Auction Theory
Class 9 – Multi-unit auctions: part 2
1
Final problem set
• Will be put on the web/email on January 23th, Noon.
• Should be submitted by February 1st, 23:59.
– By email to me (CC Assaf) – preferred.
• If sending handwriting, make sure it is clear.
• Contact me if not acknowledged within 24 hours.
– Or in the mazkirut (in its operation hours).
– If you have miluim etc, notify me in advance.
• (I am planning it as if you take the exam for 3 days, but this is
practically hard to do.)
• Shorter questions than in the problem set.
All issues covered in class may be included.
• Might be a good idea to learn the material in advance.
2
Outline
• Pricing methods
• Core
• Ascending Proxy Auction
• Proxy Auction vs. VCG
• Summary
• Mega Summary
Pricing methods
• A key design issue in auctions is the pricing method
to be used.
• There are two main criteria for pricing methods:
– Item prices vs. bundle prices.
• Also known as linear vs. non-Linear prices.
– Anonymous vs. Non anonymous prices.
Pricing methods
Item prices
Bundle prices
vs.
p(S) = Σi S pi
$
$
$
$1
$2
Arbitrary p(S)
$5
$13
$5
$10
Advantage of item prices:
simplicity, scalable to many items, quick termination.
Disadvantages: limited expressiveness.
$13
Pricing methods
Anonymous prices
Same price for
everyone
$ $ $ $
vs.
Non-anonymous
prices
Individual prices
$ $ $ $
$ $ $ $
$ $ $ $
Advantage of anonymous prices: “fairness”, easier to implement.
Disadvantages: limited expressiveness.
$ $ $ $
Pricing methods
Item prices
p(S) = Σi S pi
Bundle prices
vs.
Arbitrary p(S)
vs.
Non-anonymous
prices
Anonymous prices
Same price for
everyone
Individual prices
• Any combination of the above methods is possible.
– Each has pluses and minuses.
• The Simultaneous Ascending Auction is an anonymous itemprice auction.
• We will present a non-anonymous bundle-price auction.
– Maximum expressiveness.
Outline
• Pricing methods
• Core
• Ascending Proxy Auction
• Proxy Auction vs. VCG
• Summary
• Mega Summary
Auction design
• So far in the course, we learnt two main auction
techniques for selling multiple units:
– Simultaneous Ascending Auctions (SAA).
– VCG
• Today we will describe another type of auctions:
ascending proxy auctions
– Or just “proxy auctions”
• First, lets recall some of the properties of the SAA
and VCG?
Simultaneous ascending auction
• Properties of the Simultaneous Ascending Auction:
– Uses item prices.
– Uses anonymous prices.
– Efficient for substitutes valuations.
• Assuming straightforward bidding.
– Simple and fast.
– Exposure problems.
– Ends with VCG payments for unit-demand bidders.
VCG
• Properties of the VCG mechanism:
– Dominant-strategy truthful.
– Needs no distributional knowledge.
– Is not:
• Revenue monotone
– Adding more bidders may reduce revenue.
• Generating high revenue
– Sometimes revenue is extremely low (0)
• Shill-bidding proof
– Creating artificial bidders may be beneficial for bidders.
• Collusion proof
– Bidders can benefit from bidding together.
Core
• There is a sub-field of game theory, called cooperative
game theory.
– Focuses on the power and payoffs of coalitions.
• A central concept in cooperative game theory: the core
• Main idea: a stable solution where no coalition of
players has an incentive to deviate into a separate
arrangement.
• We will look at core solution in auctions.
Notations and definitions.
• Consider n players N={1,…,n}
• The seller is called player 0.
• Let the surplus for each bidder be denoted by πi.
– When the allocation/outcome is x=x1,…,xn:
• πi = vi(xi)-pi
for i=1,…,n
• π0 = ∑pi
• Let w(S) be the maximal social welfare achievable
from a coalition S:
– W(S)=
maxx ∑iS vi(xi)
0
if 0  S
if 0 not in S
Blocking coalition and the core
• A surplus vector π0 ,π1 ,…, πn is considered unstable if
a coalition can “block” this solution.
– That is, gain more than it gets by forming a new
coalition.
– Formally, S is a “blocking coalition” if w(S) > ∑iS πi
• (Note the π0 includes payments from all players)
• Definition: Core.
A surplus vector π0 ,π1 ,…,πn is in the core if:
– (Feasibility)
∑iN πi = W(N)
– (No Blocking Coalitions)
For every subset S of players, w(S) ≤ ∑iS πi
Core
• Is the core efficient?
– Yes. Feasibility=efficiency.
• Does an element in the core always exist?
– In general games, no.
– In our model, yes.
• For example:
the efficient outcome + payments pi(S)=vi(S) is a core
outcome.
Efficiency, core and VCG
All outcomes
Efficient outcomes
Core
outcomes
VCG
• Are the VCG outcomes in the core?
Core
• Theorem (Ausubel & Milgrom 2002):
– For substitute valuations, the VCG outcome is in
the core.
– For other valuations, the outcome is not in the
core.
• The formal claim: if values can be drawn from a class V that contains
all the additive valuations and even a single non-substitute
valuation, then for some profile of valuations from this class the
outcome is not in the core.
Revenue in core outcome
• One advantage of core outcome relative to VCG
outcomes is a greater revenue.
• Intuition:
– In some VCG setting revenue can be 0 (examples to come).
– In core outcomes this is not reasonable, since a coalition of
the seller and some losing bidders can block.
– Payment must be “sufficiently high” such that no blocking
coalition exists.
• Next: we will see an auction that finds a core
outcome.
Outline
• Pricing methods
• Core
• Ascending Proxy Auction
• Proxy Auction vs. VCG
• Summary
• Mega Summary
The ascending proxy auction
• The auction is based on work by Ausubel and
Milgrom (2002), and on a previous design by Parkes
and Ungar (1999).
• The auctions maintains non-anonymous bundle
prices.
– Recall: this means personalized price for each bidder, and
for all bundles.
• The auction finds a core outcome.
The ascending proxy auction
Initialization: set all prices to zero.
– That is, pi(S)=0 for all i,S.
Repeat:
• Let:
– Di(p) = all bundles demanded by i at price level p.
– T1,…,Tn = a revenue maximizing allocation under prices p.
• i.e., for every allocation S1,…,Sn we have ∑pi(Ti)≥ ∑pi(Si)
• T1,…,Tn is the provisional allocation.
• Terminate if: Di(p)=Φ for every losing bidder i
• that is, when Ti= Φ.
• For every losing bidder i, and for all his bundles Si  Di(p), set:
pi(Si)=pi(si)+ε
Why proxy?
• Players are asked before the auction to describe their
preferences to a proxy
– E.g., a computer program.
• Then the proxy plays on their behalf.
• Main point: commit to a single type of bidder.
– Bidding in first stages as bidder X and later as bidder Y is
not allowed.
Proxy auction and the core
• Theorem:
the proxy auction terminates at a core outcome, with
respect to the preferences reported to the proxy.
Equilibrium in the Proxy Auction
• Definition: a strategy in the proxy auction is semitruthful, if there is a constant c such that bidder
reports a value of vi(S)-c for every bundle S.
– Actually, max(0, vi(S)-c).
• Theorem: There is a Nash equilibrium in the auction
where each bidder plays a semi-truthful strategy.
– Specifically, if π is a bidder-optimal point in the core (i.e., no
other point in the core gains her a better surplus), then the
constant for the semi-truthful equilibrium strategy of each
bidder is πi.
– Note: the outcome is a core allocation with respect to
bidder’s actual preferences. (In particular, efficient)
Outline
• Pricing methods
• Core
• Ascending Proxy Auction
• Proxy Auction vs. VCG
• Summary
• Mega Summary
An alternative to VCG?
• The auction selects a core outcome.
• The result of the proxy auction can be viewed as
alternative to VCG.
– Has advantages and disadvantages compared to VCG.
• Main problems with VCG:
–
–
–
–
Low revenue despite high valuations.
No revenue monotonicity
False-name bids may be profitable
Collusion may be profitable.
Computational aspects
• Both in the proxy auction and in VCG we need to
solve hard computational problems.
– But in the proxy auction we solve a “np-hard” problem at each stage.
• Proxy auction maintains a set of bundle prices per
each bidder
– Can be n∙2n to maintain. Heavy communication load.
• Proxy auction is a reasonable alternative when the
number of items for sale is small.
– For example, 5 spectrum licenses.
• SAA and its variants are usually used for complex
numerous item auctions.
Revenue monotonicity
Alice
Bob
v(a)
0
2
v(b)
0
0
v(ab)
2
2
Carol
David
0
0.5
2
0.5
2
1
• VCG:
– Alice+ bob:
Revenue=2
– Alice + Bob + Carol:
Revenue=0
• Proxy:
– Alice + Bob + Carol:
Revenue=2
• Bob, Carol pay 1.
• VCG outcome is outside
the core!
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False-Name Bids
Alice
Bob
v(a)
0
2
v(b)
0
0
v(ab)
2
2
Carol
David
0
0.5
2
0.5
2
1
• VCG:
– Alice+ David:
Alice wins both items.
– David pretends to be Bob
and Carol:
Wins both items, pays 0.
– VCG is vulnerable to shill
bidding
• Proxy:
– When David pretends to
be Bob and Carol:
• Bob and Carol pay 1 ->
false-name bids are nonbeneficial.
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Collusion
Alice
Bob
v(a)
0
2
v(b)
0
0
v(ab)
2
2
Carol
David
0
0.5
2
0.5
2
1
• VCG:
– Alice + David x 2:
Alice wins both items.
– The 2 Davids bid like Bob
and Carol:
Each bidder wins an item
and pay 0.
• Proxy:
– If 2 Davids bid like bob
and Carol:
Each pays 1 hence
deviation is not
beneficial.
• Collusion even among losers
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Summary
• The proxy auction provides an alternative outcome
to VCG:
Property
VCG
Proxy
Truthful
Yes
Substitutes
only
Equilibrium outcome is in the core
Substitutes Yes
only
No profitable false-name bids
Substitutes Yes
only
No profitable collusion of losing bidders Substitutes Yes
only
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Summary
• Pricing methods are an important decision in the
auction design.
• Some hybrid methods are sometimes in use.
– Start with item prices, then continue with bundle bidding.
• Major complexity issues with bundle prices.
• Direct vs. indirect mechanism: indirect mechanisms
are usually preferred.
– For example, ascending-price auction over VCG.
32
Course Summary (1)
• Single item auction crystallizes the main auction
ideas.
– A fundamental microeconomic environment: probably
the simplest market, isolated from external influences.
• A problem of asymmetric information:
–
–
–
–
Private values
Common values
Interdependet values
Correlated values, affiliated values (not in this course)
• Some very influential ideas:
– Revenue equivalence, revelation principle, Bayes-Nash
equilibrium, implementation, monotonicity, etc.
33
Course Summary (2)
• The design of complex, multi unit auction is still an
art.
– Based on important theoretical insights.
• In real auction, there are many external details that
are important to learn.
– Specific to each scenario.
• Important notions: ascending auctions,
iterative/indirect auctions, competitive equilibrium,
exposure problems, substitutes and complements,
core, pricing methods.
34
Course Summary (2)
• If I had more than a 2-point course:
– Dynamic auctions.
• Bidders arrive/join the market sequentially.
– Double auctions
• E.g., stock markets, information markets.
– Digital goods.
• Goods with 0 marginal cost (e.g., software, songs).
– Mechanism design without money
• Matching: doctors to hospitals, students to schools, kidneys to
patients,
• Elections, choosing committees.
– Empirical results, experimental results.
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• Thanks!
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