game theory12 博弈论 英文 例题详细解析
game theory12 博弈论 英文 例题详细解析
Beer or Quiche gameIn this handout, we discuss signaling games of Beer or Quiche type and the way to find the plausible outcome of this type of games – weak perfect Bayesian equilibrium.The basic idea is that we try all the combinations of actions of all players one by one see if any of them is an equilibrium. In general, we work in the following steps:Step 1: Start with actions of Player 1 (both types)Step 2:Find Player 2’s beliefs ba sed on actions of Player 1 (in both information sets)Step 3: Find optimal response of Player 2 (in both information sets)Step 4: Check if both types of Player 1 play optimal response:a.yes, no type of Player 1 wants to deviate => we have WPBEb.no, at least one type of Player 1 wants to deviate => no WPBEStep 5: Move to the next possible actions of Player 1Example: consider the following game.We will analyze these options:1.Player 1: Beer if Weak, Quiche if Strong (Separating equilibrium 1)2.Player 1: Quiche if Weak, Beer if Strong (Separating equilibrium 2)3.Player 1: Quiche if Weak, Quiche if Strong (Pooling equilibrium 1)4.Player 1: Beer if Weak, Beer if Strong (Pooling equilibrium 2)Option 1: Player 1: Beer if Weak, Quiche if Strong (Separating equilibrium 1)Step 1: Start with actions of Player 1 (both types)Beer if Weak, Quiche if StrongStep 2:Find Player 2’s beliefs based on actions of Player 1 (in both information sets)Information sets are reached, so beliefs are calculated based on actions of Player 1.Step 3: Find optimal response of Player 2 (in both information sets)In the left information set, Player 1 knows that Player 2 is of Weak type and Fighting is optimal (3>0).In the right information set, Player 2 knows that Player 1 is of Strong type and Not Fighting is optimal(2>0).Step 4: Check if both types of Player 1 play optimal responseWeak type of Player 1 is satisfied, but Strong type of Player 1 wants to deviate and get higher profit.So this is not WPBE.Step 5: Move to the next possible actions of Player 1Option 2: Player 1: Quiche if Weak, Beer if Strong (Separating equilibrium 2) Step 1: Start with actions of Player 1 (both types)Quiche if Weak, Beer if StrongStep 2:Find Player 2’s beliefs based on actions of Player 1 (in both information sets)Information sets are reached, so beliefs are calculated based on actions of Player 1.Step 3: Find optimal response of Player 2 (in both information sets)In the left information set, Player 1 knows that Player 2 is of Strong type and Fighting is optima (4>1).In the right information set, Player 2 knows that Player 1 is of Weak type and Fighting is optimal(1>0).Step 4: Check if both types of Player 1 play optimal responseNo type of Player 1 wants to deviate because the payoff would be lower. So this is WPBE.Step 5: Move to the next possible actions of Player 1Option 3: Player 1: Quiche if Weak, Quiche if Strong (Pooling equilibrium 1) Step 1: Start with actions of Player 1 (both types)Quiche if Weak, Quiche if StrongStep 2:Find Player 2’s beliefs based on actions of Player 1 (in both information sets)Right information set is reached, so beliefs are calculated based on actions of Player 1. Left information set is not reached, so we are free to choose any beliefs, any set of beliefs is consistent with Player 1’s actions. Denote beliefs p and 1-p.Step 3: Find optimal response of Player 2 (in both information sets)In the right information set, Player 2 calculates expected payoff of playing Fight and NotFigt:EP(Fight) = 0.3*1+0.7*0 = 0.3EP(NotFight) = 0.3*0+0.7*2 = 1.4So NotFight is optimal response.In the left information set, optimal response of Player 2 generally depends on value of p (beliefs). However, in this case, Fight dominates NotFight in the left information set and hence Fight is optimal.Step 4: Check if both types of Player 1 play optimal responseActually, no matter what action Player 2 chooses in the left information set, Weak type of Player 1 will always want to deviate. So this is not WPBE.Step 5: Move to the next possible actions of Player 1Option 4: Player 1: Beer if Weak, Beer if Strong (Pooling equilibrium 2)Step 1: Start with actions of Player 1 (both types)Beer if Weak, Beer if StrongStep 2:Find Player 2’s beliefs based on actions of Player 1 (in both information sets)Left information set is reached, so beliefs are calculated based on actions of Player 1. Beliefs in the right information set are denoted p and 1-p because this information set is not reached and hence any set of beliefs is consistent with actions of P1..Step 3: Find optimal response of Player 2 (in both information sets)In the left information set, Player 2 calculates expected payoff of playing Fight and NotFigt:EP(Fight) = 0.3*3+0.7*4 = 3.7EP(NotFight) = 0.3*0+0.7*1 = 0.7So Fight is optimal response.In the right information set, optimal response of Player 2 generally depends on value of p (beliefs).There are two options, either the optimal response is Fight, but then Weak Player 1 would want to deviate; or the optimal response is NotFight. In this case no type of Player 1 would want to deviate.So the beliefs have to be set such that expected payoff from NotFight is larger than from Fight.EP(Fight) < EP(NotFight) => 1*p + 0*(1-p) < 0*p + 2*(1-p) => p<2/3Step 4: Check if both types of Player 1 play optimal responseNo type of Player 1 wants to deviate because the payoff would be lower. So this is WPBE.Step 5: Move to the next possible actions of Player 1. There are no more possibilities.There are two WPBE in this game:1.Weak P1 –Quiche, Strong P1 –Beer; P2 Fight if Beer and Fight if Quiche and p=0 in the leftinformation set and p=1 in the right information set.2.Weak P1 – Beer, Strong P1 – Beer; P2 – Fight if Beer and NotFight if Quiche and p<2/3.。
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.。
game theory12 博弈论 英文
8 / 28
Review
Beer or Quiche
Modified Beer or Quiche
Summary
1, 1
P2 Beer P1 Quiche P2
Weak
0, 1
1
0
3, 0
0.1
2, 0
Chance
7 / 28
Review
Beer or Quiche
Modified Beer or Quiche
Summary
• P1: Strong->Quiche; Weak->Beer • P2: optimal response to that is: if Beer->Fight; if Quiche->NoFight
0, 0
1 Beer P2
Strong
1, 0
Quiche0
0.9
P1
P2
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3, 1
11 / 28
Review
Beer or Quiche
Modified Beer or Quiche
Summary
• P1: Strong->Quiche; Weak->Quiche • P2: optimal response to that: NoFight if Quiche Fight if Beer (so P1 - Weak does not deviate) • WPBE: if Quiche, Pr(Weak)=0.1; Pr(Strong)=0.9 if Beer, Pr(Weak)=p; Pr(Strong)=1-p; where p>1/2
博弈论答案（Gametheoryanswer）
博弈论答案（Game theory answer）Game theory, exercises, reference answers (second assignments)First, the multiple-choice question1.B,2.C,3.A,4.A,5.B,6.ABCD7.C 8.B 9.CTwo, judge and explain the reason1.F best balance is an equilibrium more rigorous than the Nash equilibrium2.T best balance is an equilibrium more rigorous than the Nash equilibrium3.T game types are divided into single game, double game and multiplayer game according to the number of players in the gameUnder the condition that both sides of the 4.F game have different preferences, there may be 2 Nash equilibria in a game model, such as the sex war5.T zero sum game refers to the participation of all parties in the game, under strict competition, one side of revenue is equal to the other party's loss, the sum of gains and losses of the game is always zero, so there is no possibility of cooperation between the two sides6.T is strictly dominated equilibrium through the worstelimination method (excluding repeat decision) the dominant strategy, there is only one Nash equilibrium7.F Nash equilibrium is a collection of best policies, which means that in the case of a given strategy, the game side always chooses a relatively large strategy, and does not guarantee the outcome to be the best.In the 8.F game, people always choose their own strategies to maximize their interests and not aim at the change of the other's earnings9.T Nash equilibrium is a collection of best policies, which means that when given someone else's strategy, no one changes his strategy to reduce his earningsIn the 10.F game, people always choose their own strategies to maximize their interests and not aim at the change of the other's earningsIn the 11.F game, people always choose their own strategies to maximize their interests and not aim at the change of the other's earnings12.T although Berg Stagg model profit is less than the sum of the Cournot model, but the profit model of high Bigunuo leaderThree, calculation and analysis questions1, (1) draw A, B two enterprise profit and loss matrix.B enterpriseAdvertise without advertisingA enterprises advertise 20, 825, 2No advertising 10, 1230, 6(2) pure strategy Nash equilibrium.(advertising, advertising)2, draw two enterprise profit and loss matrix, seek Nash equilibrium.(1) draw the profit and loss matrix of A and B two enterprisesPepsi ColaOriginal price increaseCoca-Cola's original price is 10, 10100, -30Price increases -20, 30140, 35(2) seeking Nash equilibrium.Two: (the original price, the original price), (prices, prices)3, suppose the payoff matrix of a game is as follows:Methyl ethylLeft and rightOn a, B, C, DNext, e, F, G, H(1) if (on, left) is the best balance, then, a>, b>, g<, f>?Answer: a>e, b>d, f>h, g<c(2) what inequalities must be satisfied if (upper, left) is the Nash equilibrium?Answer: a>e, b>d4, answer: (1) this market is represented by the game of prisoner's dilemma.Northern AirlinesCooperative competitionXinhua Airlines cooperation 500000500000090000Competition 900000, 06000060000(2) explain why the equilibrium result may be that both companies choose competitive strategies.Answer: if Xinhua chooses "competition", then the north will choose "60000>0"; if Xinhua chooses "cooperation", the north will still choose "900000>500000".If the North chooses "competition", Xinhua will choose "60000>0"; if the North chooses "cooperation", Xinhua will still choose "900000>0".Because the competition is the dominant strategy of both sides, the equilibrium result is that both companies choose competitive strategy.5. The payoff matrix of the game is shown as follows:BLeft and rightA, a, B, C, DNext, e, F, G, H(1) if the (top, left) is the dominant policy equilibrium, what relation must be satisfied between a, B, C, D, e, F, G, and H?Answer: starting from the definition of dominant strategy equilibrium:For the one, the strategy "g" (a) is better than "C" (E);For B., the policy "left" (B, f) is superior to the policy"right" (D, H).So the conclusions are: a>e, b>d, f>h, c>g(2) if the (upper, left) is Nash equilibrium, what relation must be satisfied in (1)?Answer: Nash equilibrium only needs to meet: a>e, b>d,(3) if the (top, left) is the best balance, then is it necessarily a Nash equilibrium? Why?Answer: the equilibrium of dominant strategy must be Nash equilibrium, because the equilibrium condition of dominant strategy contains the condition of Nash equilibrium.(4) under what circumstances does the pure strategy Nash equilibrium exist?A: when each of these strategies does not satisfy the Nash equilibrium, the pure strategic Nash equilibrium does not exist.7, seek the Nash equilibrium.PigPress waitBig pigs press 5, 14, 4Wait 9, -1 0, 0The Nash equilibrium is: big pig, press, pig, etc., namely (press, etc.)6,BLow priceA low price of 10080050, 50High priced -20, -30 900600(1) what are the results of Nash equilibrium?Answer: (low price, low price), (high price, high price)(2) what is the result of the cooperation between the two firms?Answer: (high price, high price)8. The pure Nash equilibrium of the following games is obtained by using the reaction function method and the marking method.Participants 1 participants 2A, B, C, DingA, 2,3, 3,2, 3,4, 0,3B, 4,4, 5,2, 0,1, 1,2C, 3,1, 4,1, 1,4, 10,2D, 3,1, 4,1, -1,2, 10,1Participant 1's response function:R1 (2) =B, if 2 chooses a=B, if 2 chooses B.=A, if 2, choose C=C or D, if 2, choose DingParticipant 2's response function:R2 (1) = C, if 2, select A= a, if 2, select B= C, if 2, select C= C, if 2, select DFor the common set, the pure strategy Nash equilibrium is (B, a) and (A, c)9, the following game Nash equilibrium (including pure strategyand mixed strategy).Methyl ethylL RU 5,0 0,8D 2,6 4,5Solution: (1) pure strategy Nash equilibrium: we can see from the scratch method that there is no pure strategy Nash equilibrium in the matrix game.(2) mixed strategy Nash equilibriumThe probability of setting a "U" is P1, and the probability of "D" is 1-P1B. the probability of selecting "L" is P2, and the probability of "R" is 1-P2For a, the best policy is to choose "U" and "D" by a certain probability, so that the second choice of "L" and "R" is equal to the expected valueThat is, P1*0+ (1-P1), *6=, P1*8+ (1-P1), *5Xie P1=1/9That is, (1/9,8/9) Nash policy is chosen according to 1/9probability, U and 8/9 probability, and D is chosen as a mixed strategyFor B, the best strategy is to choose "L" and "R" by a certain probability, so that the second is equal to the expected value of "U" and "D"That is, P2*5+ (1-P2), *0=, P2*2+ (1-P2), *4Xie P2=4/7That is, (4/7,3/7) according to the probability of 4/7, "L", "3/7", "R" is chosen as "B", the mixed strategy Nash equilibrium10, answer the question according to the profit and loss matrix of two player game:Methyl ethylLeft and rightGo to 2,3 0,0Lower 0,0 4,2(1) write out all the strategies of the two men.Answer: all strategies: (upper, left), (upper, right), (lower, left), (lower, right)(2) find all the pure strategy Nash equilibrium of the game.A: by the scratch method, we can see that the matrix game is purely strategic and the Nash equilibrium is(upper, left) and (lower, right) two(3) the mixed strategy Nash equilibrium of the game is obtained.Solution: the probability of setting a "up" is P1, and the probability of selecting "down" is 1-P1B. the probability of "left" is P2, and the probability of "right" is 1-P2For a, the best strategy is to choose "upper" and "lower" according to a certain probability, so that the left and right of the second are equal to the expected valueThat is, P1*3+ (1-P1), *0=, P1*0+ (1-P1), *2Xie P1=2/5That is, (2/5,3/5) a mixed strategy Nash equilibrium based on the "2/5 probability", "upper", "3/5" probability, and "next"For b.,The best strategy is to choose "left" and "right" according to a certain probability, so that the candidate's "upper" and "lower" expectations are equalThat is, P2*2+ (1-P2), *0=, P2*0+ (1-P2), *4Xie P2=2/3That is, (2/3,1/3) Nash policy is chosen by the 2/3 probability "left" and "1/3", and the "right" is b11, an oligopoly market has two manufacturers, the total cost is 20 times the output of their own, the market demand letterThe number is Q=200-P.Answer: (1) if two manufacturers decide the output at the same time, how much is the output?(2) if the two firms reach an agreement to monopolize the market and arrange production together, what about their respective profits?(3) use the case to explain the prisoner's dilemma.Answer: (1) by the known conditions Q=200-P, P=200-QTC1=20q1, TC2=20q2, q1+q2=QThe profit functions obtained by 1,2 manufacturers are:K1=Pq1-TC1= (200- (q1+q2)) q1-20q1=180q1-q12-q1q2K2=Pq2-TC2= (200- (q1+q2)) q2-20q2=180q2-q22-q1q2The dK/dq1=0's 1 response function is 180-2Q1-Q2=0,The dK/dq2=0's 2 response function is 180-Q1-2Q2=0,The joint solution can be obtained by q1=q2=60K1=K2=3600(2) by the known condition Q=200-P, P=200-QTC=TC1+TC2=20q1+20q2 =20QThe total profit function of the 1,2 manufacturer is:K=PQ-TC= (200-Q) Q-20Q=180Q-Q2Order dK/dQ=0, Q=90, q1=q2=45K=PQ-TC= (200-Q) Q-20Q=180Q-Q2=8100K1=K2=4050(3) q1=45, q2=60 and q1=60, q2=45, respectively, into the profit function of 1,2 manufacturersThe profits of the 1,2 manufacturers are:K1 (q1=45, q2=60) =Pq1-TC1= (200- (q1+q2))q1-20q1=180q1-q12-q1q2=3375K1 (q1=60, q2=45) =Pq1-TC1= (200- (q1+q2))q1-20q1=180q1-q12-q1q2=4500K2 (q1=45, q2=60) =Pq2-TC2= (200- (q1+q2))q2-20q2=180q2-q22-q1q2=4500K1 (q1=60, q2=45) =Pq1-TC1= (200- (q1+q2))q1-20q1=180q1-q12-q1q2=3375Vendor 2Cooperation (q2=45), non cooperation (q2=60);Vendor 1 Cooperation (q1=45) 4050405033754500Non cooperative (q1=60) 4500337536003600According to the marking method, the best way for the manufacturer is 1.2 (non cooperation, non cooperation), that is, (36003600)The profits of both sides were lower than (cooperation, cooperation). (40504050) obviously it belonged to the prisoner's dilemma"13, consider the following (market deterrence) a dynamic game: first of all, the potential in a market entrants to choose whether or not to enter, and then on the market for enterprise (incumbent) is selected to compete with the new enterprise. The incumbent may have two types of gentle type (left) and cruel type (right), answer the following questions..Left: gentle right: cruel type(1) find the corresponding Nash equilibrium for two types of incumbent, and the sub game perfect Nash equilibrium(1) the Nash equilibrium of the gentle type of incumbent is (access, acquiescence)The Nash of the cruel type is balanced (not entering, entering, struggling)(2) when the existing enterprise is tender, at least how many times will the new enterprise be willing to enter?Four. Discussion questions1, explain the prisoner's Dilemma and explain the business case.(1) assumptions for example: two prisoners were accused of a crime is an accomplice. They were kept in separate cells, unable to communicate information. Prisoners are required to confess crimes. If two prisoners confess, each shall be sent to prison for 5 years; if two men do not confess, two prisoners may expect to be sent from prison to prison for 2 years; if a prisoner confesses, another prisoner does not confess,Frankly, the prisoner will only go to prison for 1 years, and the prisoner without confession will be sentenced to 10 yearsin prison.(2) the strategy matrix of prisoners' dilemma. Each prisoner has two strategies: to confess or not to confess. The numbers in the table represent the benefits of prisoner a and B.Prisoner BConfessPrisoner frank, -5, -5, -1, -10Don't confess, -10, -1, -2, -2(3) analysis: through the marking method, we can see that in the model of prisoner's dilemma, Nash equilibrium is that both sides confess". Given a frank case, the best strategy for B. is to confess; the optimal policy given by B. is also frank. And here both sides confess, not only is the Nash equilibrium, but also is a best balance, that is, regardless of how the other side of the choice, the individual's best choice is to confess. As a result, both sides confess.(4) business cases: oligopoly firms often find themselves ina prisoner's dilemma. When the oligarchic manufacturer chooses the output, every manufacturer can gain more profits if the oligopoly firms combine to form cartels and choose monopoly profits to maximize the output. But the cartel agreement is not a Nash equilibrium, because given both comply with the agreement, each firm to increase production, the result is that each vendor has only been Nash equilibrium yield profits, itis far less than the yield of profit under the cartel.2. Explain and discuss the Nash equilibrium of Cournot duopoly model. Why is balance a prisoner's dilemma?See class notesOr calculation questions eleventh3, use the game of thief and guard to explain the paradox of encouragement (regulation)".(1) assume the conditions for example: stealing and preventing theft is a game between thieves and guards. The guard can sleep or sleep. Thieves can take two tactics: stealing and stealing. If the thief knows that the guard is sleeping, his best bet is to steal. If the guard doesn't sleep, he'd better not steal. For the doorman, if he knows the thief wants to steal, his best choice is not to sleep, and if the thief take it without stealing, he'd better go to sleep.(2) the payment matrix of the thief and the doorman (assuming that the thief must have succeeded in stealing when the guard sleeps, and that the thief will be caught when the guard does not sleep.):GuardGo to bed without sleepThieves steal 1, -1 -2, 0Do not steal 0, 20, 0(3) analysis: through the marking method, we can see that there is no Nash equilibrium in this game. The thieves do not steal, do not sleep, neither gains nor loss; the guard did not sleep, the thief, because the job is not to reward, the thief was sentenced to 2 unit failure loss; guard sleeping, thieves do not steal, the sleeping happily get 2 utility unit, the thief did not return no loss of sleep; the guard, the thief, the guard was punished because of dereliction of duty and his failure in 1 units, 1 units of utility thieves to steal success.(4) "incentive (regulatory) paradox" shows: in reality, we can see that when the doorman without sleep, stealing a crackdown of the convergence of molecules; time, molecular theft began to make waves, the thief can not tolerate when too rampant, the guard had to begin again. The more the thief, so the guard will not sleep more, steal the thief less, not sleeping guard will be less; in turn, the more don't sleep, steal the thief less, do not sleep the less, the more the thief stole. If you steal group selection is out in force, so the guard all don't sleep, but the once all don't sleep, the best choice not to steal all the thief, the thief stole all the guard once chose not to, all the best choose to sleep.(5) conclusion: increasing penalties for thieves can not prevent theft in the long run (but only to make the guard lazy); Aggravating Punishment, dereliction of duty is just to reduce the probability of theft. This game of gatekeeper and thief reveals that the unexpected relationship between policyobjectives and policy outcomes is often called the paradox of motivation".。
博弈论介绍 Game Theory
2. 生活中的“囚徒困境”例子
例子1 商家价格战 例子1
出售同类产品的商家之间本来可以 通过共同将价格维持在高位而获利，但 实际上却是相互杀价，结果都赚不到钱。 当一些商家共谋将价格抬高，消费 者实际上不用着急，因为商家联合维持 高价的垄断行为一般不会持久，可以等 待垄断的自身崩溃，价格就会掉下来。
表2 智猪博弈 小猪 按 按 大猪 等待 5,1 9, -1 等待 4,4 0,0
这个博弈大猪没有劣战略。但是，小猪有 一个劣战略“按”，因为无论大猪作何选择， 小猪选择“等待”是比选择“按”更好一些 的战略。 所以，小猪会剔除“按”，而选择“等 待”；大猪知道小猪会选择“等待”，从而 自己选择“按”，所以，可以预料博弈的结 果是(按,等待）。这称为“ 重复剔除劣战略 的占优战略均衡 ”，其中小猪的战略“等待” 占优于战略“按”，而给定小猪剔除了劣战 略“按”后，大猪的战略“按”又占优于战 略“等待”
表4 有补贴时的博弈 空中客车 开发 开发 波音 不开发 -10,10 0, 120 不开发 100,0 0,0
这时只有一个纳什均衡，即波音公司 不开发和空中客车公司开发的均衡（不 开发,开发），这有利于空中客车。 在这里，欧共体对空中客车的补贴就 是使空中客车一定要开发（无论波音是 否开发）的威胁变得可置信的一种“承 诺行动”。
类似的例子还有： 渤海中的鱼愈来愈少了，工业化中的大气 及河流污染，森林植被的破坏等。解决公共 资源过度利用的出路是政府制订相应的规制 政策加强管理，如我国政府规定海洋捕鱼中， 每年有一段时间的“休渔期”，此时禁止捕 鱼，让小鱼苗安安静静地生长，大鱼好好地 产卵，并对鱼网的网眼大小作出规定，禁用 过小网眼的捕网打鱼，保护幼鱼的生存。又 如在三峡库区，为了保护库区水体环境，关 闭了前些年泛滥成灾的许多小造纸厂等。 问题：1、为什么在城市中心道路上禁止汽车鸣 喇叭？
第三节博弈论（GameTheory）
第三节博弈论(Game Theory)在国际关系的研究过程中，我们时常会运用到博弈论这样一个工具。
博弈论在英语中称之为“Game Theory”。
很多人会认为这是一种所谓的游戏理论，其实不然，我们不能把Games 与Fun 同论，而应该将博弈论称之为是一种“Strategic interaction”（策略性互动）。
“博弈”一词现如今在我们的生活中出现的已经很频繁，我们经常会听说各种类型的国家间博弈（如：中美博弈），“博弈论”已经深刻的影响了世界局势和地区局势的发展。
在iChange创设的危机联动体系中，博弈论将得到充分利用，代表也将有机会运用博弈论的知识来解决iChange 核心学术委员会设计的危机。
在这一节中，我将对博弈论进行一个初步的介绍与讨论，代表们可以从这一节中了解到博弈论的相关历史以及一些经典案例的剖析。
（请注意：博弈论的应用范围非常广泛，涵盖数学、经济学、生物学、计算机科学、国际关系、政治学及军事战略等多种学科，对博弈论案例的一些深入分析有时需要运用到高等数学知识，在本节中我们不会涉及较多的数学概念，仅会通过一些基本的数学分析和逻辑推理来方便理解将要讨论的经典博弈案例。
）3.1 从“叙利亚局势”到“零和博弈”在先前关于现实主义理论的讨论中，我们对国家间博弈已经有了初步的了解，那就是国家是有目的的行为体，他们总为了实现自己利益的最大化而选择对自己最有利的战略，其次，政治结果不仅仅只取决于一个国家的战略选择还取决于其他国家的战略选择，多种选择的互相作用，或者策略性互动会产生不同的结果。
因此，国家行为体在选择战略前会预判他国的战略。
在这样的条件下，让我们用一个简单的模型分析一下发生在2013年叙利亚局势1：叙利亚危机从2011年发展至今已经将进入第四个年头。
叙利亚危机从叙利亚政府军屠杀平民和儿童再到使用化学武器而骤然升级，以2013年8月底美国欲对叙利亚动武达到最为紧张的状态，同年9月中旬，叙利亚阿萨德政府以愿意向国际社会交出化学武器并同意立即加入《禁止化学武器公约》的态度而使得局势趋向缓和。
博弈论game theory
CH2 完全信息静态博弈 §2.1 博弈的解法
2.1.1占优策略 局中人的最优策略不依赖于其他局中人的选择，则称该 局中人有占优策略。如果某策略组合中的每一个策略都是局 中人各自的占优策略，此策略组合称为占优均衡。 囚徒困境中的（坦白，坦白）就是占优均衡，坦白是每 个囚徒的占优策略。但并不是每个博弈，每个局中人都有占 优策略。
博弈论
Game Theory
焦未然
目
完全信息静态博弈 不完全信息静态博弈
录
完全且完美信息动态博弈
重 复 博 弈
基本概念
进化博弈
合作博弈
完全但不完美信息动态博弈
不完全信息动态博弈
1.1.1博弈的基本描述——博弈即游戏 游戏的基本特征：有规则、有结果、有策略、相互依赖性 例子 三人决斗，开枪射杀对手，以保存自己。命中率和 每一轮的开枪次序如下。 命中率 次序 A 30% 1 B 70% 2 C 100% 3 A在第一轮的策略是什么？A最怕什么？首轮之后谁的存 活几率最高？
2.3.3反应函数 对于其他局中人的每一个可能的决策，局中人i都选定 自己的最佳决策。建立这种最佳决策与其他局中人的每一个 可能的决策的映射关系，我们称之为反应函数。局中人反应 函数的交点就是NE。
q2
a b a 2b
a bq 2 q1 2b
E
q2
R1 q2
a bq1 q2 2b a 2b a b
按钮 4，8
0
2.1.3划线法
A B 坦 白
坦
白不 坦 白
猜硬币者 正 盖硬币者 面 反 面
-8，-8 -10，0
0，-10 -1，-1
正
反
面
面
-1，1
博弈论 Game theory (全)
博弈论 Game Theory博弈论亦名“对策论”、“赛局理论”，属应用数学的一个分支, 目前在生物学，经济学，国际关系，计算机科学, 政治学，军事战略和其他很多学科都有广泛的应用。
在《博弈圣经》中写到：博弈论是二人在平等的对局中各自利用对方的策略变换自己的对抗策略，达到取胜的意义。
主要研究公式化了的激励结构间的相互作用。
是研究具有斗争或竞争性质现象的数学理论和方法。
也是运筹学的一个重要学科。
博弈论考虑游戏中的个体的预测行为和实际行为，并研究它们的优化策略。
表面上不同的相互作用可能表现出相似的激励结构（incentive structure），所以他们是同一个游戏的特例。
其中一个有名有趣的应用例子是囚徒困境（Prisoner's dilemma）。
具有竞争或对抗性质的行为称为博弈行为。
在这类行为中，参加斗争或竞争的各方各自具有不同的目标或利益。
为了达到各自的目标和利益，各方必须考虑对手的各种可能的行动方案，并力图选取对自己最为有利或最为合理的方案。
比如日常生活中的下棋，打牌等。
博弈论就是研究博弈行为中斗争各方是否存在着最合理的行为方案，以及如何找到这个合理的行为方案的数学理论和方法。
生物学家使用博弈理论来理解和预测演化（论）的某些结果。
例如，约翰·史密斯（John Maynard Smith）和乔治·普莱斯（George R. Price）在1973年发表于《自然》杂志上的论文中提出的“evolutionarily stable strategy”的这个概念就是使用了博弈理论。
其余可参见演化博弈理论（evolutionary game theory）和行为生态学（behavioral ecology）。
博弈论也应用于数学的其他分支，如概率，统计和线性规划等。
历史博弈论思想古已有之，我国古代的《孙子兵法》就不仅是一部军事著作，而且算是最早的一部博弈论专著。
博弈论最初主要研究象棋、桥牌、赌博中的胜负问题，人们对博弈局势的把握只停留在经验上，没有向理论化发展。
博弈论期末考试(英文)
- Alice decides whether to accept the ofer;
- if ofer is accepted the game ends with payof vector (x, 1 - x); otherwise we proceed to the next round. Consider the strategy profle s* : Alice always ofers (1 - xB , xB ) and accepts an ofer (x, 1 - x) if and only if x 2 xA . Bob always ofers (xA , 1 - xA ) and accepts an ofer (1 - x, x) if and only if x 2 xB . (a) (10 points) Explain precisely why xB = 1, xA = O is not a subgame perfect equilbirium. (b) (20 points) Find a subgame-perfect equilibrium of the game, and use the single deviation principle to prove that it is indeed a subgame perfect equilibrium.
- Alice ofers a division (1 - x, x), where x is the share of Bob,
- Bob decides whether to accept the ofer;
英语第一章阅读 game theory 原文及翻译
The essence of a game is the interdependence of player strategies. There are two distinct types of strategic interdependence: sequential and simultaneous. In the former the players move in sequence, each aware of the others’ previous actions. In the latter the players act at the same time, each ignorant of the others’ actions.
Game theory was pioneered by Princeton mathematician john von Neumann. In the early years the emphasis was on games of pure conflict (zero-sum games). Other games were considered in a cooperative form. That is, the participants were supposed to choose and implement their actions jointly. Recent research has focused on games that are neither zero sum nor purely cooperative. In these games the players choose their actions separately, but their links to others involve elements of both competition and cooperation.
game-theory1--博弈论-英文
/100
2
Introduction
Economic Models Game Theory Models Games Summary
This course is about …
… role of economic models … models of strategic behavior … models of social interactions
• real-life cousin Homo Sapiens
• guided by (sub)conscious psychological (irrational) drives; affected by herd instincts
• not well-defined goals
• lack of ability to achieve them, lack of motivation
/100
3
Introduction
Economic Models Game Theory Models Games Summary
Plan for Today
• Economic models
• What and Why?
• Game Theory Models
• Examples
• Games
• players, strategies, information, outcomes
• guided by emotions rather than bconomic Models Game Theory Models Games Summary
Economic Models - Problem
• models do not tell us how people truly behave people do not really maximize utility
gametheory博弈论
An example of classic game
Example 1.1 (Rousseau game卢梭游戏)
If a group of hunters set out to take a stag,
they are fully aware that they would all
have to remain faithfully at their posts in
stag
order to succeed; but if a hare happens to
pass near one of them, there can be no
— F. Carmichael
“Game theory is the mathematical study of strategic interactions among participants.”
— E. Maskin
“A game is being played whenever people have anything to do with each other.” — K. Binmore
doubt that he pursued it without qualm
（疑虑）, and that once he had caught his
prey（猎物）, he cared very little whether
or not he had made his companions miss
Starting Point of modern Game Theory
lecture12(博弈论讲义(Carnegie Mellon University))
June 4, 2003
73-347 Game Theory--Lecture 12
8
Game tree
A game tree has a set
of nodes and a set of edges such that
a path from x0 to x4
x0
a node
each edge connects two nodes (these two nodes are said to be adjacent) for any pair of nodes, there is a unique path that connects these two nodes
Dynamic Games of Complete Information
Extensive-Form Representation Game Tree
June 4, 2003
73-347 Game Theory--Lecture 12
1
Outline of dynamic games of complete information
June 4, 2003 73-347 Game Theory--Lecture 12 6
Definition: extensive-form representation
The extensive-form representation of a
game specifies:
the players in the game when each player has the move what each player can do at each of his or her opportunities to move what each player knows at each of his or her opportunities to move the payoff received by each player for each combination of moves that could be chosen by the players
微观经济学 第八章 博弈论 习题
第八章博弈论一、重点和难点（一）重点1.博弈论及其基本概念2.纳什均衡3.占优策略均衡4.囚徒困境博弈（二）难点1.最小最大值（或最大最小值）策略2.子博弈精炼纳什均衡3.动态博弈战略行动4.不完全信息静态博弈5.不完全信息动态博弈二、关键概念博弈零和博弈非常和博弈囚徒困境纳什均衡支付子博弈精炼纳什均衡完全信息静态博弈占优策略均衡重复博弈战略移动可信威胁豪尔绍尼转换三、习题（一）单项选择题1.博弈论中，局中人从一个博弈中得到的结果常被称为（）。
A. 效用B. 支付C. 决策D. 利润2.博弈中通常包括下面的内容，除了（）。
A.规则B.占优战略均衡C.策略D.结局3.在具有占优战略均衡的囚徒困境博弈中（）。
A.只有一个囚徒会坦白B.两个囚徒都没有坦白C.两个囚徒都会坦白D.任何坦白都被法庭否决了4.在多次重复的双头博弈中，每一个博弈者努力（）。
A.使行业的总利润达到最大B.使另一个博弈者的利润最小C.使其市场份额最大D.使其利润最大5.一个博弈中，直接决定局中人支付的因素是（）。
A. 策略组合B. 策略C. 信息D. 行动6.对博弈中的每一个博弈者而言，无论对手作何选择，其总是拥有惟一最佳行为，此时的博弈具有（）。
A.囚徒困境式的均衡B.一报还一报的均衡C.占优策略均衡D.激发战略均衡7.如果另一个博弈者在前一期合作，博弈者就在现期合作；但如果另一个博弈者在前一期违约，博弈者在现期也违约的战略称为（）。
A.一报还一报的战略B.激发战略C.双头战略D.主导企业战略8.在囚徒困境的博弈中，合作策略会导致（）。
A.博弈双方都获胜B.博弈双方都失败C.使得先采取行动者获胜D.使得后采取行动者获胜9.在双寡头中存在联合协议可以实现整个行业的利润最大化，则（）。
A.每个企业的产量必须相等B.该行业的产出水平是有效的C.该行业的边际收益必须等于总产出水平的边际成本D.如果没有联合协议，总产量会更大10.在什么时候，囚徒困境式博弈均衡最可能实现（）。
系统工程第12讲：冲突分析方法
行动：美 —— ① 空袭 ② 封锁
苏 —— ① 撤除 ② 升级
可行结局
确定优先序：
尽可能避免冲突紧张化而导致核战争是双方的共同原 则，在此基础上，美国力图使前苏联撤出导弹基地， 而前苏联则极希望维持现状
（3）确定优先序
美： 4 6 5 7 2 1 3 0 11 9 10 8
建立冲 突分析 模型
稳定性 分析
结果分 析与评 价
决策
冲突分析的要素 （1）时间点：是说明“冲突”开始发生时刻的标志，对于 建模而言，则是能够得到有用信息的终点。因为冲突总是一个 动态的过程，各种要素都在变化，这样很容易使人认识不清， 所以需要确定一个瞬间时刻，使问题明朗化，但时间不直接进 入分析模型。 （2）局中人（Players）：是指参与冲突的集团或个人（利 益主体），他们必须有部分或完全的独立决策权（行为主体）。 冲突分析要求局中人至少有两个或两个以上。局中人集合记作 N， N n 2 。
一.博弈论(Game Theory)
(四)博弈的分类
2.策略的数量 有限博弈：每一博弈方的可选策略都有限 无限博弈：至少一个博弈方具有无限的可选择策略 3.利益 零和博弈：博弈方始终对立(战争) 常和博弈：有对立也有妥协，和平共处(分配奖金) 变和博弈：社会总得益的角度(有效/低效/无效)
一.博弈论(Game Theory)
-5，-5 -8，0
0，-8 -1，-1
反映了博弈问题的根本特征，
是解释众多经济现象、研究
经济效率问题的非常有效的基本模型和范式。
一.博弈论(Game Theory)
小 猪 按 不按
大 按
5，1 9，-1 4，4 0，0
ch12博弈论
CH12 博弈论 game theory
2015年3月19日星期四
Page 23
i pmi [20 (m1 m2 m3 )] mi
可算出在产量组合为（3，9，6）时，市场价格为2，三厂商的 利润分8，16和12，再作其它产量组合时亦会有不同的结果，如 表12-2． 表12-2 三厂商离散产量结合对应价格和利润 m1 3 3 5 5 3 6 m2 9 8 5 5 3 3 m3 6 6 6 5 3 3 p 2 3 4 5 11 8 π1 6 9 20 25 33 48 π2 18 24 20 25 33 24 π3 12 18 24 25 33 24
亚 贸
高价
100，100 150，20
20，150 70，70
低价
类似地，广告投资、采用新技术等方面，厂商之间常常耗资 巨大，但不一定有利可图的争夺战；对公共资源的掠夺式使 用等问题。 我们的目的是如何利用这种困境达到有利于社会，合理利 用和开发公共资源，保护环境。
12.1 引言
CH12 博弈论 game theory
12.2 纳什均衡 Nash Equilibrium
CH12 博弈论 game theory
2015年3月19日星期四
Page 24
12.2.2混合策略纳什均衡
北线
南线
12.1 引言
CH12 博弈论 game theory
2015年3月19日星期四
Page 11
【解】局中人：盟军、日军；双方策略：北线、南线，记为:
S1＝ 1， 2 ；S 2＝1， 2
盟军的赢得矩阵如下： 日军
盟军 北线（ 1 ）
南线（ ）
2
北线 （ 1 ） 2 1
game-theory1--博弈论-英文
• models are NOT designed to tell us what people will do, they are not designed to be descriptive
Introduction to Game Theory
Lecture 1
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.
6
Introduction Economic Models Game Theory Models Games Summary
Economic Models - Problem
• based on Homo Rationalis
• acts purposefully and logically • has well-defined goals • has motivation and ability to approach them • does not exist!
• 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%
Game theory博弈论
that is:
* * i, xi Si : f i ( xi* , x ) f ( x , x i i i i )
To prove the existence of a Nash Equilibrium, we need using the Kakutani fixed point theorem, John Nash have proved it in 1951.
The end,thank you!
General and applied uses
As a method of applied mathematics, game theory has been used to study a wide variety of human and animal behaviors. It was developed in economics to understand a large collection of economic behaviors. Game theory in the social sciences has been applied to political, sociological, and psychological behaviors as well.
let xi be a strategy profile of player i and
x i be a strategy profile of player i , When each player i {1,, n},
xi resulting in strategy profile x ( x1 xn),
The games studied in game theory are welldefined mathematical objects. To be fully defined, a game must specify the following elements: the players of the game the information the actions available to each player at each decision point the payoffs for each outcome
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Beer or Quiche gameIn this handout, we discuss signaling games of Beer or Quiche type and the way to find the plausible outcome of this type of games – weak perfect Bayesian equilibrium.The basic idea is that we try all the combinations of actions of all players one by one see if any of them is an equilibrium. In general, we work in the following steps:Step 1: Start with actions of Player 1 (both types)Step 2:Find Player 2’s beliefs ba sed on actions of Player 1 (in both information sets)Step 3: Find optimal response of Player 2 (in both information sets)Step 4: Check if both types of Player 1 play optimal response:a.yes, no type of Player 1 wants to deviate => we have WPBEb.no, at least one type of Player 1 wants to deviate => no WPBEStep 5: Move to the next possible actions of Player 1Example: consider the following game.We will analyze these options:1.Player 1: Beer if Weak, Quiche if Strong (Separating equilibrium 1)2.Player 1: Quiche if Weak, Beer if Strong (Separating equilibrium 2)3.Player 1: Quiche if Weak, Quiche if Strong (Pooling equilibrium 1)4.Player 1: Beer if Weak, Beer if Strong (Pooling equilibrium 2)Option 1: Player 1: Beer if Weak, Quiche if Strong (Separating equilibrium 1)Step 1: Start with actions of Player 1 (both types)Beer if Weak, Quiche if StrongStep 2:Find Player 2’s beliefs based on actions of Player 1 (in both information sets)Information sets are reached, so beliefs are calculated based on actions of Player 1.Step 3: Find optimal response of Player 2 (in both information sets)In the left information set, Player 1 knows that Player 2 is of Weak type and Fighting is optimal (3>0).In the right information set, Player 2 knows that Player 1 is of Strong type and Not Fighting is optimal(2>0).Step 4: Check if both types of Player 1 play optimal responseWeak type of Player 1 is satisfied, but Strong type of Player 1 wants to deviate and get higher profit.So this is not WPBE.Step 5: Move to the next possible actions of Player 1Option 2: Player 1: Quiche if Weak, Beer if Strong (Separating equilibrium 2) Step 1: Start with actions of Player 1 (both types)Quiche if Weak, Beer if StrongStep 2:Find Player 2’s beliefs based on actions of Player 1 (in both information sets)Information sets are reached, so beliefs are calculated based on actions of Player 1.Step 3: Find optimal response of Player 2 (in both information sets)In the left information set, Player 1 knows that Player 2 is of Strong type and Fighting is optima (4>1).In the right information set, Player 2 knows that Player 1 is of Weak type and Fighting is optimal(1>0).Step 4: Check if both types of Player 1 play optimal responseNo type of Player 1 wants to deviate because the payoff would be lower. So this is WPBE.Step 5: Move to the next possible actions of Player 1Option 3: Player 1: Quiche if Weak, Quiche if Strong (Pooling equilibrium 1) Step 1: Start with actions of Player 1 (both types)Quiche if Weak, Quiche if StrongStep 2:Find Player 2’s beliefs based on actions of Player 1 (in both information sets)Right information set is reached, so beliefs are calculated based on actions of Player 1. Left information set is not reached, so we are free to choose any beliefs, any set of beliefs is consistent with Player 1’s actions. Denote beliefs p and 1-p.Step 3: Find optimal response of Player 2 (in both information sets)In the right information set, Player 2 calculates expected payoff of playing Fight and NotFigt:EP(Fight) = 0.3*1+0.7*0 = 0.3EP(NotFight) = 0.3*0+0.7*2 = 1.4So NotFight is optimal response.In the left information set, optimal response of Player 2 generally depends on value of p (beliefs). However, in this case, Fight dominates NotFight in the left information set and hence Fight is optimal.Step 4: Check if both types of Player 1 play optimal responseActually, no matter what action Player 2 chooses in the left information set, Weak type of Player 1 will always want to deviate. So this is not WPBE.Step 5: Move to the next possible actions of Player 1Option 4: Player 1: Beer if Weak, Beer if Strong (Pooling equilibrium 2)Step 1: Start with actions of Player 1 (both types)Beer if Weak, Beer if StrongStep 2:Find Player 2’s beliefs based on actions of Player 1 (in both information sets)Left information set is reached, so beliefs are calculated based on actions of Player 1. Beliefs in the right information set are denoted p and 1-p because this information set is not reached and hence any set of beliefs is consistent with actions of P1..Step 3: Find optimal response of Player 2 (in both information sets)In the left information set, Player 2 calculates expected payoff of playing Fight and NotFigt:EP(Fight) = 0.3*3+0.7*4 = 3.7EP(NotFight) = 0.3*0+0.7*1 = 0.7So Fight is optimal response.In the right information set, optimal response of Player 2 generally depends on value of p (beliefs).There are two options, either the optimal response is Fight, but then Weak Player 1 would want to deviate; or the optimal response is NotFight. In this case no type of Player 1 would want to deviate.So the beliefs have to be set such that expected payoff from NotFight is larger than from Fight.EP(Fight) < EP(NotFight) => 1*p + 0*(1-p) < 0*p + 2*(1-p) => p<2/3Step 4: Check if both types of Player 1 play optimal responseNo type of Player 1 wants to deviate because the payoff would be lower. So this is WPBE.Step 5: Move to the next possible actions of Player 1. There are no more possibilities.There are two WPBE in this game:1.Weak P1 –Quiche, Strong P1 –Beer; P2 Fight if Beer and Fight if Quiche and p=0 in the leftinformation set and p=1 in the right information set.2.Weak P1 – Beer, Strong P1 – Beer; P2 – Fight if Beer and NotFight if Quiche and p<2/3.。