week 91 cos 444 internet auctions: theory and practice spring 2009 ken steiglitz...
TRANSCRIPT
Myerson 1981optimal, not efficientasymmetric bidders
the bigger picture, all single item …
week 9 3
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 …
Inefficiency in FP with asymmetric bidders
week 9 5
New setup: Myerson 81*, also BR 89
• 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
*wins Nobel prize for this and related work, 2007
week 9 6
New setup: Myerson 81
• Definition: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 7
Revelation Principle
Proof: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 9
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 10
Asymmetric bidders
Notation: v−i = vector v with the i – th
Value omitted. Then the prob. that i wins is
Where V-i is the space of all v’s except vi and
F (v-i ) is the corresponding distribution
)(),()( -iiV ii vFdvzQzQi
week 9 11
Asymmetric bidders
Similarly for the expected payment of bidder i :
Expected surplus is then
)(),()( iiV ii vdFvzPzPi
)()()( zPzQvzS iiii
week 9 12
A yet more general RET
Differentiate wrt z and set to zero when z = vi
as usual:
But now take the total derivative wrt vi when z = vi :
And so
0)()( '' iiiii vPvQv
)()()()( '''iiiiiiiii vPvQvQvvS
)()(' iiii vQvS
week 9 13
yet more general RE
Integrate:
Or, using S = vQ – P ,
In equilibrium, expected payment of every bidder depends only on allocation function Q !
iv
iii dxxQSvS0
)()0()(
dxxQvQvPvPiv
iiiiiii )()()0()(0
week 9 14
Optimal allocation
Average over vi and proceed as in RS81:
where
)()()()0(])(E[ vdFvQvMRPvP ii
V
iii
)(
)(1)(
ii
iiiii vf
vFvvMR
←no longer a common F
week 9 15
Optimal allocation, con’t
The total expected revenue is
For participation, Pi (0 ) ≤ 0, and seller chooses Pi (0) = 0 to max surplus. Therefore
)()()()0(R vdFvQvMRP ii
V ii
ii
)()()(R vdFvQvMR i
V iii
week 9 16
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 17
Optimal allocation
The optimal allocation can now be seen by inspection!
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)
)()()(R vdFvQvMR ii
V ii
Optimal allocation (inefficient!)
week 9 19
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 20
Vickrey ’61 yet again
week 9 21
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 22
• 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 23
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 highest-value 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 24
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 Vickrey-Clark-Groves (VCG) mechanisms. … More later.
week 9 25
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 behaviorProblem: people learn from experience
week 9 26
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 ofExperimental Research", in The Handbook of ExperimentalEconomics, J. Kagel and A. Roth (eds.), Princeton Univ. Press, 1995.
week 9 27
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 28
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 29
• First-Price > Dutch. These are strongly strategically equivalent. But recall Lucking-Reiley’s pre-eBay internet test with Magic cards, where Dutch > FP by 30%!
What’s going on here?
week 9 30
• See also Kagel & Levin 93 for experiments 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:
• One explanation, as we’ve seen, is
risk aversion• But is here is an alternative explanation…
First-price > Nash
week 9 31
Spite [MSR 03 MS 03]
• Suppose bidders care about the surplus of other bidders as well their own.
Simple example: Two bidders, second-price, 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 32
Spite
• Now bidder 1 wants to choose her bid b1 to maximize the expectation of
where I is the indicator function, 1 when true, 0 else.
• Taking expectation over v2 :
2121)()(),( 122121 vbvb IbvIvvvv
21111
21
1
2220 1
2/1
)()(1
1
bbvb
dvbvdvvvb
b
week 9 33
Spite• Maximizing wrt b1 yields best response to
truthful bidding:
• Intuition?
2
111
v
b
week 9 34
Spite• Maximizing wrt b1 yields best response to
truthful bidding:
• 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.
2
111
v
b
week 9 35
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 risk-averse CRRA case with ρ = ½ (utility is t1/2 ). Thus there is overbidding.
SP equilibrium is to overbid according to
2
1 2
))(1(
))(1()(
vF
dyyFvvb v
week 9 36
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 37
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:
• This is a hawk-dove game.
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!
2/10
12/1
D
H
DH
week 9 38
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
and to a D
• 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!
))(1()( 12/1
))(1()( 2/10