Auction Design for
Atypical Situations
Overview
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General review of common auctions
Auction design for agents with hard valuation
problems
Auction design for goods in unlimited supply
Auction Design
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Many different protocols
Major auction types:
Ascending Price (English)
 Descending Price (Dutch)
 First Price, Sealed Bid
 Second Price, Sealed Bid (Vickrey)
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Essentially mixing and matching certain
properties, but some combinations work better
than others
Auction Evaluation
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Revenue for the sellers
Profit for the bidders
Avoidance of “winner’s curse” helps both
Winner in most auction protocols is the participant
who made the biggest overvaluation mistake
 Auctions that decouple an agent’s bid from the
actual price paid encourage higher bids
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Ascending Price (English)
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Most commonly known protocol
Auctioneer starts with an opening bid and
successively raises the price as participants are
willing
Allows for dynamic adjustment of bidders’
valuations by giving information about other
bidders
Buyers bid at most their utility, which may be
adjusted on the fly
Descending Price (Dutch)
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Biddings starts at an extremely high price and
descends until someone claims the item
High degree of Winner’s Curse
Intuitively, raises seller’s revenue in cases when
the high bidder wants an item badly
First Price, Sealed Bid
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Buyers submit bids once
No knowledge of one another’s bids
Auctioneer opens bids and sells the item to the
highest bidder at the price he submitted
Encourages buyers to bid conservatively (shade
down from utility) to maximize profit versus
probability of winning
Second Price, Sealed Bid (Vickrey)
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Designed to alleviate Winner’s Curse in the First
Price, Sealed Bid protocol
Same sealed bid format, but item is awarded to
the highest bidder at the second highest price
Buyers can bid their utility to increase
probability of winning, but are guaranteed a
price closer to market consensus
Optimal Auction Design for Agents
with Hard Valuation Problems
David C Parkes
Motivation
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Standard auction theory assumes that either
agents know own their utility for an item
(private value) or that there is a common utility
that is unknown to the agents (common value)
As transactions are increasingly turned over to
software agents, the cost of obtaining this utility
value may be significant
Certain auction designs can simplify this
problem
Paper Goals
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Compare the performance of agents with hard
valuation problems within three auction designs
Posted Price, sequential
 Second Price, sealed bid
 Ascending Price
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What the paper isn’t about: Actual methods for
refining beliefs about values
Examples of Hard Valuation
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Agents bidding for components on behalf of a
manufacturer
Agents bidding for collectibles or other rarities
Problem Formulation
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Agents with hard valuation problems operate in
three phases:
Metadeliberation: Decide how much effort to spend
refining the valuation of the item
 Valuation: The refinement process – solve an
optimization problem, interact with a human expert,
etc
 Bidding
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Agent Model
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Each agent has an unknown true value for an
item
Each agent maintains an upper and lower
bound, in which the true value is assumed to be
somewhere, uniformly distributed, in between
Expected true value is then the average of the
upper and lower bounds
The deliberation process refines the upper and
lower bounds
Agent Model
Upper Bound
D
New Upper Bound
Expected
True Value
New Expected
True Value
aD
New Lower Bound
Lower Bound
Deliberation
Incurs cost C
Parameters
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Cost of Deliberation (C)
Computational Effectiveness of Deliberation (1a)
Metadeliberation
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Solve the tradeoff between reducing uncertainty
and incurring the cost of deliberation
Deliberation is only worthwhile when:
It changes the agent’s bid
 The new bid has a greater expected utility than the
old bid
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Metadeliberation strategies vary by auction type
Metadeliberation Strategies
Vickrey Auction
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Need distributional information about the bids of
other agents
Paper assumes such information is somehow obtained
by the agents (eg learning)
Metadeliberation strategy:
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Deliberate until utility of placing a bid now is greater than the
estimated utility of placing a bid after another deliberation
step
Bid utility is a nonlinear function of expected value
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Higher bid decreases profit but raises probability of winning
Metadeliberation Strategies
Vickrey Auction
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Length of time that an agent spends deliberating
depends on:
The number of agents in the auction
 The agent’s current upper and lower bounds on the
value of the item
 The computational effectiveness of deliberation (1a)
 The cost of deliberation (C)
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Metadeliberation Strategies
Posted Price Sequential
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No uncertainty about the actions of other
agents
Need only to worry about the cost of the good
Deliberate only when the ask price is within the
bounds ofv some threshold function g*(a, C, D)
Reject Price
g*D
v
Deliberate
Accept Price
Metadeliberation Strategy
Ascending Price
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Third action available in addition to bid and
deliberate: wait
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Agents that wait benefit from the deliberation of
others
Optimal Strategy:
v
Leave auction
Wait, or Deliberate if
auction will close, with
probability 1/(Na -1)
v
Bid
Evaluation Metrics
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Efficiency:
True Value for the Good for the Winning Agent
/ Maximum True Value over All Agents
Revenue:
Price Paid for the Good / Maximum True Value
over All Agents
Utility of Participation:
(Surplus to Winning Agent – Total Deliberation
Cost for All Agents) / Number of Agents
Evaluation Arms
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Variance of:
Number of Agents
 Computational Effectiveness (1-a)
 Cost of Deliberation (C)
 Agent “experience” – Adjust C and a for fractions
of the agent population
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Varying the Number of Agents
+ Ascending Price
X Sealed Bid
O Posted Price Sequential
Varying the Bidding Increment
+ Ascending Price
Varying the Computational
Effectiveness of Deliberation
+ Ascending Price
X Sealed Bid
O Posted Price Sequential
Varying Agent Experience
+ Ascending Price
X Sealed Bid
O Posted Price Sequential
Competitive Auctions and Digital
Goods
Andrew Goldberg
Jason D Hartline
Andrew Wright
Motivation
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Looking for an optimal way to sell goods in
unlimited supply
Downloadable music
 Pay per view movies
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Examines auctions as an alternative to fixed
pricing, which requires expensive and probably
inaccurate market research
Scary implication: Charge more for media
created by entities with small, rabid followings?
The Optimal Threshold Function
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Given a set of bids, determine the single price
that maximizes revenue
3
3
3
2
2
Sell… • 3 at 3 (9)
• 5 at 2 (10)
• 6 at 1 (6)
1
Paper Goals
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Examine classes of single round, sealed bid
auctions for products with no marginal cost of
reproduction
Need to solve tradeoff between selling a lot at a
low price versus a few at a high price
Need to ensure that participants bid their
utilities
Would like an auction mechanism that compares
well to optimal fixed pricing
Terminology
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Truthful auctions
Encourage participants to bid their utility
 More formally:
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Bidder’s profit (bid – price if wins, or 0 otherwise) is
maximized when bid is the same as utility for any fixed
values for the bids of other participants
Example: Vickrey
 Counterexample: First price sealed bid
 Why is this important?
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Revenue is maximized in truthful auctions
Truthful Auction Example
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Imagine participating in an auction for a jar of
100 pennies that you (and only you) have
counted
In a first price sealed bid auction, you cannot
hope to profit by bidding 100
In a Vickrey auction, by bidding 100 you will at
worst break even, but most likely pay less
Terminology
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Competitive auctions
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Produce revenue within a constant factor of optimal
fixed pricing
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Must vary the number of items sold based on the bids
received
Why is this important?
Matching optimal fixed pricing is the best possible result
 Being within a constant factor is a reasonable
approximation
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Evaluated Auction Designs
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All auctions in this paper are single round, sealed
bid
Auction mechanisms studied:
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Deterministic
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Deterministic Optimal Threshold
Randomized
Single Price
 Dual Price
 Weighted Pairing
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Bid Independence
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Agent’s bid determines whether or not he wins
the auction, but not the price
Typically multi-price, but not always
Example: Vickrey
Why is this important?
Bid independence allows for one to bid her utility
and still hope for a profit
 Deterministic auctions must be bid independent in
order to be truthful
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Truthful Deterministic Auctions
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Consider the deterministic optimal threshold auction
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To determine if bid bi wins
Remove bi from the set of bids (ensure bid
independence/truthfulness)
 Determine the threshold price at which maximal revenue
is attained in the remaining set of bids
 If bi >= this threshold, accept bi at the threshold price
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Note that this removal of bi is the only thing that
differentiates this auction from optimal fixed pricing
Deterministic Optimal Threshold
Auction Example
5
3
3
3
2
2
Step 1: Remove the bid to be evaluated:
3
1
Step 2: Compute the optimal threshold on the remaining bids:
5
3 3 2 2 1
Sell 1 at 5: 5
3 at 3: 9
5 at 2: 10
6 at 1: 6
Optimal threshold is 2
Step 3: Compare the removed bid to the optimal threshold 3
Step 4: Accept bid 3
at price 2
>2
Truthful Deterministic Auctions
Theoretical Results
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Truthful Deterministic Bid-Independent
Auctions are not Competitive in the worst case
Removing bi causes bigger problems than one
would expect
Consider an input set where the high bid h
occurs r times, and there are (h – 1) (r – 1) other
bids at 1
Proof Sketch
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Example: h = 5, r = 3
5
5
5
1
1
1
1
1
Sell…
• 11 at price 1 (11 revenue)
• 3 at price 5 (15 revenue)
1
1
1
Proof Sketch, Continued
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Deterministic Optimal Threshold Auction on
this input:
5
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5
5
1
1
1
1
1
1
1
1
To determine if b1 wins, remove it and compute
the threshold on the rest of the input
5
5
5
1
1
1
1
1
1
1
1
Sell 2 units at 5, or 10 units at 1
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Threshold falls to 1, so end up selling to the
high bids at the low price
More Results
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This result generalizes to all Deterministic
Auctions due to the theorem that all Truthful
Deterministic Auctions are Bid Independent
Proof intuition:
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Can’t profit by bidding utility if you’ll pay it upon
winning
Paper goes on to present empirical evidence that
on realistic data, the worst case for these
auctions doesn’t come up often
Random Sampling Auctions
Motivation
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Auction mechanisms need not only be resistant
to bad inputs, but also to attack
By using some nondeterminism in deciding who
wins and at what price, we can avoid dominance
by worst case inputs
Random Sampling Auctions
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Choose a random sample from the set of bids
Run the optimal threshold function on the sample and
use the result on the bids not in the sample
Single price
Nondeterministic
Dual price variant:
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Choose roughly half of the bids for the sample
Calculate thresholds on both sets
Use the threshold from one set on the other, and vice versa
Avoids having to throw out bids from the sample
Random Sampling Example
5
3
3
3
2
2
1
Step 1: Choose subset at random:
3
2
2
Step 2: Compute optimal threshold:
3
2
2
Sell 1 at 3: 3
Sell 3 at 2: 6
Optimal threshold is 2
Step 3: Apply optimal threshold to those not in the sample:
5
3
3
1
Step 4:
Accept 5
3
3
at price 2
Random Sampling Auctions
Theoretical Results
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Random sampling auctions are competitive
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General flavor of the proof:
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Revenue generated is within a constant factor of optimal
fixed pricing with arbitrarily high (but not 1.0) probability
The higher the probability, the lower the constant
The chosen subset is a good representation of the whole
most of the time
The revenue lost from losing the sampled bids is a constant
factor of the whole
The dual price version performs even better
Weighted Pairing Auctions
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To determine if a particular bidder i wins with bid bi:
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Choose another bid b with probability proportional to its
value –higher bids get picked more often
Compare b with bi. If bi >= b, bidder i wins and pays b
Multi-price
Nondeterministic
High bidders likely to win and pay high prices, but
some low bidders sneak in as well
Weighted Pairing Example
5
3
3
3
2
2
Step 1: Choose a bid
3
Step 2: Choose another bid with
probability proportional to its value
2
Step 3: Compare
3
Step 4: Bid 3
wins and pays price 2
1
>
2
Weighted Pairing Auctions
Theoretical Results
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Weighted Pairing auctions are not quite
competitive
Within a logarithmic factor of fixed pricing, so
not bad either
Perform well on inputs that random sampling
does not