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teoria de juegos
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Game Theory
“A Cynic Knows the Price of Everything and the Value of Nothing”
-Oscar Wilde, Lady Windemere’s Fan
Mike ShorLectures 11 & 12
Game Theory - Mike Shor 2
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]
e
(ôk’sh n)
Game Theory - Mike Shor 3
Examples of Auctions FCC Spectrum Procurement Auctions Treasury Bills Internet Wine Options Quota Rights
Game Theory - Mike Shor 4
Classifying Auctions Private value auction
• Each bidder’s value is known to him or her• Values of other bidders unknown
Common value auction• Item has single, true value• Value unknown at time of bidding
Game Theory - Mike Shor 5
Private Value Dinner.
Game Theory - Mike Shor 6
Common Value Unproven oil fields.
Game Theory - Mike Shor 7
Basic Auction Types Open Auctions (sequential)
• English Auctions• Dutch Auctions• Japanese Auctions
Sealed Auctions (simultaneous)• First Price Sealed Bid• Second Price Sealed Bid
Game Theory - Mike Shor 8
English Auction 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
Game Theory - Mike Shor 9
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
Game Theory - Mike Shor 10
Sample Dutch AuctionMinimum Bid: $10
$120$110$100$ 90$ 80$ 70
$ 10$ 20$ 30$ 40$ 50$ 60
Game Theory - Mike Shor 11
Sealed Bid Bidders submit secret bids to the
auctioneer The auctioneer determines the
highest and second highest bids
Highest bidder gets the object and• Pays own bid (First-Price)• Pays the next highest bid
(Second-Price Vickrey)
Game Theory - Mike Shor 12
Strategic Equivalence of Auction Types
Dutch Auction~ First Price Sealed Bid
English Auction ~ Second Price Sealed Bid
Game Theory - Mike Shor 13
Other Auction Formats Double auction
• Buyers and sellers bid• Stock exchanges
Reverse auction• Single buyer and multiple sellers• Priceline.com
Multiunit auction• Seller has multiple items for sale• FCC spectrum auctions
Game Theory - Mike Shor 14
The Vickrey Second Price Auction Bidding strategy is easy!
Bidding one’s true valuation is a (weakly) dominant strategy
• Proxy bidding on eBay
Intuition: the amount a bidder pays is not dependent on her bid
Game Theory - Mike Shor 15
Optimal Bidding StrategySecond Price Auctions
You Lose
You Win
hig
her
Your bid
Others’ bids
Your value
Game Theory - Mike Shor 16
Bidding Higher Than My Valuation
Case 1 Case 2 Case 3
No difference No difference Lose money
Game Theory - Mike Shor 17
Bidding Lower Than My Valuation
Case 1 Case 2 Case 3
No difference No difference Lose money
Game Theory - Mike Shor 18
First Price Auction Second price auction
• Always bid your true valuation
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 biddersDepends on your informationOptimal bidding strategy complicated!
Game Theory - Mike Shor 19
Revenue Equivalence Assume an auction with the
following rules:• The prize always goes to the person with
the highest valuation• A bidder with the lowest possible valuation
expects zero surplus
All such auctions yield the same expected revenue!
Game Theory - Mike Shor 20
Revenue in the Real World Risk Aversion
• Does not influence 2nd price auctions• Risk averse bidders are more aggressive in
first-price auctions
Non-familiarity with auctions• More overbidding in second-price auctions• More overbidding in sealed-bid auctions
Game Theory - Mike Shor 21
More Bidders More bidders leads to higher prices More bidders leads to less surplus
Example (second price auction): Each bidder has a valuation of either $20 or $40, each with equal probability
Game Theory - Mike Shor 22
Number of Bidders Two biddersPr{20,20}=Pr{20,40}=Pr{40,20}=Pr{40,40}= ¼
Expected price = ¾ (20)+ ¼ (40) = 25
Three biddersPr{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 = (4/8) (20) + (4/8) (40) = 30
Game Theory - Mike Shor 23
Number of Bidders Assume more generally that valuations
are drawn uniformly from [20,40] Expected price:
20
25
30
35
40
1 10 100 1000
Game Theory - Mike Shor 24
Designing Auction Rules Bidding
• Who is allowed to bid?• How are bids presented?• How much must bids be beaten by?• Is bidding anonymous or favored?• Is there a reserve price?
Information• Are current bids revealed?• Are winners identified?
Clearing• Who gets what?
Game Theory - Mike Shor 25
“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 next bid be?
Game Theory - Mike Shor 26
“Tweaking the Rules” IIFCC Spectrum Auctions… Want to encourage minority and
female-owned firms to bid but licenses are very expensive.
Reserve “blocks C & F” for smaller bidders.
Allow 10% down, low interest, remaining principal owed in 7 years.
What happens?
Game Theory - Mike Shor 27
FCC Spectrum Auctions Bid high – if licenses end up being worth
less, default! 83 winners: Of the four largest, … … went bankrupt and defaulted … got $1B reduced to $66M in
bankruptcy court … was a front for Qualcom … was sold to Siemens
Game Theory - Mike Shor 28
“Tweaking the Rules” IIIFCC Spectrum Auctions… Largest potential problem: Collusion
• Bid rotation (construction)• Subcontracting (Treasury Bills)
Discouraging Collusion• Sealed-bid Auctions• Anonymity – do not release results
“I bid $16,000,483” “I bid $11,000,212”
Game Theory - Mike Shor 29
Sources of Uncertainty Private value auction
• Inherent differences among bidders• Item is for personal use – not resale• Values of other bidders unknown
Common value auction• Item has single, true value• Value unknown at time of bidding
Game Theory - Mike Shor 30
Common Value Auctions Example: Offshore oil leases
• Value of oil is same for every participant• No bidder knows value for sure• Each bidder has some information• Exploratory drilling
Different auction formats are not equivalent
• Oral auctions provide information• Sealed-bid auctions do not
Game Theory - Mike Shor 31
Levels of Thinking
What would I be willing to pay given
what I know before submitting my bid
versus
what I know before submitting my bid, and
that I will only win if no one else is willing to bid higher than me
Game Theory - Mike Shor 32
Common Value Auction Learn about value from other bids
Estimates on average correct
Winner not picked randomly – highest estimate -> too high
Game Theory - Mike Shor 33
Avoiding Winner’s Curse
COMMANDMENT
The expected value of the object is irrelevant.
To bid: Consider only the value of the object if you win!
Game Theory - Mike Shor 34
Avoiding Winner’s Curse Bidding with no regrets:
Since winning means you have the highest signal, always bid as if you have the highest signal
If you have highest signal – what is the object worth?
Use that as the basis of your bid
Game Theory - Mike Shor 35
Private or Common Value? Most Auctions are not purely private
value or purely common value
Private Value• Resale component (art)• Reputation / conspicuous consumption
Public Value• Differing abilities (mining)
Game Theory - Mike Shor 36
Multiple Unit Auctions If products are distinct and
valuations are independent, can hold distinct auctions
If values are not independent…• Objects interrelated• Substitutes• Complements
… then things get very messy!
Game Theory - Mike Shor 37
Dependent Multi-Unit (Combinatorial)
PCS Spectrum Auctions
Game Theory - Mike Shor 38
3x108 m 0 HZ
3x10-7 Å 1025 HZ
Very Low Frequency
(VLF)
LF HF UHF EHF MF VHF SHF
Infrared Visible Ultraviolet X-Ray Gamma Ray Cosmic Ray
3 KHZ 300 GHZ
Marine AM Radio Amateur VHF UHF Misc Satellite
UNLICENSED B E F C D A B E F C D A
1850 MHZ 1910 1930 1990 MHZ
300 MHZ 3 GHZ
TV Broadcasting Aeronautical Channels 14-36 Channels 38-69 Radionavigation Misc Misc
18 winning bidders, 99 licenses
$7.7 billion dollars
Game Theory - Mike Shor 39
Summary Participating
• Bid true valuation in 2nd price auctions• Shade bids in 1st price auctions• In common value auctions, assume that you
have the most optimistic estimate Designing
• Preclude cooperation among bidders• Announced reserve price to gain efficiency
and relieve doubts• In common value auctions, provide
information to help with winner’s curse