Econ 805 Advanced Micro Theory 1 Dan Quint Fall 2009 Lecture 2 Today Common auction formats The Independent Private Values model 2 Common Auction Formats and Strategic Equivalences 3 Dutch Auction Auctioneer begins at a high price and lowers it until a buyer claims the object at the current price A slightly abstracted view the price falls continuously on a clock instead of in increments In a literal sense a bidder s strategy can be thought of as a choice of whether or not to buy at each price but for practical purposes it can be reduced to a decision of at what price to shout mine if the item s still available 4 Sealed Tender or First Price Auction Each bidder submits a sealed bid The object goes to the bidder with the highest bid at that price A strategy is simply a choice of how much to bid 5 Dutch Auction First Price Auction If all the matters is who wins the object and how much they pay then the Dutch Auction and the FirstPrice Auction are equivalent In each a bidder s strategy is reduced to picking a number the highest number wins and pays that much 6 English Ascending Auction Think of art auctions Price begins low auctioneer solicits bids at the next price keeps naming higher prices until no one is willing to raise their bid Or bidders name their own prices until no one is willing to outbid the high bidder think of online auctions without proxy bidding High bidder pays what he bid 7 Simplified English Auction or Button Auction Price begins low rises continuously At each price bidders can remain active hold down a button or drop out permanently Bidders only know the current price not who has dropped out and at what price When the second to last bidder drops out the last man standing pays the current price A bidder s strategy can be reduced to choosing a price at which to drop out if he hasn t won 8 Second Price or Vickrey Auction Each bidder submits a sealed bid The object goes to the highest bidder but the price they pay is the second highest bid 9 Simplified English Second Price Again if we reduce the game to the question of who wins and how much they pay the Simplified English Auction and Second Price Auction are equivalent Strategies are reduced to picking a number Highest number wins payment is second highest number But the Simplified English Auction changes if bidders can see who is still active at each price If I m unsure of the exact value of the object I may revise my estimate depending on how other bidders bid Then strategies can no longer be reduced to picking a single number and the equivalence breaks down 10 All Pay Auctions and Wars of Attrition In an All Pay Auction bidders submit sealed bids the high bid wins the object but everyone pays what they bid All Pay Auctions are sometimes used to model lobbying attempts to buy political influence and patent races the losers already made their contributions or incurred their costs War of Attrition is the same but dynamic like an all pay button auction where bidders can see who s still active Great game for an undergrad game theory class auction off a 20 bill highest bid wins highest two bids both pay what they bid 11 Multi Unit Auctions with Unit Demand Suppose there are k 1 identical items for sale but each bidder can only have one Pay as bid auction is like a first price auction the k highest bidders win and pay their bids Analog to the second price auction is the k 1st price auction Button auction works similarly ends when the k 1st bidder left drops out 12 The Independent Private Values Model 13 Baseline model of an auction as a Bayesian Game Symmetric Independent Private Values N 1 bidders in an auction for a single object Nature moves first assigning each bidder a private valuation vi for the object Each bidder s value vi is an independent draw from a common probability distribution F Each bidder knows his own value vi but not that of his opponents F is common knowledge Bidder i s payoff is vi p if he wins 0 if he loses where p is the price he pays for the object Like the Cournot game i s payoff depends on j s type only through j s action this is what s meant by private values 14 Note all the implicit assumptions we re making The number of bidders is fixed there is no decision over whether or not to participate Each bidder knows his own valuation perfectly does not care what the other bidders think of the object The bidders are symmetric ex ante valuations are drawn from the same distribution which is common knowledge Valuations are statistically independent Bidders are risk neutral 15 Auctions to sell versus auctions to buy Suppose the government holds an auction for a contract to provide some service Bids are now offers to provide the service at a given price and the lowest bid wins Where buyers were distinguished by their valuations for winning their object firms can be thought of as distinguished by their cost of providing the service So firm i s payoffs would be p ci where p is the price received and all the same analysis goes through 16 Solving for Equilibrium in the First and Second Price Auctions 17 Second price Vickrey auctions in the IPV world Claim In a second price sealed bid auction submitting a bid equal to your value is a weakly dominant strategy Proof Let B be the highest of your opponents bids When B v you could only win the object at price B for a payoff of v B 0 bidding b v gives you 0 which is as good as you can do When B v any bid b B gives the same payoff v B 0 which is payoff from bidding b v and the best you can do When B v any bid gives the same payoff 0 Corollary Every bidder playing the strategy bi vi vi is a Bayesian Nash Equilibrium of the secondprice auction 18 Similarly In a button auction it s a dominant strategy to drop out when the price reaches your private value vi Doesn t matter if you can observe who s already dropped out or not In an open outcry ascending auction Equilibrium strategies are not clear But it is a dominant strategy to never bid above your private value vi nor to let the auction end at price below vi d where d is the minimum bid increment So any equilibrium will involve the highest value bidder winning unless the highest two are within d of each other and paying within d of the secondhighest value So with private values as d gets small second price or button auctions give approximately the same outcome as ascending auctions Also similar is a first price auction with proxy bidding a la eBay Bidders can name a …