Econ 805 Advanced Micro Theory 1

Econ 805Advanced Micro Theory 1 Dan Quint Fall 2007 Lecture 1 – Sept 4 2007

I’m Dan Quint, welcome to Econ 805 • You are… • Class website • http://www.ssc.wisc.edu/~dquint/econ805 • Syllabus online, with links to papers • Lectures (no class next Thursday) • Office hours • Mondays 11-12, Wednesdays 10-11, other times by appointment • Grading • Problem sets (35%), final exam (65%). Midterm? • Readings • I’ll try to highlight which are most important

This class will be about auction theory • Popular auction formats • Independent private values and revenue equivalence • The mechanism design approach, optimal auctions • The “marginal revenue” analogy, reserve prices • Risk averse buyers or sellers • Auctions with strong and weak bidders • Interdependent values • Pure common values, symmetry in asymmetric auctions • Endogenous information acquisition • Endogenous entry • Collusion, shill bidding • Sequential auctions • Multi-unit auctions • Other topics

Today • Why study auctions? • Review of Bayesian games and Bayesian Nash Equilibrium

Why study auctions?

A whole lot of money at stake… • Christie’s and Sotheby’s art auctions – $ billions annually • Auctions for rights to natural resources (timber, oil, natural gas), government procurement, electricity markets • eBay: $52 Billion worth of goods traded in 2006 • eBay itself had $6 Bn in 2006 revenues, current market cap. of $46 Bn • European 3G spectrum auctions raised over $100 Bn in 2000-2001; upcoming U.S. FCC auction expected to raise $20 Bn • U.S. treasury holds auctions for $4 TRILLION in securities annually • “Dark pools” gaining share of trade in U.S. stocks

…and outcomes may be very sensitive to the details of the auction • One of our first results will be revenue equivalence… • …but this fails under a wide variety of conditions • Yahoo! vs. Google • Adjusting for the size of each market, revenues in European 3G auctions varied widely • Over 600 € per capita in the UK and Germany • 20 € per capita in Switzerland later the same year • Rules in Swiss auction discouraged marginal bidders/new entrants from participating, allowed for easy collusion among the primary competitors

Auctions can be seen as a useful microcosm for bigger markets • “Rules of the game” and price formation are explicit, allowing for theoretical analysis • Most relevant data can be captured, allowing sharp empirical work • Auctions lend themselves to lab experiments • Results on auctions may offer insight (or intuition) into behavior in less structured markets

Insights from auction theory may be valuable in other areas • P. Klemperer, “Why Every Economist Should Learn Some Auction Theory”: analogies in • Comparison of different litigation systems • “All-pay” tournaments such as lobbying, political campaigns, patent races, and some oligopoly situations • Market frenzies and crashes • Online auto sales versus dealerships • Monopoly pricing and price discrimination • Rationing of output • Patent races • Value of new customers under oligopoly

And finally, • Auction theory gives us a platform to introduce a number of important mathematical tools/techniques • Envelope theorem • Supermodularity and monotone comparative statics • Constraint simplification (necessary and sufficient conditions for equilibrium strategies)

But with all that said… • Auctions have been a hot topic in micro theory for over 25 years • Basic theory of single-unit auctions is pretty well developed • Multi-unit auctions are less well understood • Very difficult theoretically • Some partial results, experimental results

Quick Review ofGame Theory andBayesian Games

Games of complete information • A static (simultaneous-move) game is defined by: • Players 1, 2, …, N • Action spaces A1, A2, …, AN • Payoff functions ui : A1 x … x AN R all of which are assumed to be common knowledge • In dynamic games, we talk about specifying “timing,” but what we mean is information • What each player knows at the time he moves • Typically represented in “extensive form” (game tree)

Solution concepts for games of complete information • Pure-strategy Nash equilibrium: sÎA1 x … x AN s.t. ui(si,s-i) ³ ui(s’i,s-i) for all s’iÎAi for all iÎ{1, 2, …, N} • In dynamic games, we typically focus on Subgame Perfect equilibria • Profiles where Nash equilibria are also played within each branch of the game tree • Often solvable by backward induction

Games of incomplete information • Example: Cournot competition between two firms, inverse demand is P = 100 – Q1 – Q2 • Firm 1 has a cost per unit of 25, but doesn’t know whether firm 2’s cost per unit is 20 or 30 • What to do when a player’s payoff function is not common knowledge?

John Harsanyi’s big idea(“Games with Incomplete Information Played By Bayesian Players”) • Transform a game of incomplete information into a game of imperfect information – parameters of game are common knowledge, but not all players’ moves are observed • Introduce a new player, “nature,” who determines firm 2’s marginal cost • Nature randomizes; firm 2 observes nature’s move • Firm 1 doesn’t observe nature’s move, so doesn’t know firm 2’s “type” “Nature” make 2 weak make 2 strong Firm 2 Firm 2 Q2 Q2 Firm 1 Q1 Q1 u1 = Q1(100 - Q1 - Q2 - 25) u2 = Q2(100 - Q1 - Q2 - 30) u1 = Q1(100 - Q1 - Q2 - 25) u2 = Q2(100 - Q1 - Q2 - 20)

Bayesian Nash Equilibrium • Assign probabilities to nature’s moves (common knowledge) • Firm 2’s pure strategies are maps from his “type space” {Weak, Strong} to A2 = R+ • Firm 1 maximizes expected payoff • in expectation over firm 2’s types • given firm 2’s equilibrium strategy “Nature” make 2 weak make 2 strong Firm 2 Firm 2 p = ½ p = ½ Q2 Q2W Q2 Q2S Firm 1 Q1 Q1 u1 = Q1(100 - Q1 - Q2 - 25) u2 = Q2(100 - Q1 - Q2 - 30) u1 = Q1(100 - Q1 - Q2 - 25) u2 = Q2(100 - Q1 - Q2 - 20)

Other players’ types can enter into a player’s payoff function • In the Cournot example, this isn’t the case • Firm 2’s type affects his action, but doesn’t directly affect firm 1’s profit • In some games, it would • Poker: you don’t know what cards your opponent has, but they affect both how he’ll plays the hand and whether you’ll win at showdown • Either way, in BNE, simply maximize expected payoff given opponent’s strategy and type distribution

Formally, for N = 2 and finite, independent types… • A static Bayesian game is • A set of players 1, 2 • A set of possible types T1 = {t11, t12, …, t1K} and T2 = {t21, t22, …, t2K’} for each player, and a probability for each type {p11, …, p1K, p21, …, p2K’} • A set of possible actions Ai for each player • A payoff function mapping actions and types to payoffs for each player ui : A1 x A2 x T1 x T2 R • A pure-strategy Bayesian Nash Equilibrium is a mapping si : Ti Ai for each player, such that for each potential deviation aiÎAi for every type tiÎ Ti for each player i Î {1,2}

Ex-post versus ex-ante formulations • With a finite number of types, the following are equivalent: • The action si(ti) maximizes “ex-post expected payoffs” for each type ti • The mapping si : Ti  Ai maximizes “ex-ante expected payoffs” among all such mappings • I prefer the ex-post formulation for two reasons • With a continuum of types, the equivalence breaks down, since deviating to a worse action at a particular type would not change ex-ante expected payoffs • Ex-post optimality is almost always simpler to verify

Going back to our Cournot example, with p = ½ that firm 2 is strong… • Strong firm 2 best-responds by choosing Q2S = arg maxqq(100-Q1-q-20) Maximization gives Q2S = (80-Q1)/2 • Weak firm 2 sets Q2W = arg maxqq(100-Q1-q-30) giving Q2W = (70-Q1)/2 • Firm 1 maximizes expected profits: Q1 = arg maxq½q(100-q-Q2S-25) + ½q(100-q-Q2W-25) giving Q1 = (75 – Q2W/2 – Q2S/2)/2 • Solving these simultaneously gives equilibrium strategies: Q1 = 25, (Q2W, Q2S) = (22½ , 27½)

Auctions are typically modeled as Bayesian games • Players don’t know how badly the other bidders want the object • Assume nature gives each bidder a valuation for the object (or information about it) from some ex-ante probability distribution that is common knowledge • In BNE, each bidder maximizes his expected payoffs, given • the type distributions of his opponents • the equilibrium bidding strategies of his opponents • Thursday: some common auction formats and the baseline model

Econ 805 Advanced Micro Theory 1