4820–11
Actions
Geir B.
Asheim
Auctions
Introduction
Analysis
4820–11
Revenue
equalization
Reservation
price
Geir B. Asheim
Relaxing
assumptions
Department of Economics, University of Oslo
ECON4820
Spring 2010
Last modified: 2010.04.27
Why are auctions important?
4820–11
Actions
Geir B.
Asheim
Introduction
Types
Outline
Analysis
Revenue
equalization
Practical: Much used
In financial markets
Procurement auctions by governments and firms
Houses, agricultural products, art, etc.
Radio spectrum licences, electricity, transport
Reservation
price
Relaxing
assumptions
Theoretical: Basis for important theoretical work
Empirical: Testing-ground for economic theory
What is an auction?
4820–11
Actions
Geir B.
Asheim
Introduction
Types
Outline
Analysis
One seller
Small number of potential buyers
Bidding by buyers determines allocation and price
Revenue
equalization
Reservation
price
Relaxing
assumptions
Important variant for procurements and contracts:
Reverse auction
One buyer
Small number of potential sellers
Bidding by sellers determines allocation and price
Application, alternatives, and concerns
4820–11
Actions
Geir B.
Asheim
Introduction
Types
Outline
Analysis
Revenue
equalization
Reservation
price
Relaxing
assumptions
When are auctions used?
A unique object
Uncertainty about who should get the object
Uncertainty about the object’s value
Seller can commit to the procedure
Alternatives to auctions
Market — decides who gets the object,
but how to determine the price?
Bargaining — determines the price,
but how to decide who is the counterpart?
Handing out for free
Two concerns for an auction
For society: Is the object bought by the bidder with the
highest willingness to pay? (Efficiency )
For the seller: Is the price the highest possible?
Types of auctions
4820–11
Actions
Geir B.
Asheim
Introduction
Types
Outline
Analysis
Revenue
equalization
Reservation
price
Relaxing
assumptions
The descending-bid (Dutch) auction
The price is successively lowered until the first bidder enters
Used for flowers in the Netherlands
The ascending-bid (English) auction
The price is successively raised until only one bidder remains
Used for art and collectibles
First-price sealed-bid auction
Simultaneous bids, highest bid win, pays highest bid
Used for real estate, government procurement
Second-price sealed-bid (Vickrey) auction
Simultaneous bids, highest bid win, pays second-highest bid
Used for stamps etc.
Outline
4820–11
Actions
Geir B.
Asheim
Introduction
Types
Outline
Analysis of bidding behavior under standard assumptions
Bidders are risk-neutral
Private values
Symmetric bidders
Analysis
Revenue
equalization
Reservation
price
Relaxing
assumptions
Important result: Revenue equalization theorem
Auction form doesn’t matter for efficiency & exp. price
Does the seller want to impose a reservation price?
Relaxing the standard assumptions
Bidders are risk-averse
Common values
Asymmetric bidders
Standard assumptions
4820–11
Actions
Geir B.
Asheim
Introduction
Analysis
English
Vickrey
1st price
Bidders are risk-neutral
Each bidder’s valuation is i.i.d.
Revenue
equalization
Reservation
price
Relaxing
assumptions
Each bidder knows only her own valuation
Seller knows only the c.d.f. from which the values are drawn
Bidding behavior in English auctions
4820–11
Actions
Geir B.
Asheim
Introduction
Analysis
English
Vickrey
1st price
Revenue
equalization
Reservation
price
Relaxing
assumptions
Continued bidding is profitable as long as
own valuation > current price
This strategy is a best response ind. of others’ behavior
(weakly dominates any other strategy)
The winner is the one with the highest valuation
(⇒ efficiency)
Price is (just above) second highest valuation
Bidding behavior in sealed-bid 2nd price auctions
4820–11
Actions
Geir B.
Asheim
Introduction
Analysis
English
Vickrey
1st price
Revenue
equalization
Reservation
price
Relaxing
assumptions
v Bidder’s valuation
b Bidder’s bid
a Largest bid from others
If v < a, the bidder does not want to win,
and bidding b ≤ v ensures this.
If v > a, the bidder wants to win,
and bidding b ≥ v ensures this.
Conclusion: Bidding b = v is a best response, weakly dominating any other strategy. Everyone bidding their true value is a
Bayesian Nash equilibrium. This equilibrium is characterized by:
The winner is the one with the highest valuation
(⇒ efficiency)
The price equals the second-highest valuation
2nd price auction corresponds to English auction
(w.r.t. bidding strategy, winner and price)
Bidding behavior in sealed-bid 1st price auctions
4820–11
Actions
Geir B.
Asheim
Bidder trades off two concerns: Bidding b < v
reduces her chances to win; not good
reduces the paid price if she wins; good
Introduction
Analysis
English
Vickrey
1st price
Revenue
equalization
Reservation
price
Relaxing
assumptions
This trade-off makes the optimal bid smaller than v .
Others behaving likewise, makes the opt. bid even smaller.
Conclusion: A symmetric equilibrium is characterized by:
The winner is the one with the highest valuation
(⇒ efficiency)
The price equals the highest bid,
which is less than the highest valuation
1st price auction corresponds to Dutch auction
(w.r.t. bidding strategy, winner and price)
Calculating the equilibrium bid strategies is difficult
Formal analysis of sealed-bid 1st price auctions
4820–11
Actions
Geir B.
Asheim
Introduction
Analysis
English
Vickrey
1st price
Revenue
equalization
Reservation
price
Relaxing
assumptions
n bidders, vi ∈ [v , vh ], i ∈ {1, . . . , n}; F (vi ): c.d.f.
Consider a symmetric equilibrium (bidders are ex ante
identical): Bi (vi ) = B(vi ) for all i
Each player believes that Bj (vi ) = B(vi ) for all j = i
Assume that B > 0 ⇒ Bid = bi implies vi = B −1 (bi )
The probability that bi is the winning bid: [F (B −1 (bi ))]n−1
Bidder i’s expected profit: πi = (vi − bi )[F (B −1 (bi ))]n−1
i
FOC: 0 = ∂π
∂bi , implying that
∂πi dbi
∂πi
−1 (b ))]n−1 = [F (v )]n−1
i
= ∂π
i
i
∂vi + ∂bi dvi = ∂vi = [F (B
Integrating, setting πi = 0 if vi = v (since zero prob. of
v
winning then): π(vi ) = vi [F (x)]n−1 dx
Two expressions for bidder i’s profit must
be equal:
v
i
(vi − B(vi ))[F (vi )]n−1 = πi (vi ) = v [F (x)]n−1 dx
dπi
dvi
⇒ B(vi ) = vi −
v
n−1 dx
i
v [F (x)]
[F (vi )]n−1
Common for all four kinds of auctions
4820–11
Actions
Efficiency: Object to the bidder with highest valuation
Geir B.
Asheim
Revenue equivalence: All four kinds give the seller the
same expected revenue under the standard assumptions
Introduction
Analysis
Revenue
equalization
Reservation
price
Relaxing
assumptions
Result (Revenue equalization theorem)
Any auction mechanism in which (i) the object always goes to
the buyer with the highest signal, and (ii) any bidder with the
lowest-feasible valuation expects zero surplus, yields the same
expected revenue (and results in each bidder making the same
expected payment as a function of her valuation)
An increase in # of bidders increases the expected price
More bidders increases the expected 2nd-highest valuation
Difference: Bid more difficult to calculate in sealed-bid 1st-price
(Dutch) auctions than in sealed-bid 2nd-price (English) auctions
Does the seller want to impose a reservation price?
4820–11
Actions
Geir B.
Asheim
Introduction
Analysis
Revenue
equalization
Reservation
price
Relaxing
assumptions
A parallel situation: The monopolist’s problem
A monopolist trades off two concerns:
Wants to sell large quantities ⇒ Low price
Wants to earn a profit per unit sold ⇒ High price
Optimal trade-off: Price above marginal cost
Auction: Seller trades off the same two concerns:
Wants to sell the object ⇒ Low reservation price
Wants to earn a profit if the object is sold
⇒ High reservation price
Optimal trade-off: Reservation price above own valuation
Three cases (v1 , v2 two highest valuations; r reserv. price):
(i) v1 > v2 > r : Increasing r has no effect
(ii) v1 > r > v2 : Increasing r increases the price
(iii) r > v1 > v2 : r prevents sale
Optimal reservation price with 1 bidder
4820–11
Actions
Geir B.
Asheim
Introduction
Analysis
Bid r or nothing
Seller’s own valuation: v0
Seller’s expected profit: π(r ) = r [1 − F (r )] + v0 F (r )
Reservation
price
FOC: 0 = [1 − F (r )] − rf (r ) + v0 f (r )
(r )
⇒ v0 = r − 1−F
f (r ) ≡ J(r )
i.e., Marginal cost = Marginal revenue
Relaxing
assumptions
Equivalently: r = J −1 (v0 )
Revenue
equalization
Efficiency with a reservation price:
With a reservation price, the object may not be sold,
even if there exists a bidder with v > v0
Ex ante efficiency vs. ex post efficiency
Relaxing the standard assumptions
4820–11
Actions
Geir B.
Asheim
Introduction
Analysis
Revenue
equalization
Reservation
price
Relaxing
assumptions
Risk-averse
bidders
Common
values
Asymmetric
bidders
Bidders are risk averse
Common values
Asymmetric bidders
Risk-averse bidders
4820–11
Actions
Geir B.
Asheim
Introduction
Analysis
Revenue
equalization
Reservation
price
Relaxing
assumptions
Risk-averse
bidders
Common
values
Asymmetric
bidders
In a sealed-bid 1st price auction, risk-averse bidders bid
higher than risk-neutral ones. An increase in the bid
(1) increases the chance of winning
(2) reduces earning in case of winning
Risk-aversion makes (1) more important relative to (2)
No effect of risk-aversion in a sealed-bid 2nd price auction
The seller gains more in a sealed-bid 1st-price auction
than in a sealed-bid 2nd price auction
Common value auctions
4820–11
Actions
Geir B.
Asheim
Introduction
Analysis
Revenue
equalization
Reservation
price
Relaxing
assumptions
Risk-averse
bidders
Common
values
Asymmetric
bidders
All bidders have identical valuations
Each bidder does not know the value, but receives a signal
Each bidder does not know the other bidders’ signals
Examples:
A jar of coins
Buying for resale
Buying a project with common uncertainty for all bidders
(an unexplored petroleum field w/equally capable bidders)
Winner’s curse
Bidder receiving the most optimistic signal wins
Bidders must take into account that winning is “bad news”
(since it means that the true value is less than the received
signal indicates) by scaling down their bids
In an English auction, bidders learn from each other during
the bidding process. Reduces the winner’s-curse problem.
Asymmetric bidders
4820–11
Actions
Geir B.
Asheim
Introduction
Example:
Public procurement – domestic vs. foreign firms. Suppose
foreign firms are more cost effective than domestic ones
Analysis
Revenue
equalization
English and sealed-bid 2nd-price auctions are still efficient
Reservation
price
Sealed-bid 1st-price auction is no longer efficient
Relaxing
assumptions
Risk-averse
bidders
Common
values
Asymmetric
bidders
It is optimal to discriminate between bidder groups;
⇒ possible that winner is not the lowest-cost bidder
In the example: It is optimal to discriminate in favor of the
domestic firm. This favoring
increases the change of getting an inefficient supplier
but also lowers the bid from the efficient firms