Transcript Auctions -1

Auctions -1
Debasis Mishra
QIP Short-Term Course on Electronic Commerce
Indian Institute of Science, Bangalore
February 15, 2006
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Outline
Single-item auctions
 Models of bidder behavior
 Multi-item auctions
 References
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Auctions - Introduction

Auction - comes from Latin word auctus to
mean increase.
– Not every auction has increasing prices.
Among one of the first engaging tales Sale of Roman empire to the highest
bidder in 1764.
 A market institution that works on the
concept of competition.
 Natural discovery of price and buyers.
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Auctions - Why and Why Not?

Why auctions?
– Seller unsure about how much the price should be.
– It can be used to sell almost anything -universal.
– Buyers learn, in some auctions, about the information
of other buyers - leads to more efficient and revenuegenerating markets.

Why not auctions?
– Overhead of time and infrastructure.
– Fixed price methods are simple.
– Values of bidders are almost known.
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Auction Settings
Forward auctions: a seller selling items to buyers
(bidders).
 Reverse auctions: a buyer buying items from
sellers/suppliers (bidders).
 Both the settings are natural transpose of each
other:

– Bidders compete in both settings.
– At low (high) price many buyers (sellers) demand
(supply) items in forward (reverse) auctions.
– Highest (lowest) price buyer (seller) wins in forward
(reverse) auctions.
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Auctions in Practice
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Selling of flowers (Holland), tobacco, fish, tea, art objects
and antique pieces (Sotheby's).
Transfer of assets from public to private: Sale of
industrial enterprises in Eastern Europe, transportation
system in Britain, timber rights all over the world, and
off-shore oil leases.
Auction of spectrum rights worldwide - US, Europe, and
even India.
Internet auctions of consumer goods (amazon.com,
ebay.com etc.). Google's Adword auctions. Procurement
auctions - freemarkets.com (now Ariba), GM and IBM's
sourcing solutions.
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Valuations
Valuation: The maximum amount a bidder is
willing to pay.
 In procurement auctions, the value is negative of
cost of procurement - the minimum price a
bidder is demanding.
 Auctions are used mainly because the auctioneer
is unsure about the valuations (or simply,
values) of bidders.
 Two models: (i) private values (ii) common or
interdependent value.
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Private Value Model
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Each bidder knows his own value of the item
exactly at the time of bidding, but knows nothing
about the values of other bidders.
– Value of other bidders do not influence his own value.
– Suitable for: auctions for paintings, stamps etc. (a
bidder knows the value of a painting exactly),
procurement auction settings (a supplier's cost
depends only on his own production technology).

Most plausible when the value of the item to a
bidder is derived from its use alone and the
bidder knows the item well.
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Interdependent Value Model (1 of 2)

Worth of an item unknown at the time of bidding
to bidders.
– Examples: oil field (depth of oil wells not well known),
second-hand products (quality of the product is not
known).
In such cases, a bidder will have an estimate or a
privately known signal (an expert's opinion or a
test result) that is correlated with the true value.
 Formally, every bidder has a signal xi and the
value of bidder i is vi(x1, x2,..., xn)
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Interdependent Value Model (2 of 2)
Information, such as estimates or signals, of
other bidders will influence the value of a bidder.
 Values are unknown to bidders at the time of
bidding and may be affected by information
available to other bidders.
 A special case is common values - every bidder
has the same value ex post (i.e., once they know
everyone’s signals). Example: oil field auction.
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– v(x1, x2,..., xn)
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Single Item Auctions
Two formats: (i) sealed-bid (ii) open-cry
 Sealed-bid: Bidders submit bids once (in a
sealed envelope to the auctioneer)
 Open-cry: Bidders submit bids in rounds,
bids result in increase in prices (commonly
termed as iterative auctions)
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– Bids reflect if bidders are willing to participate
further in the auction.
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Single Item Sealed-Bid Auctions (1 of 3)
First-price sealed-bid: Every bidder submits a
bid; the highest bid bidder wins and pays his bid
amount.
 Second-price sealed-bid (Vickrey auction): Every
bidder submits a bid; the highest bid bidder wins
but pays an amount equal to the second highest
bid.
 First-price auctions are common in practice.
 Second-price auctions are rare: but see examples
of stamp auctions and others in
http://www.u.arizona.edu/~dreiley/pa
pers/VickreyHistory.pdf
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Single Item Sealed-Bid Auctions (2 of 3)
Example: Four bidders with values 10,8,6, and 4.
 First-price: Bidders bid 8,6,5, and 3 respectively
(bid value not equal to valuation). Highest
bidder wins and pays 8.
 Second-price/Vickrey: Bidders bid 10,8,6, and 4
(bid value equals valuation). Highest bidder wins
but pays 8.
 Neither the revenue equivalence in the two
auctions nor the bid=value in Vickrey auction in
this example is a coincidence.
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Single Item Sealed-Bid Auctions (3 of 3)
The best strategy for a bidder, irrespective
of what other bidders have bid, is to bid
his value. This is also called a dominant
strategy equilibrium in game theory.
 Though economically robust, Vickrey
auction is less transparent to bidders –
transparency in auction design is
important.
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Single-Item Open-Cry Auctions (1 of 3)
English auction: Auction starts from low price. A
bidder bids by indicating if he is willing to buy
the item at the current price. If more than one
bidder bids, then the price is raised by a finite
amount ε (bid increment), else the auction stops.
The last bidder to bid wins at the final price.
 Consider the same example (values 10,8,6,4). Let
the starting price be 0 and bid increment ε. At
price < 4+ ε, only 3 bidders will be interested …
at price < 8+ ε, only 1 bidder will be interested.
Auction stops at price < 8+ ε.
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Single-Item Open-Cry Auctions (2 of 3)
It can be shown that staying in the auction till
price reaches value is the best strategy for
bidders.
 Further, the outcome of English auction is
equivalent to (as ε reaches zero) the Vickrey
auction.
 English auction is popular in practice – more
transparent – and has similar economic
properties as the Vickrey auction.
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Single-Item Open-Cry Auctions (3 of 3)

Dutch auction (popular in Holland to sell
flowers): The auction starts from high price
where there is no demand for the item; bidders
bid indicating if they are interested in the item at
the current price; if no bidder bids then the price
is decreased by ε (bid decrement), else the
auction stops. The only bidder to bid wins at the
final price.
– In case, more than one bidder bids, then the item is
allocated at random to either of them.
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Dutch auction is strategically equivalent to the
first-price sealed-bid auction.
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Strategic Considerations
Strong requirement: dominant strategy Irrespective of the bidding strategy of other
bidders, a bidder's best strategy (one that
maximizes utility over all strategies) is to be
truthful.
 Weak requirement: (ex post) Nash equilibrium Given that all bidders bid truthfully, a bidder's
best strategy is to be truthful.
 Given an auction design, is bidding truthfully the
best strategy?
 Design an auction in which truthful bidding is
the best strategy.
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Dominant Strategy in
Vickrey Auction
Consider bidder 1. Let the bid amount of any bidder i
(not 1) be bi (need not equal value). What is the best
amount to bid for 1?
 Without loss of generality, assume b2 to be the highest
bid among bids of bidders other than 1.
 Losing the auction by bidding untruthfully gives zero
payoff. To win the auction and make positive payoff,
bidder 1 should bid more than b2.
 His payment will be b2 always, independent of his bid
amount, if he wins. His payoff is v1 - b2, where v1 is his
value. So, own bidding strategy does not influence payoff
implying truthful bidding is a dominant strategy.
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Equivalence of Auction Forms
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Dutch auction - Where should a bidder respond? That
price is the payment. First-price sealed-bid auction What bid should a bidder submit? That bid price is the
payment. So, same decision in both auctions.
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English auction - best strategy is to remain interested till
price reaches value. This terminates the auction
(approximately) at the second-highest value. This is the
outcome in the Vickrey auction.
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Dutch auction = first-price sealed-bid auction. English
auction = Vickrey auction.
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Revenue in Auctions (1 of 3)
Values are drawn from uniform distribution with range
[0,a] for n bidders.
 Expected revenue in the Vickrey auction
= 0∫a n(n-1)F(x)(n-2) [1-F(x)] x f(x) dx
= a(n-1)/(n+1).
 Expected highest value
= 0∫a nF(x)(n-1) x f(x) dx = a n/(n+1).
 In the first-price sealed-bid auction, we will find an
equilibrium in which every bidder bids k times his value
(0 <= k <= 1). Such an equilibrium is called a symmetric
equilibrium.
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Revenue in Auctions (2 of 3)
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Let b be the bid amount. Expected profit for a bid b with
value v is
(v-b)b(n-1)/(ka)(n-1).
Maximizing expected profit,
-b(n-1)+(n-1)(v-b)b(n-2)=0.
We get b=v(n-1)/n.
So, if every bidder except i bids a fraction (n-1)/n of his
value, then the best strategy for i is to bid a fraction
(n-1)/n of his value.
So expected revenue (in a symmetric equilibrium) from a
first-price auction = a (n-1)/(n+1)= expected revenue
from Vickrey auction (revenue equivalence theorem). In
fact, this is the highest possible revenue in ANY auction
for single-item private values model.
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Revenue in Auctions (3 of 3)
So expected revenue (in a symmetric
equilibrium) from a first-price auction = a (n1)/(n+1)= expected revenue from Vickrey
auction (revenue equivalence theorem). In fact,
this is the highest possible revenue in ANY
auction for single-item private values model.
 In fact, we can say more: with independently and
identically distributed private values, the
expected revenue in a first-price auction is the
same as the expected revenue in a second-price
auction.
 We assumed risk neutral bidders: payoff=valueprice.
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Multi-Item Auctions (1 of 3)
Number of items more than one.
 Items may be of same type (homogeneous) or
different type (heterogeneous).
 Examples: Sale of different components of a
computer, sale of 1000 memory chips etc.
 Bidders may have value on bundles: value for 10
memory chips need not equal 10 times value of a
single memory chip; value of a monitor and a
keyboard may be more than their combined
value.
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Multi-Item Auctions (2 of 3)
If there are n items, a bidder can have values on
2n number of bundles - exponential number of
bundles.
 Simultaneous sale of multiple items is also
known as combinatorial auctions.
 Examples of combinatorial auctions:
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– Sale of airport slots: a bidder will be interested in
Mumbai 6 AM to 7 AM slot together with Bangalore 8
AM to 9 AM slot; but less interested in Mumbai 6 AM
to 7 AM slot with Bangalore 1 PM to 2 PM slot.
– Sale of train tracks in Europe, spectrum rights in
different countries.
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Multi-Item Auctions (3 of 3)
Two buyers and two items (a,b). Values are:
v1(a)=5, v1 (b)=7, v1(a+b)=15;
v2(a)=7, v2 (b)=6, v2(a+b)=12.
 Assuming truthful bidding and conducting a
sequential auction (selling one item after
another) using the Vickrey auction yields: item 1
is awarded to buyer 2 and item 2 to buyer 1.
 This is not efficient - does not maximize total
value of the system.
 Does not maximize the revenue of the seller also.
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Design Objectives (1 of 2)
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Efficiency: Maximize the total value of
bidders and the seller. These are called
efficient auctions.
– If p is the price paid by a bidder, then v-p is
his payoff and the seller gets a payoff of p.
– Thus, total payoff of the system (buyers and
seller) due to that buyer is v-p+p=v.
– So, total payoff of the system is maximized by
maximizing the total value.
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Design Objectives
Revenue: Maximize the total revenue of the
seller.
 These are called optimal auctions.
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– Generally, have to assume some distributions on
valuations.
– Much difficult than designing efficient auctions.
– Analysis is intractable for many practical multiple
items settings.
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Note: Optimal auctions maximize the payoff of
seller only, whereas efficient auctions maximize
the total payoff of the seller and the buyers.
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Other Auction Design Issues (1 of 2)
Reserve price: Sellers generally set a minimum
price below which they do not sell items.
 Bundling issues: Sellers generally do not allow
for exponential number of bundles but decide on
bundles before the auction.
 Information feedback in iterative auctions: What
bid information should be communicated to
bidders?
 Bid increments: Tradeoff between length of
auction and efficiency/revenue loss.
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Other Auction Design Issues (2 of 2)
Collusion: Bidders form groups (called
bidding rings) and act as one to bid in
auctions.
 Privacy: Depending on the information
released by the auctioneer to the bidders,
the privacy of bidders can be at stake.
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– Example: In English auction, by bidding
truthfully, all losing bidders reveal their value.
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References
Vijay Krishna, Auction Theory, Academic
Press, 2002.
 Paul Klemperer, Auctions: Theory and
Practice, Online book
http://www.paulklemperer.org/,
Also Princeton University Press, 2004
(gives outlines for undergraduate and
graduate courses – in economics and
management departments).
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