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 i' s unit (1  i  n) 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 -
xi = ( x ,..., x )  X i
1
i
k
i
• X is a subset of  k
i
• Bidders are assumed to have pure private
values for the commodities .
• Bidder i’s value is given by the function
Ui : X i  
• The price vector -p = ( p1 ,..., p k )   k
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 p * and a consumption vector
*
*
* n
x

Q
(
p
) For
{xi }i =1 for every bidder s.t.
i
i
n
*
x
i = 1,..., n and
i=1 i = S
• T is a finite time ,so that with every
we associate a price vector p (t )
t  [0, T ]
Sincere Bidding
• Bidder i is said to bid sincerely ifxi (t ) = qi ( p(t ))
t  [0, T ]
for all
.
• the functionqi () is a measurable selection
from the demand correspondence
.
Qi ()
• and xi (t ) is the desired vector by bidder i at
t
the time
.
Gross Substitutability
•
U i ( xi )
satisfies gross substitutability if
'
p
for any
and
two price vectors
p
'
such
xi  Qi ( p)
p  that
p
there
xi'  Qi ( p ' )and for any
xi'k  xik
exists
k
p 'k = p k
such that
for any
commodity
such that
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)
x i (t ) =  j i x j (t )
ci (t ) = sup ci (t )
t[ 0 ,t ]
• the payment of bidder i
T
yi (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 where p i is defined Implicitly by
i

j i
q j ( p i ) = S
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 ) = ( p1 (t ),..., p k (t ))
• bidder i bids according to the vector valued
function xi (t ) = ( xi1 (t ),..., xik (t )) from [0, T ] to X i
• the K commodity case payment equation is:
T
ai (T ) = k =1{ p k (0)  [ s k  xki (0)]   p k (t )dxki (t )}
K
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
and
for all t  [0, T ] then the integral
T
0 p(t )dxi (t ) is independent of the path
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.
p  Pi
notice that any attainable final price
implies
consisting
j allocation
i
q j ( p ) an associated
of:
i
S  q i ( p )
for each bidder
K heterogeneous commodities (5)
• THEOREM 1. If each bidderj  i
bids
sincerely and if bidder i’s bidding is
constrained so as to generate piecewise
[0, T ]
smooth price paths from
tok

then bidder i maximizes her payoff by
maximizing social surplus over
allocations associated
with
.
Pi
K heterogeneous commodities (6)
• THEOREM 1(cont) - Moreover, if a
Walrasian price vector w is attainable by
bidder i (i.e., if w  Pi ) 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).
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
Equilibrium of the auction(2)
• Example -Let Z ( p(t )) = S  i=1 qi ( p(t ))
denote the vector of excess demands at
time t and let (A) be a continuous and
k
H
()
sign-preserving transformation
so
k
that the price adjustment process
p k (t ) = Hwould
( Z k ( p(t )))
be :k = 1,..., K
for
.
Let (B) include the assumption of gross
substitutability plus additional
assumptions on the economy
n
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
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.,  j i qj ( p(0)) = S ), 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
 if a Walrasian price vector w is attainable by
, and
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.
k

An n+1 steps algorithm for
calculating payoffs
Step 1
• Run the auction procedure of naming a
price p(t), allowing bidders j1 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 jn 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:
T
ai (T ) = k =1{ p k (0)  [ s k  xki (0)]   p k (t )dxki (t )}
K
0
for bidder n: pn  w
for bidder i ( 1  i  n 1 ):
( pi  p(0))  ( p(0)  pn )  ( pn  w)
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
k
k
commodity,
q
(
p
(
0
))

S
j
j i
i.e.,

for all i and all k.
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-