An Efficient Dynamic Auction for Heterogeneous Commodities Lawrence M'Ausubel september 2000

1
An Efficient Dynamic Auction for Heterogeneous
Commodities(Lawrence M.Ausubel - september 2000)

• Authors
• Oren Rigbi
• Damian Goren

2
The problem

• An auctioneer wishes to allocate one or more
  units of each of K heterogeneous commodities to n
  bidders.

3
The Lectures Contents

• Preface Example
• Presentation of the Model
• Equilibrium of the dynamic auction
• Relationship with the Vickrey auction
• Conclusions

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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.

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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.

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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 wouldnt have participated the auction.

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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.
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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.

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The Model (1)

• A seller wishes to allocate units of each of K
  heterogeneous commodities among n bidders.
• N 1,..., n.
• The sellers available supply will be denoted by
  S (S1 ,...,Sk ) .

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The model (2)

• Bidder is consumption vector -
• 
  
• is a subset of
• Bidders are assumed to have pure private values
  for the commodities .
• Bidder is value is given by the function
• The price vector -

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The model (3)

• Bidder is net utility function
• Bidder is demand correspondence

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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

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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 .

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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

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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) .

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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) .

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Crediting Debiting (1)

• In the next few slides we will develop the
  payment equation for the case of 1 commodity.
  
• Example

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Crediting Debiting (2)

• For k1 (one commodity)
• the payment of bidder i

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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.

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Crediting Debiting (4)

• Another way of defining the payment is
• suppose that is monotonic, and define
  where is defined Implicitly
  by

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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 . . .

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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

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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 ...

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K heterogeneous commodities (3)
equals
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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 .

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K heterogeneous commodities (5)

• THEOREM 1. If each bidder bids
  sincerely and if bidder is 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 .

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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).

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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).

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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

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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)

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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),

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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).

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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 is 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.

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An n1 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).

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... 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).

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Step n1

• Run the auction procedure of naming a price p(t),
  allowing all bidders i1...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).

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Payoffs Computation

• Payment equation
• for bidder n
• for bidder i ( )

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Theorem 4

• Suppose that the triplet (A)-(B)-(C) is a
  stable price adjustment combination for
  competitive economies. Consider the (n1) 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
  (n1) step auction game, the price vector
  converges to a Walrasian equilibrium price, and
  the outcome is exactly that of a Vickrey auction.

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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.

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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.