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. The price of 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.

si' )1( ni

Example with 2 commoditiesSuppose 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 1The vector demanded was (4,2)

A units:

p1: +1

p2: +1

p3: +1

p4: +1

Sums to 4.

B units:

p1: +1

p2: +1

p3: -1

p4: 0

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 = (S1 ,...,Sk

The model (2)

• Bidder i’s consumption vector -

• is a subset of • Bidders are assumed to have pure private

values for the commodities .• Bidder i’s value is given by the function

• The price vector -

ikiii Xxxx ),...,( 1

ii XU :kkppp ),...,( 1

iXk

The model (3)

• Bidder i’s net utility function

• Bidder i’s demand correspondence

})({)( iiiXx

i xpxUpVii

)}()(:{)( pVxpxUXxpQ iiiiiii

Walrasian equilibrium

• A price vector and a consumption vector

for every bidder s.t. For

and

• T is a finite time ,so that with every we associate a price vector

*p

niix 1

*}{ )( ** pQx ii

ni ,...,1 Sxn

i i 1

*

],0[ Tt )(tp

Sincere Bidding

• Bidder i is said to bid sincerely if for all .

• the function is a measurable selection from the demand correspondence .

• and is the desired vector by bidder i at the time .

))(()( tpqtx ii

],0[ Tt

)(iq

)(iQ

)(txi

t

Gross Substitutability

• satisfies gross substitutability if for any and two price vectors such that

and for any there exists

such that for any commodity such that

) (i ix U

pp '

p 'p)( pQx ii

)( '' pQx ii ki

ki xx '

k kk pp '

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.

• Example: Bidder 1 Bidder 2 Bidder 3

p0 = 4 2 2 2

p1 = 5 2 2 1

p2 = 6 2 1 1

Crediting & Debiting (2)

• For k=1 (one commodity)

• the payment of bidder i

ij ji txtx )()(

)(sup)(],0[

tctc itt

i

dttctptdctpTy ii

T

i'

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

• suppose that is monotonic, and define where is defined Implicitly by

T

iii tdxtpxSpTa0

)()()]0()[0()(

)(tx i

ipp )0(

Spqij ij )(

ip

Crediting & Debiting (5)

• The case of debiting occurs only when is not monotonic and this can be only when we are talking about heterogeneous commodities . . .

)(tx i

K heterogeneous commodities (1)

• K ascending clocks described continuous, piecewise smooth vector valued function-

such that

• bidder bids according to the vector valued function from to

• the K commodity case payment equation is:

kTtp ],0[:)( ))(),...,(()( 1 tptptp k

i

))(),...,(()( 1 txtxtx kiii ],0[ T iX

K

k

Tki

kki

kki tdxtpxspTa

10

)}()()]0([)0({)(

K heterogeneous commodities (2)

• Lemma 1: If the price is any piecewise smooth function from to and if each bidder ( ) bids sincerely for all and for all then the integral

is independent of the path

from to and ...

)(tp

],0[ T

ij

k

ij ],0[ Tt

T

i tdxtp0

)()(

)0(p )(Tp

K heterogeneous commodities (3)

)))0((()))((( pqUTpqU iiii

ij

jjjj pqUTpqU )))]0((()))((([

equals :

K heterogeneous commodities (4)

• DEFINITION 1: The set of all final prices attainable by i,

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 implies an associated allocation consisting of:

for each bidder and for bidder .

iP

iPp

)( pq jij

)( pqS i i

K heterogeneous commodities (5)

• THEOREM 1. If each bidder bids sincerely and if bidder i’s bidding is constrained so as to generate piecewise smooth price paths from to then bidder i maximizes her payoff by maximizing social surplus over allocations associated with .

ij

],0[ T k

iP

K heterogeneous commodities (6)

• THEOREM 1(cont) - Moreover, if a Walrasian price vector w is attainable by bidder i (i.e., if ) then bidder i maximizes her payoff by selecting the derived demand from Walrasian price vector, and there by receives her payoff from a Vickrey auction with a reserve price of p(0).

iPw

Equilibrium of the auction(1)• DEFINITION 2: The triplet (A)–(B)–(C) will be said to be a stable price

adjustment combination for competitive economies if: (A) is a price adjustment process, (B) is a set of assumptions on bidders’ preferences, (C) is a condition on the initial price for convergence (e.g., local, global or universal stability).

and price adjustment process (A) for an economy satisfying bidder assumptions (B) is guaranteed to converge to a Walrasian equilibrium along a piecewise smooth path in accord with initial condition (C).

Equilibrium of the auction(2)

• Example -Let denote the vector of excess demands at time t and let (A) be a continuous and sign-preserving transformation so that the price adjustment process would be : for . Let (B) include the assumption of gross substitutability plus additional assumptions on the economy

n

i i tpqStpZ1

))(())((

)(kH

)))((()( tpZHtp kkk Kk ,...,1

Equilibrium of the auction(3)

• Example(cont) - guaranteeing that the excess demand for each commodity is a continuous function and that a positive Walrasian price vector exists. Let (C) be the condition of global convergence. Then (A)–(B)–(C) is a stable price adjustment combination .

(Arrow,Block and Hurwicz, 1959)

Equilibrium of the auction(4)

• THEOREM 2. Suppose that the triplet (A)–(B)–(C) is a stable price adjustment combination for competitive economies. Consider the auction game where price adjustment is governed by process (A) and bidders have pure private values satisfying assumptions (B). Then, for initial prices p(0) in accord with (C),

Equilibrium of the auction(5)• THEOREM 2(cont) – and if participation in the auction is mandatory: (i) sincere bidding by every bidder is an equilibrium of the auction game; (ii) with sincere bidding, the price vector converges to a Walrasian equilibrium price; (iii) with sincere bidding, the outcome is that of a Vickrey auction with reserve price of p(0).

Theorem 3

It the initial price p(0) is chosen such that the market clears without bidder i at price at p(0)

(i.e., ), if each bidder j i bids sincerely, if bidder i’s bidding is constrained so as to generate piecewise smooth price paths from [0,T] to , and if a Walrasian price vector w is attainable by bidder i (i.e., if w Pi), then bidder i maximizes her payoff by selecting a Walrasian price vector and thereby receives exactly her Vickrey auction payoff.

ijj Spq ))0((

k

An n+1 steps algorithm for calculating payoffs

Step 1

• Run the auction procedure of naming a price p(t), allowing bidders j1 to respond with quantities xj(t) while imposing x1(t)=0, and adjusting price according to adjustment process (A) until such price p-1 is reached that the market clears (absent bidder 1).

... Step n

• Run the auction procedure of naming a price p(t), allowing bidders jn to respond with quantities xj(t) while imposing xn(t)=0, and adjusting price according to adjustment process (A) until such price p-n is reached that the market clears (absent bidder n).

Step n+1

• Run the auction procedure of naming a price p(t), allowing all bidders i=1...n to respond with quantities xi(t), and adjusting price according to adjustment process (A) until such price w is reached that the market clears (with all bidders).

Payoffs Computation

Payment equation:

for bidder n:

for bidder i ( ):

wp n

11 ni

)())0(())0(( wppppp nni

K

k

Tki

kki

kki tdxtpxspTa

10

)}()()]0([)0({)(

Theorem 4

Suppose that the triplet (A)-(B)-(C) is a stable price adjustment combination for competitive economies. Consider the (n+1) step auction game where price adjustment is governed by process (A) and bidders have pure private values satisfying assumptions (B). Then for initial prices p(0) in accord with (C), sincere bidding by every bidder is an equilibrium of the (n+1) step auction game, the price vector converges to a Walrasian equilibrium price, and the outcome is exactly that of a Vickrey auction.

Replicating the outcome of Vickrey Auction

Select any p(0) that has the property that with any one bidder removed, there is still excess demand for every commodity,

i.e.,

for all i and all k.

k

ij

kj Spq

))0((

Conclusions The primary objective of the current design was

really to introduce efficient auction procedures sufficiently simple and practicable that they might actually find themselves adopted into widespread use someday.

For the case of K heterogeneous commodities: * A full Vickrey auction requires bidders to report their utilities over the entire K-dimensional space of quantity vectors. * The current design only requires bidders to evaluate their demands along a one-dimensional path of price vectors.