Game Theory
“A cynic knows the price of everything
and the value of nothing”- Oscar Wilde, Lady Windemere’s Fan
Mike ShorLecture 11
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What is an Auction?
auc•tion1. A public sale in which property or
merchandise are sold to the highest bidder.
2. A market institution with explicit rulesdetermining resource allocation and prices on the basis of bids from participants.
3. Games: The bidding in bridge
[Latin: auctiō, auctiōn – from auctus, past participle of augēre, to increase]
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Examples of Auctions 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
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Sample Auction
“Mistakes are the portals of discovery”
- James Joyce
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Going Once, Going Twice, …
Bidding starts at $1
Who will make the first bid?
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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 winner’s curse
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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
of the object e.g., FCC spectrum, drilling,
disciplinary corporate takeovers
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Basic Auction Types
Open Auctions (sequential)English AuctionsDutch AuctionsJapanese Auctions
Sealed Auctions (simultaneous)First Price Sealed BidSecond Price Sealed Bid
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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
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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
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Sealed-Bid First Price Auctions All buyers submit bids Buyer submitting the highest bid wins
and pays the price he or she bid
$700
$400
$500
$300
WINNER! Pays $700
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Sealed-Bid Second Price Auctions All buyers submit bids Buyer submitting the highest bid wins
and pays the second highest bid
$700
$400
$500
$300
WINNER! Pays $500
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Why Second Price?
It is strategically equivalent to an English Auction
$300$400
$500
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Why Second Price?
Bidding strategy is easy Bidding one’s true valuation
is a dominant strategy
Intuition: The amount a bidder pays is not
dependent on her bid
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Optimal Bidding Strategyin Second Price Auctions
You Lose
You Win
hig
her
Your bid
Others’ bids
Your value
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Bidding Higher Than My Valuation
Case 1 Case 2 Case 3
No difference No difference Lose money
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Bidding Lower Than My Valuation
Case 1 Case 2 Case 3
No difference No difference Lose money
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Second Price Auction
In a second price auction, always bid your true valuation
Winning bidder’s surplusDifference between the winner’s valuation and
the second highest valuationSurplus decreases with more bidders
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First Price Auction
First price auction presents tradeoffs If bidding your valuation – no surplus
Lower your bid below your valuationSmaller chance of winning, lower price
Bid shadingDepends on the number of biddersDepends on your informationOptimal bidding strategy is complicated!Rule of thumb: bid (N-1)/N of your value
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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
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Which is Better?
Draw 4 keep highest
Draw 4 keep 2nd highest
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Revenue Equivalence
All common auction formats yield the same expected revenue (in theory)
In fact, any auction satisfying: 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
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Revenue Equivalence in the Real World Risk Aversion
Does not influence 2nd price auctionsRisk averse bidders are more aggressive
in first price auctionsRisk aversion 1st price or Dutch are better
Non-familiarity with auctionsMore overbidding in second-price auctionsMore overbidding in sealed-bid auctionsInexperience 2nd price sealed bid is better
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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?
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Number of Bidders Two bidders
Each has a value of 20 or 40 There are four value combinations:
Pr{20,20}=Pr{20,40}=Pr{40,20}=Pr{40,40}= ¼
Expected price = ¾ (20)+ ¼ (40) = 25
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Number of Bidders Three bidders
Each has a value of 20 or 40 There are eight value combinations:
Pr{20,20,20}=Pr{20,20,40}=Pr{20,40,20}= Pr{20,40,40}=Pr{40,20,20}=Pr{40,20,40}= Pr{40,40,20}=Pr{40,40,40}= 1/8
Expected price = ½ (20)+ ½ (40) = 30
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Number of Bidders Assume more generally that valuations are drawn
uniformly from [20,40]:
20
25
30
35
40
1 10 100 1000Number of Bidders
Ex
pe
cte
d P
ric
e
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Designing Auction Rules
Every rule may have unintended consequences
What is the minimum bid for a new bidder?
What kind of financing can bidders present?
How much must bids be beaten by?
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“Tweaking the Rules” IeBay …
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?
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“Tweaking the Rules” IIFCC 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?
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“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
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“Tweaking the Rules” III FCC Spectrum Auctions… Discouraging Collusion
Do not identify highest bidders
Capturing SurplusDo not set a bidding increment
“I bid $8,000,483”
“I bid $3,000,395”