6.853: Topics in Algorithmic Game Theory
Lecture 24
Fall 2011
Matt Weinberg
Recap
• Myerson’s Lemma: The Expected Revenue of
any BIC Mechanism is exactly its Expected
Virtual Surplus
• Virtual Value φ(v): (defined on board)
• Virtual Surplus: The virtual valuation of the
winner
A simple Corollary
• Upper Bound on Optimal Revenue: Always
choose the allocation that maximizes virtual
surplus
– Proof: Any BIC allocation rule clearly obtains some
virtual surplus, this virtual surplus cannot possibly
exceed this bound
– Is this bound attainable?
Virtual Second Price Auction
• Let φi be any monotonically increasing functions.
• Let winner = argmax_i {φi(vi)} (or no one if φi(vi) <
0 for all i)
• Charge the winner price φi-1(max{0,argmax_{j ≠ i}
{φj(vj)}})
• Incentive Compatible! (Not just BIC)
– Proof: Lowering your bid cannot affect the price you
pay, may lose the item when willing to buy. Increasing
bid cannot affect the price you pay, may win the item
when unwilling.
Virtual Second Price Auction
• Candidate function for φi?
– Virtual Valuation function? If Φi is monotonically
increasing, valid choice.
– Always gives item to highest virtual value
– Achieves maximum possible expected virtual
surplus
• What if φi isn’t monotonically increasing?
– “Ironed Virtual Valuations” Picture on Board.
Virtual Second Price Auction
• Myerson’s Theorem: When bidders are singledimensional, and each vi is sampled independently
from a known distribution, the auction that maximizes
expected revenue over all BIC auctions is to:
– 1) Map each vi to φi(vi) using (ironed) virtual valuations
– 2) Award the item to the bidder with the highest nonnegative φi(vi), at price φi-1(max{0,argmax_{j ≠ i} φj(vj)})
• Furthermore, this auction is Incentive Compatible (not
just BIC).
Regular Distributions
• What does it mean for φi(vi) to be monotonically
increasing?
• If there were no other bidders and we were only selling
the item to bidder i, then:
– There exist unique prices p and q (might have p = q) such
that:
– 1) Selling the item at any price p ≤ x ≤ y ≤ q E[R(p)] = E[R(x)]
= E[R(y)] = E[R(q)].
– 2) y < x < p OR y > x > q implies that E[R(y)] < E[R(x)] <
E[R(p)].
– Proof: On Board.
– Such distributions are called Regular
An Example
• 2 bidders, v1 uniform in [0,1]. v2 uniform in [0,100].
– Φ1(x) = 2x-1, Φ2(x) = 2x-100
– Intuition behind optimal auction:
– If there was only bidder 2, would sell the item at price 50. Only bidder
1, sell at ½.
– When v1 > ½ , v2 < 50, sell to 1 at price ½.
– When v1 < ½, v2 > 50, sell to 2 at price 50.
– When 0 < 2v1 -1 < 2v2 – 100, sell to 2 at price:
• (99+2v1 )/2, a tiny bit above 50
– When 0 < 2v2 -100 < 2v1 -1, sell to 1 at price:
• (2v2 -99)/2, a tiny bit above ½.
– Intuition: For each bidder, pretend they are the only bidder.
Sometimes we want to give them both the item. In this case, we
should increase the price charged to the winner. Myerson’s Lemma
tells us exactly who should win.
I.I.D. Bidders
• Corollary of Myerson: When bidders are I.I.D.,
the revenue optimal auction is a second price
auction with reserve.
– Proof: Everyone can have the same (ironed) virtual
valuation function, so the winner will always be
the highest bidder.
When Does Myerson Apply?
• Single Item for sale.
• k copies of the same item (such as a CD) for sale, each
bidder only wants one.
– Myerson chooses the k bidders with the highest non-negative
virtual values
• k copies of the same item (such as a CD) for sale, each
bidder has value xi vi if they receive xi copies of the item.
– Myerson chooses the bidder with the highest non-negative
virtual value and gives her all k.
• Either of the above + combinatorial constraints on who can
receive how many of each item.
– Myerson chooses the feasible allocation that maximizes virtual
surplus
What if we don’t know the
distributions?
• Single Item, bidders I.I.D.?
– Bulow-Klemperer Theorem (Next Slide)
• Infinitely many copies of the same item (IE:
digital copy of a song), no prior information
– Digital Goods Auctions (After)
Bulow-Klemperer Theorem
• Motivation: In the real world, bidders are
usually i.i.d.
– Have you ever seen an auction that sets a
different price for different people?
• In the real world, learning distributions is a
hard problem
Bulow-Klemperer Theorem
• Let VCG(F,n) denote the expected revenue of
running the VCG (second price) auction with n
bidders sampled independently from F.
• Let OPT(F,n) denote the expected revenue of
running Myerson’s auction with n bidders
sampled independently from F.
• Theorem (Bulow-Klemperer): If F is regular,
VCG(F,n+1) ≥ OPT(F,n)
• Big Picture: With i.i.d. bidders, rather than learn
the distributions, try to attract more bidders.
Bulow-Klemperer Theorem
• Proof of Theorem consequence of two simple
claims.
• Let COPT(F,n) denote the maximum attainable
expected revenue by an auction that always
awards the item to someone.
• 1) COPT(F,n+1) ≥ OPT(F,n)
• 2) VCG(F,n+1) = COPT(F,n+1)
Bulow-Klemperer Theorem
• 1) COPT(F,n+1) ≥ OPT(F,n)
– Proof: Run Myerson’s auction on the first n
bidders. If Myerson doesn’t award the item, give it
to bidder n+1 for free. Revenue is exactly OPT(F,n),
item is always given away.
Bulow-Klemperer Theorem
• 2) VCG(F,n+1) = COPT(F,n+1)
– Proof: We still know that expected revenue is
exactly expected virtual surplus. If F is regular,
then the highest bidder has the highest virtual
value. So VCG chooses the highest virtual value
always.
– The only difference between VCG and Myerson is
that Myerson sometimes withholds the good from
sale if all virtual values are negative.
Worst-Case Guarantees
• Absolutely impossible to get any
approximation with no prior, even with a
single bidder and a single item.
• Value could be 1, could be 10, could be 1000000...
• Need a different benchmark
Digital Good Auctions
• Infinitely many copies of a single item (think
digital copies of a song)
• Want a worst-case guarantee
– Optimal Revenue is wrong benchmark: impossible to
attain (highest bid could be 1, 10, 10000000…)
• F(2) benchmark: maxp|(#bidders with v > p) > 1
{p*(#bidders with v > p)}
– In English: Allowed to choose any price where at least
two bidders are willing to purchase.
– Why F(2)? Anything stronger unattainable in worstcase. Simple approximation to F(2) possible (next slide)
Digital Good Auctions
• Random Sample Auction: For each bidder, flip a
coin. Denote by T the set of bidders with tails,
and H the set of bidders with heads.
– Find the optimal price for bidders in T, p(T) and the
optimal price for bidders in H, p(H). Charge price p(T)
to bidders in H, and p(H) to bidders in T.
– Truthful? Bidders are offered a price that they can’t
control. Dominant strategy to tell the truth.
– Optimal? Theorem: Random Sample Auction has
expected revenue equal to at least F(2)/15. There are
instances where the expected revenue is at most
F(2)/4.
Summary
• Independent bidders, single item:
– Myerson’s Lemma provides upper bound on possible expected
revenue.
– It is attainable using the virtual second price auction with
(ironed) virtual valuations
– Generalizes to single-dimensional settings with combinatorial
constraints
• I.I.D. bidders, unknown distribution:
•
Bulow-Klemperer: Recruiting an extra bidder is better than learning the
distribution if distributions are regular
• No Prior, worst case guarantee:
– Optimal revenue completely inapproximable in worst-case
– F(2)/15 attainable by simple auction (Random Sample Auction)