Multiple auctions

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Multiple auctions

• Lecture Series 06David Yuen

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Overview

• Multiple auctions
• Multiunit or multiple single-unit
• Characteristics of multiple single-unit auctions
• Simultaneous second price auctions
• Theoretical analysis
• With Enrico Gerding and Raj Dash
• Unrestricted auction heuristics
• Auction format and timing
• Simulation results
• With Andrew Byde (HP)

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Multiunit or multiple single unit

• Multiunit auction
• Allow to bid for multiple units
• US Treasury Bill auction
• Format
• Discriminatory (Paid you own bid)
• Uniform-Price
• Strategic behaviour
• Demand reduction
• Tacit collusion
• Not the focus of this presentation

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Multiunit or multiple single unit

• Similar items are being sold in many auctions
• Second hand car auction
• Tens of cars in each auction session
• Popular items in eBay
• More than 1000 auctions for iPod nano at any
  moment
• Participate in multiple single auctions
• Global bidder participate in all available
  auctions
• Improved expected profit
• Possibility to hunt for bargain

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Multiunit or multiple single unit
Car Auction
eBay Auction
1332
450
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Why there still exists local bidders

• Local bidders bid in a single auctionbid true
  valuation
• Participation costs
• Information
• Budget constraint
• Risk attitude
• Bounded rationality

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Characteristics of multiple auctions

• Demand from bidder
• One unit or more
• Disposal assumption
• Nature of the goods
• Substitute internet broadband contracts
• Complementary game console and games
• Timing structure
• Sequential
• Simultaneous
• Unrestricted

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Timing structure

• Sequential
• Start after last auction finishes
• Auction outcomes provides extra info
• Impossible to exceed purchase quota
• Example second hand car auction
• Optimal bidding strategy
• Second price auction
• Winner leaves (N10)
• No bidder replacement
• Increasing optimal bid

Bid fraction
Auction
Auction Theory, Ch 15, Vijay Krishna
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Timing structure

• Simultaneous
• Start at the same time
• Decision made based on little info
• Risk of exceeding purchase quota
• Example FCC spectrum auction
• Unrestricted
• The most general case
• Start/ finish at any time
• Example online auction sites

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What can be (have been) done?

• Simultaneous auctions
• 2nd price auctions
• Bidder wants only 1 unit
• Complete substitute
• Optimal and bidding strategy
• Theoretical analysis with simulation results
• Unrestricted auctions
• Any standard single unit auction format
• Bidder wants 1 or more units
• Complete substitute
• Heuristic approach

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Simultaneous second price auctions

• Settings
• Second-price (Vickrey) auctions (no reserve
  price)
• Each seller/auction sells 1 item
• Each buyer wants 1 item
• Free disposal
• Risk neutral buyers

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Global Bidder Expected Utility
v Bidder valuation G(b) Probability of
winning auction given bid b g(b) dG(b)/db

• Static local bidders exactly N bidders per
  auction
• Dynamic local bidders model bidders determined
  by Poisson with average N

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Bidding in Multiple Auctions

• Optimal to bid strictly positive in all available
  auctions, even if only 1 item is required
• Better to participate in all available auctions

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Finding Optimal Bid

• Arduous task in large settings using numerical
  methods
• Reduction of search space
• In most cases, optimal set of bids consists
  either of two different bid values (a high bid
  and a low bid) or all bids are equal
• Proof for non-decreasing bidder density functions
  (e.g. uniform and logarithmic)
• Holds empirically for most common distributions
• Bids are below the true valuation

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Optimal Bidding Strategy (cont)

• Single global bidder
• Static local bidders (N5) in M auctions
• Empirical observation
• Low valuation equal bids
• High valuation 1 high bid, (M-1) low bids
• Bifurcation phenomenon
• Expected utility increases w.r.t. M
• Most beneficent for midrange valuation

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Optimal Bidding Strategy
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Multiple Global Bidders

• Computational simulation approach
• A mix of global and local bidders
• Iteratively finding best response
• Discretize bid space initially
• Utility maximisation for each bidder type
• Next iteration based on latest bid distribution
• Stable solution ? symmetric Nash equilibrium

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Multiple Global Bidders (cont)

• Global bidders only
• No stable state is found
• Low valuations stable
• High valuations fluctuating between 2 states
• Global local bidders
• Very stable solution
• Bifurcation also occurs
• Best strategy is also to bid in all auctions

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Multiple Global Bidders (cont)
3 global bidders 10 local bidders in 2 auctions
3 global bidders in 2 auctions
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Market Efficiency (cont)

• Market efficiency reduces if
• All local bidders Highest valuation individuals
  bid in the same auction
• Dynamic local bidders items may remain unsold
• Global bidders win more than 1 item
• Against static local bidders
• Always improves efficiency
• Against dynamic local bidders
• Improves efficiency when M is small
• Reduces efficiency when M is large

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Market Efficiency
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Unrestricted auction heuristics

• Settings
• Standard single unit auction formats
• Dutch, English, first and second price
• Any combination
• Each seller/auction sells 1 item
• Each buyer wants 1 or more item (k1)
• Free disposal
• Risk neutral buyers

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Unrestricted auction

• Degree of Overlap
• of progressive auctions

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Why use heuristics?

• Long prediction horizon
• Practical time constraints
• Modelled as a Markov Decision Process
• Proved to be intractable (Boutilier 99)
• Limited to small number of auctions (Mlt6)
• Heuristics is prevalent (Anthony 03, Dumas 05)
• Neglect difference between auctions
• Never bid in more than k auctions

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Existing benchmarks

• Random (RND)
• Randomly pick k auctions
• Bid as if it is a local bidder
• Greedy (GRD)
• Calculate extra item required
• nExtra k nObtained
• Pick nExtra auctions with least bidders
• No chance to exceed purchase quota

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Two-stage heuristics

• Aim to reduce the search space
• Threshold heuristics
• Set the maximum bid for each auction
• Actual bid depends on progress in an auction
• Auction selection heuristics
• Decide whether to participate in an individual
  auction
• Allows mix-and-match

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Threshold strategies

• Single auction dominant heuristics (DOM)
• True value for second price mechanisms
• Affected by nBidder for first price mechanisms
• Equal threshold heuristics (EQT)
• Same threshold for all auctions
• Estimate average nBidder with harmonic mean
• Approximate expected utility byassuming
  identical auction format
• Find threshold that maximises utility

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Auction selection heuristics

• Exhaustive search selection (ES) (Byde 02)
• Knapsack utility approximation (KS)
• Significant loss if the demand limit is exceeded
• Find best number of auctions to participate in
• With simplified multiple auction model
• Given thresholds are fixed

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Auction selection heuristics (cont)

• Knapsack utility approximation (cont)
• Estimate the optimal number of wins, x
• Suppose it is the best to place bid in 3 out of
  4 auction and the pwin0.7 each,nOpt3,
  xOpt3?0.72.1
• Apply knapsack algorithm
• Objective maximise item value, i.e. minimise
  expected payment
• Sack weight limit xOpt
• Item weight pwin if placing bid b(a) for auction
  a
• Item value (-1)? expected payment for a

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Scenario 1 simultaneous auction

• For a set of 8 simultaneous Vickrey auctions
• Compare with optimal results

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Scenario 2 unrestricted auction

• Increasingly better than benchmarks when
• degree of overlap is high
• progressive auctions (Dutch, English) is high

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Complexity

• Acceptable speed at least gt 200 auctions

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Any Questions?

• Thank You!