Multiple auctions

Lecture Series 06David Yuen

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)

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

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

Multiunit or multiple single unit

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450

Car Auction eBay Auction

Why there still exists local bidders

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

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

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 (N=10) No bidder replacement Increasing optimal bid

“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

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

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

Global Bidder Expected Utility

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

Optimal Bidding Strategy (cont)

Single global bidder Static local bidders (N=5) 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

Optimal Bidding Strategy

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

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

Multiple Global Bidders (cont)

3 global bidders in 2 auctions3 global bidders + 10 local bidders

in 2 auctions

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

Market Efficiency

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 (k≥1) Free disposal Risk neutral buyers

Unrestricted auction

Degree of Overlap % of progressive auctions

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 (M<6) Heuristics is prevalent (Anthony 03, Dumas 05)

Neglect difference between auctions Never bid in more than k auctions

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

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”

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 by

assuming 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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Different subset of possible auctions=2|S|

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 pwin=0.7 each,nOpt=3, xOpt=30.7=2.1

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

Scenario 1: simultaneous auction

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

Scenario 2: unrestricted auction

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

Deg of overlap

%progressive auctions

low

low low

lowhigh

high

high

high

Complexity

Acceptable speed at least > 200 auctions

Any Questions?

Thank You!