Deterministic Auctions and
(In)Competitiveness
Theorem:
Let Af be any symmetric deterministic auction
defined by the bid-independent function f.
Then Af is not competitive.
Proof sketch:
Show that for any 1mn there exists a bid
vector b such that
RA f (b)  F
( m)
m
(b) 
n
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Deterministic Auctions and
(In)Competitiveness (Cont.)
Proof sketch (cont.):
 Fix m and n
 Consider bid vectors whose bids are all n and 1
 Show that there is a bid vector b with k+1 ns
such that
if bi  1
0
f(b i )  
pr  1 if bi  n
Or else the solution is trivial.
Thus: R A f (b)  (k  1)  pr  (k  1)
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Deterministic Auctions and
(In)Competitiveness (Cont.)
Proof sketch (cont.):
Now:

If k+1<m we have: F(m) (b)  n
and:
m
F (b)   m  k  1  R A f (b)
n
(m)

Else - k+1m and: F(m) (b)  (k  1)  n
resulting with
m
F (b)   m  (k  1)  (k  1)  R A f (b)
n
(m)
which proves the theorem

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Competitive Auctions via
Random Sampling
Randomly partition the bid vector b into two
sets b’ and b’’
 Compute p’ based on b’ and p’’ based on b’’
 Assign the price p’ for b’’ and p’’ for b’

Two algorithms:
 DSOT
 SCS
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Dual-Price Sampling Optimal
Threshold Auction (DSOT)


Uses opt(b)  argmax vi i  vi
as the price setting mechanism
Constant competitive against F (2)
• The bound is weak for the general case
• Significantly better performance for some
interesting special cases
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DSOT – the Algorithm


Input:
Output:
bid vector b
Allocation vector x, price vector p
 Randomly partition b into b’ and b’’

Compute p’=opt(b’) and p’’ = opt(b’’)

Use p’ as a threshold for b’’

Use p’’ as a threshold for b’
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DSOT - Example
b = 10 7 12 4 18 9
b’ = 7 18 9
opt(b’) = 7
x’ = 0
b’’ = 10 12 4
opt(b’’) = 10
0
x’’ = 1
1
0
p’ = 0 10 0
p’’ = 7
7
0
1
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DSOT – Performance Analysis

In the general case – DSOT is constant
competitive against F (2); this bound is weak

For some interesting special cases DSOT’s
performance is much better
Example:
If b is bounded-range bid vector (bi [1, h])
then
F (b)
lim max
1
n 
b
DSOT(b)
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Sampling Cost-Sharing
Auction (SCS)


Uses CostShareC for setting the price
At least 4-competitive against F (2)
Definition (CostShareC):
Given a cost C and bids b, find the largest k
such that the highest k bid’s value  C/k.
Charge each C/k.


CostShareC is truthful
If (CF (b) then CostShareC has revenue of C;
Otherwise it has no profit
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SCS – the Algorithm





Input:
Output:
bid vector b
Allocation vector x, price vector p
Randomly partition b into b’ and b’’
Compute F ’=F (b’) and F ’’=F (b’’)
Compute the auction results by running
CostShareF’(b’’) and CostShareF’’(b’)
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SCS - Example
b = 10 7 12 4 18 9
b’ = 7 18 9
F (b’)=21
x’ = 1
1
b’’ = 10 12 4
F (b’’)=20
1
p’ = 6 2 3 6 2 3 6 2 3
x’’ = 0
0
0
p’’ = 0
0
0
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SCS – Performance Analysis
Theorem:
SCS is 4-competitive and this bound is tight
Proof:

Assume F (b)=k·p
then F ’(b’)=k’·p’k’·p and
F ’’(b’’)=k’’·p’’k’’·p
F ’=F ’’ then F ’+F ’’F (b) and we are done

If

Otherwise
min(F(b' ), F(b' ' )) min(p  k' , p  k' ' ) min(k' , k' ' )


(2)
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F (b)
pk
k
SCS – Performance Analysis
Proof (continue):


Expected value of min(k’, k’’):
k
1
Emin(k' , k' ' )  k   min(i, k - i)   
2 i 0.. k
i
Thus, the competitive ratio
k 

k
  1  k  k
ER 
1 1
  k   min(i, k - i)           2
(2)
F (b' ' ) k 2 i 0.. k
 i  2   2  
achieves its minimum of ¼ at k=2,3.
as k increases, the ratio approaches ½

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Bounded Supply

We may sell no more than k items

We wish to be competitive against
F (m,k):
F(m, k)  max i  vi
m i  k
Reduction to the unlimited supply case:
 Reject any bid that is not among the k highest bids
 Run the unlimited supply auction on the rest
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Another look at
Competitive Analysis


Thus far we have compared performance to F, the
optimal single-price auction
Is it “fair” to compare a dual-price auction to the
optimal single-price auction?
Theorem:
for any monotone (truthful) randomized auction A,
and for all bid vectors b, RA(b)=ipi satisfies
E[R]  F(b)
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Monotonicity
Definition (monotone auction):
An auction is monotone if for any pair of bidders
i and j with bi  bj and for any t  bi, we have
Pr[(xi=1)  (pit)] Pr[(xj=1)  (pjt)]
Intuition:
 Since bi  bj then b-i looks like a higher set of bids
than b-j
 We would expect a higher set of bids to yield a
higher price
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Hard-Coded Auctions
For any bid vector b, there exists a truthful
auction A that satisfies A (b)= T (b)
n2




 Example: b  (1,...,1, h,..., h )
n2

Consider the following auction
bid-independent function
A given by the
f:
1 if more hs than 1s in b -i
f (bi )  
otherwise
h
Where’s the catch?
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Monotonicity (Cont.)
DSOT, SCS, Vickrey auctions are all monotone; so is
F
Theorem:
Let A be any monotone truthful randomized auction.
For all bid vectors, the revenue of A satisfies
E[R]  F(b)
Thus
F is the optimal monotone function
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Summary

The notion of competitive auction was introduced

Justification for using
F was given
It was shown that no deterministic auction may can
be competitive
 2 novel randomized auctions for the unbounded
supply scenario, DSOT and SCS were introduced
 Reduction to the bounded supply was shown

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Related Work
Cancelable auctions
 Envy-free auctions
 Almost truthful auctions
 Online auctions

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Thank you!
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