Lecture #6: Auctions: Theory and Applications. Prof. Dr. Sven Seuken
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1 Lecture #6: Auctions: Theory and Applications Prof. Dr. Sven Seuken
2 Housekeeping Questions? Concerns? BitTorrent homework assignment? Posting on NB: do not copy/paste from PDFs Game Theory Homework: Formal/mathematical reasoning + write-up Discussing homework assignments Study the standard solution!! Registration deadline: Friday, 17:00 MSc vs. BSc level If in doubt, talk to me (e.g., today from 2:30-3:00pm) Course will be required elective with 99.9%
3 Outline 1. Recap of Algorithmic Game Theory 2. Today s topic: Auctions 3. Discussion + Questions
4 Recap: Algorithmic Game Theory Why do we care? Two-player Zero-Sum Games? Two-Player General-sum Games? Three-Player Games? Complexity of Find-Nash? Correlated Equilibrium?
5 Today: Auctions
6 What is an Auction
7 Auctions: More Formally
8 Quasi-Linear Utility Function
9 The Independent Private Values Model Bidder i s value is a random variable on[0, ] Bidder iknows realization with only probabilistic information about others Each is independently and identically distributed (IID) on 0, according to cumulate distribution function : 0, 0,1, with =Pr., thus =.
10 In contrast to Not independent E.g., correlated values Not identical Every agent draws from a different distribution Not private values Common value auctions Affiliated value auctions
11 Questions in Auction Design Efficiency Revenue
12 First-Price Auction
13 Second-Price Auction
14 Equilibrium Analysis The outcome is determined by the equilibrium A strategy [0, ]defines an agent s bid for every possible. Strategy vector: Bid vector:
15 Dominant-Strategy Equilibrium
16 Equilibrium of the 2 nd -price Auction
17 Bayes-Nash Equilibrium
18 Equilibrium of the 1 st -Price Auction
19 Relationship of Equilibria SPNE NE DSNE BNE
20 Revenue Equivalence What auction design is best for the seller? Revenue will be determined by the equilibrium! Need equilibrium analysis
21 Order Statistics Given n independent draws from distribution F, order them as! # %. Define the first-order statistic as!, the second-order statistic as #,and so on. Thus, order statistics are new random variables, and we can take their expectations, e.g., E[! ] For a uniformdistribution on [0,1], the expected value of the k-thorder statistic is
22 Revenue Equivalence for Uniform Distribution Remember, for 1 st -price auction: bidding %&! % was a symmetric BNE. For 2 nd -price Auction: expected value of the second highest bid from remaining n-1 bids that are lower than : %&! '!&! %&! '! = %&! % Thus, the auctioneer can expect the same revenue
23 Revenue-Maximizing Auctions? So, what if a seller wants to maximize his revenue? Anything he can do?
24 English Auction
25 Clock Auction
26 Dutch Auction
27 Strategic Equivalences
28 The ebay Auction First-price Auction? Second-price Auction?
29 Proxy Agents bid proxy bid bidder proxy
30 Closing Rule Soft vs. Hard Closing Rule Effect? Irrationality?
31
32 Bidding Dynamics
33
34
35 If you were the designer What auction rule would you choose for ebay?
36 Seller Design choices start price reserve price block buyers duration, start time what to sell (!) fixed price vs. auction
37 Trust Who do we need to trust? Do you trust the seller? (shill bids) Do you trust ebay?
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