【博弈论基础】(吉本斯)课后习题答案英文版

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1、2.1Proof:weshouldshowthatAmaximizesI()AIA+(),thisisequivalenttocpshowIAIA′′()+=()0.cpSolvingthegamebybackwardinduction:Atstage2,givenIandI,parentchoosesBtomaximizehispayoff:cpmax(VIBkUIB−+)(+)pcBFOC:kUI′′()()+−BVI−=B0*cpAtstage1,childchoosesAtomaximizehispayoff:max(()U

2、IAB+)cA∂BFOC:UIABIA′(())(()++i′)0=cc∂AFrom(*),wehave:∂∂BB∂B=+iiIA′′()IA()cp∂∂AI∂IcpkUI′′()+B−−VIB′′()=−c′′−piiI()AI()AcpkUI′′()()++BVI′′−BkUI′′()()++BVI′′−BcpcpkUI′′()+BVIB′′()−=−c′′+piiI()AI()AcpkUI′′()()++BVI′′−BkUI′′()()++BVI′′−Bcpcp∂BUIABIA′(())(()++i′)cc∂A−+kUI′′(

3、)BVIB′′()−=++′′′′c+pUIABIA(())(()iiiIA()IA())cccpkUI′′()()++BVI′′−BkUI′′()()++BVI′′−BcpcpVIB′′()−=+′p′′+=UIAB(())ii(()IAIA())0ccpkUI′′()()++BVI′′−Bcp⇒IAIA′′()+=()0(SinceU′′()0,()0ii>

4、parentchoosesBtomaximizehispayoff:cpmax(VIBkUIB−+)(+)pcBFOC:kUI′′()()+−BVI−=B0*cpAtstage1,childchoosesAtomaximizehispayoff:max(()UIAB+)cA∂BFOC:UIABIA′(())(()++i′)0=cc∂ABecauseUisincreasingandstrictlyconcave,so∂BUIAB′(())0(()+>⇒IA′+=)0（1）cc∂A∂∂BB(*)对A求偏导：kUIABIA′′(())((

5、)++′)−−−VIABIA′′(())(()′)0=ccpp∂∂AA∂∂BB∂B⇒VIABIA′′(())(()pp−−′)=++kUIABIA′′(())(()cc′)0=(Since(())0IAc′+=)∂∂AA∂A∂BBecauseVisstrictlyconcave,soVIAB′′(())0−>⇒(()IA′−=)0(2)pp∂A(1)+(2)⇒IAIA′′()+=()0cp2.2Proof:At,first,solvingthegamebybackwardinduction:Atstage2,givenS,paren

6、tchoosesBtomaximizehispayoff:max(VIBkUISUSB−+)((−+)(+))pc12BFOC:−−VIBkUSB′()()++=′0(*)p2Atstage1,childmaximizeshispayoff:maxUISUSB(−+)(+)12cS∂BFOC:−−UISUSB′′()()+++=(1)012c∂SFrom(*),wehave∂BkU′′()S+B2=−∂SVIBkUSB′′()()−+′′+p2kU′′()S+B−−UISUSB′′()()++−(12)12cVIBkUSB′′()(

7、)−+′′+p2VIB′′()−=−′′−++pSoUISUSB()()i12cVIBkUSB′′()()−+′′+p2=0*Hence,inthegame,childchoosesSsuchthat:VIB′′()−−−++′′**p=UISUSB()()i012cVIBkUSB′′()()−+′′*+p2Ontheotherhand,ifchildchoosesS′tomaximizeUISUSB()()−++,12cwhereBisexogenous,thenS′satisfies:−−UISUSB′′()()′′++=012

8、c*WeneedtoshowSS<′.DenotefSUISUSBfSUISUSB()()()=−′−+′+,′()()()=′′−+′′+<012cc12fS()**=−UIS′′(−)+USB(*+)12cVIB′′()−kU′′