Transcript LECTURE 6 - ComLabGames

Chapter 20
Violating Equivalence
1. Preferences for Risk
2. Asymmetric Valuations
3. Relaxing Independence
4. Differential Information
5. Collusion and Entry
1. Preferences for Risk
We now relax the conditions for
revenue equivalence to apply,
seeking to show the effects on
bidding
behavior
and
auction
revenue. First we investigate what
happen if bidders are not risk
neutral.
When does revenue equivalence fail?
Bidders might be risk averse or risk loving rather
than risk neutral.
The private valuations of bidders might be drawn
from probability distributions that are not identical.
The theorem does not apply when bidders receive
signals about the value of the object to them that are
correlated with each other.
If bidders are differentially informed, the format of
the auction also matters.
Attitudes towards risk in second price
sealed bid auctions with private values
It remains a weakly dominant strategy for
each player to bid his or her valuation.
The optimal bidding strategy for the second
price sealed bid auction (and also the
Japanese and English auctions) is independent
of a bidder's attitude towards risk and
uncertainty when private values are drawn
from a common probability distribution.
Attitudes towards risk in first price
sealed bid auctions with private values
Bidding your valuation guarantees exactly zero surplus.
If you place a lower bid than your valuation your
expected surplus initially increases until it reaches the
maximum for a risk neutral bidder, and then falls, but the
variance of the surplus increases as well.
A risk averse gambler is willing to trade a lower
expected value to reduce the amount of uncertainty, he
accordingly bids higher than a risk neutral bidder.
Comparing first and second price
sealed bid auctions
Revenue generated by a second price auction is
independent of the bidders' preferences over
uncertainty, since bidding is unaffected.
The revenue generated by the first price auction
is the same as the revenue generated by a second
price auction when bidders are risk neutral.
Therefore risk averse bidders generate more
revenue in a first price auction than they would in
a second price auction, and they generate more
revenue in a first price auction than do risk neutral
bidders.
Theorem on optimal bidding players
with constant relative risk aversion
Suppose there are two bidders with constant
relative risk aversion. In a first price auction the
utility the nth bidder obtains from winning is

v n b 

We show the optimal bidding function is
vn
1
b
vn vn  F v n 
v
 F vdv
where
1/ 
F 
v
F
v
a probability distribution function that is first order
stochastically dominated by F(v).
Corollary: risk averse players make
higher bids in first price auctions
Since
F
vF
vn for
all v < vn and  1 it follows that
vn
b risk averse v n v n 
v
vn
v n 
v
F v
dv
F v n 
Fv
Fv n 
1/ 
dv
vn
Fv
1
v n 

Fv n  v Fv n 
vn
1
v n 
 Fvdv
Fv n  v
1/ 1
Fvdv
b risk neutral v n 
and vice versa if the bidder is risk loving. Thus risk averse
players “buy insurance” against losing the auction.
Proof of theorem
Let b(v) denote the optimal bidding function as
a mapping from a player’s valuation and consider
the optimization problem
max 
F
v
vn b
v

v
v,v 
which is solved by setting v = vn. The first order
condition for this problem is
F 
v
vn b
v F
v
vn b
v1 b 
v0


or
F 
v
vn b
vF
v
b 
v0

a probability distribution function that is first order
stochastically dominated by F(v).
Concluding the proof
We rewrite the line on the previous slide as:
F 
w
w F 
w
b
wF
w
b 
w
Rescaling:
1F
w1/1 F 
w
w  1 F
w1/1 F 
w
b
wF
w1/b 
w
By definition this is:


F
w
w
F
w
b
w

F
w
b
w







Or, upon integrating from v to vn we obtain the
desired result using the facts that
v
v
vn F wwdw vn F vn vn F wwdw
v
F
w
b
wF 
w
b 
w
dw F 
vn 
b
vn 
vn 

2. Asymmetric Valuations
What happens if the private
valuations of bidders are not drawn
from
the
same
probability
distribution function?
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.
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.
Instead of there being a single distribution F(v) from
which the bidders draw their valuations, 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).
A conceptual experiment
Suppose each bidder sees his valuation, but does
not immediately learn whether he comes from the
high or low probability distribution. She only know v
comes from F(v).
At that point the bidding strategy cannot depend
on which probability distribution the valuation comes
from. She forms hew bid b(v).
Then each bidder is told what type of person they
are, type i ∈ {1,2}, that is the probability distribution
their valuations were drawn from F₁(v) and F₂(v).
How should he revise his 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.
3. Relaxing Independence
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.
4. Differential Information
We have discussed several types of
information structures in auctions, but
one important case we have not touched
yet, is when some bidders know more
about the common value of the object
than other bidders do.
Bidding with differential information
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
5. Collusion and Entry
Collusion and entry deterrence are also
considerations that auctioneers should
account for. The auctioneer also affects
the number of bidders by promoting
the auction.
The field of bidders
The last part of study on auctions discusses two
issues outside the bidding process itself that
nevertheless may affect the outcome of the auction.
They are:
1. Collusion amongst bidders
2. Determining the number of bidders
The degree of collusion and entry deterrence may
affect how the auctioneer and the bidders rank
different types of auctions that are revenue
equivalent, or even strategically equivalent.
Collusion
Collusion between bidders is only possible if there is a
mechanism for determining within a designated bidding
ring which bidder has the highest valuation, and then
ensuring members of the ring do not break the collusive
agreement in the bidding.
We focus on the second point. Sometimes determining
which member has the highest valuation is trivial (for
example in common value auctions), and sometimes it
can be resolved through an auction within the ring for the
right to present the only serious bid to the auctioneer.
Evidence about collusion in auctions
Section 1 of the Sherman Act pertains to trusts
and illegal constraints to trade.
Over three quarters of the criminal cases filed
in the 1980s under its provisions were in auctions
markets.
Given the difficulty in finding sufficient grounds
to prosecute this illegal activity, one can safely
conclude that collusive behavior in auctions is
often a serious concern for the seller.
What are the gains
from collusive behavior?
Consider a bidding ring of R members out of a total
of N bidders.
The goal of the ring is to internally solve which of the
players have the highest valuation, and then make one
(serious) bid instead of R bids.
With fewer effective bidders in the auction, the
expected price of the object is lower.
If all the bidders are able to collude, meaning N = R,
the auction reduces to a game of bilateral bargaining
between the auctioneer and the single bidder.
Bidding rings in
second price sealed bid auctions
For example, suppose the player with the highest
valuation in the ring bids it, the other bidders in the ring
submit the auctioneer’s reservation price, and the bidders
outside the ring respond optimally but individually by bidding
their true valuations.
Then the ring benefits from colluding if:
1.
The high bid from the ring wins the auction
2.
The second highest valuation of bidders in the
ring exceeds the highest valuation of all the
bidders outside the ring.
Enforcing collusion in first and
second price sealed bid auctions
One reason why second price auctions are rarely used in
practice is because they are more susceptible to collusion.
In a first price sealed bid auction, a member of the bidding
ring must submit a bid over the ring’s low price to win the
auction. Providing his valuation exceeds the ring’s bid, there is
an opportunity to make profits by deviating from the ring’s
decision. This makes collusive bidding harder to enforce.
Contrast this with a second price auction. In order to win
the auction a bidder in the ring who breaks the collusive
agreement must pay the reported value of the ring. The
agreement by the ring is self enforcing!
Ascending auctions encourage
communication amongst the bidders
A key issue for firms attempting to collude is
agreeing how to share the spoils.
In a multiunit ascending auction, bidders can
use the early stages when prices are still low to
signal their views about who should win which
object, and then when consensus has been
reached, tacitly agree to stop pushing prices up.
By contrast, bidders cannot easily achieve the
same coordination in simultaneous sealed-bid or
descending auctions, in which each player
simultaneously makes at most a single “best and
final” offer to each object.
Low reservation prices
encourage collusion
Reducing the reserve price increase the
potential gains from joint-bidding or colluding,
because the gains from colluding are greater.
Therefore the auctioneer should set a
reservation price that is higher than the opportunity
cost of failing to sell the auctioned item if it reduces
the probability that bidders will collude.
Entry and the provision of information
about the auctioned object
Should the auctioneer encourage more players to enter
the auction?
Potential bidders could, for example, be encouraged to
bid in an auction, by providing them with services that
help them to value the object for sale.
Note that simply paying people to participate in the
auction would not achieve any useful purpose, because
anyone could accept the payment and then make a very
low bid.
Encouraging entry
in private valuation auctions
If another bidder enters a private valuation auction,
the the level of the highest or the second highest
valuation might increase.
In either case, the revenue from the auction would
increase.
Therefore the expected revenue from holding a
private valuation auction increases with the number of
bidders.
Encouraging entry
to avoid collusion
The lower the number of bidders, the easier
it is for them to reach a collusive agreement.
Thus another reason to encourage entry is to
reduce the probability for facing a bidding
cartel.
Summary and synthesis
When comparing different auction mechanisms,
we should consider the following questions:
1.
2.
3.
4.
5.
6.
7.
8.
Are the auctions strategically equivalent?
Are the auctions revenue equivalent?
Are the bidders risk neutral?
Are bidders with private valuations drawing from the same
probability distribution?
Are the valuations of bidders affiliated with each other?
Do some bidders have more information than others?
Is collusion between bidders likely?
Should information about the auction be given to bidders?