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Econ 805 Advanced Micro Theory 1 Dan Quint Fall 2007 Lecture 2 – Sept 6 2007

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1 Today Common auction formats The Independent Private Values model

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2 Common Auction Formats and Strategic Equivalences

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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

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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

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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 First- Price Auction are equivalent In each, a bidder’s strategy is reduced to picking a number; the highest number wins, and pays that much

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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

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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

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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

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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

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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

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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+1 st -price” auction Button auction works similarly, ends when the k+1 st bidder left drops out

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12 The Independent Private Values Model

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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 v i for the object Each bidder’s value v i is an independent draw from a common probability distribution F Each bidder knows his own value v i but not that of his opponents Bidder i’s payoff is v i – 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”

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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

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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 – c i, where p is the price received, and all the same analysis goes through

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16 Solving for Equilibrium in the First- and Second-Price Auctions

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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 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 b i (v i ) = v i is a Bayesian Nash Equilibrium of the second-price auction

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18 Similarly… In a button auction, it’s a dominant strategy to drop out when the price reaches your private value v i 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 v i, nor to let the auction end at price below v i – d 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 second-highest 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 maximum, then the computer raises their bid to the minimum required to win until that maximum is reached

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19 Sadly, “everyone bids their value” is not the only equilibrium of the second-price auction Suppose bidder values were drawn from a distribution with support [0,10] The following is an equilibrium of the second-price auction: Bidder 1 bids 15 regardless of his type All other bidders bid 0 regardless of their type b i (v i )=v i is “nearly” the only symmetric equilibrium; and it involves bidders playing a strict best-response at nearly every type

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20 First-price auctions in the symmetric IPV world We’ll look for “nice” equilibria: Symmetric (bidders all play the same strategy) Bids are increasing in valuations Tomorrow, we’ll learn a trick that makes finding this type of equilibrium much easier Suppose such an equilibrium exists, and let b : [0,V] R + be the common bid function; then at a given type v, b(v) must be a solution to max x R + (v – x) Pr(win | bid x, opponents bid b(-)) = max x R + (v – x) Pr(b(v j ) < x j i) = max x R + (v – x) Pr(v j < b -1 (x) j i) = max x R + (v – x) F N-1 (b -1 (x))

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21 If there is a symmetric, increasing equilibrium in a first-price auction… First-order condition (b -1 )’ = 1/b’ x = b(v) in equilibrium b(v) must solve so integrating from 0 to v,

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22 So if there is a “nice” equilibrium, it must be b(v) = 0 v sd(F N-1 (s)) / F N-1 (v) What is this? Well, if a random variable y has cumulative distribution G with positive support, then 0 v s dG(s) / G(v) = E(y | y < v) And F N-1 (v) is the cumulative distribution function of the highest of N-1 independent draws from F So if we let v 1 and v 2 refer to the highest and second- highest valuations in a symmetric IPV model, then b(v) = E(v 2 | v 1 = v)

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23 Now here’s where it gets cool… In the symmetric equilibrium of the second-price auction, the price paid is v 2, so the seller’s expected revenue is simply E(v 2 ) In the symmetric, increasing equilibrium in the first-price auction (if it exists), The bidder with the highest value wins If the highest value is v, the winner pays E(v 2 | v 1 = v) So the seller’s expected revenue is E v1 E(v 2 | v 1 ) = E(v 2 ) So the seller’s expected revenue is the same in both auctions!

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24 And similarly… In the first-price auction… A bidder with type v expects to win with probability F N-1 (v), and to pay b(v) = E(v 2 | v 1 = v) when he wins So his expected payoff isF N-1 (v) [ v – E(v 2 | v 1 = v) ] In the second-price auction… A bidder with type v expects to win whenever he has the highest value (v 1 = v), and to pay v 2 when he wins So his expected payment, conditional on winning, is E(v 2 | v 1 = v) And so his expected payoff is F N-1 (v) [ v – E(v 2 | v 1 = v) ] So each type of bidder gets the same expected payoff in the two auctions

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25 This turns out not to be a fluke This is exactly what we’ll prove more generally next class: With independent private values, any two auctions in which, in equilibrium, the player with the highest value wins the object, and any player with the lowest possible type gets expected payoff of 0 will give the same expected payoff to each type of each player, and the same expected revenue to the seller So, (a) this is pretty interesting, and (b) once we’ve proven this, we can use it to calculate equilibrium strategies much more easily

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