清华大学经济博弈论期末考试04

0420041.(20points)LucyofferstoplaythefollowinggamewithCharlie:“Letusshowpenniestoeachother,eachchoosingeitherheadsortails.Ifwebothshowheads,Ipayyou$3.Ifwebothshowtails,Ipayyou$1.Ifthetwodon’tmatch,youpayme$2.”Charliereasonsasfollows.“Theprobabilityofbothheadsis1/4,inwhichcaseIget$3.Theprobabilityofbothtailsis1/4,inwhichcaseIget$1.Theprobabilityofnomatchis1/2,andinthatcaseIpay$2.Soitisafairgame.”Isheright?Ifnot,(a)whynot,and(b)whatisLucy’sexpectedprofitfromthegame?gametable,5points;solutions,7points;(a),4points;(b),4points203121/431/411/22ab57a4b45CHARLIEHeadTailHead-32LUCYTail2-113/85/86a1/24b1/84-1/82.(20points)Youhavetodecidewhethertoinvest$100inafriend’senterprise,whereinayear’stimethemoneywillincreaseto$130.Youhaveagreedthatyourfriendwillthenrepayyou$120,keeping$10forhimself.Butinsteadhemaychoosetorunawaywiththewhole$130.Anyofyourmoneythatyoudon’tinvestinyourfriend’sventure,youcaninvestelsewheresafelyattheprevailingrateofinterestr,andget$100(1+r)nextyear.(a)Drawthegametreeforthissituationandshowtherollbackequilibrium.(8points)Nextsupposethisgameisplayedrepeatedlyinfinitelyoften.Thatis,eachyearyouhavethe104opportunitytoinvestanother$100inyourfriend’senterprise,andtheagreementistosplittheresulting$130inthemanneralreadydescribed.Fromthesecondyearonward,yougettomakeyourdecisionofwhethertoinvestwithyourfriendinthelightofwhetherhemadetheagreedrepaymentalltheprecedingyears.Therateininterestbetweenanytwosuccessiveperiodsisr,thesameastheoutsiderateofinterestandthesameforyouandyourfriend.(b)Forwhatvalueofrcantherebeanequilibriumoutcomeoftherepeatedgame,inwhicheachperiodyouinvestwithyourfriendandherepaysasagreed?(Hint:onlycheckasingledeviationofyourfriendwhenbothofyoualwaysagreedpreviously.)8points(c)Iftherateofinterestis10%peryear,cantherebeanalternativeprofit-splittingagreement,whichisanequilibriumoutcomeoftheinfinitelyrepeatedgame,whereeachperiodyouinvestwithyourfriendandherepaysasagreed?(4points)2010013012010130r100(1+r)a8100130rbr8c104(a)4elsewhereRunaway100(1+r)04204(b)130-10=12010-0=1010/(1+r)2+10/(1+r)3+……=10/r12010/rr1/12≈8.3%.r1/128.3%8(c)b(b)(130-b)b/r,(130-b)b/0.1,b130/11≈11.8.130/1111.843.(20points)Supposeaclassof100studentsiscomparingthechoicebetweentwocareers–lawyersorengineers.Anengineergetsatake-homepayof$100,000peryear,irrespectiveofthenumberswhochoosethiscareer.Lawyersmakeworkforoneanother;so,asthetotalnumberoflawyersincreases,theincomeofeachlawyerincreases–uptoapoint.Ultimately,thecompetitionbetweenthemdrivesdowntheincomeofeach.Specifically,ifthereareNlawyers,eachwillget100N-N2thousanddollarsayear.Theannualcostofrunningalegalpractice(officespace,secretary,paralegals,accesstoonlinereferenceservices,andsoforth)is$800,000.Soeachlawyertakeshome100N-N2-800thousanddollarsayearwhenthereareNofthem.(a)Drawagraphshowingthetake-homeincomeofeachlawyerontheverticalaxisandthenumberoflawyersonthehorizontalaxis.(2points)(b)Whencareerchoicesaremadeinanuncoordinatedway,whatarethepossibleequilibriumoutcomes?(9points)(c)Nowsupposethewholeclassdecideshowmanyshouldbecomelawyers,aimingtomaximizethetotaltake-homeincomeofthewholeclass.Whatwillbethenumberoflawyers?(9points)20100100,000N100N-N2800,000100N-N2-800Na2b9c9(a)2(b)N*=0,N*=90.:N*=10.3304(d)3T(N)=Np(N)+(100-N)s(N)=N(100N-N2-800)+(100-N)100=10,000-900N+100N2-N3.3T’(N)=-900+200N-3N2=0⇒N≈62,orN≈5.N=62N=5T(62)≈100,272623420ThiefYouVc1c2c1V,c2Va6bV,c1,c2V2EvolutionaryGamepopulationphenotypesWFThcth+t-c404de-dh-c-edh-d-c-dc3dAll-WinvadesuccessfullyAll-F2ehc–d-t02fprimarycriterionAll-T2g(e)(f)TESS3a3-c2,-c1V-c2,-Vp0,-c10,01-pq1-qp=c1/V1-p=1-c1/Vq=1-c2/V1-q=c2/V20-c11bVqpc1c22c3COLUMNTWFT-d,-d-d,h-c-eh-d,-cWh-c-e,-dh+t-c,h+t-ch+t-c,h+t-cROWF-c,h-dh+t-c,h+t-ch+t-c,h+t-cdprimarycriterionh+t-csecondarycriterionh+t-c2e-dh+t-ch+t-c-dhc-t-dh-dh+t-ch-dh+t-cc-t-d0504hc-t-d02fh-c-e-dh-c-e-d-c-c-dcd(ii)c-t-d02gx(1-x)x(-d)+(1-x)(h-d)=(1-x)h-dh+t-ch+t-c(1-x)h-dx(c-t-d)/hhc-t-d0(c-t-d)/h3520-10,-2,2,52,-4,-8,-10a4bstrategicmoves32,-6,-8,-10cabb4dpBrinkmanshipeffectivenessacceptabilityp7e2604a22,-10-10,-85,-4-2,2-222b15-42c22,-10-10,-65,-8-2,2(a)1b1dq1370481-42-10qq0.61p(-2-8q)+(1-p)5-2,q0.875(1-p)/pp=0.6q=0.61eqq1-8-6-101