Auction Theory an introduction

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Auction Theoryan introduction

• DAI Hards
• October 16th

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Introduction

• Auctions are the most widely-studied economic
  mechanism.
• Auctions refer to arbitrary resource allocation
  problems with self-motivated participants
  Auctioneer and bidders
• Auction (selling item(s)) one buyer, multiple
  bidders)
• e.g. selling a cd on eBay
• Reverse Auction (buying item(s)) one buyer,
  multiple sellers
• e.g. procurement
• Well discuss auction, though the same theory
  holds for reverse auction

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Historical note

• Reports that auctions was used in Babylon 500
  B.C.
• 193 A.D. After having killed Emperor Pertinax,
  Prætorian Guard sold the Roman Empire by means of
  an Auction

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Where auctions are used nowadays?

• Treasury auctions (bill, notes, Treasury bonds,
  securities)
• Has been used to transfer assets from public to
  private sector
• Right to drill oil, off-shore oil lease
• Use the EM spectrum
• Government and private corporations solicit
  delivery price offers of products
• Private firms sell products (flowers, fish,
  tobacco, livestock, diamonds)
• Internet auctions

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Questions

• Information problem the seller has usually
  incomplete information about buyers valuations
  (else, he just need to set the price as the
  maximum valuation of the buyer) ? what pricing
  scheme performs well even in incomplete
  information setting (is auction better suited for
  a given problem? Does a type of auction yield
  greater revenue?)
• For the buyer, what are good bidding strategies?

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Terminology

• Criterion of comparison
• Revenue expected selling price
• Efficiency the object ends up in the hands of
  the person who values it the most (resale does
  not yield to efficiency)
• Private Value no bidder knows with certainty the
  valuation of the other bidders, and knowledge of
  the other bidders valuation would not affect the
  value of the particular bidder
• Pure common value the actual value is the same
  for ever bidders but bidders have different
  private information about the what that value
  actually is (e.g. auction of an oil field and the
  amount of oil is unknown, different bidders have
  different geological signals, learning another
  signal would change the valuation of a bidder).
• Correlated value agents value of an item
  depends partly on its own preferences and partly
  on others values for it

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Agents care about utility, not valuation

• Auctions are really lotteries, so you must
  compare expected utility rather than utility.
• Risk attitude speak about the shape of the
  utility function
• linear utility function refers to
  risk-neutrality
• ? optimize her/his expected payoff
• Concave utility function refers to risk-aversion
  (ugt0 and ult0)
• convex utility function refers to risk-seeking
  (ugt0 and ugt0)
• The types of utility functions, and the
  associated risk attitudes of agents, are among
  the most important concepts in Bayesian games,
  and in particular in auctions. Most theoretical
  results about auction are sensitive to the risk
  attitude of the bidders.

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Outline

• Single-item Auctions
• Common auctions forms
• Equivalence between auctions
• Revenue equivalence
• Multi-unit Auction
• Multi-item Auction

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Single Item Auction
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English(first-price open-cry ascending)

• Protocol Each bidder is free to raise his bid.
  When no bidder is willing to raise, the auction
  ends, and the highest bidder wins the item at the
  price of his bid
• Strategy Series of bids as a function of agents
  private value, his prior estimates of others
  valuations, and past bids
• Best strategy In private value auctions,
  bidders dominant strategy is to always bid a
  small amount more than current highest bid, and
  stop when his private value price is reached
• Variations
• In correlated value auctions, auctioneer often
  increases price at a constant rate or as he
  thinks is appropriate (japonese auction)
• Open-exit Bidder has to openly declare exit
  without re-entering possibility gt More info to
  other bidders about the agents valuation

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First-price sealed-bid

• Protocol Each bidder submits one bid without
  knowing others bids. The highest bidder wins
  the item at the price of his bid
• Single round of bidding
• Strategy Bid as a function of agents private
  value and his prior estimates of others
  valuations
• Best strategy No dominant strategy in general
• Strategic underbidding counterspeculation
• Can determine Nash equilibrium strategies via
  common knowledge assumptions about the
  probability distributions from which valuations
  are drawn
• Variant kth price

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Example

• Values are uniformly distributed on 0,1
• The equilibrium bid is (N-1)x/N
• Where
• x is the valuation of the bidder
• N is the number of bidders
• (proof)

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Dutch (descending)

• Protocol Auctioneer continuously lowers the
  price until a bidder takes the item at the
  current price
• Strategically equivalent to first-price
  sealed-bid protocol in all auction settings
• Strategy Bid as a function of agents private
  value and his prior estimates of others
  valuations
• Best strategy No dominant strategy in general
• Lying (down-biasing bids) counterspeculation
• Possible to determine Nash equilibrium strategies
  via common knowledge assumptions regarding the
  probability distributions of others values
• Requires multiple rounds of posting current price
• Dutch flower market, Ontario tobacco auction,
  Filenes basement, Waldenbooks

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Vickrey ( second-price sealed bid)

• Protocol Each bidder submits one bid without
  knowing (!) others bids. Highest bidder wins
  item at 2nd highest price
• Strategy Bid as a function of agents private
  value his prior estimates of others valuations
• Best strategy In a private value auction with
  risk neutral bidders, Vickrey is strategically
  equivalent to English. In such settings,
  dominant strategy is to bid ones true valuation
• No counterspeculation
• Independent of others bidding plans, operating
  environments, capabilities...
• Single round of bidding
• Widely advocated for computational multiagent
  systems
• Old Vickrey 1961, but not widely used among
  humans
• Revelation principle --- proxy bidder agents on
  www.ebay.com, www.webauction.com, www.onsale.com

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All Pay (e.g.lobbying activity)

• Protocol Each bidder is free to raise his bid.
  When no bidder is willing to raise, the auction
  ends, and the highest bidder wins the item. All
  bidders have to pay their last bid
• Strategy Series of bids as a function of agents
  private value, his prior estimates of others
  valuations, and past bids
• Best strategy ?
• In private value settings it can be computed (low
  bids)
• Potentially long bidding process
• Variations
• Each agent pays only part of his highest bid
• Each agents payment is a function of the highest
  bid of all agents

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In a Nutshell
Sealed Bid Format
Open Format
Weak
Second-Price Sealed Bid i.e Vickrey
English Auction
Private Value
First-Price Sealed Bid
Dutch Descending Price
Strong
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Setting for Private Value Auctions

• N potential bidders. Bidder i is assigned a value
  of Xi to the object
• Each Xi is i.i.d. on some interval 0,?
  according to the cumulative distribution F
• Bidders I knows her/his xi and also that other
  bidders values are i.i.d. according to F
• Bidders are risk neutral (seek to maximize their
  expected payoffs)
• The number of bidders and the distribution F are
  common knowledge.
• Symmetry
• The distribution of values is the same for all
  bidders. WE can consider that all bidders are
  alike, hence an optimal bidding strategy for one
  should also be an optimal strategy for the others
  ? symmetric equilibrium

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Results for private value auctions

• Dutch strategically equivalent to first-price
  sealed-bid
• Risk neutral agents gt Vickrey strategically
  equivalent to English
• All four protocols allocate item efficiently
  (assuming no reservation price for the
  auctioneer)
• English Vickrey have dominant strategies ? no
  effort wasted in counterspeculation
• Which of the four auction mechanisms gives
  highest expected revenue to the seller?
• Assuming valuations are drawn independently
  agents are risk neutral The four mechanisms have
  equal expected revenue!

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Reserve Price in Private Values

• A seller can reserve the right to not sell the
  object if the price is below a reserved price r
• Bidders with value xltr are excluded from the
  auction
• Bidders change their strategy (can be computed)
• A revenue maximizing seller should always set a
  reserve price r that exceeds her or his
  valuation x0
• Proof compute the expected payoff of the seller,
  differentiate it with respect to the reserve
  price and observe it the derivative is positive
  at x0
• Entry Fees the auctioneer can use an entry fee
  a nonrefundable amount that the bidder has to pay
  to participate to the auction.
• Note there is a way to fix the entry fee such
  it is equivalent to using a reserved price as far
  as the agents that are excluded are concerned.
• Trade-off may improve revenue at the expense of
  efficiency (if seller set a reservation price
  which is too high)

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Revenue Equivalence Theorem

• In all auctions for k units with the following
  properties
• Buyers are risk neutral
• Private Value, with values independently and
  identically distributed over a,b (technicality
  distribution must be atomless)
• Each bidder demands at most 1 unit
• Auction allocates the units to the k highest bids
  (efficiency)
• The bidder with the lowest valuation has a
  surplus of 0 (i.e. a bidder with a value of 0 has
  an expected payment of 0)
• ?a buyer with a given valuation will make the
  same expected payment, and therefore all such
  auctions have the same expected revenue

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Application of the Revenue Equivalence Theorem

• Helps to find some equilibrium strategy
• Ex compute the equilibrium bid in an all pay
  auction or in a third price auction
• In the case where the number of bidders is
  uncertain, we can compute the equilibrium bid
  strategy for a first price auction

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Revenue equivalence ceases to hold if agents are
not risk-neutral

• Risk averse Agents
• for bidders
• Dutch, first-price sealed-bid Vickrey, English
• Compared to a risk neutral bidder, a risk averse
  bidder will bid higher (buy insurance against
  the possibility of loosing)
• (utility of winning with a lower bid ltutility
  consequence loosing the object)
• For auctioneer auctioneer
• Dutch, first-price sealed-bid Vickrey, English
• Risk-Seeking Agents
• The expected revenue in third-price is greater
  than the expected revenue in second-price
  (English)
• Under constant risk-attitude (k1)-price is
  preferable to k-price

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Revenue equivalence ceases to hold if it is not
Private Value

• Results for non-private value auctions
• Dutch strategically equivalent to first-price
  sealed-bid
• Vickrey not strategically equivalent to English
• All four protocols allocate item efficiently
• Winners curse each bidder must recognize that
  she/he wins the objects only if she/he has the
  highest signal, failure to take into account the
  bad news about others signal can lead the bidder
  to pay more than the prize it is worth.
• Common value auctions
• Agent should lie (bid low) even in Vickrey
  English Revelation to proxy bidders?
• Thrm (revenue non-equivalence ). With more than 2
  bidders, the expected revenues are not the same
• English Vickrey Dutch first-price sealed
  bid

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Results for non-private value auctions

• Common knowledge that auctioneer has private info
  
• Q What info should the auctioneer release ?
• A auctioneer is best off releasing all of it
• No news is worst news
• Mitigates the winners curse

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The revelation principle(mechanism Design)

• In a revelation mechanism agents are asked to
  report their types (e.g.valuations for the good),
  and an action (e.g. decision on the winner and
  his/her payment) will be based the agents
  announcement.
• In general, agents may cheat about their types,
  but
• Any mechanism that implements certain behavior
  (e.g. a good is allocated to the agent with the
  highest valuation,v, and he pays (1-1/n)v) can be
  replaced by another mechanism that implements the
  same behavior and where truth-revealing is in
  equilibrium.

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Multi-unit Auction
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Auctions with multiple indistinguishable units
for sale

• Examples
• IBM stocks
• Barrels of oil
• Pork bellies
• Trans-Atlantic backbone bandwidth from NYC to
  Paris

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Setting for sealed bid auctions

• Each bidder sends a bid vector indicating how
  much she/he is willing to pay for each additional
  unit
• ? Can be understood as a demand function

Value of the bid
Number of units
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Pricing rules

• Auctioning multiple indistinguishable units of
  an item
• The discriminatory (or pay your bid) auction
• The uniform price auction
• The Vickrey auction

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Discriminatory auction

• Each bidder pays an amount equal to the sum of
  his bids that are among the K highest of the NK
  bids submitted.

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Uniform-price Auction

• Any price between the highest loosing bid and the
  lowest winning bid is possible
• ? can choose the highest losing bid

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Vickrey Auction

• Basic principle is the same as the
  Vickrey-Clarke-Groves mechanism (see Mechanism
  Design)
• A bidder who wins k units pays the k highest
  losing bids of the other bidders
• For bidder i to win the kth unit, is kth highest
  bid must defeat the kth lowest competing bid

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Some Open Auctions

• Dutch Auctions
• English Auctions
• Ausubel Auctions

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Multi-item auctions

• multiple distinguishable items for sale

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Bundle bidding scenario
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Bundle bidding scenario
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Bundle bidding scenario
(console, television, cd player 1000)
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Bundle bidding scenario
(television, music system, computer, 1600)
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Bundle bidding scenario
(cd player, console, music system 400)
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Bundle bidding scenario
((console, television, cd player
1000), (television, music system, computer,
1600), (cd player, console, music system 400))
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Bundle bidding scenario
((Computer, television, cd player
1000), (television, music system, console,
600), (cd player, console, music system 400))
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Bundle bidding scenario
((Computer, television, cd player
1000), (television, music system, console,
600), (cd player, console, music system 400))
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Bundle bidding scenario
((Computer, television, cd player
1000), (television, music system, console,
600), (cd player, console, music system 400))
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Multiple-item auctions

• Auction of multiple, distinguishable items
• Bidders have preferences over item combinations
• Combinatorial auctions
• Bids can be submitted over item bundles
• Winner selection combinatorial optimization
• NP-complete

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Source

• Vijay Krishna Auction Theory (Academic Press)
• Paul Klemperer Auction Theory A guide to the
  literature (Journal of Economics Survey)
• Elmar Wolfstetter Auctions An Introduction
• Tuomas Sandholm COURSE CS 15-892 Foundations
  of Electronic Marketplaces (CMU)