Lecture 7
Auctions
Auctions serve the dual purpose of eliciting
preferences and allocating resources between
competing uses. A less fundamental but more
practical reason for studying auctions is that the
value of goods exchanged each year by auction
is huge. We describe the main kinds of auctions,
define strategic and revenue equivalence,
analyze optimal bidding behavior, and compare
the outcomes from using different types.
Read Chapters 19 and 20 of Strategic Play.
Auctions
Auctions are widely used by companies, private
individuals and government agencies to buy and sell
commodities.
They are also used in competitive contracting between
an (auctioneer) firm and other (bidder) firms up or down
the supply chain to reach trading agreements.
Merger and acquisitions often have an auction flavor
about them.
To begin our discussion we compare two sealed bid
auctions, where there are two bidders with known
valuations of 2 and 4 respectively.
First price sealed bid auction
In a first price sealed auction,
players simultaneously submit
their bids, the highest bidder
wins the auction, and pays
what she bid for the item.
Highway contracts typically
follow this form.
For example, if the valuation 4
player bids 5 and the valuation
2 player bids 1, then the former
wins the auction, pays 5 and
makes a net loss of 1.
Dominance for the row player
The column player
does not have a
dominant strategy.
There is a dominant
strategy for the row
player to bid 1.
By iterative
dominance the column
player bids 2.
Iterative dominance
in a first price auction
The column player does
not have a dominant
strategy.
There is a dominant
strategy for the row
player to bid 1.
By iterative dominance
the column player bids
2.
Second price sealed bid auction
In a second priced sealed bid
auction, players simultaneously
submit their bids, the highest
bidder wins the auction, and
pays the second highest bid.
This is similar to Ebay, although
Ebay is not sealed bid.
For example, if the valuation 4
player bids 5 and the valuation 2
player bids 1, then the former
wins the auction, pays 1 and
makes a net profit of 3.
Dominance
in a second price auction
The row player has a
dominant strategy of
bidding 2.
The column player has
a dominant strategy of
bidding 4.
Thus both players have
a dominant strategy of
bidding their (known)
valuation.
Comparing auction mechanisms
Comparing the two auctions, the bidder place
different bids but the outcome is the same, the
column player paying 2 for the item.
How robust is this result, that the form of the
auction does not really matter?
1. Do people know their own valuations?
2. If not, are the valuations related?
3. What do bidders know about the valuations
of their rivals?
4. Is everyone risk neutral?
5. How many items are up for sale, and how
many units is each player bidding on?
6. Is collusion an issue?
Bidding strategies
Recall that a strategy is a complete description of
instructions to be played throughout the game, and
that the strategic form of a game is the set of
alternative strategies to each player and their
corresponding expected payoffs from following them.
Two games are strategically equivalent if they share
the same strategic form.
In strategically equivalent auctions, the set of bidding
strategies that each potential bidders receive, and the
mapping to the bidder’s payoffs, are the same.
Descending auctions are strategically
equivalent to first-price auctions
During the course of a descending auction no
information is received by bidders.
Each bidder sets his reservation price before the
auction, and submits a market order to buy if and when
the limit auctioneer's limit order to sell falls to that
point.
Dutch auctions and first price sealed bid auctions share
strategic form, and hence yield the same realized
payoffs if the initial valuation draws are the same.
Rule 1: Pick the same reservation price in Dutch auction
that you would submit in a first price auction
English auction
In an English auction bidders compete against each
other by successively raising the price at which they
are willing to pay for the auctioned object.
The bidding stops when nobody is willing to raise
the price any further, and the item is sold to the
person who has bid the highest price, at that price.
Second-price versus ascending auctions
When there are only 2 bidders, an ascending auction
mechanism is strategically equivalent to the second price
sealed bid auction (because no information is received
during the auction).
More generally, both auctions are (almost) strategically
equivalent if all bidders have independently distributed
valuations (because the information conveyed by the other
bidders has no effect on a bidder’s valuation).
In common value auctions the 3 mechanisms are not
strategically equivalent if there are more than 2 players.
Rule 2: If there are only two bidders, or if valuations are
independently distributed, choose the same reservation
price in English, Japanese and second price auctions.
Bidding in a second-price auction
If you know your own valuation, there is a general result
about how to bid in a second price sealed bid auction, or
where to stop bidding in an ascending auction.
Bidding should not depend on what you know about the
valuations of the other players, nor on what they know
about their own valuations.
It is a dominant strategy to bid your own valuation.
A corollary of this result is that if every bidder knows his
own valuation, then the object will be sold for the second
highest valuation.
Rule 3 : In a second price sealed bid auction, bid your
valuation if you know it.
Proving the third rule
Suppose you bid above your valuation, win the auction,
and the second highest bid also exceeds your valuation.
In this case you make a loss. If you had bid your valuation
then you would not have won the auction in this case. In
every other case your winnings would have been identical.
Therefore bidding your valuation dominates bidding above
it.
Suppose you bid below your valuation, and the winning
bidder places a bid between your bid and your valuation.
If you had bid your valuation, you would have won the
auction and profited. In every other case your winnings
would have been identical. Therefore bidding your
valuation dominates bidding below it.
The proof is completed by combining the two parts.
Relaxing strategic equivalence
In strategically equivalent auctions, the strategic
form solution strategies of the bidders, and the
payoffs to all them, are identical.
This is a very strong form of equivalence.
Can we show that such bidders might be
indifferent to certain auctions which lack
strategic equivalence?
Revenue equivalence defined
The concept of revenue equivalence provides a useful
tool for exploring this question.
Two auction mechanisms are revenue equivalent if, given
a set of players their valuations, and their information
sets, the expected surplus to each bidder and the
expected revenue to the auctioneer is the same.
Revenue equivalence is a less stringent condition than
strategic equivalence. Thus two strategic equivalent
auctions are invariably revenue equivalent, but not all
revenue equivalent auctions are strategic equivalent.
Preferences and Expected Payoffs
Let:
U(vn) denote the expected value of the nth bidder
with valuation vn bidding according to his equilibrium
strategy when everyone else does too.
P(vn) denote the probability the nth bidder will win
the auction when all players bid according to their
equilibrium strategy.
C(vn) denote the expected costs (including any fees
to enter the auction, and payments in the case of
submitting a winning bid).
An Additivity Assumption
We suppose preferences are additive,
symmetric and private, meaning:
U(v) = P(v) v - C(v)
So the expected value of participating in
the auction is additive in the expected
benefits of winning the auction and the
expected costs incurred.
A revealed preference argument
Suppose the valuation of n is vn and the valuation of j is vj.
The surplus from n bidding as if his valuation is vj is U(vj),
the value from participating if his valuation is vj, plus the
difference in how he values the expected winnings
compared to a bidder with valuation vj, or (vn – vj)P(vj).
In equilibrium the value of n following his solution strategy
is at least as profitable as deviating from it by pretending
his valuation is vj. Therefore:
U(vn) > U(vj) + (vn – vj)P(vj)
Revealed preference continued
For convenience, we rewrite the last slide on the
previous page as:
U(vn) - U(vj) > (vn – vj)P(vj)
Now viewing the problem from the jth bidder’s
perspective we see that by symmetry:
U(vj) > U(vn) + (vj – vn)P(vn)
which can be expressed as:
(vn– vj)P(vn) > U(vn) - U(vj)
A fundamental equality
Putting the two inequalities together, we obtain:
(vn – vj) P(vn)> U(vn) - U(vj) > (vn – vj) P(vj)
Writing:
yields:
vn = vj + dv
dU v  Pvdv
which, upon integration, yields:
U vn   U v    Pv dv
vn
v v
Revenue equivalence
This equality shows that in private value
auctions, the expected surplus to each bidder
does not depend on the auction mechanism itself
providing two conditions are satisfied:
1. In equilibrium the auction rules award the bid
to the bidder with highest valuation.
2. The expected value to the lowest possible
valuation is the same (for example zero).
Note that if all the bidders obtain the same
expected surplus, the auctioneer must obtain the
same expected revenue.
A theorem
Assume each bidder:
- is a risk-neutral demander for the auctioned object;
- draws a valuation independently from a common,
strictly increasing probability distribution
function.
Consider auction mechanisms where:
- the buyer with the highest valuation always wins
- the bidder with the lowest feasible signal expects
zero surplus.
Then the same expected revenue is generated by the
auctions, and each bidder makes the same expected
payment as a function of her valuation.
Steps for deriving expected revenue
The expected revenue from any auction
satisfying the conditions of the theorem, is the
expected value of the second highest bidder.
To obtain this quantity, we proceed in two
steps:
1. derive the probability distribution of the
second highest valuation
2. obtain its density and integrate to find
the mean.
Probability distribution of the
second highest valuation
Since any auction satisfying the conditions for the
theorem can be used to calculate the expected
revenue, we select the second price auction.
The probability that the second highest valuation is
less than x is the sum of the the probabilities that:
1. all the valuations are less than x, or: F(x)N
2. N-1 valuations are less than x and the other
one is greater than x. There are N ways of
doing this so the probability is:
NF(x)N-1[1 - F(x)]
The probability distribution for the second highest
valuation is therefore: NF(x)N-1 - (N - 1) F(x)N
Expected revenue
from Private Value Auctions
The probability density function for the second
highest valuation is therefore:
N(N –1)F(x)N-2 [1 - F(x)]F‘(x)
Therefore the expected revenue to the auctioneer, or
the expected value of the second highest valuation
is:
Using the revenue equivalence theorem
to derive optimal bidding functions
We can also derive the solution bidding strategies
for auctions that are revenue equivalent to the
second price sealed bid auction.
Consider, for example a first price sealed bid
auctions with independent and identically
distributed valuations.
The revenue equivalence theorem implies that
each bidder will bid the expected value of the next
highest bidder conditional upon his valuation being
the highest.
Bidding in a
first price sealed bid auction
The truncated probability distribution for the
next highest valuation when vn is the highest
valuation is:
In a symmetric equilibrium to first price sealed
bid auction, we can show that a bidder with
valuation vn bids:
Comparison of bidding strategies
The bidding strategies in the first and second
price auctions markedly differ.
In a second price auction bidders should submit
their valuation regardless of the number of
players bidding on the object.
In the first price auction bidders should shave
their valuations, by an amount depending on the
number of bidders.
The derivation
The probability of the remaining valuations being
less than w when the highest valuation is v(1) is:
1N
N1
Pr
v 2 w|v 1F
v 1 F
w
Therefore the probability density for the second
highest valuation when vn = v(1) is:
N 1
F
vn 1N F
wN2 F 
w

This implies the expected value of the second
highest valuation, conditional on vn = v(1) is:
N  1F vn  v
1 N
wF w
vn
N 2
F ' wdw
Integrating by parts we obtain the bidding function:
b
vF
vn 
1N
N1
vF
v
vn
vv n
vv
vn F
vn 1N  F
vN1 dv
v
vn
F
vn   F
vN1 dv
1N
v
An example: the uniform distribution
Suppose valuations are uniformly distributed within a
closed interval, with probability distribution:
F
v
vv
v v
Then in equilibrium, a player with valuation v bids a
weighted average of the lowest possible valuation and
his own, where the weights are 1/N and (N-1)/N:
vn
b
vn vn F
vn   F
vN1 dv
1N
v
vn
vn 
vn v   
v v N1 dv
1N
v
 v /N v
N 1
/N
When does revenue equivalence fail?
The private valuations of bidders might be
drawn from probability distributions that are
not identical.
The theorem does not apply when values are
not private, such as when bidders receive
signals about the value of the object to them
that are correlated with each other.
Bidders might be risk averse or risk loving
rather than risk neutral. (This is of less
importance value maximizing firms.)
Affiliated Valuations
The revenue equivalence theorem
applies to situations where the valuation
of each is bidder is independently
distributed.
This is not always a valid assumption,
because how a bidder values the object
on the auction block for his own use
might depend on information that
another bidder has.
An example: Value of the object not
known to bidders
Consider a new oil field tract that drillers bid for after
conducting seismic their individual explorations.
The value of the oil field is the same to each bidder, but
unknown. The nth bidder receives a signal sn which is
distributed about the common value v, where
sn = v + n
and n  E[v| sn] – v is independently distributed across
bidders.
Notice that each drilling company would have more
precise estimates of the common valuation from
reviewing the geological survey results of their rivals.
The expected value of the item upon
winning the auction
If the nth bidder wins the auction, he realizes his signal
exceeded the signals of everybody else, that is
sn ≡ max{s₁,…,sN}
so he should condition the expected value of the item on
this new information.
His expected value is now the expected value of vn
conditional upon observing the maximum signal:
E[vn| sn ≡ max{s₁,…,sN}]
This is the value that the bidder should use in the auction,
because he should recognize that unless his signal is the
maximum he will receive a payoff of zero.
The Winner’s Curse
Conditional on the signal, but before the bidding starts,
the expectation of the common value is
E
v|sn E
sn 
s1 ,  , sN 
n |s n s n max
We define the winner’s curse as
maxs1,  , sN E
v|sn 
Although bidders should take the winner's curse into
account, there is widespread evidence that novice
bidders do not take this extra information into account
when placing a bid.
Symmetric valuations
We relax independence and consider the class of
symmetric valuations, which have two defining features:
1. All bidders have the same utility function.
2. Each bidder only cares about the collection of
signals received by the other bidders, not who
received them.
Thus we may write the valuation of bidder n as:
u n s1 ,,sn1 ,sn ,sn1 ,sN usn ,s1 ,,sn1 ,sn1 ,,sN 
usn ,sN,,sn1 ,sn1 ,,s1 
Revenue Comparisons for
Symmetric Auctions
We can rank the expected revenue generated in
symmetric equilibrium for auctions where valuations are also
symmetric.
There are two basic results. In a symmetric auction:
1. The expected revenue from a Japanese auction is
higher than what an English auction yields.
2. The expected revenue from an English auction
exceeds a first price sealed bid auction.
Asymmetric valuations
In a private valuation auctions suppose the bidders
have different uses for the auctioned object, and this
fact is common knowledge to every bidder.
Each bidder knows the probability distributions from
which the other valuations are drawn, and uses that
information when making her bid.
In that case the revenue equivalence theorem is not
valid, and the auctioneer's prefers some types of
auctions over others.
What happens if the private valuations of bidders are
not drawn from the same probability distribution
function?
An example of asymmetry
Instead of assuming all bidders appear the same to
the seller and to each other, suppose that bidders fall
into two recognizably different classes.
Suppose there are two cumulative distributions, F₁(v)
and F₂(v) with probability p₁ and p₂ respectively.
Thus:
F(v) = p₁F₁(v) + p₂F₂(v)
Bidders of type i ∈ {1,2} draw their valuations
independently from the distribution Fi(v).
We assume the supports are continuous and overlap,
but that they do not coincide.
An example of asymmetry
Instead of assuming all bidders appear the same to
the seller and to each other, suppose that bidders fall
into two recognizably different classes.
Suppose there are two cumulative distributions, F₁(v)
and F₂(v) with probability p₁ and p₂ respectively.
Thus:
F(v) = p₁F₁(v) + p₂F₂(v)
Bidders of type i ∈ {1,2} draw their valuations
independently from the distribution Fi(v).
We assume the supports are continuous and overlap,
but that they do not coincide.
A conceptual experiment
Suppose each bidder sees his valuation, but does
not immediately learn whether other bidders comes
from the high or low probability distribution.
At that point the bidding strategy cannot depend
on which probability distribution the other
valuations come from. She forms her bid b(v).
Then each bidder is told what type of person the
other bidders are, type i ∈ {1,2}, that is whether
their valuations were drawn from F₁(v) and F₂(v).
How should she revise her bid?
Intuition
Suppose F₁(v) and F2(v) have the same
support, but F1(v) first order stochastically
dominates F2(v), that is F₁(v) 6 F2(v) .
When a bidder learns that his valuation is
drawn from F2(v) (respectively F₁(v)), he
deduces the other one is more likely to draw
a higher (lower) valuation than himself,
realizes the probability of winning falls (rises),
so adjusts his bid upwards (downwards).
The intuition is in first price auctions to bid
aggressively from weakness and vice versa.
Bidding with differential information
Another type of asymmetry occurs when one
bidder knows more about the value of the
object being auctioned than the others.
What happens if they are asymmetrically
informed about a common value?
An extreme form of dependent signals occurs
when one bidder know the signal and the others
do not. How should an informed player bid?
What about an uninformed player?
Second price sealed bid auctions
The arguments we have given in previous
lectures imply the informed player
optimally bids the true value.
The uninformed player bids any pure or
mixed distribution. If he wins the auction
he pays the common value, if he loses he
pays nothing, and therefore makes
neither gains or losses on any bid.
This implies the revenue from the auction
is indeterminate.
Perspective of the less informed
bidder in a first price auction
Suppose the uninformed bidder always makes the
same positive bid, denoted b. This is an example of a
pure strategy.
Is this pure strategy part of a Nash equilibrium?
The best response of the informed bidder is to bid a
little more than b when the value of the object v is
worth more than b, and less than b otherwise.
Therefore the uninformed bidder makes an
expected loss by playing a pure strategy in this
auction. A better strategy would be to bid nothing.
A theorem on
first price sealed bid auctions
The argument in the previous slide shows that the
uninformed bidder plays a mixed strategy in this game.
One can show that when the auctioned item is worth v
the informed bidder bids:
(v) = E[V|V  v]
in equilibrium, and that the uninformed bidder chooses a
bid at random from the interval [0, E[V]] according to the
probability distribution H defined by
H(b) = Prob[(v)  b]
Return to the uninformed bidder
If the uninformed player bids more than E[V], then
his expected return is negative, since he would win
the auction every time v < E[V] but less frequently
when v > E[V].
We now show that if his bid b < E[V], his expected
return is zero, and therefore any bid b < E[V] is a best
response to the informed player’s bid.
If the uniformed bids less than E[V] and loses the
auction, his return is zero. If he bids less than E[V]
and wins the auction, his return is
E[V| (V) < b] – b = E[V| V < -1(b) ] – b
= (-1(b) ) – b
= 0
Return to the informed bidder
Since the uninformed player bids less than E[v] with unit
probability, so does the informed player.
Noting that (w) varies from v to E[v], we prove it is better
to bid (v) rather than (w). Given a valuation of v, the
expected net benefit from bidding (w) is:
H((w))[v - (w)] = Pr{V  w}[v - (w)] = F(w)[v - (w)]
Differentiating with respect to w, using derivations found on
the next slide, yields F’(w)[v - w] which is positive for all v > w
and negative for all v < w, and zero at v = w. Therefore
bidding (v) is optimal for the informed bidder with valuation v.
The derivative
Noting
w
F
w

w F
w
E
V|V w   tF 
t
dt
v
it follows from the fundamental theorem of calculus that
d
dw
F
w

w F 
w
w

and so the derivative of F(w)[v - (w)] with respect to w is :
d
F 
w

v 
w
F
w

wF 
w
v  dw
F
w

w 

F 
w
v F 
w
w
Lecture Summary
We described several types of auction mechanisms.
We showed conditions under which auctions are
strategically equivalent, revenue equivalent, and
compared revenue from different auctions when
these conditions do not hold.
We showed how to derive the optimal bidding rule
for several private value auctions, and explained the
nature of the winner’s curse in common value
auctions.