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Econ 805 Advanced 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 1 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 2 Today Why study auctions? Review of Bayesian games and Bayesian Nash Equilibrium 3 Why study auctions? 4 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 20002001; 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 5 …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 6 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 7 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 8 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) 9 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 10 Quick Review of Game Theory and Bayesian Games 11 Games of complete information A static (simultaneous-move) game is defined by: Players Action spaces Payoff functions 1, 2, …, N A1, A2, …, AN 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) 12 Solution concepts for games of complete information Pure-strategy Nash equilibrium: s A1 x … x AN s.t. ui(si,s-i) ui(s’i,s-i) for all s’i Ai for all i {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 13 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? 14 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 “Nature” make 2 weak make 2 strong Firm 2 Firm 2 Q2 Q2 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” Firm 1 Q1 Q1 u1 = Q1(100 - Q1 - Q2 - 25) u1 = Q1(100 - Q1 - Q2 - 25) u2 = Q2(100 - Q1 - Q2 - 30) u2 = Q2(100 - Q1 - Q2 - 20) 15 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 Firm 2 make 2 strong p=½ p=½ Q2W Firm 2 Q2S Firm 1 Q1 Q1 u1 = Q1(100 - Q1 - Q2 - 25) u1 = Q1(100 - Q1 - Q2 - 25) u2 = Q2(100 - Q1 - Q2 - 30) u2 = Q2(100 - Q1 - Q2 - 20) 16 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 17 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 t kj T j p kj ui si (ti ), s j (t kj ), ti , t kj t T p kj ui ai , s j (t kj ), ti , t kj k j j for each potential deviation ai Ai for every type ti Ti for each player i {1,2} 18 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 Et j T j ui si (ti ), s j (t j ), ti , t j Et j T j ui ai , s j (t j ), ti , t j The mapping si : Ti Ai maximizes “ex-ante expected payoffs” among all such mappings Eti Ti ,t j T j ui si (ti ), s j (t j ), ti , t j Eti Ti ,t j T j ui si ' (ti ), s j (t j ), ti , t j 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 19 Going back to our Cournot example, with p = ½ that firm 2 is strong… Strong firm 2 best-responds by choosing Q2S = arg maxq q(100-Q1-q-20) Maximization gives Q2S = (80-Q1)/2 Weak firm 2 sets Q2W = arg maxq q(100-Q1-q-30) giving Q2W = (70-Q1)/2 Firm 1 maximizes expected profits: Q1 = arg maxq ½q(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) 20 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 21