Auction Theory Class 9 Multiunit auctions part 2

Auction Theory Class 9 – Multi-unit auctions: part 2 1

Final problem set • Will be put on the web/email on January 23 th, Noon. • Should be submitted by February 1 st, 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 vs. p(S) = Σi S pi $ $ $ Bundle prices $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 Anonymous prices Same price for everyone vs. Bundle prices Arbitrary p(S) 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 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=x 1, …, 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. – T 1, …, Tn = a revenue maximizing allocation under prices p. • i. e. , for every allocation S 1, …, Sn we have ∑pi(Ti)≥ ∑pi(Si) • T 1, …, 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∙ 2 n 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 Carol v(a) 0 2 0 v(b) 0 0 2 v(ab) 2 2 2 David 0. 5 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! 28

False-Name Bids Alice Bob Carol v(a) 0 2 0 v(b) 0 0 2 v(ab) 2 2 2 David 0. 5 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. 29

Collusion Alice Bob Carol v(a) 0 2 0 v(b) 0 0 2 v(ab) 2 2 2 David 0. 5 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 30

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 31

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. 35

• Thanks! 36