 7.30pm – 8.00pm, NB339 or by email appointment