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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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Optimal Auctions

A gauge for measuring auction performance

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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

And Other Definitions…

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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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Measuring Performance

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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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2

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Hhh

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