Auctions -1

1
Auctions -1

• Debasis Mishra
• QIP Short-Term Course on Electronic Commerce
• Indian Institute of Science, Bangalore
• February 15, 2006

2
Outline

• Single-item auctions
• Models of bidder behavior
• Multi-item auctions
• References

3
Auctions - Introduction

• Auction - comes from Latin word auctus to mean
  increase.
• Not every auction has increasing prices.
• Among one of the first engaging tales - Sale of
  Roman empire to the highest bidder in 1764.
• A market institution that works on the concept of
  competition.
• Natural discovery of price and buyers.

4
Auctions - Why and Why Not?

• Why auctions?
• Seller unsure about how much the price should be.
• It can be used to sell almost anything
  -universal.
• Buyers learn, in some auctions, about the
  information of other buyers - leads to more
  efficient and revenue-generating markets.
• Why not auctions?
• Overhead of time and infrastructure.
• Fixed price methods are simple.
• Values of bidders are almost known.

5
Auction Settings

• Forward auctions a seller selling items to
  buyers (bidders).
• Reverse auctions a buyer buying items from
  sellers/suppliers (bidders).
• Both the settings are natural transpose of each
  other
• Bidders compete in both settings.
• At low (high) price many buyers (sellers) demand
  (supply) items in forward (reverse) auctions.
• Highest (lowest) price buyer (seller) wins in
  forward (reverse) auctions.

6
Auctions in Practice

• Selling of flowers (Holland), tobacco, fish, tea,
  art objects and antique pieces (Sotheby's).
• Transfer of assets from public to private Sale
  of industrial enterprises in Eastern Europe,
  transportation system in Britain, timber rights
  all over the world, and off-shore oil leases.
• Auction of spectrum rights worldwide - US,
  Europe, and even India.
• Internet auctions of consumer goods (amazon.com,
  ebay.com etc.). Google's Adword auctions.
  Procurement auctions - freemarkets.com (now
  Ariba), GM and IBM's sourcing solutions.

7
Valuations

• Valuation The maximum amount a bidder is willing
  to pay.
• In procurement auctions, the value is negative of
  cost of procurement - the minimum price a bidder
  is demanding.
• Auctions are used mainly because the auctioneer
  is unsure about the valuations (or simply,
  values) of bidders.
• Two models (i) private values (ii) common or
  interdependent value.

8
Private Value Model

• Each bidder knows his own value of the item
  exactly at the time of bidding, but knows nothing
  about the values of other bidders.
• Value of other bidders do not influence his own
  value.
• Suitable for auctions for paintings, stamps etc.
  (a bidder knows the value of a painting exactly),
  procurement auction settings (a supplier's cost
  depends only on his own production technology).
• Most plausible when the value of the item to a
  bidder is derived from its use alone and the
  bidder knows the item well.

9
Interdependent Value Model (1 of 2)

• Worth of an item unknown at the time of bidding
  to bidders.
• Examples oil field (depth of oil wells not well
  known), second-hand products (quality of the
  product is not known).
• In such cases, a bidder will have an estimate or
  a privately known signal (an expert's opinion or
  a test result) that is correlated with the true
  value.
• Formally, every bidder has a signal xi and the
  value of bidder i is vi(x1, x2,..., xn)

10
Interdependent Value Model (2 of 2)

• Information, such as estimates or signals, of
  other bidders will influence the value of a
  bidder.
• Values are unknown to bidders at the time of
  bidding and may be affected by information
  available to other bidders.
• A special case is common values - every bidder
  has the same value ex post (i.e., once they know
  everyones signals). Example oil field auction.
• v(x1, x2,..., xn)

11
Single Item Auctions

• Two formats (i) sealed-bid (ii) open-cry
• Sealed-bid Bidders submit bids once (in a sealed
  envelope to the auctioneer)
• Open-cry Bidders submit bids in rounds, bids
  result in increase in prices (commonly termed as
  iterative auctions)
• Bids reflect if bidders are willing to
  participate further in the auction.

12
Single Item Sealed-Bid Auctions (1 of 3)

• First-price sealed-bid Every bidder submits a
  bid the highest bid bidder wins and pays his bid
  amount.
• Second-price sealed-bid (Vickrey auction) Every
  bidder submits a bid the highest bid bidder wins
  but pays an amount equal to the second highest
  bid.
• First-price auctions are common in practice.
• Second-price auctions are rare but see examples
  of stamp auctions and others in
  http//www.u.arizona.edu/dreiley/papers/VickreyHi
  story.pdf

13
Single Item Sealed-Bid Auctions (2 of 3)

• Example Four bidders with values 10,8,6, and 4.
• First-price Bidders bid 8,6,5, and 3
  respectively (bid value not equal to valuation).
  Highest bidder wins and pays 8.
• Second-price/Vickrey Bidders bid 10,8,6, and 4
  (bid value equals valuation). Highest bidder wins
  but pays 8.
• Neither the revenue equivalence in the two
  auctions nor the bidvalue in Vickrey auction in
  this example is a coincidence.

14
Single Item Sealed-Bid Auctions (3 of 3)

• The best strategy for a bidder, irrespective of
  what other bidders have bid, is to bid his value.
  This is also called a dominant strategy
  equilibrium in game theory.
• Though economically robust, Vickrey auction is
  less transparent to bidders transparency in
  auction design is important.

15
Single-Item Open-Cry Auctions (1 of 3)

• English auction Auction starts from low price. A
  bidder bids by indicating if he is willing to buy
  the item at the current price. If more than one
  bidder bids, then the price is raised by a finite
  amount e (bid increment), else the auction stops.
  The last bidder to bid wins at the final price.
• Consider the same example (values 10,8,6,4). Let
  the starting price be 0 and bid increment e. At
  price lt 4 e, only 3 bidders will be interested
  at price lt 8 e, only 1 bidder will be
  interested. Auction stops at price lt 8 e.

16
Single-Item Open-Cry Auctions (2 of 3)

• It can be shown that staying in the auction till
  price reaches value is the best strategy for
  bidders.
• Further, the outcome of English auction is
  equivalent to (as e reaches zero) the Vickrey
  auction.
• English auction is popular in practice more
  transparent and has similar economic properties
  as the Vickrey auction.

17
Single-Item Open-Cry Auctions (3 of 3)

• Dutch auction (popular in Holland to sell
  flowers) The auction starts from high price
  where there is no demand for the item bidders
  bid indicating if they are interested in the item
  at the current price if no bidder bids then the
  price is decreased by e (bid decrement), else the
  auction stops. The only bidder to bid wins at the
  final price.
• In case, more than one bidder bids, then the item
  is allocated at random to either of them.
• Dutch auction is strategically equivalent to the
  first-price sealed-bid auction.

18
Strategic Considerations

• Strong requirement dominant strategy -
  Irrespective of the bidding strategy of other
  bidders, a bidder's best strategy (one that
  maximizes utility over all strategies) is to be
  truthful.
• Weak requirement (ex post) Nash equilibrium -
  Given that all bidders bid truthfully, a bidder's
  best strategy is to be truthful.
• Given an auction design, is bidding truthfully
  the best strategy?
• Design an auction in which truthful bidding is
  the best strategy.

19
Dominant Strategy in Vickrey Auction

• Consider bidder 1. Let the bid amount of any
  bidder i (not 1) be bi (need not equal value).
  What is the best amount to bid for 1?
• Without loss of generality, assume b2 to be the
  highest bid among bids of bidders other than 1.
• Losing the auction by bidding untruthfully gives
  zero payoff. To win the auction and make positive
  payoff, bidder 1 should bid more than b2.
• His payment will be b2 always, independent of his
  bid amount, if he wins. His payoff is v1 - b2,
  where v1 is his value. So, own bidding strategy
  does not influence payoff implying truthful
  bidding is a dominant strategy.

20
Equivalence of Auction Forms

• Dutch auction - Where should a bidder respond?
  That price is the payment. First-price sealed-bid
  auction - What bid should a bidder submit? That
  bid price is the payment. So, same decision in
  both auctions.
• English auction - best strategy is to remain
  interested till price reaches value. This
  terminates the auction (approximately) at the
  second-highest value. This is the outcome in the
  Vickrey auction.
• Dutch auction first-price sealed-bid auction.
  English auction Vickrey auction.

21
Revenue in Auctions (1 of 3)

• Values are drawn from uniform distribution with
  range 0,a for n bidders.
• Expected revenue in the Vickrey auction
• 0?a n(n-1)F(x)(n-2) 1-F(x) x f(x) dx
• a(n-1)/(n1).
• Expected highest value
• 0?a nF(x)(n-1) x f(x) dx a n/(n1).
• In the first-price sealed-bid auction, we will
  find an
• equilibrium in which every bidder bids k times
  his value
• (0 lt k lt 1). Such an equilibrium is called a
  symmetric equilibrium.

22
Revenue in Auctions (2 of 3)

• Let b be the bid amount. Expected profit for a
  bid b with value v is
• (v-b)b(n-1)/(ka)(n-1).
• Maximizing expected profit,
• -b(n-1)(n-1)(v-b)b(n-2)0.
• We get bv(n-1)/n.
• So, if every bidder except i bids a fraction
  (n-1)/n of his value, then the best strategy for
  i is to bid a fraction (n-1)/n of his value.
• So expected revenue (in a symmetric equilibrium)
  from a first-price auction a (n-1)/(n1)
  expected revenue from Vickrey auction (revenue
  equivalence theorem). In fact, this is the
  highest possible revenue in ANY auction for
  single-item private values model.

23
Revenue in Auctions (3 of 3)

• So expected revenue (in a symmetric equilibrium)
  from a first-price auction a (n-1)/(n1)
  expected revenue from Vickrey auction (revenue
  equivalence theorem). In fact, this is the
  highest possible revenue in ANY auction for
  single-item private values model.
• In fact, we can say more with independently and
  identically distributed private values, the
  expected revenue in a first-price auction is the
  same as the expected revenue in a second-price
  auction.
• We assumed risk neutral bidders
  payoffvalue-price.

24
Multi-Item Auctions (1 of 3)

• Number of items more than one.
• Items may be of same type (homogeneous) or
  different type (heterogeneous).
• Examples Sale of different components of a
  computer, sale of 1000 memory chips etc.
• Bidders may have value on bundles value for 10
  memory chips need not equal 10 times value of a
  single memory chip value of a monitor and a
  keyboard may be more than their combined value.

25
Multi-Item Auctions (2 of 3)

• If there are n items, a bidder can have values on
  2n number of bundles - exponential number of
  bundles.
• Simultaneous sale of multiple items is also known
  as combinatorial auctions.
• Examples of combinatorial auctions
• Sale of airport slots a bidder will be
  interested in Mumbai 6 AM to 7 AM slot together
  with Bangalore 8 AM to 9 AM slot but less
  interested in Mumbai 6 AM to 7 AM slot with
  Bangalore 1 PM to 2 PM slot.
• Sale of train tracks in Europe, spectrum rights
  in different countries.

26
Multi-Item Auctions (3 of 3)

• Two buyers and two items (a,b). Values are
  v1(a)5, v1 (b)7, v1(ab)15
• v2(a)7, v2 (b)6, v2(ab)12.
• Assuming truthful bidding and conducting a
  sequential auction (selling one item after
  another) using the Vickrey auction yields item 1
  is awarded to buyer 2 and item 2 to buyer 1.
• This is not efficient - does not maximize total
  value of the system.
• Does not maximize the revenue of the seller also.

27
Design Objectives (1 of 2)

• Efficiency Maximize the total value of bidders
  and the seller. These are called efficient
  auctions.
• If p is the price paid by a bidder, then v-p is
  his payoff and the seller gets a payoff of p.
• Thus, total payoff of the system (buyers and
  seller) due to that buyer is v-ppv.
• So, total payoff of the system is maximized by
  maximizing the total value.

28
Design Objectives

• Revenue Maximize the total revenue of the
  seller.
• These are called optimal auctions.
• Generally, have to assume some distributions on
  valuations.
• Much difficult than designing efficient auctions.
• Analysis is intractable for many practical
  multiple items settings.
• Note Optimal auctions maximize the payoff of
  seller only, whereas efficient auctions maximize
  the total payoff of the seller and the buyers.

29
Other Auction Design Issues (1 of 2)

• Reserve price Sellers generally set a minimum
  price below which they do not sell items.
• Bundling issues Sellers generally do not allow
  for exponential number of bundles but decide on
  bundles before the auction.
• Information feedback in iterative auctions What
  bid information should be communicated to
  bidders?
• Bid increments Tradeoff between length of
  auction and efficiency/revenue loss.

30
Other Auction Design Issues (2 of 2)

• Collusion Bidders form groups (called bidding
  rings) and act as one to bid in auctions.
• Privacy Depending on the information released by
  the auctioneer to the bidders, the privacy of
  bidders can be at stake.
• Example In English auction, by bidding
  truthfully, all losing bidders reveal their value.

31
References

• Vijay Krishna, Auction Theory, Academic Press,
  2002.
• Paul Klemperer, Auctions Theory and Practice,
  Online book http//www.paulklemperer.org/, Also
  Princeton University Press, 2004 (gives outlines
  for undergraduate and graduate courses in
  economics and management departments).