1

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 )()( )(

2

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

3

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:

4

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)

6

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’’ =

8

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

9

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’’ =

12

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!