Competitive Auctions and
Digital Goods
Andrew Goldberg, Jason Hartline,
and Andrew Wright
presenting: Keren Horowitz,
Ziv Yirmeyahu
Introduction
•
•
•
•
Selling a large number of items
every customer wants a single item
unlimited supply
examples:
– downloadable software
– pay-per-view movies
Motivation
• Mechanism Design purpose:
– to maximize seller’s revenues!
• Using:
–
–
–
–
–
Fixed price auctions
Multiple price auctions
Deterministic auctions
Randomized auctions
All single round, sealed bids
The economic view
Price
Quantity
The economic view
Price
Revenues =
Price X Quantity
Quantity
Notations
•
•
•
•
•
•
•
Set of Bids, B= b1...bn, sorted: bi  bi 1,
so b1  bl & bn  bh
ui = utility of bidder I
n
T =  i 1 ui
F = Optimal fixed price revenues
R = current game revenues
assumption: h is small compare to F
Notations 2
• Truthful auction: encourage bidders to bid
their utility value
• Competitive auction: yields revenues within
a constant factor of optimal fixed price.
• We use computer science analysis similar to
online algorithms analysis where we check
if R
 1
F
For example:
• The k-item vickery auction:
– truthful, single price
• Worst case scenario:
– k bidders bid at h, n-k bidders bid at 1.
• Vickery revenues (R) = k
• optimal fixed price revenues (F)  k  h
• R
1
  
F
h
How T relates to F
• Theorem: T/(2log h)  F
• First, see that F uses the price opt(B):
opt ( B)  arg max bi B bi  (n  i  1)
How T relates to F (2)
• Proof:
X
• 1. Divide the bids into log h bins.
• 2. the sum of the bids = T => there exists a bin, say
X, that sums to at least T/(log h).
• 3. the lowest bid in X is at least 1/2 the highest.
• 4. choose price(B) to be the lowest bid in that bin.
• 5. Each bid in that bin will contribute at least half
its value.
1
1
1 T
• 6.
b 
b 
2
bi bin
i

2 bi bin
i
2 log h
Random Sampling Auctions
• Select a subset B’ of B (the set of bids),
B m
• Compute f(B’) to be the optimal fixed price
for B’
• Use f(B’) as a threshold to B\B’
• Bidder I wins the auction at price f(B’) if
bi  f (B' )
Performance analysis
• Theorem: assume ah  F, then R  F / 6
with a probability  1  e  a / 36  40e  a / 72
• Proof is basing in the Chernoff bound.
• By this Theorem: R
 1
F
Weighted Pairing Auction
• A bid-independent, truthful, multi price
auction:
f ( B)  b  B
W .P.
b
b'B b'
Bids
distribution
Weighted pairing performance
• if 4h  F, then for the weighted pairing
auction E(R)=(T/(log h)).
• But still, weighted pairing isn’t competitive,
in the worst case, it has a tight bound of:
( F / log h )
An upper bound
• We can see that no truthful auction, even a
multi price one, can do better then F in the
average case:
• Theorem: for any truthful auction: E[ R ]  F
An upper bound - proof
• Let’s define:
– pi the probability a bid I is satisfy
– ci expected cost to winning bidder I
– g i expected profit (gain) for bidder I
• (1) gi  pi (ui  ci )
An upper bound - proof(2)
• Lemma: in a truthful auction:
•
•
•
•
•
bi  b j  pi  p j
Proof:
if u  u  b , then p (b  c )  p (b  c )
i i
i
j i
j
i
j
i
if
, then p (b  c )  p (b  c )
ui  u j  b j
i j
i
j j
j
together:
p j (b j  bi )  pi (b j  bi )
since
, we have
.
bi  b j
pi  p j
An upper bound - proof(3)
• Theorem: for any truthful auction:
E ( R)  F
• Proof:
• if bidder I-1 had the utility value of bi his gain will
not exceed: (2) gi  pi 1 (bi  ci 1)
• so: gi  pi 1(bi  bi 1  bi 1  ci 1)


pi 1(bi  bi 1)  pi 1(bi 1  ci 1)
pi 1(bi  bi 1)  gi 1
An upper bound - proof(4)
• We can recursively extend in the same way until:
(3)
gi 
i 1
 p j (b j 1  b j )
j 1
• define the total expected revenue from bidder I as:
Ri  pi ci
• rewrite (1) gi  pi (ui  ci ) as: Ri  pi bi  gi
• using equation (3) we get:
i 1
Ri  pi bi   p j (b j 1  b j )
j 1
An upper bound - proof(5)
E[ R] 
i 1
 Ri
j 1
n
  pi bi 
i 1
 pnbn 
 pnbn 
n
n i 1
i 1
i 1 j 1
  pi bi    p j (b j 1  b j )
n 1
 p j (b j 1  b j )( n  j )
i 1
n 1
n 1
 p j b j   p j (b j 1  b j )( n  j )
j 1
n 1
j 1
 p j b j  (b j 1  b j )( n  j )
j 1
 E[ R ]  pnbn 
n 1
 p j b j (n  j  1)  b j 1(n  j )
i 1
An upper bound - proof(6)
E[ R ]  pnbn 
n 1
 p j b j (n  j  1)  b j 1(n  j )
i 1
• define V j  b j (n  j  1) , this is also the revenues
from a fixed price auction using b j as the price.
• Note that V j  F .
E[ R ]  pnVn 
n 1
 p j V j  V j 1 
i 1
• Rearrange the sum over V j and define p0  0 :
n
E[ R ]   ( p j  p j 1 )V j
i 1
An upper bound - proof(7)
n
E[ R ]   ( p j  p j 1 )V j
i 1
• but V j  F and by the lemma in the beginning we
showed that p j  p j 1  0 so:
n
E[ R ]  F  ( p j  p j 1 )
i 1
• this sum telescopes to pn  p0 but p0  0 so:
E[ R]  pn F  F
Deterministic Optimal Threshold
Auction
• A bid-independent auction, meaning bidder i‘s bid
value should only determine whether bidder i wins
or losses. The bid value should not determine
bidder i’s price.
• Uses optimal threshold function opt (Bi) ,to be
used as a threshold for bidder i, which on set of
bids B\bi returns the fixed price at which items
should be sold to achieve revenue F.
• This action has a worst case input that causes it
not to be competitive.
Deterministic Optimal Threshold
Auction (cont.)
• An example for worst case input :
– r bids with value h.
– (h-1)r-1 bids with value 1.
– Optimal single price auction take r bid with price h s.t
F= rh => threshold is h.
(the second best threshold is 1 taking all bid for
revenue of hr-1).
– Deterministic optimal threshold auction takes r high
bids with price 1 s.t R=r. (the high are taken in price 1
because removing one h bids causes the opt. threshold
to switch to 1)
– So R/F=1/h.
Upper bound for deterministic
Bid-independent action
• Theorem:
For any truthful deterministic
bid-independent auction and any constant α
there exist an input for which R/F =O(1/h)
and αh  F.
Truthful deterministic Auction
are Bid-independent
• Lemma: Any truthful deterministic auction is bid
independent
• Proof:
– Bi = B\{i}
– Bix = bi replaced with value x.
– A(B) = the result of running auction A on set of
bids B.
– Ai(B) – the result for bidder i( if i is rejected).
– We will define function g as: g(x) = Ai(Bix).
Proof(cont.)
• It is left to show that g is  except for some region (v,
), [v, ) or {} where it has the value v. This
would imply A is bid-independent.
• If g(x)=  for all x than we have empty interval
{}.
• Otherwise, let b = inf{x:g(x)  } and v=minx g(x).
It is left to show b=v and for all b’>b, g(b’)=v
Proof(cont.)
1. v=b: v  b since g(x) is defined as  (loss) or
g(x)  x (a bid will win at price necessarily at
most x). v  b since else there might exist b’ s.t
v<b’<b. The bidder with utility value b’ would
be better off biding greater than b winning the
auction and pay v which is less that b’. This is a
contradiction of A been truthful. Thus, v=b.
2. For all b’>b, g(b’)=v: since that if g(b’) will be
bigger than v, a bidder with utility b’ will be
better off bidding b. This is a contradiction of A
been truthful.
Upper bound for deterministic
Bid-independent action
• Theorem: For any truthful deterministic
bid-independent auction and any constant α
there exist an input for which R/F =O(1/h)
and αh  F.
Why Simulation is needed?
• In terms of asymptotic worst-case performance,
deterministic auctions are significantly worse that
randomized auction => this is not to say
deterministic auctions are bad to use for all input
families.
• Constant factors (not necessarily tight) obtained in
the analysis do not enable determination which
auction does better.
• Theoretical analysis of the presented auctions for
specific distributions is non-trivial.
Some simulation results
• On uniform and normal distribution (“average
case” families) for large n the dual-price sampling
optimal threshold and the deterministic optimal
threshold have the best results.
• The above families have the property that any
uniformly chosen random subset of the bids has
the same distribution as the original. Because of
this random sampling auction perform very good
on such families.
• Weighted pairing is the worst auction for average
case families.
Some simulation results (cont.)
• Even on contrived worst-case families the auction
revenue is a large constant fraction of F.
• Sampling about a square root of the number of
bids for the sampling auction seems to balance out
the loss due to rejecting all of the sample and the
loss due to having a non representative sample.
Extension for Bounded Supply
• The bounded supply case is a generalization of the
unlimited supply case as items are available in
unlimited supply when the number of available
items is the same as the number of bidders.
• Denote the number of items available by k.
• Denote by Fk the revenue in the bounded supply
case.
• Let optk be the function that given a set of bids
return the optimal threshold that sells k item or
less.
Extension for Bounded
Supply(cont.)
• The single price sampling optimal threshold will
be modified to use threshold function optmk/(n-m)
on sample of size m. If more than k bids are
satisfied, we arbitrary reject bids until there are
only k left.
• The dual-price auction with sample size of m=n/2
use optk/2 so that about k/2 bids are selected from
each of the sample and the non-sample.
• Both auction can be shown to be competitive.
Conclusion
• There exist truthful auction for unlimited supply
market.
• Randomized auction are competitive in that they
yield revenue that is within a constant factor of
optimal fixed pricing.
• No deterministic auction is competitive in the
worst case.
• The presented auctions compare favorably to fixed
pricing with market analysis (simulation result)
Appendix: Competitive Auction
for Multiple Digital Goods.
Problem Definition
• The problem of selling several items , with each
item available in unlimited supply. Each bidder
wants only a single item. Example: concurrent
broadcast of several movies.
• The input for an auction:
– Number of bidders n.
– Number of items m.
– Set of bids {aij}.
Problem Definition (cont.)
• Auction outcome: an assignment of a subset of
wining bidders to items. Each bidder i in the
subset is assigned a single item j and a sales price
of at most aij.
• Stable auction: bidding Uij is a dominant strategy
for bidder i. It maximize the bidder profit at the
bidder’s utility value.
Note: the set of bidders strategies where bidder i
bids uij for item j is a Nash equilibrium.
“Fixed Price” Auction
• Bidders supply bids aij, seller supplies sale prices
rj 1 j  m.
• Definition cij=aij-rj.
• The auction assigns each bidder i to the item j with
the maximum cij. The sale price is rj.
• Lemma: Suppose the sale prices are set
independently of input bids. Then fixed price
auction is stable.
Optimal Sale Prices
• The problem of finding for a given set of bids and
a set of prices an assignment such that the fixed
price auction brings the highest revenue can be
view as a matching problem.
• Translating this to an integer program we get the
following linear program:
max j iXij(aij-rj) s.t.
jXij  1
1 in
Xij  0
1in,1j m
Xij is one exactly when bidder i get item j.
Optimal Sale Prices (cont.)
• The dual problem is expressed it terms of pi,
the profit of the corresponding bidders:
min ipi
s.t.
pi  aij –rj 1  i  n , 1  j  m
pi  0
1 in
• If we combine both of these optimization
problems we get:
Optimal Sale Prices(cont.)
Find Xij and Pi s.t
jXij  1
1 in
Xij  0
1in,1j m
pi + rj  aij 1  i  n , 1  j  m
ipi = j i Xij (aij - rj)
Optimal Sale Prices(cont.)
• Treating rj as variable the optimal sale price
problem as the following mathematical
programming problem:
max j i Xij rj
s.t.
rm=0
jXij  1
1 in
Xij  0
1in,mj m
pi + rj  aij 1  i  n , m  j  m
ipi = j i Xij (aij - rj)
Extension of the random
sampling auction
• Pick a random sample S of the set of
bidders. Let N be the set of bidders not in
the sample.
• Compute the optimal sale prices for S.
• The result of the random sampling auction
is then just the result of running the fixed
price auction on N using the sale prices
computed in the previous step.
Extension of the Deterministic
Auction
• Delete i from B.
• Compute optimal prices for the remaining
bidders
• Choose the most profitable item for i under
these prices.
• This is done independently for each bidder.
Conclusion
• The random sample auction is competitive.
• In order to studies this auction using
simulation there is a need for a fast
algorithm for calculation the optimal fixed
pricing problem.