Game Theory

1
Game Theory
Topic 8Auctions

• Everything is worth what its purchaser will pay
  for it.

- Publilius Syrus (Maxim 847, 42 B.C.)
2
What is an Auction?

• Definition
• A market institution with rules governing
  resource allocation on the basis of bids from
  participants
• Over 30 of US GDP moves through auctions

• Wine
• Art
• Flowers
• Fish
• Electric power
• Treasury bills

• IPOs
• Emissions permits
• Radio Spectrum
• Import quotas
• Mineral rights
• Procurement

3
Sample Auction
Mistakes are the portals of
discovery
- James Joyce
4
Going Once, Going Twice,

• Bidding starts at 1
• Who will make the first bid?

5
Overview of Auctions

• Auctions are a tricky business
• Different auction mechanisms
• sealed vs. open auctions
• first vs. second price
• optimal bidding care in design
• Different sources of uncertainty
• private vs. common value auctions
• the winners curse

6
Private Value Auction

• Dinner

7
Common Value Auction

• Unproven oil fields

8
Sources of Uncertainty

• Private Value Auction
• Each bidder knows his or her value for the object
• Bidders differ in their values for the object
• e.g., memorabilia, consumption items
• Common Value Auction
• The item has a single though unknown value
• Bidders differ in their estimates of the true
  value
• e.g., FCC spectrum, drilling,
  disciplinary corporate takeovers

9
Basic Auction Types

• Open Auctions (sequential)
• English Auctions
• Dutch Auctions
• Japanese Auctions
• Sealed Auctions (simultaneous)
• First Price Sealed Bid
• Second Price Sealed Bid

10
English Auctions
(Ascending Bid)

• Bidders call out prices (outcry)
• Auctioneer calls out prices (silent)
• Bidders hold down button (Japanese)
• Highest bidder gets the object
• Pays a bit over the next highest bid

11
Dutch (Tulip) Auction Descending Bid

• Price Clock ticks down the price
• First bidder to buzz in and stop the clock is
  the winner
• Pays price on clock

12
Sample Dutch AuctionMinimum Bid 10
13
Sealed-Bid First Price
Auctions

• All buyers submit bids
• Buyer submitting the highest bid wins
  and pays the price he or she bid

14
Sealed-Bid Second
Price Auctions

• All buyers submit bids
• Buyer submitting the highest bid wins
  and pays the second highest bid

15
Why Second Price?

• It is strategically equivalent
  to an English Auction

16
Why Second Price?

• Bidding strategy is easy
• Bidding ones true valuation
  is a
  (weakly) dominant strategy
• Intuition
• The amount a bidder pays is not dependent on her
  bid

17
Bidding True Valuation

• Say your value is 100
• Why not bid 500?
• If others all bid under 100, no difference
• If someone bids gt 500, no difference
• If someone bids 300, you overpay!
• Why not bid 50?
• If someone bids 80, you lose (but would have
  made money bidding 100)

18
First Price Auction

• First price auction presents tradeoffs
• If bidding your valuation no surplus
• Lower your bid below your valuation
• Smaller chance of winning, lower price
• Bid shading
• Depends on the number of bidders
• Depends on your information
• Optimal bidding strategy is complicated!

19
Which is Better?

• In a second price auction
• bidders bid their true value
• auctioneer receives the second highest bid
• In a first price auction
• bidders bid below their true value
• auctioneer receives the highest bid

20
Revenue Equivalence

• All common auction formats yield the same
  expected revenue (in theory)
• Any auctions in which
• The prize always goes to the person
  with the highest valuation
• A bidder with the lowest possible valuation
  expects zero surplus
• yield the same expected revenue

21
Revenue Equivalence in the
Real World

• Risk Aversion
• Does not influence 2nd price auctions
• Risk averse bidders are more aggressive
  in first price auctions
• Risk aversion ? 1st price or Dutch are better
• Non-familiarity with auctions
• More overbidding in second-price auctions
• More overbidding in sealed-bid auctions
• Inexperience ? 2nd price sealed bid is better

22
Designing Auction Rules

• Every rule may have unintended consequences
• What is the minimum bid for a new bidder?
• How much must bids be beaten by?

23
Importance of RuleseBay

• Three laptops for sale
• Top three bidders pay the third
  highest bid
• Opening bid 1
• Current high bids
• 50, 80, 400
• How high should the next bid be?

24
Importance of Rules FCC Spectrum Auctions

• Discouraging Collusion
• Do not identify highest bidders
• Capturing Surplus
• Do not set a bidding increment
• I bid 8,000,483
• I bid 3,000,395

25
Summary

• Bidding
• Bid true valuation in 2nd price auctions
• Shade bids in 1st price auctions
• Designing
• Take advantage of inexperience
• Take advantage of risk aversion
• Do sweat the little stuff

26
Sources of Uncertainty

• Private Value Auction
• Difficult to lose money
• Do not bid more than your value
  (or less than your cost)
• Common Value Auction
• The item has a single though unknown value
• Bidders differ in their estimates
• The winner might be wrong!

27
Common Value Auctions

• Example Offshore oil leases
• Value of oil is roughly the same
  for every participant
• No bidder knows value for sure
• Each bidder has some information
• Auction formats are not equivalent
• Oral auctions provide information
• Sealed-bid auctions do not

28
Hypothetical Oil Field Auction
4
5
2
3
1
10
8
9
6
7
10 tracts for sale
each with four bidders
29
Hypothetical Oil Field Auction

• Each tract has four bidders
• Each bidder knows the amount of
  oil in his or her quadrant
• Each quarters value is evenly distributed
  between 200,000 and 800,000

• Total value of oil field
  Sum of the values of
  the four quarters
• Type of auction
  First price sealed bid

30
Oil Field Auction

• How much do you bid?

31
The Winners Curse
40
70
50
60
80
60

• The estimates are correct, on average
• What happens if everyone bids his or her estimate?

32
The Winners Curse Defined

• If the average estimate is generally correct, the
  highest estimate is usually too high
• If bids are based on estimates, the highest
  bidder overpays
• To avoid the winners curse, estimate the average
  of the object conditional on winning the auction

33
Avoiding the Winners Curse

• Given that I win an auction
• All others bid less than me
• Thus the objects value must be lower than I
  thought
• Winning the auction is bad news
• One must incorporate this into ones bid
• Assume that your estimate is the most optimistic

34
Avoiding the Winners
Curse

• Bidding for a company
  of uncertain value

35
Avoiding the Winners Curse
The expected value of the object
is irrelevant. To bid
Consider only the
value of the object if you win!
36
Avoiding the Winners Curse

• Bidding with no regrets
• Since winning means you have the most optimistic
  signal, always bid as if you
  have the highest signal
• If your estimate is the most optimistic
  what is the object worth?
• Use that as the basis of your bid

37
Summary

• Average value of an object is irrelevant
• Consider only the value if you win
• In common value auctions, assume that you have
  the most optimistic estimate

38

• The greatest auction in history
• - New York Times, March 16, 1995, p.A17

39
More Bidders

• More bidders lead to higher prices
• Example
• Second price auction
• Each bidder has a valuation of either
  20 or 40, each with equal probability
• What is the expected revenue?

40
Number of Bidders

• Two bidders
• Each has a value of 20 or 40
• There are four value combinations
• Pr20,20Pr20,40Pr40,20Pr40,40 ¼
• Expected price ¾ (20) ¼ (40) 25

41
Number of Bidders

• Three bidders
• Each has a value of 20 or 40
• There are eight value combinations
• Pr20,20,20Pr20,20,40Pr20,40,20
• Pr20,40,40Pr40,20,20Pr40,20,40
• Pr40,40,20Pr40,40,40 1/8
• Expected price ½ (20) ½ (40) 30

42
Number of Bidders

• Assume more generally that valuations are drawn
  uniformly from 20,40

• Example
• New Zealand 1993 UHF License Auction
• Second price auction
• Four lots won by Sky Network

Lot High Bid (k) Second Bid (k) price/high
1 2,371 401 17
2 2,273 401 18
3 2,273 401 18
4 1,121 401 36
43
Importance of RulesFCC Spectrum Auctions

• Want to encourage minority and female-owned firms
  to bid but licenses are very expensive.
• Reserve several frequency blocks for smaller
  bidders.
• Allow 10 down, low interest, remaining principal
  owed in 7 years.
• What happens?

44
Tweaking the Rules II(continued)

• Bid high!
• If licenses end up being worth less, default!
• Of the four largest winners,
• one went bankrupt and defaulted
• one had 1B reduced to 66M in bankruptcy court
• one was a front for Qualcomm
• one was sold to Siemens