An Efficient Dynamic Auction for Heterogeneous Commodities Lawrence M Ausubel september 2000 Authors Oren Rigbi Damian Goren The problem An auctioneer wishes to allocate one or more units of each of K heterogeneous commodities to n bidders The Lecture s Contents Preface Example Presentation of the Model Equilibrium of the dynamic auction Relationship with the Vickrey auction Conclusions Situations abound in diverse industries in which heterogeneous commodities are auctioned On a typical day the U S Treasury sells Some 8 billion in three month bills Some 5 billion in six month bills Vickrey Auction 1 The second price auction is commonly called the Vickrey auction named after William Vickrey For one commodity The item is awarded to highest bidder at a price equal to the second highest bid Vickrey Auction 2 For K homogenous Commodities The items are awarded to the highest bidders 1 i nThe price of i s the unit is calculated by the price that would have been paid for this unit in case that the bidder that won this unit wouldn t have participated the auction Example with 2 commodities Suppose that the supply vector is 10 8 i e 10 commodities of A are available and 8 commodities of B and suppose that there are n 3 bidders Price Vector Bidder 1 Bidder 2 Bidder 3 p1 3 4 5 4 5 4 5 4 p2 4 5 4 4 5 4 4 3 p3 5 7 4 3 4 4 4 1 p4 6 7 4 3 4 4 3 2 p5 7 8 4 2 3 4 3 2 For Example Bidder 1 The vector demanded was 4 2 A units p1 1 p2 1 p3 1 p4 1 B units p1 1 p2 1 p3 1 p4 0 Sums to 4 Sums to 2 The Model 1 A seller wishes to allocate units of each of K heterogeneous commodities among n bidders N 1 n The seller s available supply will be denoted by S S S 1 k The model 2 Bidder i s consumption vector 1 k xi xi xi X i k X i is a subset of Bidders are assumed to have pure private values for the commodities Bidder i s value is given by the function Ui X i 1 k k p p p The price vector The model 3 Bidder i s net utility function Vi p U i xi p x i xi X i Bidder i s demand correspondence Qi p xi X i U i xi p xi Vi p Walrasian equilibrium A price vector n i i 1 p and a consumption vector x xi Qi p n for every bidder s t x i 1 i S i 1 n and T is a finite time so that with pevery t associate a price vector For t 0 T we Sincere Bidding xi t Bidder i is said to bid sincerely ifqi p t t 0for T all qi the function is a measurable Qi selection from the demand correspondence xi t and t is the desired vector by bidder i at the time Gross Substitutability satisfies gross substitutability if for any and p two vectors xsuch that p price p i Qi p U i xi p xi Qi p xi k xik and k for anyp k p k there exists such that any commodity such that for 2 commodities that are not substitutable Assume that there are 5 left shoes and 5 right shoes The utility function is U R L 10 for the first couple 8 for the second couple etc then for p 4 3 the demand would be 2 2 but for p 4 5 the demand would be 1 1 2 commodities that are substitutable Assume that there are 5 red shirts and 5 blue shirts The utility function is U R B 10 for the first shirt 8 for the second shirt etc then for p 6 4 the demand would be 0 4 but for p 6 8 the demand would be 3 0 Crediting Debiting 1 In the next few slides we will develop the payment equation for the case of 1 commodity Bidder 1 Bidder 2 Bidder 3 Example p0 4 2 2 2 p1 5 2 2 1 p2 6 2 1 1 Crediting Debiting 2 For k 1 one commodity x i t j i x j t ci t max 0 S x i t ci t sup ci t t 0 t the payment of bidder i T y i T p t dci t p t ci t dt 0 Crediting Debiting 3 every time it becomes a foregone conclusion that bidder i will win additional units of the homogeneous good she wins them at the current price and with sincere bidding she gets the same outcome of Vickery auction so sincere bidding by every bidder is an efficient equilibrium of the ascending bid auction for homogeneous goods Crediting Debiting 4 Another way of defining the payment is T ai T p 0 S x i 0 p t dx i t 0 suppose that x i t is monotonic and define p 0 p p i where is defined Implicitly by q p S j i j i i Crediting Debiting 5 The case of debiting occurs only when x i t is not monotonic and this can be only when we are talking about heterogeneous commodities K heterogeneous commodities 1 K ascending clocks described continuous piecewise smooth vector valued functionp t 0 T k such that p t p 1 t p k t bidder i bids according to the vector valued 1 k x t x t x function i i i t from 0 T to X i T the K commodity K case payment equation is k k k k k ai T k 1 p 0 s x i 0 p t dx i t 0 K heterogeneous commodities 2 Lemma 1 If the price p t is any piecewise k 0 T smooth function from to and if each bidder j i bids sincerely for all j i t 0 T and for all then the integral T p t dx i t is independent of the path 0 from p 0 to p T and K heterogeneous commodities 3 equals U i q i p T U i q i p 0 U j i j q j p T U j q j p 0 K heterogeneous commodities 4 DEFINITION 1 The set of all final prices attainable by i Pi denoted is the set of all prices at which the auction may terminate given that all bidders j i bid sincerely the specified price adjustment process and all constraints on the strategy of bidder i notice that any attainable final price p P i implies an associated allocation consisting of for each bidder j i q j p and for bidder S q i p i K heterogeneous commodities 5 THEOREM 1 If each bidder j i bids sincerely and if bidder i s bidding is constrained so as to k 0 T price …
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