Auction Theory

Class 9 – Multi-unit auctions: part 2

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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

vs.Item prices

p(S) = Σi S pi

Bundle prices

Arbitrary p(S)

$$$ $5$2$1 $13 $5 $10 $13

Advantage of item prices: simplicity, scalable to many items, quick termination.

Disadvantages: limited expressiveness.

Pricing methods

vs.Anonymous prices

Same price for everyone

Non-anonymous prices

Individual prices

$$$$$$$$$$$$ $$$$$$$$

Advantage of anonymous prices: “fairness”, easier to implement.

Disadvantages: limited expressiveness.

Pricing methods

vs.Item prices

p(S) = Σi S pi

Bundle prices

Arbitrary p(S)

vs.Anonymous prices

Same price for everyone

Non-anonymous prices

Individual prices

• Any combination of the above methods is possible.– Each has pluses and minuses.

• The Simultaneous Ascending Auction is an anonymous item-price 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 ∑iS vi(xi) if 0 S

0 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) > ∑iS πi

• (Note the π0 includes payments from all players)

• Definition: Core.A surplus vector π0 ,π1 ,…,πn is in the core if:– (Feasibility)

∑iN πi = W(N) – (No Blocking Coalitions)

For every subset S of players, w(S) ≤ ∑iS 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 auctionInitialization: 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 semi-

truthful, 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

• VCG:– Alice+ bob:

Revenue=2– Alice + Bob + Carol:

Revenue=0• VCG outcome is outside

the core!

• Proxy:– Alice + Bob + Carol:

Revenue=2• Bob, Carol pay 1.

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v(a) v(b) v(ab) Alice 0 0 2Bob 2 0 2Carol 0 2 2David 0.5 0.5 1

False-Name Bids

• 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 non-beneficial.

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v(a) v(b) v(ab) Alice 0 0 2Bob 2 0 2Carol 0 2 2David 0.5 0.5 1

Collusion

• 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 losers30

v(a) v(b) v(ab) Alice 0 0 2Bob 2 0 2Carol 0 2 2David 0.5 0.5 1

Summary• The proxy auction provides an alternative outcome

to VCG:

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Property VCG Proxy

Truthful Yes Substitutes

onlyEquilibrium outcome is in the core Substitutes

only

Yes

No profitable false-name bids Substitutes

only

Yes

No profitable collusion of losing bidders Substitutes

only

Yes

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.

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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.

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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.

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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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