Advanced Auction Theory

1
Advanced Auction Theory

• IEOR 298 Supply Chain Management
• April 22, 2005
• Nick Rosic
• Shehzad Wadalawala
• Juan Wang

2
Presentation Outline

• Introduction to Auctions
• Reverse Auctions
• Rules and Procedures
• Case Smart Markets
• BREAK
• Advanced Forward Auctions
• Vickrey Clarke Grove Mechanism
• Simultaneous Ascending Auction
• Case FCC Spectrum
• Double Auction

3
Introduction to Auctions

• An auction is a method of allocating scarce
  goods,
• A seller wishes to obtain as much surplus as
  possible,
• A buyer wants to pay as little as necessary.
• An auction is a simple way of determining
  allocations and prices.
• Auctions have the quality of being fair
• Any bidder has a chance to win if his bid is
  sufficient
• An auction is efficient when objects are
• Allocated to those bidders who value them most
• The price is determined by the bids.

4
Application of Auctions

• Auctions are useful when
• the goods do not have a fixed or determined
  market value, in other words, when a seller is
  unsure of the price he can get.
• Choosing to sell an item via auction is
• more flexible than fixing the price
• less time-consuming and expensive than
  negotiating a price.
• Usually earns greater revenue
• Auctions can be used
• for single items such as real estate or works of
  art
• and for multiple units of a homogeneous item such
  as gold or Treasury securities.

5
Information Extraction

• The price is determined by the bidders.
• The seller controls the auction type.
• The auctioneer is usually a third party that acts
  as an agent for an object owner.
• The buyers frequently know more than the seller
  about the value of the item.
• A seller, not knowing the true value of an object
  would rather let the informed bidders determine
  price than suggest a price out of fear that his
  ignorance will prove costly.

6
Bidder Valuations

• Private valuation
• Bidders valuations do not depend on others, in
  literature often the bids are assumed as iid
  (independent and identically distributed), when
  is this a reasonable assumption?
• All bidders have private valuations and tend to
  keep that information private.
• The surplus that a bidder obtains through an
  auction is sometimes referred to as his
  information rent.
• Most models assume IPV (Independent Private
  Valuation)

7
Bidder Valuation (cont)

• Common valuation
• Goods are acquired goods for resale or commercial
  use
• An individual bid is based not only upon a
  private valuation but also upon an estimate of
  market value of the object after the auction.
  Each bidder tries to guess the ultimate price of
  the item.
• The item is really worth the same to all, but the
  exact amount is unknown
• Example
• Purchasing land for oil drilling
• Each bidder has different information and a
  different valuation, but the value of the oil is
  common.

8
Winners Curse

• In common value auctions, bidders must be
  concerned about the "Winners curse."
• Bidders go to auctions to win, but in common
  value auctions, the "lucky" winner pays more for
  an item than it is worth. Auction winners are
  faced with the sudden realization that their
  valuation of an object is higher than that of
  anyone else.
• How can bidders adjust for the winners curse?
• Bidders who estimate value correctly do not win
  in common value auctions

9
Auction Formats

• Traditional Auction Formats
• English
• Dutch
• First Price Sealed Bid
• Vickrey Second Price Sealed Bid

10
Auction Formats (cont)

• Alternative Auction formats
• Reverse Auction
• Simultaneous Ascending Auction
• Double Auction
• Anglo-Dutch
• Clock Auction

11
Introduction to Reverse Auctions

• In contrast to many consumer auctions, a reverse
  auction involves one buyer and many sellers.
• A reverse auction is descending in price. Sellers
  place bids on the desired item, and the lowest
  bid wins. In order to achieve a positive payoff,
  each seller cannot bid below its valuation of the
  item.

12
Reverse Multi-item Auctions for Industrial
Procurement

• In this type of auction, an industrial
  organization seeks to buy a bundle of goods from
  many different suppliers.
• As a reverse auction, the auction is descending
  in price. Suppliers place bids on the desired
  bundle of goods, and the lowest bid for each item
  wins.
• The buyer can purchase different goods from
  different suppliers. Because of capacity
  constraints, the buyer may also need multiple
  suppliers to provide a single good.
• In order to receive a positive profit, suppliers
  cannot bid below their production cost for a
  particular good. Therefore, the supplier with the
  lowest production cost can outbid its
  competitors.

13
Smart Markets for Auctions

• Definition A smart market is an exchange
  institution in which a computer uses an
  optimization algorithm to solve the allocation
  problem associated with each set of bids.
• Bidding Process
• Before making any bids, suppliers submit their
  production costs to the computer program.
• Suppliers place initial bids on the given bundle
  of goods.
• The computer program inputs these bids, then
  computes the allocation which minimizes the
  buyers total cost.
• Each supplier is informed of what its allocation
  would be given the current bids.
• The program also outputs to each supplier a best
  response bid for the next round.
• The next round of bidding begins, and suppliers
  are free to change their bids.
• The auction continues until all suppliers bids
  are unchanged in consecutive rounds.

14
Quantitative Auction Model

• Basic problem A large manufacturing company
  would like to buy a certain quantity of m
  different components, and n suppliers compete to
  offer the lowest selling prices for these
  components.
• Each supplier has a unique set of production
  constraints a given supplier may not be able to
  offer the entire desired quantity of a good.
• Variables
• qj buyers requested quantity of component j
• ci total amount of production resource
  available to supplier i
• aij amount of resource needed for supplier i to
  produce one unit of component j
• bij (t) unit price bid by supplier i for
  component j in round t
• (this bid can be for any quantity between
  0 and qj)
• xij (t) supplier is potential allocation of
  component j in round t
• vij supplier is unit production cost for
  component j

15
Bidding Rules

• 1. Non-reneging rule The supplier may not
  increase a previous bid for any component.
• Mathematically, this means that bij (t) (t) for any t
• 2. Common multiple rule All bids must be integer
  multiples of e for some e 0.
• Therefore, there is a minimum bid decrease of e.

16
The Buyers Minimization Problem

• Given a particular set of bids b(t) and desired
  set of quantities q, the buyer seeks to minimize
  the total cost of purchasing the bundle q.
• Mathematically, we have
• min Si Sj bij(t)xij
• s.t. Sj aijxij
• Si xij qj for all j
• xij 0 for all (i,j)

17
Suppliers Payoff Problem

• Each supplier seeks to maximize its profits given
  its own capacity and production costs and the
  bids of its competitors.
• Supplier is potential payoff in a given round of
  bidding is
• pi (b(t)) Sj (bij(t) vij)xij(t)
• Unfortunately for suppliers, the quantities
  xij(t) are unknown when bids are submitted for
  round t. Under the MBR assumption, however, the
  supplier can calculate estimates of these
  quantities based on the current bids of
  competitors.

18
Myopic Best Response Model

• The best response bids calculated in the smart
  market are called myopic best response (MBR).
• Under MBR, a suppliers potential profit in the
  next round of bidding is maximized, given that
  all bids of competitors are unchanged.
• This assumption is extremely unrealistic, given
  that many suppliers will probably change their
  bids in order to improve their payoffs. So, an
  MBR bid can often produce a suboptimal result for
  a given supplier.

19
Smart Market Information Structure

• The buyer knows
• Supplier identities
• Capacity constraints of all suppliers
• All bids
• Potential allocation at the end of each round
• The buyer does not know
• Suppliers production costs
• MBR suggested bids
• Each supplier knows
• Desired quantities of the buyer
• Its own potential allocation
• Its own MBR suggested bid
• Each supplier does not know
• Identities of other suppliers
• Capacity constraints and production costs of
  competitors
• Competitors bids

20
Deficiencies of the Model

• As explained previously, the MBR assumption will
  most likely not provide the optimal bids for a
  given supplier.
• The model assumes that production costs are
  linear. This is quite inaccurate, since suppliers
  can often use economies of scale to lower their
  average production costs.
• In addition, the costs of producing one component
  may be affected by whether company produces other
  components. The model does not consider
  interactions between goods.

21
Basic Results

• At first, we make only a weak behavioral
  assumption if a supplier receives a potential
  allocation of zero for all components, it will
  lower its bid, unless a decreased bid would
  produce a negative profit.
• With this simple assumption, the following result
  holds
• Proposition Let T be the final round of the
  auction, let v1nj, , vnnj be the order
  statistics of (v1j, , vnj), and define PC i ?
  1,, n, xi(T) 0. Provided that PC 1 and
  under the weak behavioral assumption, we have
• xij(T) 0 ? bij(T) e for all j ? 1, , m.
• So, the buyer is guaranteed a maximum level of
  bids given the production costs of the suppliers.
  This upper bound depends heavily on how many
  suppliers are shut out of the final allocation.

22
MBR Behavioral Model

• Suppose that each bidder follows its MBR
  suggestion in every round of bidding.
• This means that bids in every round are
  completely determined by the initial set of bids
  and the suppliers production costs, since the
  suppliers do not actually make any decisions once
  they have submitted their first round bids.
• Proposition Let (b(t))t?N be a myopic best
  response bidding sequence defined by a set of
  initial bids b(0) and the recursive relation
  b(t1) Fb(t). Then, there exists an integer T
  0 such that b(t) b(T) for all t T.
• Proof All bids are non-decreasing and have a
  lower bound of zero. The common multiple rule
  implies that bids can only take on a finite set
  of values. So, the bidding sequence must converge
  in a finite number of steps.

23
A Sample Auction

• Consider the case where n2, m1. That is, there
  are two suppliers competing for one component.
• If the suppliers constraints are low enough, the
  buyer will need to purchase some amount from both
  suppliers.
• In this case, if the initial bids are very far
  apart, there may be a premature equilibrium. If
  the higher bidder outbids the lower one, the
  increase in volume might not make up for the loss
  in price.
• The buyer may be able to prevent premature
  equilibria by requiring a maximum initial bid,
  since this requirement will encourage suppliers
  to bring their bids closer together.

24
Dynamic 2x2 Auction
25
Summary

• 1. For industrial procurement auctions in which
  the capacity constraints of suppliers are known,
  a smart market computer program can find the
  optimal allocation of components to minimize the
  buyers total cost.
• With an effective information structure in place,
  the smart market is able to limit collusion.
• 2. Even when we make a very simple behavioral
  assumption, we can derive upper bounds for the
  suppliers bids, so the buyers total cost is
  also bounded.
• 3. If all suppliers follow the myopic best
  response suggestions, the outcome of the
  procurement auction is fully determined by the
  opening bids.
• When opening bids are very far apart, the auction
  may converge quickly to a premature equilibrium.
• The buyer can impose a maximum initial bid to
  prevent this occurrence.

26
Possible Improvements

• Nonlinear production costs
• More complicated capacity constraints
• Alternative behavioral models
• Inclusion of decision-making factors other than
  price
• Switching costs between suppliers
• Historical preferences and supplier reputations
• Quality measure for components

27
Other Examples of Reverse Auctions

• Sears Logistic Services (SLS) writes contracts
  with trucking companies to provide shipping
  services for Sears department stores.
• In 1992, SLS worked with the consulting firm JSCO
  in attempts to lower the costs of its trucking
  contracts.
• JSCO designed a reverse auction in which
  suppliers could place all-or-nothing bids on
  bundles of lanes.
• SLS shipping costs dropped from 190 million per
  year to 165 million per year, a savings of 25
  million, or 13.

28
Other Examples

• The U.S. Navy hired Freemarkets in 2000 to hold a
  procurement auction for the parts of airplane
  ejection seats. The Navy was able to save 29
  from its traditional costs.
• PriceLine.com uses reverse auctions to find
  inexpensive hotels and flights for travelers.

29
Vickrey-Clarke-Groves (VCG) Auction

• A Lovely in Theory but lonely in Practice
  Mechanism

30
Background

• Generalization of the second price sealed bid
  auction of Vickrey Auction (1961) by Clarke
  (1971)and Groves (1973)
• Vickrey Auction
• a single type of goods, bidders report demand
  schedule for all possible quantities
• Auctioneers then select the allocation to
  maximize the total value
• Each bidder pays the lowest total bid that buyer
  could have made to win its part of the final
  allocation, given the other bids
• Vickrey proved Dominant strategy property,
  efficient allocation
• VCG
• Dominant strategy property still holds when
  extending to many types of goods
• bidders make bids on all possible packages
• The allocation still assigns goods efficiently
• charges bidder opportunity costs of the item they
  win

31
Uniqueness and Equivalence Results

• Uniqueness Green and Laffont (1979) and
  Holmstrom (1979)
• Any efficient mechanism with the dominant
  strategy property, and in which losers have zero
  payoffs is equivalent to the VCG mechanism, in
  the sense of leading to identical equilibrium
  outcomes.
• Revenue Equivalence Williams (1999)
• All Bayesian mechanisms that yield efficient
  equilibrium outcomes and in which losers have
  zero payoffs lead to same expected equilibrium
  payments as the VCG mechanism.
• Important theoretical foundation of auction design

32
VCG Mechanism in Formal Language

• Bidder 1.N
• a vector of goods that a seller has on offer
• bidder ns value for any bundle xn
• value function bidder n reports to the
  auctioneer
• The auctioneers value maximization problem
• The Price paid by bidder n
• Vickrey discount marginal contribution to the
  auction

33
An Example of VCG Auction

• Real valuations for different bundles

• Bids for different bundles

• Winning allocation
• A-3 B-2 (XOR bids)
• Payments of Agent 2 and agent 3?
• What if bidding true value?

34
Dominant Strategy in VCG Mechanism

• Theorem Truthful reporting is a dominant
  strategy for each bidder in the VCG mechanism.
  Moreover, the outcome of the mechanism is one
  that maximizes total value.
• Proof

35
Virtues of the VCG Mechanism

• Dominant-strategy Property
• Reduce the cost of auction
• Easy for the bidders to determine their optimal
  bidding strategy
• Eliminate bidders incentive to spend resources
  learn other bidders value
• Efficient prediction is reliable
• Scope of application
• No restrictions on the possible ranking of
  different outcomes
• Allow auctioneer to impose some extra constraints
• Replaces by other
  constraints of the form x ? X without changing
  Theorem 12 in any essential way

36
Why Practical Application of VCG are rare?

• Revenues can be very low or zero
• Example
• What is the revenue?
• 0!!!
• The revenue deficiency of VSG mechanism is
  decisive to reject it for most practical
  applications

37
Shill bidding and Collusion

• Shill bidding submit additional bids under false
  identities
• Winning bid? Payment?
• Bidder 1 wins, pays 1
• Bidder 2 bids as bidders 2 and 3, submits 2 for
  one item
• Dominant strategy property depends on unlimited
  budget
• Example? Homework!!
• Privacy preservation problem

38
Simultaneous Ascending Auctions

• A Successful Practical Mechanism

39
Brief History

• First use sell licenses to use bands of radio
  spectrum in 1994 by FCC in US
• Motivation
• Reduce Federal regulation of radio spectrum
• Main rules from two detailed proposals
• Preston McAfee
• Robert Wilson and Paul Milgrom
• Success
• Employed by the FCC in almost all US radio
  spectrum auctions
• US 617 million sale of ten paging licenses in
  July 1994
• Adopted with slight variations for dozens of
  spectrum auctions worldwide, 200 billion
• Early auctions in Europe for 3G mobile wireless
  licenses 100 billion
• Extended to the sale of divisible goods in
  electricity, gas and environmental markets

40
Introduction

• Usually for a group of items with strong value
  interdependency
• Aggregation of licenses is important to achieve
  efficiency
• Natural generalization of English Auction
• Multi-round, simultaneous
• Not combinatorial auctions
• Bidders are not allowed to bid on packages
• A useful benchmark for comparison of
  Combinatorial Auctions

41
SAA Rules

• Each round, bidders simultaneously make sealed
  bids for any item
• Round results are posted after each bidding.
• identities of new bids and bidders for each item
• standing high bid and the corresponding bidder
• Minimum bids for the next round
• standing high bid predetermined bid increment
• Activity rule
• Requires bidder maintain a minimum level of
  activity to preserve its current eligibility
• Considered as active if it makes an eligible new
  bid or owns the standing high bid
• Create pressure on bidders to bid actively,
  increase pace of auction
• Increase the information available to bidders,
  improve price discovery

42
SAA Rules Continued

• Stopping rule
• Bidding on all licenses closes simultaneously
  where there is no new bidding on any license
• Payment/Allocation rule
• Allocate the standing high bids to the
  corresponding bidders at the price equal to the
  bids
• Bid withdrawal
• Withdraw penalty max0, withdrawn bid-final sale
  price

43
Performance of SAA

• Example Three US PCS broadband auction
• Revenue
• Generate market price?
• Results
• Narrowband auctions a few percent and often zero
• First broadband auction price difference minimum bid increment in 42 out of 48 markets
• Compared with Sequential Auction
• Swiss wireless-local-loop auction in March 2000
  3 nationwide licenses, first two 28 MHz blocks
  for 121 and 134 million francs, third one 56 MHz
  for 55 million francs
• Efficient license aggregation?
• bidders appear to piece together sensible license
  aggregation
• Efficient?
• Absence of resale

44
Why Success?

• Excellent Price discovery
• simultaneous rather than sequential
• In sequential auctions, bidder must guess what
  prices will be in future auctions when
  determining bids in the current auction
• Multi-round
• Bidders see tentative price info. at each round,
  winners curse reduced
• Price info. helps bidders focus valuation efforts
  in the relevant range of price space----discoverin
  g values is costly
• Bidders retain sufficient flexibility to shift
  towards their best package
• Problems?

45
Demand Reduction

• Definition
• When multiple items are sold through the use of a
  SAA, bidders can find it in their mutual
  interests to reduce their aggregate demand for
  the items while prices are still below the
  bidders' valuations
• Example
• The bidder prefers winning one unit at low prices
  than winning two items at a price high enough to
  outbid the other bidder, not efficient
  equilibrium
• 1999 German GSM spectrum auction, lasted just two
  rounds
• Nationwide broadband auction, the largest bidder,
  PageNet, reduced its demand from 3 to 2, when
  prices

46
No Competitive Equilibrium Exists

• Theorem suppose that the set of possible
  individual valuation functions includes both
  mutual substitutes in individual demand, and at
  least one other valuation function, then if there
  are at least two bidders, there is a profile of
  possible individual valuation functions such that
  no competitive equilibrium exists.
• Parking Spots Auction
• Unique value maximizing license allocation?
• Why no equilibrium?
• For it to happen, PA75 and Pb75

47
Exposure Problem

• With individual bidding, a bidder may fail to
  acquire key pieces of the desired combination,
  but pay prices based on the complementary gain.
• Exposure problem results in inefficiency
• Netherlands DCS-1800 auction in February 1998
• 18 licenses for sale, A and B efficiently scaled,
  16 too small to be useful for a mobile phone
  business alone, need to be combined
• final price per unit in A and B 2 times those
  for any of the small lots
• Parking Spots Example
• if the first bidder drops out early, result?
• Allowing package bidding partially solves the
  problem
• Result?
• Withdrawal of bids mitigates the exposure problem
• Usually complementarities are not extreme and
  competition is greater

48
Ascending Auctions with Package Bidding

• L.M. Ausubel and P.R MilgromFrontiers of
  Theoretical Economics, 1(1) 1-50, 2002

49
Double Auctions

• Double auctions are closest to reflecting natural
  marketplaces
• Many buyers and many sellers
• Examples
• Covisint
• http//www.isa.org/InTechTemplate.cfm?SectionArti
  cle_Indextemplate/ContentManagement/ContentDispl
  ay.cfmContentID8342
• 2001 Daimler Chrysler held a reverse auction with
  spending of 2.5 billion
• What is the effect on buyer supplier
  relationships?

50
Emission Permits

• The vast majority of the worlds climate
  scientists have concluded that if the countries
  of the world do not work together to cut the
  emission of greenhouse gases, then temperatures
  will rise and will disrupt the climate. In fact,
  most scientists say the process has already
  begun.
• President Clinton, October 22, 1997
• 1246 million metric tons of permits each year
• Typically marginal price is 100, equivalent to
  125 billion if auctioned efficiently
• Kyoto Protocol
• US uses grandfathering instead of auctions for
  allocation of permits

51
Types of Double Auctions

• Call Auction
• All bids are submitted to a third party that
  matches supply and demand and clears the market
• Continuous Auction
• At any time a bidder and supplier can agree to a
  price and quantity and transact
• Why one over the other?

52
Objectives of Double Auction Design

• Truthful Revelation (Strategy proofness)
• Efficiency
• Budget Balance
• Individual Rationality (Participation Constraint)
• Myerson and Sattherthwaite (1983) prove that it
  is impossible to have all four for the double
  auction

53
The allocation problem
54
Vickrey Model for the Double Auction

• Efficient Outcome
• Truthful Revelation
• Individually Rational
• NO BUDGET BALANCE
• Example
• I 3, N 3
• f (300, 250, 200), g (175, 225, 275)

55
Tug of WAR!
56
Keeping the peace

• If we relax the need to maximize efficiency, we
  can obtain truthful revelation
• Eliminate the least profitable trade from the
  allocation
• Bidders pay price equal to newly rejected
  bidders price and suppliers are paid newly
  rejected bidders ask price.

57
Multi-unit demand/Multi-unit supply

• In multi-unit demand and multi-unit supply case,
  prices are discriminatory.
• Demand Reduction
• Bidder 1 f (10, 3)
• Bidder 2 f (7)
• Supplier 1, g (2)
• Supplier 2, g (2)
• Concerns about fairness
• Supply Reduction is similar

58
Future Directions

• Combinatorial Auctions
• Multi-attribute Auctions
• Questions?

59
Sources

• An, N., Elmaghraby, W., and Keskinocak, P., 2004,
  Bidding Strategies and their Impact on Revenues
  in Combinatorial Auctions, Georgia Institute of
  Technology.
• Ausubel, L, Cramton, P., 2002, Demand Reduction
  and Inefficiency in Multi-Unit
• Auctions, University of Maryland,
  Working Paper 9607, revised July 2002.
• Chu, L, Shen, Z., 2003, Agent Competition Double
  Auction Mechanism
• Cramton, P., Shoham, Y., and Steinberg, R.
  (editors), 2006, Combinatorial Auctions,
  forthcoming, MIT Press.
• Dayama, P. and Narahari, Y., Combinatorial
  Auctions for Electronic Business.

60
More Sources

• Gallien. J., and Wein, L., 2000, Design and
  Analysis of a Smart Market for Industrial
  Procurement, Massachusetts Institute of
  Technology.
• Hardy, Michael, 2003, Reverse auctions save Navy
  millions, fcw.com.
• Milgrom, P., Putting Auction Theory to Work The
  Simultaneous Ascending Auction.
• Sunnevag, K., 2001, Auction design for the
  allocation of emission permits
• Vries, S, Vohra, R., 2001, Combinatorial
  Auctions A Survey