博弈论答案第二章

2.1Proof:weshouldshowthatAmaximizes()()cpIAIA+,thisisequivalenttoshow()()0cpIAIA′′+=.Solvingthegamebybackwardinduction:Atstage2,givencpIandI,parentchoosesBtomaximizehispayoff:max()()pcBVIBkUIB−++FOC:()()0cpkUIBVIB′′+−−=*Atstage1,childchoosesAtomaximizehispayoff:max(())cAUIAB+FOC:(())(())0ccBUIABIAA∂′′++=∂iFrom(*),wehave:()()()()()()()()()()()()()()()()()()cpcppccpcpcppccpcpcpBBBIAIAAIIVIBkUIBIAIAkUIBVIBkUIBVIBVIBkUIBIAIAkUIBVIBkUIBVIB∂∂∂′′=+∂∂∂′′−−′′+′′=−−′′′′′′′′++−++−′′−′′+′′=−+′′′′′′′′++−++−iiiiii(())(())()()(())(()()())()()()()()(())(()())0()()()()0(ccpccccpcpcppccpcpcpBUIABIAAVIBkUIBUIABIAIAIAkUIBVIBkUIBVIBVIBUIABIAIAkUIBVIBIAIAS∂′′++∂′′−′′−+′′′′=+++′′′′′′′′++−++−′′−′′′=++=′′′′++−′′⇒+=iiiiii()0,()0()0)inceUVandU′′′′′iiiAnotherapproach:Solvingthegamebybackwardinduction:Atstage2,givencpIandI,parentchoosesBtomaximizehispayoff:max()()pcBVIBkUIB−++FOC:()()0cpkUIBVIB′′+−−=*Atstage1,childchoosesAtomaximizehispayoff:max(())cAUIAB+FOC:(())(())0ccBUIABIAA∂′′++=∂iBecauseUisincreasingandstrictlyconcave,so(())0(())0ccBUIABIAA∂′′+⇒+=∂（1）(*)对A求偏导：(())(())(())(())0ccppBBkUIABIAVIABIAAA∂∂′′′′′′++−−−=∂∂(())(())(())(())0ppccBBVIABIAkUIABIAAA∂∂′′′′′′⇒−−=++=∂∂(Since(())0cBIAA∂′+=∂)BecauseVisstrictlyconcave,so(())0pVIAB′′−⇒(())0pBIAA∂′−=∂(2)(1)+(2)()()0cpIAIA′′⇒+=2.2Proof:At,first,solvingthegamebybackwardinduction:Atstage2,givenS,parentchoosesBtomaximizehispayoff:12max()(()())pcBVIBkUISUSB−+−++FOC:2()()0pVIBkUSB′′−−++=(*)Atstage1,childmaximizeshispayoff:12max()()cSUISUSB−++FOC:12()()(1)0cBUISUSBS∂′′−−+++=∂From(*),wehave22()()()pkUSBBSVIBkUSB′′+∂=−∂′′′′−++So2122122()()()(1)()()()()()()()0cppcpkUSBUISUSBVIBkUSBVIBUISUSBVIBkUSB′′+′′−−++−′′′′−++′′−′′=−−++′′′′−++=iHence,inthegame,childchooses*Ssuchthat:**12*2()()()0()()pcpVIBUISUSBVIBkUSB′′−′′−−++=′′′′−++iOntheotherhand,ifchildchoosesS′tomaximize12()()cUISUSB−++,whereBisexogenous,thenS′satisfies:12()()0cUISUSB′′′′−−++=Weneedtoshow*SS′.Denote1212()()(),()()()0ccfSUISUSBfSUISUSB′′′′′′′=−−++=−++***12****2122**22**22*2()()()()()()()()()()()()()()()()0()cpcpppfSUISUSBVIBkUSBUISUSBUSBVIBkUSBVIBkUSBkUSBUSBVIBkUSBfS′′=−−++′′′′−+′′′=−−++++′′′′′′′′−++−++′′+′=+′′′′−++′=iiiSo*SS′.Ifchildsavemore,i.e.S′,boththeparent’sandchild’spayoffscouldbeincreased.2.4Solvingthegamebybackwardinduction:InPeriodtwo:given1c,partner2choose2c,theminimizeof2ctocompletetheprojectis1Rc−.If2121(),VRccRc≥−=−,bothreceiveV.If212(),0VRcc−=,bothreceive0.InPeriodone:partner1choose1c.Considerfourcases:(1)if(1)RVδ≤−,thatis2VRVδ−≥,1cR=,partner1willcompletetheprojecthimself.So20c=.(2)If(1)VRVδ−≤,thatis2VRVδ−and2VR≥,120,ccR==.(3)If(1)VRVδ+≥,thatis2RV,and2()RVVδ−≤,theminimizeof1ctocompletetheprojectsatisfies:211(),VRccR=−,So1cRV=−,2cV=.(4)If(1)RVδ+,thatis2()RVVδ−,itisnotworthcompletingtheproject,hence120,0cc==.2.7Inthesubgame,theequilibriumis1()1iiqLawn==−+.Atstage1,theunionchooseswtomaximizeitsutility:max()()1awna−−+FOC:202aaawa−=⇒=.Thenpayoffsoftheunionare2()12aawnn−+,whichisincreasingwithn.Ifnincreases,thetotaloutputincreases,sodoesthedemandforlabor,sotheunion’utilityincreases.2.13Proof:Themonopolypriceis2acp+=.Ifthefirmsusetriggerstrategies,thenifthereisnofirmdeviate,bothget21()22ac−ioneverystagegame,andthetotaldiscountedprofitis21()221acδ−−i.Thepayofffromdeviatingonanstageis2()2ac−.ForthetriggerstrategiestobeSPNE,wemusthave221()22()12acacδ−−≥−i,thatis12δ≥.2.14Themonopolypriceis,22HLHLacacpp++==Ifthefirmsusetriggerstrategies,andthereisnofirmdeviate,inperiodwithdemandHa,thetotalpayoffis22211(()(1)())12222()221HLHacacacδππδ−−+−−+−iiii,inperiodwithdemandLa,thetotaldiscountedpayoffis22211(()(1)())12222()221HLLacacacδππδ−−+−−+−iiii.InperiodwithdemandHa,payofffromdeviatingis2()2Hac−;inperiodwithdemandLa,payofffromdeviatingis2()2Lac−.ForthetriggerstrategiestobeSPNE,wemusthave222211(()(1)())12222()()2212HLHHacacacacδππδ−−+−−−+≥−iiiiAnd222211(()(1)())12222()()2212HLLLacacacacδππδ−−+−−−+≥−iiii22221()22111(()(1)())()222222HHLHacacacacδππ−⇒≥−−−+−+iiiiiThelowestofδsuchthatthefirmscanusetriggerstrategiestosustainthesemonopolypricelevelsinaSPNEis:22221()22111(()(1)())()222222HHLHacacacacππ−−−−+−+iiiiiForeachvalueofδbetween1/2and*δ,supposethehighestpriceis()pδwhenthedemandishigh.Ifthefirmsusetriggerstrategies,andthereisnofirmdeviate,inperiodwithdemandHa,thetotalpayoffis211((())(())(1)())1222(())(())21LHHacappcappcδπδδπδδδ−−−+−−−+−iiii,inperiodwithdemandLa,thetotaldiscountedpayoffis2211((())(())(1)())1222()221LHLacappcacδπδδπδ−−−+−−+−iiii.InperiodwithdemandHa,payofffromdeviatingis(())(())Happcδδ−−;inperiodwithdemandLa,payofffromdeviatingis2()2Lac−.ForthetriggerstrategiestobeSPNE,wemusthave211((())(())(1)())1222(())(())(())(())21LHHHacappcappcappcδπδδπδδδδδ−−−+−−−+≥−−−iiiiAnd22211((())(())(1)())1222()()2212LHLLacappcacacδπδδπδ−−−+−−−+≥−iiiiThenwecangetthehighestprice()pδfromthesetwoinequalities.2.15Themonopolyoutputis2acn−.Ifthefirmsusetriggerstrategies,whenthereisnofirmdeviate,totaldiscountedpayoffofeachfirmis:2()2(1)acnδ−−.Givenotherfirms’equilibriumstrategies,payofffromdeviating:22()(1)()121acacnnnδδ−−+++−.Fort