lecture_10(博弈论讲义game theory(mit))

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1、1IPVandRevenueEquivalence:KeyassumptionsAuctions2:•Independenceofvalues.•Risk-neutrality.Departuresfrom•Nobudgetconstraints.symmetricIPV•Symmetry(Sameallocationrule!).Multi-unitauctions•Otherconsiderations:—Collusion—Resalepossibilities2Risk-aversebiddersFormally,supposeγ:[0,w]→R+

2、isanequilibriumstrategy(incr.diff.).maxEU=maxG(z)u(x−γ(z)).•Eachbidderhasu:R+→Rwithu(0)=0,zzu0>0,andu00<0.FOC:g(z)u(x−γ(z))−G(z)u0(x−γ(z))γ0(z)=0.Insymmetriceqm:Proposition:Withrisk-aversesymmetricbidderstheexpectedrevenueinafirst-priceauctionisgreater0u(x−γ(z))g(x)γ(z)=,thaninaseco

3、nd-priceauction.u0(x−γ(z))G(x)0g(x)β(z)=(x−β(x)).G(x)Intuition:Considerabidderinthefirst-priceauction.u(y)Byreducingcurrentbidbbysome∆,abiddergainsNotethatforally>0,u0(y)>y.Therefore,∆whenwins,butincreasesaprobabilityoflosing,0g(x)whichhasagreatereffectonexpectedutility.γ(z)>(x−γ(x)

4、).G(x)Result:moreaggressivebiddinginthefirst-priceauc-Nowβ(x)>γ(x)⇒γ0(x)>β0(x).Togetherwithtion.β(0)=γ(0)=0weobtainβ(x)<γ(x).Nochangeinstrategiesinthesecond-priceauction.Proof:Inthesecond-priceauction:3Budget-constrainedbiddersβII(x,w)=min{x,w}.Define(effectivetype)xII∼(x,w)asthetype

5、that•Everybidderobtainsvalue(signal)Xi∈[0,1]iseffectivelyunconstrainedandsubmitsthesamebidandabsolutebudgetWi∈[0,1].as(x,w).CanbefoundasasolutiontoβII(x,w)=βII(xII,1)=xII.•(Xi,Wi)areiidacrossbidders.(XiandWineedII(N)LetYbethesecondhighestoftheequivalentnotbeindependent.)2values,xII

6、i,amongNbidders.Itsdistributionis³´N−1GII(z)=FII(z),Proposition:Withbudget-constrainedbidderstheex-pectedrevenueinafirst-priceauctionisgreaterthanwhereFII(z)istheprobabilitythatβII(x,w)=βII(xII,1)=inasecond-priceauction.(providedsymmetricequi-xII

7、III(N)Intuition:Thebidsinsecond-priceauctionarehigherE[R]=EY.2onaverageandsoaremoreoftenconstrained.Inthefirst-priceauction:Supposeasymmetricin-(Notenough:playerswillreducebidsinthefirst-pricecreasingequilibriumexistswithauction).βI(x,w)=min{β(x),w}.DefinexI∼(x,w)asthesolutiontoβI(x,

8、w)=βI(xI,1)=β(xI)