Auction Theory

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

2. 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.

3. Outline • Pricing methods • Core • Ascending Proxy Auction • Proxy Auction vs. VCG • Summary • Mega Summary

4. 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.

5. Pricing methods vs. $ $ $ $1 $2 $5 $13 $5 $10 $13 Advantage of item prices: simplicity, scalable to many items, quick termination. Disadvantages:limited expressiveness.

6. Pricing methods vs. $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ Advantage of anonymous prices:“fairness”, easier to implement. Disadvantages:limited expressiveness.

7. Pricing methods vs. • 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. vs.

8. Outline • Pricing methods • Core • Ascending Proxy Auction • Proxy Auction vs. VCG • Summary • Mega Summary

9. 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?

10. 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.

11. 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.

12. 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.

13. 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Svi(xi) if 0  S 0 if 0not in S

14. Blocking coalition and the core • A surplus vector π0,π1 ,…, πnis 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Sπi • (Note the π0 includes payments from all players) • Definition:Core.A surplus vector π0 ,π1 ,…,πnis in the core if: • (Feasibility) ∑iNπi = W(N) • (No Blocking Coalitions) For every subset S of players, w(S) ≤ ∑iSπi

15. 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.

16. Efficiency, core and VCG All outcomes Efficient outcomes Core outcomes VCG • Are the VCG outcomes in the core?

17. 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.

18. 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.

19. Outline • Pricing methods • Core • Ascending Proxy Auction • Proxy Auction vs. VCG • Summary • Mega Summary

20. 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.

21. 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,…,Tnis the provisional allocation. • Terminate if:Di(p)=Φ for every losing bidderi • that is, when Ti= Φ. • For every losing bidder i, and for all his bundlesSi Di(p), set: pi(Si)=pi(si)+ε

22. 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.

23. Proxy auction and the core • Theorem: the proxy auction terminates at a core outcome, with respect to the preferences reported to the proxy.

24. 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)

25. Outline • Pricing methods • Core • Ascending Proxy Auction • Proxy Auction vs. VCG • Summary • Mega Summary

26. 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.

27. 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.

28. 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.

29. 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.

30. 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 losers

31. Summary • The proxy auction provides an alternative outcome to VCG:

32. 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.

33. 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.

34. 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.

35. 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.

36. Thanks!