lecture 1112

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)


Examples of Auctions FCC Spectrum Procurement Auctions Treasury Bills Internet Wine Options Quota Rights


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


Private Value Dinner.


Common Value Unproven oil fields.


Basic Auction Types Open Auctions (sequential)

• English Auctions• Dutch Auctions• Japanese Auctions

Sealed Auctions (simultaneous)• First Price Sealed Bid• Second Price Sealed Bid


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


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


Sample Dutch AuctionMinimum Bid: $10

$120$110$100$ 90$ 80$ 70

$ 10$ 20$ 30$ 40$ 50$ 60


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)


Strategic Equivalence of Auction Types

Dutch Auction~ First Price Sealed Bid

English Auction ~ Second Price Sealed Bid


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


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


Optimal Bidding StrategySecond Price Auctions

You Lose

You Win

hig

her

Your bid

Others’ bids

Your value


Bidding Higher Than My Valuation

Case 1 Case 2 Case 3

No difference No difference Lose money


Bidding Lower Than My Valuation

Case 1 Case 2 Case 3

No difference No difference Lose money


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!


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!


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


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


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


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


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?


“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?


“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?


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


“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”


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


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


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


Common Value Auction Learn about value from other bids

Estimates on average correct

Winner not picked randomly – highest estimate -> too high


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!


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


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)


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!


Dependent Multi-Unit (Combinatorial)

PCS Spectrum Auctions


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


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

lecture 1112

Documents

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winnerpays price

price auctionbidding

highest bidshighest

bidding ones true valuation

pricefirst bidder

bidder pays

game theorya cynic