Deterministic Auctions and (In)Competitiveness
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Deterministic Auctions and (In)Competitiveness
Proof sketch:Show that for any 1mn there exists a
bid vector b such that
Theorem:Let Af be any symmetric deterministic
auctiondefined by the bid-independent function f.Then Af is not competitive.
n
mbFbR m
A f )()( )(
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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
Or else the solution is trivial.Thus:
nb if
1b if
1pr
0)f(b
i
ii
1)(kpr1)(k(b)RfA
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Deterministic Auctions and (In)Competitiveness (Cont.)
If k+1<m we have:and:
n)(F(m) b
)(R1kmn
m)(F
fA(m) bb
Else - k+1m and:resulting with
n1)(k)(F(m) b
)(R1)k(1)(kmn
m)(F
fA(m) bb
which proves the theorem
Proof sketch (cont.):Now:
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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)
Usesas 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
iviivargmaxopt(b)
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DSOT – the Algorithm
Input: bid vector b Output: 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 = 12
7 410
18
9
opt(b’) = 7
opt(b’’) = 10
010 x’ =
010
0 p’ =
1 1 0 x’’ =
7 7 0 p’’ =
12
7 410
18
9
918
7 b’ =
10
12
4 b’’ =
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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])
then1
DSOT(b)
(b) max lim
bn
F
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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 (CF (b) then CostShareC has revenue of C; Otherwise it has no profit
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SCS – the Algorithm
Input: bid vector b Output: 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 = 12
7 410
18
9
F (b’)=21 F (b’’)=20
12
7 410
18
9
111 x’ = p’ =
0 0 0 x’’ =
0 0 0 p’’ = 666 32 3
23
2
918
7 b’ =
10
12
4 b’’ =
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SCS – Performance Analysis
Proof: Assume F (b)=k·p
then F ’(b’)=k’·p’k’·p and F
’’(b’’)=k’’·p’’k’’·p If F ’=F ’’ then F ’+F ’’F (b) and we are done
Otherwise
Theorem:SCS is 4-competitive and this bound is tight
k
)'k',min(k'
kp
)'k'p,k'min(p
(b)F
))'F(b'),min(F(b'(2)
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SCS – Performance Analysis
Proof (continue): Expected value of min(k’, k’’):
Thus, the competitive ratio
achieves its minimum of ¼ at k=2,3.as k increases, the ratio approaches ½
k
kikkb
2
2
kk
2
1
i
ki)-kmin(i,
2
11
)''(F
RE
..0(2)
kik
..0 i
ki)-kmin(i,
2
1)'k',min(k'E
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Bounded Supply
We may sell no more than k items We wish to be competitive against F (m,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
ikim
k)(m, vi maxF
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Another look atCompetitive 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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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
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
havePr[(xi=1) (pit)] Pr[(xj=1) (pjt)]
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Example:
Consider the following auction A given by the
bid-independent function f :
Where’s the catch?
Hard-Coded Auctions
)hh,...,,1,...,1(b
22 nn
otherwise
bin 1s than hs more if
h
1)( i-
ibf
For any bid vector b, there exists a
truthful auction A that satisfies A (b)= T
(b)
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Thus F is the optimal monotone function
Monotonicity (Cont.)
Theorem:Let A be any monotone truthful randomized
auction.For all bid vectors, the revenue of A satisfies
E[R] F(b)
DSOT, SCS, Vickrey auctions are all monotone; so is F
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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
Summary
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Cancelable auctions Envy-free auctions Almost truthful auctions Online auctions
Related Work
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Thank you!