1 Competitive Auctions Authors: A. V. Goldberg, J. D. Hartline, A. Wright, A. R. Karlin and M. Saks Presented By: Arik Friedman and Itai Sharon
Jan 02, 2016
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Competitive Auctions
Authors:
A. V. Goldberg, J. D. Hartline, A. Wright, A. R. Karlin and M. Saks
Presented By:
Arik Friedman and Itai Sharon
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Motivation – Current Trends
Negligible cost of duplicating digital goods Emergence of the internet the problem: profit optimization for seller in an
auction Possible uses: PPV-TV, audio files
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Traditional Solution – Bayesian Auction
Example: VCG selling mechanism However:
– Accurate prior distribution unavailable or expensive
– Might be infeasible or unacceptable to consumers
Required: dynamic selling mechanism, for any market condition
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Settings
Single-round, sealed-bid, truthful auction mechanism
Performance of algorithms gauged in terms of optimal algorithm– Worst-case analysis– Success competitive algorithm
Unlimited supply– Can be extended for limited supply
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Agenda
Optimal Auctions Bid-independent auction
– Equivalent to truthful auction
No symmetric, truthful, deterministic auction is competitive
Two competitive randomized auctions:– DSOT– SCS
Justyfing optimality of F(m)
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Single Round Sealed Bid Auctions
n bidders b – vector of bids
– Maximum amount each bidder will pay
Auctioneer computes: (Randomized?)– Allocation x = (x1,x2,…,xn)
– Prices p = (p1,p2,…,pn)
• For winning bidders (xi=1): 0≤pi ≤bi
• For losing bidders (xi=0): pi=0
Profit: R(b) = Σipi
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Assumptions
Each bidder i has private utility value ui
Bidders want to maximize profit, uixi-pi
Bidders have full knowledge of auctioneer’s strategy
Bidders do not collude
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Some More Definitions
Symmetric auctions:Values of x and p are independent of order of bids
Deterministic Truthful auction:Bidder i’s profit is maximized by bidding ui
Randomized Truthful auction:May be described as a probability distribution over
deterministic truthful auctions
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Optimal Auction – First Try
The Optimal multiple-price omniscient
auction:
But: not truthful… As we will see – not a good bound…
niibbT
1
)(
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Optimal Auction – Second Try
The Optimal single-price omniscient
auction:
– vi is the ith largest bid in b
– All bidders with bi≥vk win at price vk
However – impossible to compete with...– As will be shown later
i1
vi max)( ni
bF
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Theorem (T(b) vs. F(b))
For all bid vectors b
F (b) ≥ T(b)/ln n
There exist bid vectors b for which
F(b) = Θ(T(b)/ln n)
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Optimal Auction – Final Try
The m-optimal single-price omniscient
auction:
– vi is the ith largest bid in b
– Determines k such that k≥m and kvk is maximized
– All bidders with bi≥vk win at price vk
i)( vi max)(
nim
m bF
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Competitive Auctions – Definitions A – truthful auction β - the competitive ratio of A.
A is β-competitive against F(m) if for all bid vectors b:
E[A(b)] ≥ F(m)(b) / β A is competitive against F(m) if it is
β-competitive against F(m) for constant β For m=2: A is [β-]competitive
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Bid-Independent Auctions: Definitions
1≤i≤n fi : bid vectors prices The deterministic bid-independent auction
defined by the functions fi.For each bidder i:– ti = fi(b-i) , b-i = (b1,…,bi-1,bi+1,…,bn)– if bi≥ti, bidder i wins at price ti
– Otherwise, bidder i is rejected Bid-Independent = Truthful
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Bid Independent Truthful
ui ≥ ti – bid at least ti and pay ti
– specifically, bid ui
ui < ti – can’t win without losing…
– so bid ui and lose
ui maximizes bidder i’s profit.
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Truthful Bid Independent
A – any truthful deterministic auction– We want to find f such that Af is identical to A.
bix=(b1,…,bi-1,x,bi+1,…,bn)
If x* such that in A(bix*) i wins and pays p
then: f(b-i)=p
otherwise f(b-i)=∞
Given p, We can show for A(bix) that:
– If bidder i wins, he pays p.– Bidder i wins by bidding any x≥p.
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Which Implies…
A deterministic auction is truthful if and only if it is equivalent to a bid-independent auction
Definition: a randomized bid-independent auction is a probability distribution over bid-independent auctions.
Corollary: a randomized auction is truthful if and only if it is equivalent to a bid-independent auction
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Theorem (can’t compete F(1)(b))
For any
truthful auction Af and
constant β≥1,
there is a bid vector b such that
E[R(b)] <F(b)/β
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Proof Consider a bid-independent randomized
auction on two bids, 1 and x≥1. let h be the smallest value greater or equal to 1 such that Pr[f(1)≥h] ≤ 1/2β.
Then the profit on input vector b = (1,H) with H = 4βh is at most
For any constant β≥1, no auction isβ-competitive against F =F(1)
)(
412
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