CS 3150 Paper Presentation

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CS 3150 - Paper Presentation

• Competitive Auctions and Digital Goods
• Andrew V. Goldberg
• Jason D. Hartline
• Andrew Wright
• September 1999
• Presented By Saurabh Sircar
• October 8, 2002

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Auctions
WINNER(S) AT PRICE(S)
BID 1
AUCTIONEER
BID 2
BIDDERS
BID 3
...
...
UNITS (GOODS)
BID n
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What is the Paper About?

• A formal study of auctions
• Different types of auctions that have their own
  principles
• Different ways of conducting auctions with their
  own procedures
• Different metrics to measure performance
• Upper bounds
• Lower bounds

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Auctions - Game Theory Mechanism Design

• Auction Game-theoretic implication
• Bidders (players/agents) - N
• Bids made by bidders (actions/strategies) - A
• Worth of the goods to every bidder (utility
  functions) - u
• Game form ltN,A,ugt
• Auction Mechanism design implication
• Auctioneer (mechanism designer)
• Decide who is(are) the winner(s) (outcome
  function) - g A ? C
• Determine the price of the auctioned goods
  (payment) - p
• Determine the principle of the auction (choice
  function) - f U ? 2A

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Auctions - Mechanism Design

• Auction settings
• Goods are identical and unlimited
• Single price - every winning bidder pays the same
  price
• Multiple prices - winning bidders may pay
  different prices
• Auctions with fixed-price setting using market
  analysis Why?
• Easy to implement
• Perfect knowledge of agent utilities
• Comparison baseline Fixed-price revenue F

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Auctions - Mechanism Design

• Auction principles bidder i bids bi and has
  utility ui
• Untruthful - not good for the auctioneer
• bi lt ui
• Truthful (a few characterizations) - good for the
  auctioneer
• bidding ui is the dominant strategy
• bi ui
• Let profit / gain g(bi) (ui - pi) where pi is
  the price paid by winning bidder i
• In this case, g(bi) is maximized at bi ui

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Auctions - Mechanism Design

• Auction principles bidder i bids bi and has
  utility ui
• Competitive - good for the
  auctioneer
• Perspective of revenue to the auctioneer (sum of
  all sales)
• Fixed-priced revenue F
• Revenue from the auction R
• R/F ?(1) (i.e., R is within some constant
  factor of F)
• Bid-independent - good for the auctioneer
• Price(s) for the winner(s) is(are) independent of
  the winners bid(s)
• Leads to truthful auction
• Has a price function pi f(B-i) and bidder i
  wins if bi ? f(B-i)

NEW
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Auctions - Mechanism Design

• Auction procedures
• Deterministic
• General maps sets of bids to auction outcomes
• Some specific (in this paper)
• Optimal single-price using threshold opt (B)
  argmax bi ? B bi . (n - i 1)
• yielding revenue F
• Optimal multiple-price ?i bi yielding revenue T
  Total revenue at value T

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Auctions - Mechanism Design

• Auction procedures
• Randomized
• General maps sets of bids to probability
  distributions of auction outcomes
• Some specific (in this paper)
• Random sampling Select B ? B set price
  threshold p f(B) select winners from B\ B
  with bi gt p
• Weighted pairing Set price threshold p f(B)
  b ? B-i w.p. b / ?j ? i bj
• if b ? bi i wins at cost b

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Auctions - Mechanism Design

• Auction analyses
• Inspiration analyses of on-line algorithms
• Essentials
• Correctness - Auction fills each winning bid at
  or below bid value
• Efficiency - Time needed to process bids
• Performance - In terms of revenue R, relative to
  F or T
• Mechanics
• WLOG, the bids may be sorted in ascending order
  1, 2, h
• Positive results (lower bounds) expressed as
  ratios R/F or R/T using ?
• Impossibility results (upper bounds) expressed as
  ratios R/F or R/T using O

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Auctions - Performance Results

• Existing results
• Truthful auctions are strategy-proof Truth
  revelation is the dominant strategy
• VCG mechanism maximizes total utility of agents
  (bidders perspective)
• Shapley mechanism shares cost among agents
  (bidders perspective)
• k-item Vickrey auction gives R/F O(1/h) bound
  for bipolar input - k bids at h and n-k bids at 1

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Auctions - Performance Results

• New results in this paper
• (Auctioneers perspective of revenue
  maximization)

PROCEDURES
S-price Opt. Threshold
Untruthful
Truthful
Multiple-price
Competitive
Random Sampling
Bid-Independent
Weighted Pairing
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Auctions - Performance Results
PROCEDURES
S-price Opt. Threshold
Untruthful
Truthful
Multiple-price
Competitive
Random Sampling
Bid-Independent
Weighted Pairing
F or T

• F ? (T / 2 log h) or even tighter ? (T / 4 log
  n)
• Dividing bids into log h bins

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Auctions - Performance Results
PROCEDURES
S-price Opt. Threshold
Untruthful
Truthful
Multiple-price
Competitive
Random Sampling
Bid-Independent
Weighted Pairing
F or T

• Concept of expected revenue ER instead of
  simply R
• Ideas used
• Use of Chernoff bound-like result connecting
  probability of sample size and the expected value
  of the whole set
• For some constant ? and with ?h ? F,
• R ? F/6 w.p. 1 - exp(- ?/36) - 40exp(- ?/72)
• ER / F ? (1) provided F / h is not small -
  Competitive!

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Auctions - Performance Results
PROCEDURES
S-price Opt. Threshold
Untruthful
Truthful
Multiple-price
Competitive
Random Sampling
Bid-Independent
Weighted Pairing
F or T

• ER ? (T / log h) if 4h ? T for weighted
  pairing
• Also, ER ? (T / log h) for random pairing ,
  but
• ER ? (F / (log h)1/2) - Not competitive (but
  not too off!)

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Auctions - Performance Results
PROCEDURES
S-price Opt. Threshold
Untruthful
Truthful
Multiple-price
Competitive
Random Sampling
Bid-Independent
Weighted Pairing
F or T

• Let pi probability that bid i is satisfied,
  then
• if bi ? bj then pi ? pj
• Also, ER O(F)

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Auctions - Performance Results
PROCEDURES
S-price Opt. Threshold
Untruthful
Truthful
Multiple-price
Competitive
Random Sampling
Bid-Independent
Weighted Pairing
F or T

• Difference between S-price optimal threshold and
  fixed-price is in set B-i versus B so we may
  expect competitive performance for large n
• But still R O(F/h) - bid-independent but not
  competitive!

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Auctions - Experiments

• Why experiments?
• Constant factors in analyses (like (log h)1/2
  gap) are too pessimistic
• Imperfections of fixed-price despite market
  analysis
• Experimental setup
• Auction Mechanisms
• DSO dual-price sampling optimal threshold
• SSO single-price sampling optimal threshold
• WP weighted pairing
• DOT deterministic optimal threshold
• FP- optimal fixed-price - 25
• FP optimal fixed-price 25

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Auctions - Experiments

• Experimental setup (continued)
• Input families
• Uniform Bids chosen from a uniform distribution
• Normal Bids chosen from a normal distribution
• Zipf Bids chosen from a Zipf distribution
• Equal-Revenue Based on parameter ?, h n / ?
• Bipolar Bids with high or low value only
• Simulation size
• Bidders between 10 and 100,000

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Auctions - Experimental Conclusions

• Main experimental results
• Random sampling auctions achieve R/F ? 1 as n
  becomes large
• WP does not perform as well as random sampling
  even in many contrived cases, it does not attain
  competitiveness
• DOT with O(F/h) performs well in the average case