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CS6283 Topics in Computer Science IV: Computational Social Choice. Instructor: Yair Zick 2017. Introduction to Mechanism Design. Truthful Auctions and Bidding. Auctions Around Us. Why do we want truthful reporting?. Single Item Auctions. We have a single item for sale.
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CS6283 Topics in Computer Science IV: Computational Social Choice Instructor: YairZick 2017
Introduction to Mechanism Design Truthful Auctions and Bidding
Single Item Auctions We have a single item for sale. Each bidder values the item at . Seller can decide on a price. What are the outcomes?
Single Item Auction Formats • English auction: • auctioneer sets a starting price • bidders take turn raising their bids • the person who makes the last bid wins and pays his bid • Japanese auction: • auctioneer sets a starting price and then starts raising it • a bidder can drop out, and cannot return once he dropped • the bidder who stays in last gets the object, pays the current price
Single Item Auction Formats, Continued • Dutch auction: • auctioneer sets a (high) starting price and then starts lowering it • the auction ends when some bidder accepts the price • used in the Amsterdam flower market • Sealed-bid auction: • all bidders simultaneously submit their bid • the highest bidder gets the item and pays.... • his bid (first-price auction) • 2nd highest bid (second-price, or Vickrey, auction)
English Auction • Suppose your value for the object is , the current price is , and the minimum bid increment is • It is rational to bid if and only , and your bid should be • If , and you end up winning, you will pay more than the object is worth to you. • If you bid more than , but no one else was willing to pay more than , you pay more than is necessary to win.
English Auction, Dynamics . The auction starts at . While , all bidders are submitting bids. At , player stops bidding. At , player stops bidding. If player was the one to bid , she wins and pays . If player was the one to bid , player bids and wins. Winning bid is either or .
Japanese Auction, Dynamics • The auction starts at • At , player drops out • At , player drops out • Player wins and pays • Communication: • English auction: 50 messages • Japanese auction: 2 messages
Vickrey Auction • All bidders submit bids simultaneously in sealed envelopes, • The highest bidder wins and pays the second highest price • Strategic game • players (bidders) • actions: bids (continuous action space) • payoff: if a player values the object at and the 2nd highest bid is , her payoff is • if she gets the object • if she does not get the object
In a Vickreyauction, truthful bidding is a dominant strategy
Proof: • suppose your value is • suppose other players’ bids are • Case 1: • if you bid , you win and pay ; payoff is • if you bid , you lose; payoff is .
Proof: • suppose your value is • suppose other players’ bids are • Case 2: • if you bid , you win and pay ; payoff is • if you bid , you lose; payoff is .
Is Truthful Bidding a Strictly Dominant Strategy? • is an equilibrium in dominant strategies. • is also a NE • auctioneer makes no profit • is also a NE • Player 1 gets the object and pays nothing • if player 2 increases his bid so as to beat player 1, she will end up paying at least 70, so she cannot increase her profit
Multi Unit Auctions We have a multiple, identical items for sale. Each bidder wants one item; values the item at . Assume that Seller can decide on prices. What are the outcomes?
Multi Unit Auctions We have a multiple, different items for sale. Each bidder wants one item; values item at . Seller can decide on prices. What are the outcomes?
The Revelation Principle Let be a mechanism with a (very complicated) query model; then there exists a mechanism whose inputs are users’ valuations, and whose outputs are exactly like those of .
In VCG mechanisms, truthful reporting is a dominant strategy.
Total social welfare Does not depend on ’s report Utility of player when she reports truthfully:
Further Reading • Basic Mechanism Design • Chapters 9 and 10 of Algorithmic Game Theory by Nisan et al. • Further reading: • Combinatorial Auctions • Approximate Mechanism Design