auction theory class 9 – multi-unit auctions: part 2 1
TRANSCRIPT
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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
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Outline• Pricing methods
• Core
• Ascending Proxy Auction
• Proxy Auction vs. VCG
• Summary
• Mega Summary
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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.
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Pricing methods
vs.Item prices p(S) = Σi S pi
Bundle prices Arbitrary p(S)
$$$ $5$2$1 $13 $5 $10 $13
Advantage of item prices: simplicity, scalable to many items, quick termination.
Disadvantages: limited expressiveness.
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Pricing methods
vs.Anonymous prices Same price for everyone
Non-anonymous prices
Individual prices
$$$$$$$$$$$$ $$$$$$$$
Advantage of anonymous prices: “fairness”, easier to implement.
Disadvantages: limited expressiveness.
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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.
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Outline• Pricing methods
• Core
• Ascending Proxy Auction
• Proxy Auction vs. VCG
• Summary
• Mega Summary
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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?
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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.
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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.
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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.
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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
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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
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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.
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Efficiency, core and VCG
All outcomes
Efficient outcomes
Core outcomes
VCG
• Are the VCG outcomes in the core?
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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.
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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.
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Outline• Pricing methods
• Core
• Ascending Proxy Auction
• Proxy Auction vs. VCG
• Summary
• Mega Summary
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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.
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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
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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.
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Proxy auction and the core• Theorem:
the proxy auction terminates at a core outcome, with respect to the preferences reported to the proxy.
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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)
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Outline• Pricing methods
• Core
• Ascending Proxy Auction
• Proxy Auction vs. VCG
• Summary
• Mega Summary
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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.
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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.
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Revenue monotonicity
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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
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False-Name Bids
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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
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Collusion
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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
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Summary• The proxy auction provides an alternative outcome
to VCG:
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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.
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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.33
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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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