### Document

```COS 444
Internet Auctions:
Theory and Practice
Spring 2009
Ken Steiglitz
[email protected]
week 9
1
the bigger picture, all single item …
Myerson 1981
optimal, not efficient
asymmetric bidders
Moving to asymmetric bidders
Efficiency: item goes to bidder with
highest value
• Very important in some situations!
• Second-price auctions remain efficient
in asymmetric (IPV) case. Why?
• First-price auctions do not …
week 9
3
Inefficiency in FP with asymmetric bidders
New setup: Myerson 81*, also BR 89
*wins Nobel prize for this
and related work, 2007
• Vector of values v
• Allocation function Q (v ):
Qi (v ) is prob. i wins item
• Payment function P (v ):
Pi (v ) is expected payment of i
• Subsumes Ars easily (check SP, FP)
• The pair (Q , P ) is called a
Direct Mechanism
week 9
5
New setup: Myerson 81
• Definition: When agents who participate in
a mechanism have no incentive to lie about
their values, we say the mechanism is
incentive compatible.
• The Revelation Principle: In so far as
equilibrium behavior is concerned, any
auction mechanism can be replaced by an
incentive-compatible direct mechanism.
week 9
6
Revelation Principle
Proof: Replace the bid-taker with a direct
mechanism that computes equilibrium
values for the bidders. Then a bidder can
bid equilibrium simply by being truthful,
and there is never an incentive to lie. □
This principle is very general and
includes any sort of negotiation!
week 9
7
Asymmetric bidders
• We can therefore restrict attention to
incentive-compatible direct mechanisms!
• Note: In the asymmetric case, expected
surplus is no longer vi F(z) n-1 − P(z)
(bidding as if value = z )
Next we write expected surplus in the
asymmetric case …
week 9
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Asymmetric bidders
Notation: v−i = vector v with the i – th
Value omitted. Then the prob. that i wins is
Qi ( z )   Qi ( z, vi ) d F (v -i )
Vi
Where V-i is the space of all v’s except vi and
F (v-i ) is the corresponding distribution
week 9
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Asymmetric bidders
Similarly for the expected payment of bidder i :
Pi ( z )   Pi ( z, vi ) dF(vi )
Vi
Expected surplus is then
Si ( z)  viQi ( z)  Pi ( z)
week 9
11
A yet more general RET
Differentiate wrt z and set to zero when z = vi
as usual:
viQ (vi )  Pi (vi )  0
'
i
'
But now take the total derivative wrt vi when z
= vi :
Si' (vi )  viQi' (vi )  Qi (vi )  Pi ' (vi )
And so
week 9
S (vi )  Qi (vi )
'
i
12
yet more general RE
Integrate:
vi
S (vi )  Si (0)   Qi ( x) dx
0
Or, using S = vQ – P ,
vi
Pi (vi )  Pi (0)  viQi (vi )   Qi ( x) dx
0
In equilibrium, expected payment of every
bidder depends only on allocation function Q !
week 9
13
Optimal allocation
Average over vi and proceed as in RS81:
E[ Pi (v)]  Pi (0)   MR i (vi ) Qi (v) dF (v)
V
where
1  Fi (vi ) ←no longer a common F
MRi (vi )  vi 
f i (vi )
week 9
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Optimal allocation, con’t
The total expected revenue is
R   Pi (0)    MR i (vi ) Qi (v) dF (v)
i
V
i
For participation, Pi (0 ) ≤ 0, and seller chooses
Pi (0) = 0 to max surplus. Therefore
R    MR i (vi ) Qi (v) dF (v)
V
week 9
i
15
Optimal allocation, con’t
When Pi (0 ) ≤ 0 we say bidders are
individually rational : They don’t
participate in auctions if the expected
payment with zero value is positive.
week 9
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Optimal allocation
The optimal allocation can now be seen by inspection!
R    MR i (vi ) Qi (v) dF (v)
V
i
For each vector of v’s, Look for the maximum value of
MRi (vi ). Say it occurs at i = i* , and denote it by MR* .
• If MR* > 0, then choose that Qi* to be 1 and all the
other Q’s to be 0 (bidder i* gets the item)
• If MR* ≤ 0, then hold on to the item (seller retains
item)
week 9
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Optimal allocation (inefficient!)
Payment rule
Hint: must reduce to second-price
when bidders are symmetric
Therefore: Pay the least you can
while still maintaining the highest
MR
This is incentive compatible; that is,
bidders bid truthfully! Why?
week 9
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Vickrey ’61 yet again
week 9
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Wrinkle
• For this argument to work, MR must be an
increasing function. We call F ’s with
increasing MR’s regular. (Uniform is
regular)
• It’s sufficient for the inverse hazard rate
(1 – F) / f to be decreasing.
• Can be fixed: See Myerson 81 (“ironing”)
• Assume MR is regular in what follows
week 9
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• Notice also that this asks a lot of bidders in
the asymmetric case. In the direct
mechanism the bidders must understand
enough to be truthful, and accept the fact
that the highest value doesn’t always win.
• Or, think of MRi(vi) as i’s bid
• As usual in game-theoretic settings,
distributions are common knowledge---at
least the hypothetical auctioneer must know
them.
week 9
22
In the symmetric case…Ars are
optimal mechanisms!*
• By the revelation principle, we can restrict
attention to direct mechanisms
• An optimal direct mechanism in the
symmetric case awards item to the highestvalue bidder, and so does any auction in Ars
• All direct mechanisms with the same
allocation rule have the same revenue
• Therefore any auction in Ars has the
same allocation rule, and hence revenue,
as an optimal (general!) mechanism
*Includes any sort of negotiation whatsoever!
week 9
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Efficiency
• Second-price auctions are efficient --- i.e., they
allocate the item to the buyer who values it the
most. (Even in asymm. case, truthful is dominant.)
• We’ve seen that optimal (revenue-maximizing)
auctions in the asymmetric case are in general
inefficient.
• It turns out that second-price auctions are optimal
in the class of efficient auctions. They
generalize in the multi-item case to the VickreyClark-Groves (VCG) mechanisms. … More later.
week 9
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Laboratory Evidence
Generally, there are three kinds of empirical
methodologies:
• Field observations
• Field experiments
• Laboratory experiments
Problem: people may not behave the same
way in the lab as in the world
Problem: people differ in behavior
Problem: people learn from experience
week 9
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Laboratory Evidence
Conclusions fall into two general
categories:
• Revenue ranking
• Point predictions (usually revenue
relative to Nash equilibrium)
For more detail, see J. H. Kagel, "Auctions: A Survey of
Experimental Research", in The Handbook of Experimental
Economics, J. Kagel and A. Roth (eds.), Princeton Univ. Press,
1995.
week 9
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Best revenue-ranking results
for IPV model
•
•
•
•
•
Second-Price > English Kagel et al. (87)
English  truthful=Nash Kagel et al. (87)
First-Price ? Second-Price
First-Price > Dutch Coppinger et al. (80)
First-Price > Nash Dyer et al. (89)
Thus, generally,
sealed versions > open versions!
week 9
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A violation of theory is the
scientist’s best news!
Let’s discuss some of the violations…
• Second-Price > English. These are
(weakly) strategically equivalent. But
• English  truthful = Nash.
What hint towards an explanation does the
“weakly” give us?
week 9
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• First-Price > Dutch. These are strongly
strategically equivalent. But recall LuckingReiley’s pre-eBay internet test with Magic
cards, where Dutch > FP by 30%!
What’s going on here?
week 9
29
with 3rd-price auctions that test IPV theory
• More about experimental results for
common-value auctions later
• We next focus for a while on a widely
accepted point prediction:
First-price > Nash
• One explanation, as we’ve seen, is
risk aversion
• But is here is an alternative explanation…
week 9
30
Spite [MSR 03 MS 03]
• Suppose bidders care about the surplus
of other bidders as well their own.
Simple example: Two bidders, secondprice, values iid unifom on [0,1].
Suppose bidder 2 bids truthfully, and
suppose bidder 1’s utility is not her own
surplus, but the difference Δ between
hers and her rival’s.
week 9
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Spite
• Now bidder 1 wants to choose her bid b1
to maximize the expectation of
(v1, v2 )  (v1  v2 )  Ib 1  v 2  (v2  b1)  Ib 1  v 2
where I is the indicator function, 1 when
true, 0 else.
• Taking expectation over v2 :

b1
0
1
(v1  v2 ) dv2   (v2  b1 ) dv2
b1
 b1v1  1 / 2  b1  b12
week 9
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Spite
• Maximizing wrt b1 yields best response to
truthful bidding:
v1  1
b1 
2
• Intuition?
week 9
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Spite
• Maximizing wrt b1 yields best response to
truthful bidding:
v1  1
b1 
2
• Intuition: by overbidding, 1 loses surplus
when 2’s bid is between v1 and her bid. But,
this is more than offset by forcing 2 to pay
more when he wins.
Notice that bidder 2 still cannot increase his
absolute surplus. (Why not?) He must take a
hit to compete in a pairwise knockout
tournament.
week 9
34
Spite
• Some results from MSR 03: take the case
when bidders want to maximize the difference
between their own surplus and that of their
rivals. Values distributed as F, n bidders. Then
 FP equilibrium is the same as in the riskaverse CRRA case with ρ = ½ (utility is t1/2 ).
Thus there is overbidding.
 SP equilibrium is to overbid according to

b(v )  v 
1
v
week 9
(1  F ( y ))2 dy
(1  F (v))2
35
Spite
 Revenue ranking is SP > FP.
(Not a trivial proof. Is there a simpler one?)
• Thus, this revenue ranking is the opposite of
the prediction in the risk-averse case, where
there is overbidding in FP but not in SP.
(Testable prediction.)
• This explains overbidding in both first- and
second-price auctions, while risk-aversion
explains only the first. (Testable prediction.)
• Raises a question: do you think people bid
differently against machines than against
people?
week 9
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Spiteful behavior in biology
• This model can also explain spiteful behavior in
biological contexts, where individuals fight for survival
one-on-one [MS 03]. Example:
H
D
• This is a hawk-dove game.
H 1/ 2 1
D
0
1/ 2
Winner type replaces loser type.
• In a large population where the success of an
individual is determined by average individual
payoff, there is an evolutionarily stable solution
that is 50/50 hawks and doves.
• If winners are determined by relative payoff in each
1-1 contest, the hawks drive out the doves.
• Thus, there is an Invasion of the Spiteful Mutants!
week 9
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Invasion of the spiteful mutants
• To see this, suppose in the large population there is a
fraction ρ of H’s and (1-ρ ) of D’s.
• The average payoff to an H in a contest is
 (1 / 2)  (1   )(1)
and to a D
 (0)  (1   )(1 / 2)
• The first is greater than the second iff ρ<1/2. A 50/50
mixture is an equilibrium.
• But if the winner of a contest is determined by who
has the greater payoff, an H always replaces a D!
week 9
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