• Chapter 18 Auctions
• Key Concept: Honesty is the best policy
in a private-value second price auction.
• However in a common-value auction,
winner’s curse may occur.
• Chapter 18 Auctions
• Auctions are one of the oldest form of
markets, dating back to at least 500 BC.
• Auctions to sell the right to drill in
coastal areas and the FCC auctions to sell
radio spectrum.
• Let us first look at private-value auctions.
At the end, we will mention briefly the
common-value auctions.
• Bidding rules
• Open auctions
• English auctions (ascending auction):
Bidders successively offer higher prices
until no participant is willing to increase
the bid further.
• Dutch auction (cheese and fresh flowers,
descending auctions): The auctioneer
starts with a high price and gradually
lowers it by steps until someone is willing
to buy the item.
• Sealed-bid auction
• First price (construction work)
• Second price (philatelist auction or
Vickery auction)
• Assume there are n bidders with private
values v1, v2, …, vn. Suppose seller has a
zero value for the object. We want to
design an auction (mechanism) to meet
our goal.
• Two natural goals are Pareto efficiency
and Profit maximization.
• Profit maximization is straightforward,
the seller wants to get the highest
expected profit.
• Suppose v1>v2> …>vn. Then to achieve
efficiency, the good should be sold to
person 1.
• How can we achieve effiency? Note that
the English auction will achieve this.
• Suppose v1 =100 and v2=10. And the bid
increment is 1. Then the winning price
may be 11. (So the winner will pay the
value of the second-highest bidder.
Similar to the second price auction if
bidders bid truthfully.)
• Suppose (vi, vj)=(10,10), (10,100),
(100,10), (100,100), each occurring with
probability 1/4. Then the winning bid
may be 10, 11, 11, 100. The expected
revenue to the seller is
33=(10+11+11+100)/4.
• What if the seller sets the reserve price at
say 100? The expected revenue would be
(0+100+100+100)/4=75. It is not Pareto
optimal since ¼ of the time no one gets
the good!
• This demonstrates that we might not be
able to achieve the two goals (Pareto
efficiency and Profit maximization) at the
same time.
• Now let us turn to the second price
sealed-bid auction. If bidders will bid
truthfully, then the item will be awarded
to the bidder with the highest value, who
pays the price of the second highest value.
• But will bidders bid truthfully?
• Let us look at the case with two bidders vi
and vj and bids bi and bj. When i gets the
good, his surplus is vi - bj. Now, if vi > bj,
then i would like to get the item. How can
he achieve this? He can simply bid bi = vi
> bj.
• On the other hand, if vi < bj, then i would
not like to get the item. How can he
achieve this? He can simply bid bi = vi <
bj .
• Honesty is the best policy.
• When v>bj, he wants bi> bj.
• When v<bj, he wants bi< bj.
• Does it run contrary to your intuitions?
Why?
• Vickery auctions in practice? eBay
introduces an automated bidding agent.
Users tell the bidding agent the most they
are willing to pay for an item and an
initial bid. As the bidding progresses, the
agent automatically increases a
participant’s bid by the min bid increment
whenever necessary.
• Essentially it is a Vickery’s auction. Each
user reveals to their bidding agent the
maximum price he or she is willing to
pay. In theory, the highest value bidder
wins and pays the second highest value.
And we have shown the honesty is the
best policy.
• In practice, we see late bidding. In one
study, 37% of the auctions had bids in the
last minute and 12 % had bids in the last
10 seconds.
• Story one: if you are an expert on rare
stamps, you may want to hold back
placing your bid so as not to reveal your
interest (the common value story).
• Story two: two bidders (valuations at 10)
are bidding for a Pez dispenser. The
seller’s reserve price is 2. If both bid
early, then end up paying 10. If both bid
10 in the last possible seconds, then
maybe one of the bid won’t go through,
and the winner may end up paying only 2.
• Escalation auction: The highest bidder
wins but the highest bidder and the
second highest bidders both have to pay
the amount they bid.
• A good way to earn some money in a
party…
• Analogous to international escalation?
• Lobbying may be an all-pay auction.
• A position auction is a way to auction off
positions such as a position on a web
page. Let us look at a simple case.
• Suppose there are two slots where ads
can be displayed and x1 (x2) denotes the
number of clicks an ad can receive in slot
1 (2). Assume that slot 1 is better than
slot 2 so x1> x2.
• Two advertisers bid for the two slots. The
reserve price is r. Suppose bm> bn>r.
• Then bidder m gets slot 1 and pays bn per
click. Bidder n gets slot 2 and pays r per
click. In other words, an advertiser pays a
price determined by the bid of the
advertiser below him.
• Let us look at any bidder i. When bi> bj,
he gets slot 1 and his payoff is (v-bj) x1.
On the other hand, when bi<bj, he gets
slot 2 and his payoff is (v-r) x2.
• Bidder i would like to get slot 1 (rather
than slot 2) if and only if (v-bj) x1> (v-r)
x2. This is equivalent to v(x1-x2)+rx2>bjx1.
• When v(x1-x2)+rx2>bjx1, he wants bi> bj.
• When v(x1-x2)+rx2<bjx1, he wants bi< bj.
• Thus, bidder i could just bid bix1=v(x1x2)+rx2.
• When (v(x1-x2)+rx2)/x1>bj, he wants bi>
bj .
• When (v(x1-x2)+rx2)/x1<bj, he wants bi<
bj .
• When v>bj, he wants bi> bj.
• When v<bj, he wants bi< bj.
• Notice how similar this is to the proof
where we show that honesty is the best
• But the proof breaks down when there are
three bidders and three slots. The logic is
too specific so I would not go into details.
Also mind that there are errors on page
343.
• As for the expected revenue when the
number of bidders change in a second
price auction, since everyone bids
truthfully, suppose reserve price is 0, then
expected revenue will be the expected
value of the second-largest valuation.
• As the number of bidders goes up, the
expected value of the second-largest
valuation goes up.
• Shown is the expected revenue if the
values are distributed uniformly on [0,1].
By the time there are 10 or so bidders, the
expected revenue is pretty close to 1,
illustrating that auctions are a good way
to generate revenue.
• Problems with auctions: On the buyer
side, buyers may form bidding rings. On
the seller side, sellers may take bids off
the wall (take fictitious bids). (Ask your
employees to place bids!)
• Turn to the common-value auction where
the good that is being awarded has the
same value to all bidders (off-shore
drilling rights).
• Let us assume that v+ei where v is the
common value and ei is the error term
associated with bidder i’s estimate.
• To develop intuitions, let us see what
happens when bidders bid their estimated
values.
• The person with the highest value of ei or
emax gets the good. But as long as emax>0,
the bidder pays more than v, the true
value of the good. This is called the
winner’s curse.
• So bidders should shade bids. Moreover,
the more bidders there are, the lower you
want your bid to be.
• (Swing voter’s curse)
• A slight detour to stable marriage
problem (auctions are matching persons
to goods). In general we could consider
men matched to women, interns matched
to hospitals, organ donors matched to
recipients.
• Consider n men and n women and we
need to match them up as dancing
partners. Each woman rank the men
according to her preference and the same
goes for the men. Suppose there are no
ties and that everyone prefers to dance
than to sit on the sidelines.
• What is a good way to arrange for
dancing partners? Is there a “stable”
matching?
• “Stable” here means no couple would
prefer each other to their current partners.
• The deferred acceptance algorithm could
find a stable matching.
• Step 1: Each man proposes to his most
preferred woman.
• Step 2: Each woman records the list of
proposals she receives on her dance card.
• Step 3: After all men have proposed to their
most-preferred choice, each woman rejects all
of the suitors except for her most preferred.
• Step 4: The rejected suitors propose to the next
woman on their lists.
• Step 5: Continue to step 2 or terminate the
algorithm when every woman has received an
offer.
• An example:
• m1: w1 w2
m2: w1 w2
• m1
w1
m2
w2
• m1----w1
m2 w2
match.
w1: m1 m2
w2: m2 m1
m1----w1
m2 w2
m1----w1
m2 ----w2, a stable
• Auctions are examples of economic
mechanisms. The idea is to design a game
that will yield some desired outcome.
• For instance, you may want to sell a
painting. First of all, we need to make
sure what your goal is (To max profit? To
max efficiency?). Then we should think
about which auction format (or game)
may help you achieve that.
• Thinking of things this way, mechanism
design is the “inverse” of game theory.
With game theory, we are given a game
and we want to know what the
equilibrium outcome will be. With
mechanism design, we are given an
outcome we want to achieve and we try
to design a game so that that equilibrium
of the game is the outcome.
• Let us look at the Vickery auction again
using this view.
• The seller has an item and his goal is to
award the item to the highest value
person.
• In this case, he can design a mechanism,
which is the Vickery auction. Since we
have shown that in Vickery auction, it is
an equilibrium that bidders will bid
truthfully, this equilibrium outcome will
achieve what the seller wants.
• Typically the agents participating in the
mechanism will have some private
information the mechanism designer does
not know (for instance, in auctions,
bidders know their values but the
auctioneer does not). So the agents will
report some message about their private
information to the center.
• The center then examines the messages
and determines the outcome.
• There are typically some constraints the
center has to be aware of. The resource
constraint is obvious (for instance there is
only one item to be sold), constraints
about how agents will behave are less
obvious but not hard to understand.
• First, agents will act in their self-interest,
this leads to the incentive compatible
constraint (IC).
• Second, agents will participate in the
mechanism voluntarily. They get at least
as high a payoff from participating as not
participating. This leads to the individual
rationality constraint (IR).
• Chapter 18 Auctions
• Key Concept: Honesty is the best policy
in a private-value second price auction.
• However in a common-value auction,
winner’s curse may occur.
• Chapter 18 Auctions
• Key Concept: Honesty is the best policy
in a private-value second price auction.
• However in a common-value auction,
winner’s curse may occur.