Econ 805 Advanced Micro Theory 1

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

Today • Common auction formats • The Independent Private Values model

Common Auction Formatsand Strategic Equivalences

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

“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

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

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

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

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

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

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

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

The IndependentPrivate Values Model

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 • 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”

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

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

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

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 second-price auction

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 • 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

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 • bi(vi)=vi is “nearly” the only symmetric equilibrium; and it involves bidders playing a strict best-response at nearly every type

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 letb : [0,V]  R+ be the common bid function; then at a given type v, b(v) must be a solution to max x ÎR+(v – x) Pr(win | bid x, opponents bid b(-)) = max x ÎR+(v – x) Pr(b(vj) < x " j ¹ i) = max x ÎR+(v – x) Pr(vj < b-1(x) " j ¹ i) = max x ÎR+(v – x) FN-1(b-1(x))

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

So if there is a “nice” equilibrium, it must be b(v) = ò0v sd(FN-1(s)) / FN-1(v) • What is this? • Well, if a random variable y has cumulative distribution G with positive support, then ò0v s dG(s) / G(v) = E(y | y < v) • And FN-1(v) is the cumulative distribution function of the highest of N-1 independent draws from F • So if we let v1 and v2 refer to the highest and second-highest valuations in a symmetric IPV model, then b(v) = E(v2 | v1 = v)

Now here’s where it gets cool… • In the symmetric equilibrium of the second-price auction, the price paid is v2, so the seller’s expected revenue is simply E(v2) • 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(v2 | v1 = v) • So the seller’s expected revenue is E v1 E(v2 | v1) = E(v2) • So the seller’s expected revenue is the same in both auctions!

And similarly… • In the first-price auction… • A bidder with type v expects to win with probability FN-1(v), and to pay b(v) = E(v2 | v1 = v) when he wins • So his expected payoff is FN-1(v) [ v – E(v2 | v1 = v) ] • In the second-price auction… • A bidder with type v expects to win whenever he has the highest value (v1 = v), and to pay v2 when he wins • So his expected payment, conditional on winning, is E(v2 | v1 = v) • And so his expected payoff is FN-1(v) [ v – E(v2 | v1 = v) ] • So each type of bidder gets the same expected payoff in the two auctions

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

Econ 805 Advanced Micro Theory 1