高级微观经济理论_英文参考课件-gt2.pdf

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1、Chapter 2Static Games of IncompleteInformationGame theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-Table of contents 1.Strategic form representation of static Bayesian games.2.Timing 3.Bayesian Nash equilibrium 4.Application:auctionsGame theory(2)Sta

2、tic games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-1.Strategic form representation1.1 Incomplete information The private information ofplayer i is integrated into his“type”ti Ti.T=T1 T2 Tnt=(t1,.,tn)TTi=T1 Ti1 Ti+1 Tn;ti=(t1,.,ti1,ti+1,.,tn)Tiui(a1,.,an;ti)Game the

3、ory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-Beliefs Prior probability Belief,pi(ti|ti):player is belief about the other playerstypes,ti,given player is knowledge of his own type,ti.Bayes rule:pi(ti|ti)=pi(ti,ti)PtiTipi(ti,ti)Game theory(2)Static gam

4、es of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-1.2 Strategic form representation(1)the set of players,I=1,.,n(2).the players action spaces A1,.,An(3).the players type spaces T1,.,Tn(4)the players beliefs p1,.,pn(5).the players payofffunctions u1,.,un.G=(I;Ai;Ti;pi;ui)

5、.Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-Strategies in static Bayesian gamesIn G=(I;Ai;Ti;pi;ui),astrategy for player i is a func-tion si(ti),where for each type ti,si(ti)specifies the actionai,ai Ai,that type tiwould choose.Game theory(

6、2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-ExampleExample 1.Cournot game with asymmetric informationSuppose there are two firms,firm 1 and 2.They produce ho-mogeneous good and choose outputs simultaneously.Theinverse demand function is P(Q)=1Q,Q=q1+q2.

7、Firm1s marginal cost is c.Firm 2 knows firm 1s cost,but firm1 is uncertain about firm 2s costs,firm 2s cost can be ei-ther high(H)with probability p or low(L)with probability1p.Assume that the players types are independent.Howto describe the game?Game theory(2)Static games of incomplete information(

8、c)by Foxit Software Company,2004 Foxit PDF Editor-Strategic form representationThe set of players:Types:Strategies:Beliefs:Payofffunctions:Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-How to solve a static Bayesian game?Harsanyis Trick:Imperf

9、ect information:At some move in the game theplayer with the move does not know the complete historyof the game so far.Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-2.Timing of a static game with incomplete information Stage 1:“Nature”draws a t

10、ype vector t=(t1,.,tn),ti Tiac-cording to a prior probability distribution p(t)“Nature”reveals tito player i but not to any other player.Stage 2:The players simultaneously choose actions;player i chooses ai,ai Ai.The payoffs ui(a1,.,an;ti)are received.Prior probability distribution and beliefs:Game

11、theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-ExampleExample 2.Cournot game with incomplete informationSuppose there are two firms,1&2.They produce homoge-neous good and choose outputs simultaneously.The inversemarket demand is p(Q)=1 Q,Q=q1+q2.Bot

12、h haveconstant unit costs,which may be either high cHor low,cL.The joint probability distribution is given byPr(cHcH)=Pr(cHcL)=Pr(cLcH)=Pr(cL,cL)=14.Please compute the symmetric Bayesian Nash equilibrium.Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF

13、Editor-3.Bayesian Nash EquilibriumIn G=(I;Ai;Ti;pi;ui),the strategy vector s=(s1,.,sn)is a Bayesian Nash equilibrium if for each player i and foreach of is types ti Ti,si(ti)solves:maxsiSiXtiTipi(ti|ti)ui(s1(t1),.,si1(ti1),si,si+1(ti+1),sn(tn);ti)Note:In a finite static game of incomplete informatio

14、n,there exists at least a Bayesian Nash equilibrium(perhapsin mixed-strategy).Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-Solution for Example 1Solution for Example 1:q2(cH)=1 2cH+c3+1 p6(cH cL)q2(cL)=1 2cL+c3p6(cH cL)q1=1 2c+pcH+(1 p)cL3Int

15、erpretation:Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-Solution for Example 2The symmetric Bayesian Nash equilibrium isq(cH)=4 5cH+cL12q(cL)=4 5cL+cH12Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Fox

16、it PDF Editor-ExampleExample 3.Modification of DAs brother Supposenowplayer 2 has two types.With probability p player 2 haspreferences as in DAs brother(type I),with probability 1pplayer 2 hates to rat on his accomplice(type II).In this case,he pays a psychic penalty equal to 6 years in prison for c

17、on-fessing.Player 1 does not observe player 2s type and al-ways has the preferences as in the DAs brother.Player 2DCCPlayer 1DC0,-2-10,-1C-1,-10-5,-5Type Iplayer 2DCCPlayer 1DC0,-2-10,-7C-1,-10-5,-11Type IIWhat is the BE in this game?Game theory(2)Static games of incomplete information(c)by Foxit So

18、ftware Company,2004 Foxit PDF Editor-2.4 Applications:Auctions2.4.1 IntroductionThe earliest auctions:the bidding for men and wives in Baby-lon around 500 B.C.Why auctions?ExamplesGame theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-Types of auctions

19、Single-unit and multi-unit auctions:whether there is oneunit or multi-unit of goods available for sale.Private value,affiliated value,and common value auc-tions:This is classified according to information structureThe way different bidders valuations are correlated.Game theory(2)Static games of inco

20、mplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-Popular auctions1.English(ascending-bid)auction.2.Dutch Auction.Both English and Dutch auctions are open(oral)auctions.3.First-price sealed-bid auction.4.Second-price sealed-bid auction(Vickrey auction).Differences between open and

21、 close auctions.Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-Equivalent auctions Dutch auction is strategically equivalent to the first-pricesealed bid auction.In both English and second-price sealed bid auctions,bid-ders bid truthfully,and b

22、oth have the same equilibriumpayoff.Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-Revenue equivalenceProposition 1.Suppose that valuations are independentlyand identically distributed and all bidders are risk neutral.The auctions that award th

23、e item to the highest bidder(stan-dard auctions)and the expected payment of a bidder withvaluation zero is zero,yields the same expected revenue tothe seller.Proof:See“Auction Theory by Krishna,pp30-31.Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Ed

24、itor-First-price and second price auctions.Dutch auction a first-price sealed bid auction;English auction a second-price sealed bid auction.Well focus on the first-price and second price auctions andaddress:What are symmetric equilibrium strategies in the first-price and second price auctions?What i

25、s the expected revenue of the seller?Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-2.4.2 First-price sealed-bid auction A single object is put up for sale to one of n bidders.Bidders simultaneously submit their bids(sealed-bid).The bidder with

26、 the highest bid gets the object and paysa price equal to his own bid(first-price),the others do notget anything and do not pay anything.Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-Assumptions(SIPV model)All bidders are indistinguishable(sym

27、metric),they adoptthe same strategy.Each bidder knows the realization viof his own valuationViand that other bidders valuations Vjare continuousrandom variables and independently and identically dis-tributed on the support 0,1(independent).Bidders and the seller are risk neutral.Suppose thesellers o

28、wn valuation and minimum bid are equal to zero.Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-Strategic form of the first price auction(1)playerss set:(2)types:(3)bidder is strategy:(4)Prior belief:(5)payofffunctions:u(x,v)=Prwinning(v b(x).A b

29、idders payoffwhen his valuation is v but pretends tohave valuation x.Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-Equilibrium bid functionProposition 2.In the SIPV model,the equilibrium bid func-tion(strategy)is b=b(v)=n1nv(Bayesian Nash equi

30、lib-rium).Proof:We assume that bidding function bi(vi)is strictly in-creasing(higher valuation leads to higher bid).Suppose symmetric bidders have the same equilibrium bid-ding strategy(function).Given that the other bidders employ the same equilibriumstrategy b,what is the best reply of bidder i?Ga

31、me theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-Probability of winningBidder i who has valuation v but pretends to have x bidsbi(x),he wins with probability:Prwinning=Prmaxjbj(vj)bi(x),j i=Prmaxjvj x=F(x)n1=xn1.Game theory(2)Static games of incomp

32、lete information(c)by Foxit Software Company,2004 Foxit PDF Editor-The bidders expected profit:u(x,v)=Fn1(x)(v b(x).Differentiate the right side w.r.t.x and evaluate at x=v,we obtainb(v)=1Fn1(v)Zv0 xdFn1(x)=(n 1)vn.Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004

33、 Foxit PDF Editor-2.4.3 Second-price sealed-bid auction(Vickrey auction)Bids biare made simultaneously.Valuation viis bidder is private information.The highest bidder wins and pays the second-highest bid.Proposition 3.In the second-price sealed bid auction,it isa weakly dominant strategy to bid acco

34、rding to b(v)=v.Proof:Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-2.4.4.Expected revenue of the sellerProposition 4.Both the firs-price and second-price sealed-bid auctions give the seller the same expected revenue thatequalsn1n+1.ProofThe e

35、xpected revenues for the seller from the first-price and second-price auctions are the expected value ofthe highest and the second highest bids,i.e.RI=Emaxb(v1),.,b(vn)=Eb(V(1:n);(1)RII=Eb(V(2:n)=EV(2:n);(2)Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit P

36、DF Editor-Order StatisticsConsider a sample of n iid random variables(X1,X2,.,Xn),each drawn from the continuous cdf F.Denote the largest,the second largest,etc.,down to lowest,byY(1:n)Y(2:n).Y(n:n).Y(k:n):the“k-th order statistic”or“order statistic of rank k”.Game theory(2)Static games of incomplet

37、e information(c)by Foxit Software Company,2004 Foxit PDF Editor-CDFs the highest order statistic Y(1:n)=Y1F(1:n)(y):=PrY(1:n)y=F(y)nThe associated probability density isf1(y)=nF(y)n1f(y).Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-CDF of the

38、 second highest order statisticF(2:n)(y):=PrY(2:n)y=F(y)n+nF(y)n1?1 F(y)?The associated probability density function isf2(y)=n(n 1)(1 F(y)F(y)n2f(y).Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-Expected revenue in FP and SP auctionsEb(V(1:n)=Z10b(y)f1(y)dy=n 1n+1.EV(2:n)=Z10b(y)f2(y)dy=n 1n+1.Game theory(2)Static games of incomplete information(c)by Foxit Software Company,2004 Foxit PDF Editor-