高级微观经济理论_英文参考课件-auction.pdf
2.4 Applications:Auctions2.4.1 IntroductionWhen?Why auctions?ExamplesStatic Games of Incomplete Information1/?Four common auctions oral(open)auctions:bidders hear each others bids andcan make counteroffers;each bidder knows his rivals.(1).English auction.(2).Dutch auction.written(closed)auctions:bidder submit their bids si-multaneously without revealing them to others;often bid-ders do not even know how many rival bidders participate.(3).firstprice sealedbid auction(FPA).(4).second-price sealed-bid auction(SPA or Vickreyauction).Static Games of Incomplete Information2/?Types of auctionsSingle-unit and multi-unit auctions:whether there is oneunit or multi-unit of goods available for sale.Classified according to information structure:private value auctions affiliated value auctions common value auctionsStatic Games of Incomplete Information3/?Symmetric independent private value(SIPV)model a single object is put up for sale to one of n bidders;all bidders are indistinguishable(symmetric);each Bidder knows his valuation,no one else does(privatevalue);unknown valuations are independently and identically dis-tributed(iid)and continuous random variables(indepen-dent);bidders are risk neutral,and so is the seller.Static Games of Incomplete Information4/?Some basic results Dutch auction is strategically equivalent to the first pricesealedbid auctions.English auction and secondprice sealedbid auctions Generally,the two auctions are not strategically equiv-alent.In the IPV case,the two auctions become strategicallyequivalent.In both auctions,it is a weakly dominant strategy to bidtruthfully.Well focus on FPA and SPA.Static Games of Incomplete Information5/?The auction gameA single object is put up for sale to one of n bidders.Bid-der i whose bid is biassigns a value of Vito the object.Heknows the realization viof Viand only that the other bid-ders values are i.i.d according to the distribution functionF on 0,1 with density function f.Both the bidders andthe seller are risk neutral.Static Games of Incomplete Information6/?Strategic form representation of the auction game(1)the set of the players:(2)types:(3)bidders strategies:(4)prior belief:(5)payofffunctions:Static Games of Incomplete Information7/?2.4.2.Firstprice sealed-bid auction(FPA)In a firstprice auction,a bidder who submits a bid bihasthe payoffs:ui=(vi bi,ifbi maxjibj,0,ifbi maxjibj.If bi=maxjibj,the object goes to each winning bidderwith equal probability.Static Games of Incomplete Information8/?How to bid in the FPA?bi Y(2:n).Y(n:n).Y(k:n):the“k-th order statistic”.Denote the highest order statistic among the n 1 rivalbidders.by Y1=Y(1:n1)Static Games of Incomplete Information10/?Equilibrium strategies in FPAProposition 1.Symmetric equilibrium strategies in a firstprice auction are given byI(v)=EY1|Y1 v.v:the value of the highest bidderY1:the highest value among the rivals(the second highestvalue).Static Games of Incomplete Information11/?ProofSuppose that all the other bidders employ the same equilib-rium strategy(.),and that(.)is strictly increasing,whatis the best reply of bidder i?Bidder i who has valuation v but pretends to have x bidsb=(x),what is the probability of his winning?PrWin=Pr(Y1)(x)=PrY1 x=G(x)G:the distribution function of Y1.Static Games of Incomplete Information12/?The bidders expected profit:u(v,x)=G(x)(v (x).Differentiate the right side w.r.t.x and evaluate at x=v,yields(v)=1G(v)Zv0yg(y)dy=EY1|Y1 maxjibj,0,ifbi b.In this case,its not profitable to bid lessthan vi.Case 2:vi b.In this case,it is not profitable to bid morethan vi.Convince yourself!Static Games of Incomplete Information16/?2.4.4 The expected payment by a bidderProposition 3.With independently and identically privatevalues,the expected payment by a bidder in a firstpriceauction is the same as that in a secondprice auction.Inboth cases,the expected payment by a bidder with value visPA=G(v)EY1|Y1y+nF(x)n1(1 F(y)|zPrexactly one y=F(y)n+nF(y)n1?1 F(y)?The associated probability density function isf2(y)=n(n1)(1F(y)F(y)n2f(y)=n(1F(y)g(y).Static Games of Incomplete Information20/?Proof of Proposition 4Generally,the expected revenue of the seller is just the sumof the ex-ante expected payments of the bidders:The ex ante expected payment of a particular bidder isEPA=Z10(Zx0yg(y)dy)f(x)dx=Z10(Z1yf(x)dx)yg(y)dy=Z10(1 F(y)y(n 1)Fn2(y)f(y)dyStatic Games of Incomplete Information21/?The expected revenue to the seller:ERA=nEPA=nZ10(1 F(y)y(n 1)Fn2(y)f(y)dy=Z10yf2(y)dy=EY(2:n).Static Games of Incomplete Information22/?Calculating the expected revenue:an alternativeThe expected revenue of the seller in FPA:ERI=E(V(1:n)=Z10I(v)f1(y)dyThe expected revenue of the seller in SPA:ERII=E(V(2:n)=EV(2:n)=Z10II(v)f2(y)dy.Static Games of Incomplete Information23/?ExampleExample 2.If values are uniformly distributed on 0,1,then the expected revenues of the seller in both the FPA andSPA are ERA=n1n+1.Method 1:The expected payment by a bidder with value v is:The total expected payments of the bidders is:Thus,the expected revenue of the seller is:Static Games of Incomplete Information24/?Method 2 Expected revenue in the FPA:Eb(V(1:n)=Expected revenue in the SPA:EV(2:n)=Recall that:f2(y)=n(n 1)(1 F(y)F(y)n2f(y).Static Games of Incomplete Information25/?Revenue equivalenceProposition 5.Suppose that valuations are independentlyand identically distributed and all bidders are risk neutral.The auctions that award the 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:Static Games of Incomplete Information26/?
