Auctions Cs2008

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Game TheoryAuctions

cs-Sep 2008

Ehud Lehrer [email protected]



Auctions - Introduction

Auctions have been used since antiquity for the sale of a variety

of objects:

Herodotus Book 1: Clio [190], ~500B.C.

“So when they have arrived at Babylon in their voyage and have

disposed of their cargo, they sell by auction the ribs of the boat

and all the straw …”

Today:

Fish, tobacco, flowers, horses, airplanes, art objects, collectibles.Internet auctions (e-bay).

Treasury bills.

Radio spectrum.



An Auction in General

• There is a valuable prize ( “object” ).

• People ( the “bidders” ) take an action that signals how muchthey are willing to pay for the prize ( place a “bid” ).

• There is a well-defined rule that assigns the prize to one of the

bidders according to all the bids, and all bidders know the rule(e.g., the bidder with the highest bid ).

• There is a well-defined rule that dictates how much each bidder

should pay, and all bidders know the rule

(e.g., the winner pays his/her bid ).



These are also auctions

Queueing for a scarce ticket

Prize: A ticket.

Bidders: People in the queue.Bids: Time spent waiting on line.

Corporate takeoverPrize: The company that is taken over.

Bidders: The corporate raiders.

Bids: Takeover offers.



All Bidders Pay

It might sound strange. But, in some auctions, all bidders pay anamount equal to their bid.

I.e., when queueing for a scarce ticket: time spent waiting in the queue.



Objectives of this session

1. To acquire the language

2. To understand how much bidders should bid.

3. To understand the consequences of the seller’s choice

regarding auction design on the expected revenue.



Auctions – pros and cons

Advantages of auctions Disadvantages of auctions

1) Simple and transparent games

(mechanisms). Universal rules (does

not depend on the object for

sale), anonymous (all bidders are

treated equally).

1) Costly (Sothby’s charges 15%

from buyer, and 20% fromseller of the final price; E-bay

charges seller about 5%).

2) If badly designed, the revenuemight be disappointingly low.

2) Efficient method to sell objects

when demand is not known.

3) Optimality and efficiency in

broad range of settings.

4) Reduces the potential of

corruption.



Valuation

The value of a bidder = his/her maximum willingness to pay.

If bidder’s value is v, and she pays p (if wins), her total gain is:

v - p if the bidder wins the auction0 if the bidder does not win

Private value: different bidders have different values (bidders have different tastes).

The value of each bidder is unrelated to the values of other bidders. One bidder might

have high value, while the other a low value for the same object (art, antiquity).

Interdependent values: different bidders have different, yet related, values. A bidderdoes not have a precise knowledge of the value of the object (radio spectrum, TV

broadcast rights, refinery).

A special case:

Common value: the object is worth the same to all bidders (oil leases, gold).



Basic Auction Formats

English auction (open ascending

price auction)

An auctioneer raises the price of the object aslong as there are at least two interested bidders.

The auction stops when there is only one

interested bidder. The last bidder pays an

amount equal to the price at which the second-last bidder dropped out.

Art, antiquity, E-bay




English auction (open ascending price auction)

Dutch auction (open descending price auction)

The auctioneer begins by calling out a

price high enough (presumably no

bidder is interested in buying the object

at that price). The price is gradually

lowered until some bidder indicates

his/her interest. The object is then sold

to this bidder at the given price.

Cut flowers in the Netherlands.






Sealed-bid first-price auction

Bidders submit bids in sealed envelopes. The person submitting the highest bid wins

the object and pays what he bid.

Refinancing credit and foreign exchange. Governments use them to sell treasurybills, mineral rights including oil fields.







Bidders submit bids in sealed envelopes. The person submitting the highest bid winsthe object but pays not what he bid, but the second-highest bid.

Stamps, E-Bay system of using a proxy.

Sealed-bid second-price auction







Sealed-bid second-price auction

Bidders submit bids in sealed envelopes. The person submitting the highest bid wins

the object. All bidders pay an amount equal to their bid.

E.g., queuing for a scarce ticket

Sealed-bid all-pay auction



Dutch ≈ Sealed-bid first-price

Sealed-bid first-price auction Dutch auction (open descending price auction)

1) Each bidder chooses a bid. 1) Each bidder chooses a bid.

2) The bidder “remember” his/her bid.2) The bidder places the bid in an envelope.

3) The bidder with the highest bid wins, 3) The price is lowered until the bidder with

the highest bid indicates his/her interest.

That bidder wins,4) and pays his/her bid.

4) and pays his/her bid.

The bidder chooses a bid without

knowing the bids of the other

bidders.

The bidder chooses a stopping price

without knowing the stopping prices

of the other bidders.

Stopping pricebid

These two methods are equivalent.

A l i f E li h A i



Analysis of English Auction

English auction: the price increases until only one bidder remains.

What is the optimal strategy?

Stay in as long as the current price is at or below your value.

Once the current price exceeds your value – drop out.

If all players use their optimal strategy:

The winner is the one with highest value.

The winner pays an amount equal to the second-highest value

(modulo the bid increment).

A l i f S d P i A ti



Analysis of Second-Price Auction

Sealed-bid second-price auction: the winner pays the second highest bid.


Consider the following strategy: bid your actual value.

Claim: this strategy dominates all other strategies.

One strategy dominates another strategy if the expected gain by using the latter never

exceeds the expected gain by using the former.

Reason:

Over-bidding is dominated by bidding the

actual value.

v

b = highest bid among the other bidder.

Bid actual value Over-bidding

b

Win,

pay b

Win,

pay b








b

Doesn’t

win

Doesn’t

win


v




Reason:


actual value.









bDoesn’twin Wins,pays b


v




Reason:


actual value.


But b is more than the private value, so the bidder

gains by bidding the actual value – he/she pays for

the object more than its (subjective) value.








b

Win,

pay b

Win,

pay b

Bid actual value Under-bidding

v




Reason:


actual value.


Under-bidding is dominated by bidding theactual value as well.

Anal sis of Second Price A ction







b

Doesn’t

win

Doesn’t

win

Bid actual value

v

Under-bidding




Reason:


actual value.



Analysis of Second Price Auction







bWin,

pays bDoesn’t

win

Bid actual value

v

Under-bidding




Reason:


actual value.



But b is less than the private value, so the biddergains by bidding the actual value – he/she pays for

the object less than its (subjective) worth.



In a sealed-bid, second-price

auction bidding the true valueis a dominant strategy.

When each participant bids his/her

true value:• The bidder with the highest value wins.

• The winner pays the second-highest

value.

English Sealed bid second price



English ≈ Sealed-bid second-price

English auction: the price increases until only one bidder remains.

Stay in as long as the current price is at or below your value.

Once the current price exceeds your value – drop out.


A dominating strategy: bid your actual value.

Both methods are strategically equivalent: the same strategy should be used in both.

The winner is the bidder with the highest private value.The winner pays an amount equal to the second highest private value.

ThirdThird Price AuctionPrice Auction



ThirdThird--Price AuctionPrice Auction

The highest bid wins.

The winner pays the third-highest bid.

Is bidding the true value a

dominant strategy?

ThirdThird Price AuctionPrice Auction

http://www.pages.ebay.com/

http://www.pages.ebay.com/



ThirdThird--Price AuctionPrice Auction

Anne values the painting at 60.

The bids of the others are as follows.

70

50

highest

2nd highest

Anne bids 60:

gains 0.

Anne bids 80:

60

70

50

80

Wins!Wins!

gains 10.

Wins!Wins!

Bidding the true

value is not a

dominant

strategy in a third

price auction.

Let’s play a first-price auction



Let s play a first-price auction

You are about to participate in a sealed-bid first-price auction.

• There are n other bidders whose private values• The values are independent of each other and are randomly drawn from 0 to 100 (whole

numbers

• Your private value is written on the top of the paper you receive

• Please write down your bid at the bottom

Analysis of first-price auction



Analysis of first-price auction

Sealed-bid first-price auction: the winner pays his/her bid.

The strategy “bid your actual value” is dominated !!

The strategy “bid your actual value” guarantees payoff 0:• If you do not win, you get 0.

• If you win, you pay an amount equal to the value of the object, so the overall gain is 0.

Consider the strategy “bid your actual value minus 10”:

• If you do not win, you get 0.• If you win, your overall gain is 10.

One lowers the probability to win, but in case one wins, his/her gain is positive.

The solution to this tradeoff gives the optimal bidding strategy.

As more bidders participate in the auction, you are less likely to win. To counter this

bidders tend to bid more aggressively – this is good for the seller.



Two bidders: first-price auction



Two bidders: first price auction

Two bidders.

The private value of each bidder is uniformly distributed in the interval [0,1].

That is, v1 and v2, the private values of the two bidders, are independent, and have the

uniform distribution over the unit interval.

Sealed-bid first-price auction:

An equilibrium: bid half your value.

g(b1)Let us see why:

b1

v1

1/2v1 /2

Suppose bidder 2 bids half his private value.

If bidder 1 bids b1, her expected gain is:

maximum

g(b1) = E[P(b1 > v2 /2) × (v1 – b1)]

= 2b1 × (v1 – b1) if b1<1/2,

= (v1 – b1) if b1 ≥ 1/2.

Two bidders: first-price auction



Two bidders: first price auction

Two bidders.The private value of each bidder is uniformly distributed in the interval [0,1].

That is, v1 and v2, the private values of the two bidders, are independent, and

have the uniform distribution over the unit interval.

Sealed-bid first-price auction:

An equilibrium: bid half your value.

Revenue of the seller = selling price = max{v1 /2, v2 /2} = ½ max{v1, v2}

1/3What is the expected revenue of the seller?

Example – equivalent revenues



Example equivalent revenues

Two bidders.

The private value of each bidder is uniformly distributed in the

interval [0,1].That is, v1 and v2, the private values of the two bidders, are

independent, and have the uniform distribution over the unit

interval.

Expected revenue of the seller:

Sealed-bid second-price auction: 1/3

Sealed-bid first-price auction: 1/3

More generally – n bidders



More generally n bidders

Suppose that are n bidders in the interval 0 to M (uniform)

In equilibrium bid:

1(own value)n

n

−

The Revenue Equivalence Theorem



q

Revenue Equivalence Theorem for auctions with private

values and risk-neutral bidders:

Suppose there are two auction methods that satisfy the following:1) In both, a bidder with private value 0 pays nothing.

2) The bidder with the maximal bid wins the object.

Then, the expected revenue to the seller is the same in both

auction methods.

Conclusion: First-price auction and Second-price auction generate the same

revenue.

But, first-price auction might generate different expectedrevenue than second-price auction with minimum price.

France Telecom eAuction



• A round of the eAuction referred to 3 categories

• each category was partitioned into 3 sub-categories (equipment, service-installation,

maintenance) – actually 9 items per round

• 30 min for each round unless a better bid comes at the last moment

• in this case the time is extended by 3 min• at any time and any sub-category the bidder knows its rank and the lowest bid

What actually happened?

• we placed high bids first ( we got to know that there were at least 12 other bidders).

• the prices went down rapidly: 65M, 46.5M, 25.6M, 24.9M, 24.7M (in one sub-category)

• the bids then stabilised

• we were 2nd, 3rd, 4th

• 30 min were over• the goal was to reach one of the first three highest positions in the ranking

• the prices went down at a low pace: 24.6M, 23.8M, 23.6M

• it lasted one hour and a half

• then suddenly the round was over• our ranking was 3 and in one sub-category second (with a gap of 1 Euro from the first)



Collusive behavior



Example: US West competed vigorously with McLeod for a license in Rochester with

code name 378. At the same time, McLeod faced no competition for the Iowa licenses.

US West bid $313,378 and $62,378 for two Iowa licenses.

Lesson: Multi-unit ascending auctions are very prone to collusive behavior.

Other thoughts:

• In what sense it might be that it is not a pure private-value auction?

• Shall I bid immediately the auction stage is extended, or should I

wait to the last minute?



Risk Aversion



The revenue equivalence Theorem holds since bidders are risk neutral (theymaximize expected gain).

What happens if bidders are risk averse (they do not like risk)?

In sealed-bid second price auction, it is still dominant to bid an amount equal to theprivate value.

In sealed-bid first price auction, risk neutral bidders bid less than their private value.

The less one bids, the higher chance he/she loses the object.

If the bidder is risk averse, he will bid higher than the amount he bids if he were risk neutral.

Conclusion:

Expected Revenue in sealed-bid second-

price auction with risk neutral bidders

Expected Revenue in sealed-bid first-

price auction with risk neutral bidders=

= <

Expected Revenue in sealed-bid first-

price auction with risk averse bidders

Expected Revenue in sealed-bid second-

price auction with risk averse bidders

Cournot Game - solution

Summary of today’s session



y y

Auctions:• probably the most active research field in micro-economics

• popular types: English, Dutch, first and second price,sealed-bid

Cournot Game - solution

Next time



• Auctions: a bit more on common values – forming beliefs

• Repeated interactions

Please read the first part of “Reputation and Coalitions in Medieval Trade: Evidence on

the Maghribi Traders” by A. GREIF

Auctions Cs2008

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