Competitive Auctions Review

Competitive Auctions Review

Rattapon Limprasittiporn

Feb 9, 2004 Rattapon Limprasittiporn 2

Outlines Bibliography Introduction Software seller problem Truthful Competitive Goal & Solution


Bibliography Andrew V. Goldberg

Microsoft Research MASSACHUSETTS INSTITUTE OF

TECHNOLOGY, Doctor of Philosophy degree in Computer Science, January 1987.

Digital commerce models and languages. Auctions. Algorithm Design and Analysis. Implementation and Computational

Evaluation of Efficient Algorithms. Archival Intermemory.


Bibliography Jason D. Hartline

University of Washington Postdoctoral research fellow at

Carnegie Mellon University with the ALADDIN Center

Economic aspects of algorithms Optimization problems when input is

private information of selfish agents Game theoretic


Bibliography Anna R. Karlin

Stanford University Theoretical computer Design and analysis of algorithms Probabilistic algorithms Online algorithms




Introduction Bidder: person who bid Utility Value

Max price that bidder willing to pay Not the price that bidder pays

Bidder is happy if he pay less than his utility value

Auctioneer: person who sell


Example Bidders: Alice, Bob, and Carrol

Bob wins, but ...

Alice

7

Bob

10

Carrol

4

37

Nah!

4Nah!

69


Example Bidder’s Goal

Pay minimum price which greater than all other people’s utility values

Problem Lots of bidding tactic

Single-round sealed-bid auction


Single-Round Sealed-Bid Single-round

Each bidder submits bid only once Sealed-bid

Bidder blinded from other bidder’s bid Who win? Vickrey auction

The highest bid wins Pay 2nd-highest-bid price


Example Bidders: Alice, Bob, and Carrol

Bob wins and pay $7

Alice

7

Bob

10

Carrol

4

7 410


Vickrey Auction At the end

The highest bid wins Pay 2nd-highest-bid price

k-item Vickrey auction Have k items to sell

Single price auction k highest bidders win All winners pay the k+1th highest bid


Have 2 items to sell Bidders: Alice, Bob, Carrol, Daniel, and Eve

Bob and Denial win Both winner and pay $5

Example

Alice

5

Bob

11

Carrol

4

5 411

Deniel

12

Eve

2

212


Truthfulness Should bidder bid their utility value?

Yes, at least in k-item Vickrey Auction An auction is “truthful” if it

encorages bidder to bid their utility K-item Vickrey Auction is a truthful

auction




Software Seller’s Problem

Seller has unlimitted amount of products How can he put them in auction?

Choose k that maximizes his revenue

Selling 3 items to Alice, Bob, and Deniel is better than sell 2 items to Bob and Deniel

Is this a good auction?

Alice

5

Bob

11

Carrol

4

5 411

Deniel

12

Eve

2

212



Bidders: Alice, Bob, and Carrol

To maximize revenue, seller sells software to Carrol only

Alice

10

Bob

30

Carrol

40

10 4030



What if Bob changes his bid from 30 to 11

To maximize revenue, seller sells software to Bob and Carrol at the price of 10

Hey, this is not right! This auction is thus “not truthful”

Alice

10

Bob

30

Carrol

40

10 4011


Problem Find a good way to “auction” for

software seller We are on the seller side

What is a “good auction” Truthful Yield “good” revenue

Good compared to an “ideal” case “Competitive” (to the ideal case)




Truthful (revisit) What is “truthful” auction?

Encorage bidders to bid their utility Prevent tactic and strategy

How to make an auction “truthful”? Process result of bidderi without

looking at his bid “Bid-Independent Auction”


Bid-Independent Auction b = set of all bids that bidders bid For each bidderi

Exclude bid from bidderi to get b-i , the set of all bids except the bid from bidderi

Compute “auction funtion”, f, on b-i to get threshold ti

If bidderi bids more than ti , he wins at price ti , otherwise, he loses


Example b = {5, 11, 4, 12, 2} Let “auction function”, f, = “maximum of” For bidder1

b-1 = {11, 4, 12, 2} f(b-1) = 12 Since 5 < 12, bidder1 loses the auction

For bidder2 b-2 = {5, 4, 12, 2} f(b-1) = 12 Since 11 < 12, bidder2 loses the auction

For bidder4 b-4 = {5, 11, 4, 2} f(b-4) = 11 Since 12 > 11 , bidder4 wins the auction at price 11


Auction Function Auction function, f, is a core of bid-

independent auction f is “maximum of” = 1-item Vickrey Auction f is “kth maximum of” = k-item Vickrey

Auction Must be “monotone”

If b-i > b-j then f(b-i) > f(b-j)

Every monotone bid-independent auction is truthful




Competitive Good revenue compared to an ideal (in

seller’s sense) case Ideal case: Optimal single price

omniscient auction (F) k highest bids win at price kth highest bid Find k > 1 that yields highest revenue to be

the revenue of F Ex. b = {5, 11, 4, 12, 2, 8}

“k = 3” yields max revenue of 24 Revenue of F is F(b) = 24

Seller is happy if the revenue is close to F


Competitive Competitive = good revenue

Competitive to the ideal case Gives revenue within constant factor far

away form F Auction A is competitive if

A(b) F(b) / for some constant , and for all possible bid input b (worst case analysis)


Example Is 3-item Vickrey Auction competitive? Let b = {20, 20, 20, 1, 1, 1} 3-item Vickrey Auction gives revenue of 3 Optimal single price omniscient auction F

gives revenue of 60 3-item Vickrey Auction is not competitive In fact, all deterministic auctions are not

competitive!




Goal Find a way of auction to make

software seller happy Truthful Competitive

No deterministic auction is competitive

Sol: randomized auction Compute twice might not be the same


Solutions Dual-Price Sampling Optimal

Threshold Auction (DSOT) Sampling Cost-Sharing Auction

(SCS)


DSOT Partition bids b randomly into two sets b’

and b’’ Use omniscient auction F to compute ideal

revenue of b’ and b’’ and get p’ and p’’ p’: price that each winner in b’ pay to get F(b’) p’’: price that each winner in b’’ pay to get F(b’’)

Use p’ as a threshold for all bids in b’’ All bids in b’’ less than p’ are rejected All bids in b’’ greater than p’ win at price p’

Use p’’ as a threshold for all bids in b’


Example b = {14, 21, 13, 4, 23, 15, 6, 12, 7} Random partition:

b’ = {14, 15, 21, 6, 12, 7} b’’ = {13, 4, 23}

Compute threshold F(b’) = 48 which sell 4 items at price 12 = p’ F(b’’) = 26 which sell 2 items at price 13 = p’’

Use p’’ = 13 as a threshold in b’ Bidders who bid 14, 15, 21 win at price 13

Use p’ = 12 as a threshold in b’’ Bidders who bid 13, 23 win at price 12


DSOT Analysis Truthful

Bid-Independent Auction Competitive

Get some factor of F Multiple price?


Solutions Dual-Price Sampling Optimal

Threshold Auction (DSOT) Sampling Cost-Sharing Auction

(SCS)


SCS Partition bids b randomly into two sets b’

and b’’ Use omniscient auction F to compute ideal

revenue of b’ and b’’ and get F(b’) and F(b’’)

The highest k’ bids in b’ that each bid higher than F(b’’) / k’ win at price F(b’’) / k’

The highest k’’ bids in b’’ that each bid higher than F(b’) / k’’ win at price F(b’) / k’’


Example b = {14, 21, 13, 4, 23, 15, 6, 12, 7} Random partition:

b’ = {14, 15, 21, 6, 12, 7} b’’ = {13, 4, 23}

Compute threshold F(b’) = 48 which sell 4 items F(b’’) = 26 which sell 2 items

Two bids in b’, 21 and 15, can share the cost of 26 by paying 13 each

Bidders who bid 21 and 15 win at price 13 No bid in b’’ can share the cost of 48

No bid in b’’ wins


SCS Analysis Truthful

Bid-Independent Auction Competitive

Get some factor of F


Conclusion Two models for software seller

DSOT SCS

Truthful and competitive Worst case analysis


Thank you

Question?

Documents

Competitive Auctions Review