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
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
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.
– 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)
– 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.
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
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.
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.
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)
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:
– 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)
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.
– Generally, have to assume some distributions on
valuations.
– Much difficult than designing efficient auctions.
– Analysis is intractable for many practical multiple
items settings.
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.
– 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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