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

Final problem set Will be put on the web/ on January 23 th, Noon. Should be submitted by February 1st, 23:59. – By 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) = Σ i  S p i 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) = Σ i  S p i 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 π i. – When the allocation/outcome is x=x 1,…,x n : π i = v i (x i )-p i for i=1,…,n π 0 = ∑p i Let w(S) be the maximal social welfare achievable from a coalition S: – W(S)= max x ∑ i  S v i (x i )if 0  S 0if 0 not in S

Blocking coalition and the core A surplus vector π 0,π 1,…, π n 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 π i (Note the π 0 includes payments from all players) Definition: Core. A surplus vector π 0,π 1,…,π n is in the core if: – (Feasibility) ∑ i  N π i = W(N) – (No Blocking Coalitions) For every subset S of players, w(S) ≤ ∑ i  S π i

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 p i (S)=v i (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, p i (S)=0 for all i,S. Repeat: Let: – D i (p) = all bundles demanded by i at price level p. – T 1,…,T n = a revenue maximizing allocation under prices p. i.e., for every allocation S 1,…,S n we have ∑p i (T i )≥ ∑p i (S i ) T 1,…,T n is the provisional allocation. Terminate if: D i (p)=Φ for every losing bidder i that is, when T i = Φ. For every losing bidder i, and for all his bundles S i  D i (p), set: p i (S i )=p i (s i )+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 v i (S)-c for every bundle S. – Actually, max(0, v i (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 π i. – 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 v(a)v(b)v(ab) Alice002 Bob202 Carol022 David0.5 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. 29 v(a)v(b)v(ab) Alice002 Bob202 Carol022 David0.5 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 losers 30 v(a)v(b)v(ab) Alice002 Bob202 Carol022 David0.5 1

Summary The proxy auction provides an alternative outcome to VCG: 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