Econ 805Advanced Micro Theory 1

Dan Quint

Fall 2007

Lecture 1 – Sept 4 2007

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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

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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

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Today

Why study auctions?

Review of Bayesian games and Bayesian Nash Equilibrium

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Why study auctions?

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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

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…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

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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

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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

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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)

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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

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Quick Review ofGame Theory andBayesian Games

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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)

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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

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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?

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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 1Q1 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)

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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

Q2 Q2

Firm 1Q1 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)

p = ½ p = ½

Q2W Q2

S

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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

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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 = {t1

1, t12, …, t1

K} and T2 = {t21, t2

2, …, t2K’} for each player,

and a probability for each type {11, …, 1

K, 21, …, 2

K’}

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 ai Ai

for every type ti Ti

for each player i {1,2}

jkjj

kj Tt

kji

kjjii

kjTt

kji

kjjiii

kj tttsautttstsu ,),(,,),(),(

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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

jijjiiTtjijjiiiTt tttsauEtttstsuEjjjj

,),(,,),(),(

jijjiiiTtTtjijjiiiTtTt tttstsuEtttstsuEjjiijjii

,),(),(',),(),( ,,

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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-Q2

W-25)

giving Q1 = (75 – Q2W/2 – Q2

S/2)/2

Solving these simultaneously gives equilibrium strategies:

Q1 = 25, (Q2W, Q2

S) = (22½ , 27½)

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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