Parkes Auction Theory 1. Auction Theory. Jacomo Corbo. School of Engineering and Applied Science, Harvard University

Transcription

1 Parkes Auction Theory 1 Auction Theory Jacomo Corbo School of Engineering and Applied Science, Harvard University CS 286r Spring 2007

2 Parkes Auction Theory 2 Auctions: A Special Case of Mech. Design Allocation problems finite set G of items to allocate variations possible (e.g. information goods, configurable items) 1:N settings typical, N:M possible. Agent models private values vs. common values no externalities (compare to FAA problem) quasi-linear, i.e. u i (S, p) = v i (S) p for item(s) S G at price p; i.e. risk-neutral Mechanism properties Budget-balanced ( trading mechanisms ) efficiency (maximize total value), or revenue (maximize the utility of a single agent)

3 Parkes Auction Theory 3 Private vs. Common Values Private values [e.g. antique collectors, contractors] independently distributed, according to some prior, F i (θ), for agent i; priors common knowledge [iid is special case] models of information asymmetry also possible Common values [e.g. oil drilling rights] common value, v, each agent s signal of the value is a draw from common distr., v i H(v) learning about someone else s value useful Interdependent values [e.g. FCC spectrum] e.g. inherent differences in production costs; but some shared problem difficulty Model of agent valuations changes auction prescriptions.

4 Parkes Auction Theory 4 [Vickrey 61] Single-item: Efficient English/Vickrey (second-price) Dutch/FPSB (first-price) All efficient. Also, all revenue-equivalent (if IID, quasi-linear, symmetric). Let v (k) denote the k-th order statistic. First-price Sealed-bid/Dutch equil. bid s i (v) = E[v (2) v (1) = v i ]; expected revenue, E[s i (v (1) )] = E[v (2) ] Vickrey/English revenue E[v (2) ] Thm. [Rev. Equiv.] In any efficient auction, the expected payoff to every bidder, and the seller is the same. as long as bidders receiving no allocation make zero payment

5 Parkes Auction Theory 5 Revenue Equivalence Theorem. If two auctions implement the same allocation rule in equil. and bidders with no alloc. make zero payment, then they have the same expected revenue. Proof (sketch, see notes). Focus on incentive-compatible mechanisms (by revelation principle), and substitute a term for payments to ensure that the auction is IC given an allocation rule.

6 Parkes Auction Theory 6 Optimal Auction Design [Myerson 81] Suppose the seller can set a reservation price r 0. Consider Vickrey auction, with bids ordered v 1 v 2... v n. Tradeoff: between loss of revenue when v 1 < r and gain in revenue when v 2 < r < v 1. The optimal auction is not an efficient auction. The revenue-equivalence result just says that all efficient auctions have the same revenue.

7 Parkes Auction Theory 7 Optimal Selling Mechanism (w/out symmetry) Suppose each bidder s value v i is drawn with pdf f i and cdf F i, with the virtual valuation defined as J i (v i ) = v i (1 F i (v i ))/f i (v i ). Suppose regularity of the distribution which provides that J i (v i ) is strictly increasing in v i. Revenue-optimal auction: 1. Sell to the agent whose J i (v i ) is largest, breaking ties at random, as long as this priority level is non-negative. 2. Collect max of (a) minimal value could have bid and still won the virtual valuation auction; (b) minimal value could have bid and had a non-negative virtual valuation. Note. this is truthful in a dominant-strategy equilibrium.

8 Parkes Auction Theory 8 Optimal Auction Under Symmetry As a special case, if n bidders have values i.i.d. from the same pdf f and with regularity, then this is equivalent to a Vickrey auction with reserve price r set so that J(r) = 0!! Note: The four standard auctions all yield the same revenue under symmetry, are efficient without a reserve price, and optimal with an appropriate reserve price.

9 Parkes Auction Theory 9 Example

10 Parkes Auction Theory 10 Efficiency: Multi-unit, single-item bids. m units of a homogeneous item. First, consider the special case in which each bidder demands a single unit. Let v i 0 denote the value of bidder i. Def. The VCG auction for this special case sells the items to the m highest bidders, each pays the m + 1st highest bid price. p vick,i = b i X j m b j X j m+1,j i b j 1 A = b m+1

11 Parkes Auction Theory 11 Efficiency: Multi-unit Auctions Single bid, (k i, b i ), for k i units, from each agent. Let x i {0,1} define whether bid i is accepted, and p i denote payment by agent i. Again use VCG. (1) compute x to solve (weighted knapsack) problem: X V = max x i p i x s.t. X i i x i k i m (2) compute payments, p i = b i (V V i ) if x i = 1, with p i = 0 otherwise; where V i is maximal value over subproblem induced by removing bid from agent i.

12 Parkes Auction Theory 12 Efficiency: Multiple Heterogeneous Items G items, values v i (S) for S G. Outcome S = (S 1,...,S N ) is feasible if S i S j = for all i, j. efficient: maximize i v i(s i ). Examples: course registration; take-off/landing; logistics; bus routes, etc. Can continue to use a VCG mechanism. But, Computational challenges: winner-determination (weighted set-packing), bidding languages, preference elicitation.

13 Parkes Auction Theory 13 Fast & Strategyproof Comb. Auctions Lehmann et al. 99 single-minded bidders: there is a single set S G demanded by each agent. (still NP-hard). greedy, monotonoic allocation rule: sort bids by some criterion, then take bids in order if not in conflict. e.g. scheme with norm a/ S 1/2 approximates within factor of G 1/2. strategyproof auction: charge each winner the per-item price of the first unsuccessful bid. Example: goods A, B. bidders Red (10,A); Green (19,AB); Blue (8, B).

14 Parkes Auction Theory 14 Double Auctions Multiple buyers, multiple sellers, each with private information. Suppose bids, b 1 b 2... b m, and asks, s 1 s 2...s n. Compute l, s.t. b l s l and b l +1 < s l +1. Recall Myserson-Satterthwaite impossibility. McAfee-Double auction compute candidate trading price, p 0 = 1/2(b l +1 + s l +1), if s l p 0 b l clear first l bids and asks at this price, clear first l 1 bids at price b l and first l 1 asks at price s l. strategy-proof, BB, not EFF. k-da execute first l bids and asks; for a uniform price s l + k(b l s l ), for some k [0,1]. not strategyproof or EFF, but BB and good efficiency in practice, in particular for large markets.

15 Parkes Auction Theory 15 Other issues Proxy agents Closing rules Collusion Trust Common value Interdependent value

16 Parkes Auction Theory 16 ebay proxy agents Provide an upper bid-limit to the ebay agent, which competes in an English auction until price reached. Revelation principle! English Vickrey Note: issue of trust.

17 Parkes Auction Theory 17 Closing Rules [Roth & Ockenfels 01]; ebay vs. Amazon (auctions now dead). ebay [hard closing rule] industry in sniping, favors bidders with better technology empirically, limits information revelation during the auction, many bidders do not use proxy agents [esp. experienced bidders] Amazon [soft closing rule] removes this arms race for bidding technology empirically, encourages bidding earlier in the auction Little details matter!

18 Parkes Auction Theory 18 Collusion E.g. Bidder rings. Group of bidders get together beforehand, and decide that only one will participate in the auction. Share gains afterwards. [Robinson 85] problems in reaching an agreement, sharing rewards first-price [Dutch, FPSB] collusion is not self-enforcing because the selected bidder must submit a very small bid second-price [Vickrey, English] collusion is self-enforcing, because deviators are punished. shills, pulling bids off chandelier a tool for sellers to fight collusion

19 Parkes Auction Theory 19 Trust Vickrey auction. bidders must trust the auctioneer not to submit a false bid. [without risk] computational remedies? [bid verif. mechanism, trusted 3rd party] English auction. more transparent, although the auctioneer can still use a shill to increase the bid price [some risk] how does this compare to setting a reservation price?

20 Parkes Auction Theory 20 Common Value Settings [Wilson 77; Kagel & Levin 86; Bazerman & Samuelson 83] $8 pennies in a jar; collect sealed bids average bid $5.13, winning bid $10.01 winner s curse, all get an unbiased estimate, f( ) bids increase in f( ) in equil. winner is one with most optimistic estimate, adverse selection bias Simple model; signal s i U(V ǫ,v + ǫ) should bid b i s i ǫ

21 Parkes Auction Theory 21 [Milgrom & Weber 82] Interdependent Values Model: if one agent has a high value, then other agents are more likely to have high values ascending Vickrey; because the winning bidder s surplus is due to private information the more the price is related to the information of other agents, the lower the information rent of the winning bidder Linkage principle if the seller has any private information, should precommit to releasing the information honestly same argument; better to allow competition across bidders and drive price